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Appendix

Elliptic Partial Differential Operators on a Manifold by Daisuke Fujiwara

§1. Distributions on a Torus

A local coordinate system on an n-dimensional differentiable manifold X gives a diffeomorphism of its domain Xj onto the unit ball U inU", while the unit ball in R" is diffeomorphic to a domain V in the torus T" = [R"/27rZ". Thus a section of a vector bundle B with /i-dimensional fibres over Xj can be identified with a vector-valued function with values in C^ over the domain V in T". Various function spaces of sections of B over Xj can thus be considered as those of corresponding C^-valued functions over V in T". In what follows, we shall always treat the functions defined on V which can be extended to the whole space T". Since the torus T" is compact, a function space consisting of vector-valued functions on T" has much simpler structure than that on R".

(a) Definition of Distributions

By A: = ( x \ . . . , x") we denote a point in R". If we denote by ITTZ" the set of the points in R" whose components are ITT times integers, then this becomes an additive group. The n-dimensional torus T" is defined as

T" = R727rZ".

A C°° function/ on T" can be regarded as a C°° function on R", hence we can write it as f{x) = / ( x \ . . . , x"). Then we have the multi-periodicity

/ ( x + 27r^)=/(x), ^eZ\ (1.1)

We write DjJ = l,... ,n, for the differential operator d/dx\ An n-tuple a = ( « ! , . . . , a„) of non-negative integers will be called a multi-index, and |a | = aj 4- • • • + a„ its length. Moreover we introduce the following notation:

D" = D^^ • • • D ^ .

364 Appendix. Elliptic Partial Differential Operators on a Manifold

We denote by C°°(T") the totality of complex-valued C°° functions on T". For / = 0 , 1 , . . . , the norm | |/ is defined by

\ip\i= I m a x | D > ( x ) | . (1.2)

With the topology defined by these countably many norms, C°°(T") becomes a Frechet space, which we denote by ^ (T" ) .

Definition 1.1. A continuous linear map of ^ (T") to C, namely a continuous linear functional on ^ (T" ) , is called a distribution on T". We denote the totality of distributions on T" by ®'(T").

In other words, S is a distribution on T" if and only if it satisfies the following two conditions:

(1) S\ C°°(T") 3ip^ S{(p) G C is linear with respect to <p. (2) There exist a non-negative integer /, and a positive constant C such

that for any <peC^(r)

\Si<p)\^C\cph (1.3)

holds.

The totality ^'(J'') of distributions on T" is the dual space of ^ (T" ) . We write (5, (p) for S{(p) if 5 G ^'("IT") and cp e ^ (T" ) . < , ) gives a bilinear map ^ ' (T") x ^ ( T " ) ^ C . As the dual space of ^ (T" ) , we can introduce a topology into ^ ' (T") . Recall that a sequence {Sm}Z=i of distributions converges to S with respect to the weak topology of ^ ' (T") if and only if

{S,<p)=\im{S^,cp) (1.4) m-^oo

holds for any (pe2D{J"). Let f(x) be an integrable function on T", and put

if,<p)=\ f{x)<p{x)dx (1.5)

for any (p e ^ (T" ) , where dx = (I/ITT) dx^ - - - Jx". Then { , ) is linear with respect to (p. By putting / = 0, and C = ljn \f(x)\ dx, (1.3) holds for (f, (p), hence by (1.5) / defines a distribution. Thus we have a map

LHT")^^'(ir").

This map is injective. Indeed, le t /and g be elements of L^(T"), and suppose

§1. Distributions on a Torus 365

for any cp e C°°(T"). Then we have

I (f{x)-gix))cpix)dx = 0.

Hence we have the equaHty

f{x) = gix) = 0

for almost every x. Therefore f=g in V(J"). Consequently the map of L^(T") to ^ ' (T") defined above is injective. By this injection L^(T") can be considered as a subspace of 2)'{J"): L'(T")c: ^ ' (T") .

Let a{x) be a C°° function on T", and f{x) an integrable function on T". Then the product {af)(x) = a{x)f(x) also becomes an integrable function. We want to define a multiplication of any distribution 5 by a by extending this operation. In a special case S ' = / G L ^ ( T " ) , we have the identity

(af, (p) = a(x)f(x)(p(x) dx = f(x)a{x)(p{x) dx = {f, acp) JT" JT"

holds for any (pe2)(J"), In view of this identity we shall define aS as follows:

Definition 1.2. Let 5 G ^ ' (T") and a e C°°(T"). ThenthQproductaSe 2\J'')

is defined as follows:

{aS,(p) = {S,a(p)

for any ( ; P G ^ ( T " ) .

To make this definition possible, it should be proved that the linear map

< | ) ^ : ^ ( T " ) 3 ( p ^ a ( ; p G ^ ( T " )

is continuous with respect to the topology of ^ (T" ) . For any multi-index a = (« ! , . . . , « „ ) , we have the following identity (Leibniz' formula):

D-(a(x)<p(x))= I ll)D^a(x)D-^<p(x\ (1.6)

where ^ = (/3i , . . . , i8„) is a multi-index, and by j8 ^ a we mean that Pj ^ a for every j = 1 , . . . , n. Further we have put

( ; ) = ( ; ; ) • ( ; : ) •


Using Leibniz' formula, we obtain for / = 0 , 1 , . . . ,

\a<p\i^n'\a\A<p\u (1.7)

where a C°°(T") and (p C°°(T"). Therefore O^ is a continuous linear map. Since <I>a is a continuous linear map, we can define its dual

which is continuous with respect to the weak topology. 6y the definition of the product of distributions given above, we have *^a{S) = aS. Hence the map ^\J'') 3 (p -^ aS e ^'{J'') is weakly continuous. Its linearity in S is also obvious.

L e t / 6 C^T") . Then we obtain the following identity.

This is just the formula of integration by parts, but this characterizes the partial derivative {d/dx^)f of/ as a distribution. By extending this formula, we define the derivative (d/dXj)S of any distribution 5 as follows. First we note that the map

- ^ : 2(Jn 3<p-^ - / - ( p e ^ (T" ) , J = 1 , . . . , n, (L8) oX oX

is linear and continuous. For, in fact,

U-(p ^Wh^u / = 0,1,.. . , (L9) \oX 1/

holds.

Definition 1.3. We denote the dual map of the linear map (1.8) by

^ : S ' ( T " ) 9 S ^ ^ 5 e S ' ( T " ) , j = l , . . . , n ,

and call (d/dx^)S the partial derivative of 5 in x^. The map S^(d/dx-^)S is continuous in the weak topology of ^ ' (T") .

The above definition of the partial derivative d/dx-^ coincides with that of the usual partial derivative of a C^ function / This follows from the


above formula of the integration by parts. So we always write

^ ^ • = ^

as an operator acting on distributions. Leibniz' formula (1.6) can be extended for any aeC°°(T") and Se

DâS)= I (-^)DâD"-^5, (1.10) Pâ \p/

which is called Leibniz' formula as well. Similarly we can define the translation of a distribution 5 in the direction

of Xj by h. For (p e C°°(T"), we put

r;'cp(x) = <p{x\...,x^ + K...,xn.

Clearly the map thus defined

T 7 ' ' : ^ ( T " ) - > ^ ( T " )

is continuous and linear. As its dual map,

TJ':^'(T")95->TJ'5G^Xir") (1.11)

is defined. We call TJS the translation of 5 in the direction of x' by h. The difference quotient operator Aj* is defined by

^^S = h-\T^S-S) (1.12)

for any distribution S. Especially in case S=fe L^(T"), we have

AJ'/(x) = / i - H / ( x \ . . . , x ^ + / i , . . . , x " ) - / ( x S . . . , x " ) ) .

The formula DÂJ* = AJ'Dfe holds, and for any a e ^ (T") and any S e ^'(IT"), we have

AJ'(a5) = A>T7''5 + aAJ'5. (1.13)

Let 5 be a distribution on T", and K a compact subset of T". S is said to vanish on i^ ' = r " - X if

(5,<p) = 0

holds for any cp e ^ (T") vanishing on K. The smallest one of such K is called the support ofS and is denoted by supp 5. If 5 is a function/G L ^ ( T " ) ,


the support of S coincides with the support of / as a function:

supp 5 = s u p p /

ForaGC°°(T"),wehave

supp aS = supp a n supp 5. (1-14)

(b) Vector-Valued Distributions

Let 6 be a positive integer. A vector-valued distribution S with values in C^ is defined as a /x-tuple

5 = (S\ . . . , S' ) G ^ ' (T") X . . . X ^ ' (T") .

We denote the space consisting of all such vector-valued distributions by ^ ' (T", C^). Since 2'{J", C^) = 2)'{J") x • • • x ^ ' (T") , it is a vector space.

For aGC°°(T") and 5 G ^ ' ( T " , C ^ ) , we define their product by aS = {aS\... ,aS^). Similarly we define the partial derivative DjS and the difference quotient AjS by

DjS = (DjS\...,DjSn,

A ; 5 = ( A J ' 5 \ . . . , A J ' 5 ' ^ ) .

We denote the space of vector-valued C°° functions by

^(T" , C^) = ^ (T") X . . . X ^ (T" ) .

For (p = {(p\...,(pn^^(T\C^) and 5 = ( 5 \ . . . , 5 ^ ) G ^ ' (T" ,C^) , we define the bilinear map

A

which makes 2\J", C ) and ^(T", C") dual to each other.

(c) Fourier Series Expansion of Vector-Valued Distributions

We denote by L^(T", C^) the totality of vector-valued functions f{x) = (/^(^), • •. ,f^M) with values in C^ defined on T" such that each component/ '^(x) is square-integrable. For / gG L^(T", C^), we define their inner product by

A J T " a«)=I fix)g\x)dx. (1.15)


With this inner product L^(T", C^) becomes a Hilbert space. The norm of / is given by

11/11 = «/) ' ' ' = ( i f {fixWdx) .

For any f G Z", we write i' x = ^x^ + • • • + ^„x", and put

f^=\ f{x)txp{-i^'x)dx. (1.16) JT"

We call /^ = if],... , / f ) the Fourier coefficient of / Using these /^ , we obtain for each A a Fourier series expansion

f{x)=lf, exp(i^- x), A = 1 , . . . , M, (1.17)

which converges in L^(T''). We denote by P(Z", C^) the totality of ^c-tuple of infinite sequences of complex numbers {a^} with ^e Z" and A = 1 , . . . , /i, such that

A ^

An element of /^(Z", C^) can be considered as a map of Z" to C^ which maps ^ e Z " to a^ = ( a | , . . . , a f ) e C ^ /^(Z",C^) is also a Hilbert space. The following is the fundamental theorem for Fourier series.

Theorem 1.1. The map which associates/e L^(T", C^) to its Fourier coefficients gives an isomorphism o/L^(T", C^) onto /^(Z", C^) as Hilbert spaces.

\\fr = ll\fl\' (Parseval). I (1.18) A ^

The Fourier series (1.17) converges to / with respect to the norm of L'(T",C^).

It is difficult to give the condition under which a square-integrable function on T" becomes continuous in terms of the Fourier coefficients. But for our present purpose the following theorem is sufficient.

Theorem 1.2 (Sobolev's Imbedding Theorem). Let f^ be the Fourier coefficients offe L^(T", C^), and suppose that for some s> n/2, we have

A ^


where we put \^\^ = ^?+ • • • + 1 ^ . Then the Fourier series

\ . (1.19) fix)={f\x),...,r(x)),

converge absolutely and uniformly on T", hence give continuous functions of X. Further

f{x)=f'ix)

holds almost everywhere. Moreover there exists a positive constant C^ depending on s such that

m a x l / ^ W I ^ c Y z Z d + l ^ r r i / ^ r ) . (1.20)

Proof. Putting

we have

•=(ii{iêy\n?)"\

E i/^Ni(i+i^pn/î(i+iîv

îi/L(i(i+i^rr) 1/2

< 0 0

because s> n/2. Therefore the infinite series (1.19) whose terms are continuous functions converge absolutely and uniformly in x, hence represent continuous functions of x. On the other hand, by the preceding theorem, / and / coincide as elements of L^(T", C^). Consequently on T" the equality

f\x)=f'{x)

holds except possibly for a set of measure 0, which proves the theorem. I

Theorem 1.3 (Sobolev's Imbedding Theorem). Let f) be the Fourier coefficients O / / G L^(T", C^), and k a non-negative integer. If for some s> n/2-\-k,

ll{l-^mVW<^ (1-21) A I

holds, then the Fourier series

/ ' ( x ) = 1 / ^ exp(if • x), A = 1 , . . . , /x, (1.22) A


and the infinite series obtained by term-wise differentiating them j-times for any j^k all converge absolutely and uniformly. Therefore f^{x) are C \ Moreover on T", the equality

fix)=f\x)

holds except possibly for a set of measure 0.

Proof. We have already proved that (1.22) converge absolutely and uniformly in the preceding theorem. By differentiating their general term, we have

\D"f, exp(i^- x)| = \{i$r-ff exp(i^- x)|

1 g|^n/^|, A = 1,...,M,

where we put \a\-j^k, and ^" = ^?^ • * • f ". Therefore, similarly as in the proof of Theorem 1.2, we obtain

i\^nf,mfi(i{i+m'-'Y'-

Since s>n/2-\-k, we have S^ (1 + | | |^)^"'<oo, hence

ZD" /^exp( i ^ .x ) , A = l,...,)Lt,

converge absolutely and uniformly, and the j{x) are C^. Also there is a positive constant C« independent of / such that

max |D« /^ (x ) | ^C j | / | | , . (1.23) X

The fact that f{x) =f(x) holds almost everywhere is proved similarly as in the proof of the preceding theorem. I

The vector space of all C^ functions defined on T" with values in C^ becomes a Banach space if we define the norm o f / = ( / \ ,f^)^ C^(T",C^)by

\J]jc = l I m a x | D T W | .

Let /^ be the Fourier coefficients of / Then for any multi-index a with I a I ^ /c, we have

(-i) '" 'r/^ = [ f{x)D" expi-ii • X) dx

= - I ( - D ) T ( x ) exp(-»|- x) dx.


Therefore, recalling {-D)ye L^J",^), we obtain

ii:a+\i\'r\m'=m+Di+---+Dim^c,\A,. (1.24) By the deep gap between (1.24) and Sobolev's imbedding theorem, we cannot get the precise characterization of C^(T", C^) in this way. But we can characterize C°°(T", C^) as follows.

Theorem 1.4. fe L^(T", C^) coincides with an element o/C°°(T", C^) almost everywhere if and only if its Fourier coefficients satisfy

llil + my\fe\'<^ (1-25) for any seU,

The proof runs as follows. First by (1.24) we see the necessity of (1.25), Conversely, if (1.25) holds for any 5, we see from Sobolev's imbedding theorem that / is equal to an element of C°^(T", C^) almost everywhere. I

Definition 1.4. Let cp^ be the Fourier coefficients of (p = {(j)^,... ,(p^)e C°°(T", C^). For an arbitrary 5 e R, we call

ikiis=(i;z(i+i^r)>^r)''' the Sobolev norm of degree s.

By (1.23) and (1.24), for /c = 0 , 1 , . . . , there exists a positive constant Q depending only on k such that for any (p e C°°(T", C^),

Ck^\\(p\\2k^\(p\2k^Ck\\(p\\2kHn/2-]+U (1-26)

where [n/2] denotes the integral part of n/2. Therefore the topology of the Frechet space ^(T", C^) can also be defined by the countable system of norms || H , /c = 0,1, From this fact combined with Theorem 1.4 and the proof of Sobolev's imbedding theorem, we obtain the following theorem.

Theorem 1.5. Let cp e ^ (T" , C^). The Fourier series of cp converges to (p with respect to the topology ofQ){'f, C^). Therefore {exp(i|^ • x)}^^^" forms a basis of 2(J\C^). I

Consider the Fourier series expansion of a vector-valued distribution S = (S\ .,.,S'')e S)\J\ C^) with values in C^

Definition 1.5. Put 5^ = {S\ exp(-/^• x)), ^G Z", A = 1 , . . . , M. We call S^ = ( 5 | , . . . , 5^) the Fourier coefficients of the vector-valued distribution 5 =


( S \ . . . , 5 ^ ) . The series

X S, exp(-f^- x) = (l S\cxp{-i^' x ) , . . . ,Z 5^ exp(-/^- x))

is called the Fourier series of S.

Theorem 1.6.

( r ) The map ^ ' (T", C ) BS-îS""^) is injective. (2°) 77ie sequence (5^) w the Fourier coefficient of a vector-valued distribu

tion if and only if there exists an integer kÔ such that

I I ( l + l ^ m 5 ^ r < ^ • (1.27) A ^

holds. This being the case, the Fourier series

^S'^Qxpi-i^'x)

converges weakly in ^ ' (T", C"). Let S e 2)'(J", C^) be its limit. Then the Fourier coefficients of S coincide with S^.

Proof ( r ) It suffices to show that if 5 = ( 5 \ . . . , 5^) e 2)'(T", C^) and for a n y ^ e Z ^

5^ = {S\ exp(-i^ • x)) = 0, A = 1 , . . . , /x,

holds, then 5^ =0. But this is clear since {exp(-/^- x)} forms a basis of

(2°) By the definition of 2)'(J", C^), there exist a positive integer / and a positive constant C such that for any f eZ",

K 5 \ exp(-/^- x))| C1exp(-/^- x)b

holds. Hence by taking a suitable constant C,

holds. Thus if we choose k so that /c-1 > n, (1.27) holds. Conversely, suppose that (1.27) holds for some kÔ. If we put A^ = (1 + If p)~'=Sf, then we have

II|A^P<oo.


Hence if we put A^{x) = Y.^^^ exp(~i^- x), then this becomes a square-integrable function of x Thereifore A^{x) represents ^ distribution on T", namely, for any cp e ^ (T" ) ,

{A\<p)= A\x)<p(x)dx= Y.A^^Qxp{-ii'x)(p(x)dx JT" JT" ^

holds. Since thg series I ^ A^ exp(-if • x) converges strongly in L^(T"), we can exchange the integration witlj X^, so we have

^ JT" {A\cp) = lA'^ exp(-f^-x)<^(x)^x = IA^</>_^,

^ JT" ^

where we denote the Fourier coefficients of (p{x) by (p^. Next, we put a partial sum of the series made of {5^} as

S'N= Z 5^exp(- /^-x) ,

where N>0. This is a C°° function, hence gives a distribution on T". We shall prove that if N ^ o o , 5 ^ converges to the distribution (1 - D j - • • • -Dl)^A^ with respect to the weak topology of ^ '(T")- In fact, if ( G ^ (T" ) , we have

<(1-D? Dl)'A\<p) = {A\il-D', DD'cp)

I = I A ^ exp( /^ -x ) ( l -D? D^JVWrf:^ i J T "

Applying (1.25) to <^(x) € C°^{J", C ) by putting 5 = fc, we have

ikf.=i(i+i^m.p-,r<^. Therefore we have

m-Dl Dl)''A\v)-{S%,cp}\

= \ I 5,V-

\|^|>iV / \ |^|>iV /


hence by (1.27) we obtain

lim(5^^,(p) = < ( l -D? Dl)Â\<t>). N-»oo

Thus SN converges weakly to ( 1 - D ? DD'Â'' in ©'(T"). If we put 5^ = (1 - Di DD'Â^, then the Fourier coefficients of S" are given by

{S\cxpi-ii-x)) = {A\{l-D] Di)''exp{-i^-x)}

=ii+\enA\cxp{-i^-x))

Thus the proof is completed. |

Theorem 1,7. For any S = {S\ . . . , 5^) e ^ ' (T", C^), there exist a non-negative constant / ^ 0 qndf= (f\ . . . ,f^) e L^(T", C^) such that in the sense of distributions

S' = {l-Dl DlYf (1.28)

holds. (The structure theorem of distributions.)

Proof, Let 5 " = I 5 ^ e x p ( / f - x ) .

be the Fourier expansion of S^. Then (1.27) holds for some kÔ. Therefore as in the proof of the preceding theorem, we put

/^=(i+i^r)-'s^

If we pu t /^ (x) = X^/^ exp(f|- x), then/^(x)G L^(T"), and as is shown in the proof of Theprem 1.6, we have

S' = (l-Dl dl)T

in the sense of distributions. By putting / = /c, the theorem is proved. I

Theorem 1.8. Let 5^ be the Fourier coefficients ofSeSD'iJ", C ) , and ( ^ the Fourier coefficients of cp e S)(J", C^). Then

{S,<p) = llS',<pt, (1.29)

holds. I

This formula is already obtained in the proof of the preceding theorem.


(d) Sobolev Space

We introduce a space which Hes between ^(T" , C^) and ^ ' (T", C ) .

Definition 1.6. For any 5 G IR, we put

and call it the vector-valued Sobolev space of degree s with values in G^, where S^ denotes the Fourier coefficient of 5^.

For S,Te W^(T", C'^), we define their inner product by

(S,T),=llil + \^\ysi¥,, (1.30)

which makes \y^(T", C^) a Hilbert space. Here T^ denotes the Fourier coefficient of T^. The norm of this space is just the Sobolev norm of degree s

1/2

i5L=(ii(i+i^rri5^r)

Theorem 1.9.

(V) Ifs'<s, W'{J\C^)cz W^'(T",C^).

(3°) Ifk-\-n/2<s, there exists a continuous imbedding

W'{J\ C^)-> C\J\ C^), (1.31)

namely, there exists a positive constant C^k such that for any S e W^(T", £^), there is an SeC\J\ C^) with

| 5U^C, , | |5 | | , . (1.32)

(4°) ^ (T" ,C^) is dense in each \^'(T",C'^).

Proof (1°) is clear from the definition. (2°) follows from Theorem 1.4 and Theorem 1.6. (3°) is just Sobolev's imbedding theorem, Thorem 1.3. Finally, if we put

S = (S\ ...,S'')e \¥'(T", C^), and let

5 "= i :5êxp( /^ -x )

be its Fourier series, then the partial sum X|^|siv ê exp(i^- x) is an element of ^(J\ C^) fnd converges to S in W'(T", C^) for Nôo.

(4°) follows immediately from this. I


The following inequality is often used in the following.

Lemma 1.1. Let a, b, t be positive numbers. If O^X^l,

a^b^-^Âr^/â + ( l - A ) r ' / ( ' - ^ ^ 6 (1.33)

holds, where the equality holds for t^^â = t~^^^^~^^b.

The proof is given by the comparison of the arithmetic and geometric means. I

Proposition 1.1.

( r ) Ifs"<s' < s, then for anyfe W'(J", C^) the following interpolation inequality holds:

ui'^mi'-'"'^''-'"^^^^^^^ (1.34)

(2°) Ifs"<s'<s,foranyfe T^'(T",C^) and any positive number t, the following interpolation inequality holds:

s-s"

s-s'

+ s-s"

^-u- .vu-. ; | |y | | . . . (135)

Proof By the above lemma, for any ^ > 0, we have

ii+\^\Y^^^^t^'-'"^^^''-'"\i-h\^\y

P u t / = ( / \ . . . , / ^ ) , and let/^ be the Fourier coefficients of F^. Multiplying the above inequality by |/^|^ and summing over êZ", we obtain

Taking the minimum of the right-hand side for r > 0, we obtain

Taking the square root, we obtain (1.34). (2°) follows from this and Lemma 1.1. I


Theorem 1.10. Ifs'<s, the inclusion M^'(T",C^)^ l^^'CT^C^) is compact.

Proof. It suffices to show that if a countable sequence {fm}m in W'(J", C^) is bounded, we can choose a subsequence which converges in W (¥", C^). Let fm = (fL --^Jm) and frn,^ the Fourier coefficients of fl where ^eZ\ Since {/m}m is bounded in \y^(T", C^), there exists a positive number M such that

ll/ll?=iiiA/(i+iirr<M.

Since for each , A, {/^, Jm=i is a bounded sequence, we may take a suitable subsequence {m'} so that for each , and A, {/ ,m'}m' converges. Then {f^'} converges in \^^ (T", C^). In fact, we have the following Fourier expansion:

/'^' = Z/m', ,exp(/^-x).

For any positive number e, take a sufficiently large N so that

holds. Then for any m', we have the following inequality.

Fix such N. Since there are only a finite number of lattice points ^ e Z " with 11 ^ N , we may choose a sufficiently large mo so that for any mj, m^>mo,

A|^|<7V

holds. Then we have

11/.,-/mill?' I z \fU,,-fUf(^+\^\y' A i^i<;v

+ 2E I (IA./ + l/k/)(l + l lY A l ^ l ^ i V

Therefore {/mlm' is a Cauchy sequence in W"^(T", C^), hence converges in W''(J\ C^). This completes the proof. I

Since the Sobolev space of degree 0 is nothing but L^(T", C^), we shall write II II for || ||o below. It is difficult to have an intuitive understanding


of \y^(T", C^), but if 5 = / is a positive integer, it is rather easy to understand. For this we use the following lemma.

Lemma 1.2. Let S = {S\ . . . , 5^) G £ ) ' ( T " , C ^ ) , and 5^ the Fourier coefficients ofS. If we write the Fourier coefficients ofDjS = {DjS\ ..., DjS^) as (D,5^)^ forj = 1, 2 , . . . , n, we have

iDjS'), = i^jS'^, j = l , . . . , n ; A = 1 , . . . , M- (1.36)

Proof, (DjS^)^ = {DjS\ exp(-/^- x)) = -{S\ Dj exp(-/f • x))

= i^j{S\exp{-i^'x))=i^jS',. I

From this lemma we obtain the following proposition.

Proposition 1.2. Let I be a positive integer. Then

W\J\ C^) = {Se L^(T", C^)ID'^Se L^(T", C^) for any a with \a\^ /}.

For any Se W\J", C^) we have

1/2

i iD«5i i ; Mais

n-'''\\S\uJ I | | D " 5 | | ) " ' ^ | | 5 | | , . (1.37)

Proof It suffices to show (1.37). Letting 5^ be the Fourier coefficient of 5, we have

For ^ G Z " ,

\a\^l

holds. Therefore, multiplying this by |5^|^ and summing over , we obtain

n-^\\S\\l^ I \\D-Sf^n^\\S\\l \a\^l

which proves (L37).

Theorem 1.11.

(1°) The restriction of the bilinear form { , ) on ^\J\ C^) x ^(T" , C^) to ^ (T" , C ^ ) x ^ ( T " , C^) gives the following inequality: For any (t>,il/e ^ (T" , C ) and any seU,

K(A,^)|^||eA||-slkL. (1.38)


(2°) The bilinear form ^ (T" , C^) x (T" , C^) 3ip,(p^ (lA, <p) extends uniquely to a continuous bilinear map o/\y~^(T", C^) x \y^(T", C^) to C which is continuous in if/ with respect to the norm \\ \\_s ci^d in (p with respect to \\ || . By this bilinear form, \^~^(T", C^) and W'(J", C^) become dual to each other.

(3°) Let seU.ForSe W~'(J\ C^) we have

s u p ^ ^ | | S | | _ . (1.39)

Here sup is taken over all non-zero cp eQ)(J'',C^).

Proof (1°) Let cp^ and il/^ be the Fourier coefficients of (p and ip respectively. Then we have

A ^

By Schwartz' inequaUty, we obtain

^ll'All-.lkL.

(2°) Since ^(T",C^) is dense both in W-'(J",C^) and in W'{J\C^), the inequaHty (1.38) shows that the bilinear form ( , ) extends uniquely to a continuous bilinear form on W"'(T", C^) x W'(J\ C^) to C. Then (1.39) proves that by this bilinear form, \ y ( T " , C^) and \y'(T", C^) become dual to each other.

(3°) Let S^ be the Fourier coefficients of 5. For any N>0, we define <PN = {<PN, . . . , <^^) e ^(T" , C^) by putting

Then

\{S, <PN)\ = llS',9N.-i =1 I \s'A-^+m-'. 2 \ - s | c'\\2\l/2

On the other hand, since ||<PN||. = ( I A I|«lsiv (l + l^r)"'!'^^!) > we have

\\<PN\\S \ A l^l^iv /

Taking the supremum with respect to N, we obtain (1.39). I


Proposition 1.3. Let a be a multi-index. If s> s' + \a\, then for any Se W'{r,C^), D'^SeWXl^C'^). D": \^^(T",C^)^\^ ' ' (T" ,C ' ' ) is continuous, and the inequality

||D"5i|,,^||5||,^,,«l (1.40)

holds.

Proof. It suffices to show (1.40). If we let 5^ be the Fourier coefficients of 5, the Fourier coefficients of D"5'^ are given by

{D''s)',={iirs',. Hence we have

iio"5ii^.gii(i+iinîiiî"i|s^rgiisii?.,i„|. I

Next we consider the product of a function and a distribution.

Lemma 1.3. Let f(x) e C°°(J",C''^) be a C^ function with values in {px fi)-matrices, and put fix) = if\{x),...,fj,{x)). Let g = (g\ ..., g^)e W\J'',C^) be a vector-valued distribution with values in C^. Suppose that

p = l

holds in the sense of the product of distributions. Then h = {h^,... ,h'')e \y^(T", C^). Moreover there exists a positive constant C depending only on s such that the following inequality holds:

ii/iii,^cm,,,iigii,.

Proof The above theorem is true for any seU, but its proof is complicated. Therefore we shall give proof only for the case s is an integer here. This is enough for our present purpose.

(1°) The case 5 = 0. The assertion follows from the well-known inequality

JT" ^ JT )g ' ' (x) |^^xsmax | /^(x) | | |g ' ' (x) |Vx

J T "

with C = l. The case 5 ^ 1 . For a multi-index a with | a | ^ 5 , we have by Leibniz'

formula

Pâ \p/


Hence letting D^f be the (^ X/i)-matrix function with D^/p as its (A, p)-component, and D'^'^g the C^-valued distribution with the pth component D'^'^g^, and applying the result of the case 5 = 0 to D ^ / and D'^'^g instead of / and g respectively, we obtain

D«ft||g I (f\\D'f]o\\D''-''g\\

by taking the summation with respect to j8. Therefore, by Proposition 1.2, there exists a positive constant Ci such that

(T) The case 5 = - / < 0 . Take an arbitrary «p7^0 with (p^ \^^(T",C"). Using (1.39), we have

II/i II _f = s u p - — p - = sup-7—r-, <p ||<p||/ <p 11 11/

where (A = (<A\ • • •, ^ ' ' ) and iA^(x) = S^^^/^(x)<p^(x). By ( D there exists a positive constant Cj such that

\i^C,\f\x\W\i

holds, hence cqmbining with the inequality |(g, ^)\^ ||g||_/||(/^||/, we obtain

ll^ll-/^Qiyi, | |g| |_,

which completes the proof. I

In particular in case 5 ^ 0 ,

^ ' W = Z / p W ^ ' W , A = l , . . . , ^ , (1.41) p

hold for almost every x,

Theorem 1.12. Let I be a positive integer with l^n-\-\. Then for any fe W\J\C^^)^ and geW'iJ^C''), /i = (/i\ . . . , / i ^ ) given by (1.41) is an element 0/ V^^(T", C ) . Moreover there exists a positive constant C depending only on /, n, /i, v such that the following estimate holds:

\\hh^c\\fUg\\i-

§1. Distributions oh a Toms 383

Proof. Using the estimate of ||i^"/i|| for the inulti-indice^ d with |a | ^ /, we can find an estimate of ||/i||/. By Leibniz' formiila we have

|^|^|a|/2 \ P / l^l^kl/2 \ P /

Since | a | ^ / , if | j8 |^ | a | /2 , then |j8| + (n + l ) / 2 ^ / , hence, by Sobolev*s imbedding theorem, there exists a positive constant C independent o f / such that on T"

|D^/^(x) |SC| | / | | ,

holds outside of a set of measure 0. Therefore we have

||ID^/^D"-VNC||/|M|D"-^g||, |)8|S|a|/2. P

Similarly in case |j8| > |a | /2 , since |a - / 3 | < |a | /2 , applying Sobolev's imbedding theorem to g^, w6 obtain

IIID^/^D-VNCIID^/Il ||g||„ |)8|>|a|/2.

Therefore

||D"ft|| s c(||/|M|g|||„|+||/|||„|||g||,) g c||/|M|g||,

holds. Thus by Proposition 1.2, we obtain

|i/.||,sc||/|M|g||,. I

We define a linear partial differential operator A(x, D) which operates on vector-valued distributions S = {S\ .,.,S'')e ^ ' (T", C^) by

(A(x,D)Sr=l I aâ(x)D«5^ A = l , . . . , ^ , (1.42) p |a |gm

where the coefficients Upîx) are C°° functions. If there are some a with |a | = m. A, p, and a point x such that apa(x) T 0, we say that A(x, D) is a partial differential operator of order m.lfm = 0, A(x, D) is called a multiplication operator. For a non-negative integer /, we put

Mi= I I Zmax|Dâ^«(x)|. (1.43)

Propositidn 1.4. Ler A(x, D) be a linear partial differential operator of order m with apa(x) being C°° functions. Then for any seU, there exists a positive number C depending only on s, n, m, v and fjL such that for any S e W'{J",C^),


the following inequality holds:

\\A(x,D)Sl^CM\4Sl^m-

This follows from Lemma 1.3 and Proposition 1.3. I

Let (p(x) e C°°(T", C), and (p the operator of multiplication by (p(x): For 5 6 ^ ' (T", C^), (p:S = (S\..., 5^)-^ {(pS\ . . . , (pS^). We will often use the commutator of (p and A(x, D) later:

[A(x, D), (p] = A(x, D)<p - cpA{x, D). (1.44)

In terms of components, this is written as

([A(x,D),(p])^=S Z a^„(x)[D",(p]5^A = l , . . . , / x . (1.45)

Lemma 1.4.

(1°) For a multi-index a, /le commutator [D", (p] i5 a //near partial differential operator of order \OL\ — \ whose coefficients are derivatives of(p of order up to\a\. For any seR, there exists a positive constant C depending on s and a such that for any Se iy^(T", C^), the following inequality holds:

||[D'',.p]S|l,SC|.p||,H„ll|S||,,l„l_i.

(2°) For such a linear partial differential operator A(x, D) as stated in Proposition 1.4, the commutator [A(x, D), <p] is a linear partial differential operator of order m — l. For any seU, there exists a positive number C depending on s, m, v and /JL such that for any

\\[A(x, D), cp]Sl^ C M H | ( ^ | , , , ^ ^ | | 5 L ^ ^ _ I

holds.

Proof (1°) We proceed by induction on \a\. First let |a:| = 1. Then

i[Dj,<p]Sr = Dj<pS', J = l , . . . , n ,

hence, there is a positive constant C such that

\\[Dj,<p-\S\l^C\DjcpmSlSC\<pl\.i\\Sh.

Thus in this case (1°) is true. Suppose that (1°) is proved for |a | S / - 1 . For


a multi-index a of length / - 1 , and 7 = 1 , . . . , n, we have

hence, by the hypothesis of induction we obtain

||[D,D«, <p-\S\l^ ||[D", ^>-\S\U,^\m, ^]D«5| | ,

^C|J|,4-/||*S'||, + /_i,

which proves (1°). {T) follows immediately from (1°). I

Next we examine the relation between the difference quotients and the Sobolev spaces. The results obtained will be used in §6.

Theorem 1.13.

( r ) For S e W'(J", C^), the difference quotient Af 5 is again an element

(2°) For any h with \h\ ^ 1, ifSe l ^ - ^ T " , C^), we have

| |AtS| | ,^ | |5 | | ,^i , j=h...,n. (1.46)

(3°) IfSe W'(J\C^), and for any h with 0 9^\h\^l and j = 1,... ,n,

IIAJ'SII.^M (1.47)

holds, then Se W'^\r,C^) and

| | 5 | |L i ^«M+ | | 5 | | ? (1.48)

holds.

Proof. (1°) Let

S'=lS'^cxp{i^-x)

be the Fourier expansion of 5. Translating S in the direction of Xj by -h, we obtain

r7^5^ = 1 5 ^ Qxp(i^jh) ' exp(/^- x),

hence the Fourier coefficients of rJ^S are given by S^ Qxp(i^jh). Since |Sj | = |5^exp(/^,/i)|, we have | |T7^5 | | , = ||S||,. On the other hand, the


difference quotient is given by

A^S=h-\r;'S-S\

hence for /i 7 0, we have A^Se ^^(T", C^).

(2°) ^^S' =1 S',h-\txp{ih^j)-l) exp(/^. x),

hence, using the inequahty \Qxp(ih^j)-l\^\h^j\, we obtain

siii5^r(i+iirr' A e

S\\S\\U^.

(3°) Aj"5^ = Z 5^p(ft, e ) exp(i|^- x),

where we put

pih,^j) = h-\expiiHj)-l).

By the hypothesis (1.47), we have

A i

By Fatou's lemma,

Msiiminfiii5^np(/i,f,)r(i+krr

g I I l i m i n f | 5 ^ r | p ( / , , $ ) r ( l + |^ |Y

A ^

Therefore we obtain

A e

^ 11511, +wM^ I

Proposition 1.5. Set 0<s'<l, and let 5 G iy'(T", C^). Then Se W'^^(T", C^) if and only if there exists a positive number 8 such that for


eachj=l,...,n,

I {A^SfsH'^'' dh< 00 (1.49)

holds. This being the case, there exists a positive constant C = C(s\ 8) depending on s' and 8 such that the following inequality holds:

/•oo

c-'iisii?,,,siisiiHi \\A^sr\hr'''dh^c\\s\u.. j J-oo

Proof. Since Hrj'SlI^ = \\S\\s, for h> 8,WQ have

|A;5| |?^2;I-^| |5| | , .

Hence

[ \\Afsry-'''dh^2\\srs\ h-'-'-uh=^\\s\\i J\h\>8 J\h\>8 S

Therefore the condition (1.49) is equivalent to the following inequality.

/•oo

J —oo

Let 5^ be the Fourier coefficients of S. Then we have

M S r ll\S',\'\cxp{ihij)-l\\l + \ey\hr''-'dh. J - o o A ^

By Fubini's theorem, changing the order of the integration and XA Ii » we obtain

where we put

a{$j)- \cxp{iMj)--mhr'-'dh.

Thus a{$j)>0, and for f eR, we have

ait^j) = \cxp{iht^j)- l|^|/i|-^^-' dh = I f ^ ai^j).

Therefore there is a positive constant C = €{$') such that


Consequently there is another positive constant C = C{s') such that

C(5')- ' ( l + | | | T s i + a ( | . ) + - - - + a ( ^ J g C ( 5 ' ) ( l + | f |Y' ,

C(s'nS\\U,.^ll\S',\^l + \eni + a{^,)+-- • + a(^„))

= ||si|,+i f" \\^^sfy-''dh j J —oo

^C{s')\\S\\Us'.

This completes the proof. I

The following proposition and Lemma 1.5 will be used in §2.

Proposition 1.6. Let z be a point of T", and Bîz) the ball of radius r with centrez: B^z) = {xeT"|X,- (xj-Zjf^r}. Supposes^ 1. Forcpe W^(T", C^) with supp (pc: Br{z), we put

,(z,r) = sup-7——. <P^0 \\(p\\s

Then Nîz, r) does not depend on z. If we put Ns(z, r) = Nîr), we have

l imN,(r) = 0. (1.50)

Proof. It is clear that Nîz, r) does not depend on z. Suppose that (1.50) is false. Then there is eo > 0 such that for any positive integer /, there exists <Pi e W(T" , C^) with supp (pi a B^îz) such that

lk/||s = l , \\(pi\\s-iêo'

Since {cpi} is a bounded sequence in W'{J", C^), by taking a subsequence if necessary, we may assume that {(pi} converges to cp in W*~^(T", C^). Since 5 - 1 ^ 0, we have cp e L^(T", C^), but supp cpi a B^/iiz) = {z}. Therefore ^ = 0 as an element of L^(T", C^). On the other hand, since || (Pi\\s-i = o in W^"^(T", C^), we have Hc H .i = lim/_ôo||<P/||5-i = ô, that is, (p9^0 as an element of W^"'(T", C^). Since W'-\J\ £^)^ L\J\ C ^ ) , we must have <p = 0 in L^(T", C^), too. This is a contradiction. Thus we proved (1.50). I

Lemma 1.5. Let ae C°°(T", C^), and suppose that a(z) = 0. We denote by Br(z) the ball of radius r with centre z,

(i) Let / ^ O be an integer. Then there is a positive function R{r.,l) such


that for any cp G \^'(T", C^) with supp (p ^ B,(z),

\\acp\\i^R{rJ)\\<pl

holds. Here we may choose R{r,l) so that

R{r, 0) ^ r\aU, Rir, I) r\a\^ + Ar|z|(r)|a||„, / ^ 0,

hold. Moreover we have lim^^o ^ ( ^ /) = 0. (ii) Ifl = —kisa negative integer, there exists a positive-valued function

C{r, I) such that for any (p e W\J'', C^) with supp (p a B,(z),

||a(p||/^2r|a|i||(p||f+C(r,/)|a|2|/|||^||/-i

holds.

Proof (i) Note that for XG JB,(Z),

\a{x)\^r\a\,

holds. First suppose / ^ l . Then since / is a positive integer, there is a positive constant C = C{1) such that

WacpW^^n'^' Y. \\D-a<pf = 2n'^' I ( | | a D > f + ||[a, D«](pf)

^C(r|a | i | | (p| | , + |ah||^||;_i).

Using Proposition 1.6, we obtain

\\acpl^C{r\aU + Ni(r)\ah)\\<p\\i,

hence we have

R{r,l)^C(r\a\,-hNM\ah).

In case/ = 0, ||(p||/^ r|a|i||<p||, hence we have

Rir,0) = r\aU.

(ii) First we assume that l = -kis even. Put A = D?+ • • • + D^, and let 5^ be the Fourier coefficients of 5 G W " ^ ( T " , C ^ ) . Then since the Fourier coefficients of (1 -A)"^/^5 are given by (l + I^H'^^^S^, for any integer m, we have

(*) | |S | | i _ .= | |(l-A)-'^ '^S| |L


Take a function fe C°°(T") so that 0 ^ / ( x ) ^ 1, / (x ) = 1 on B^z) ^nd that f{x) = 0 outide of B2r(z). Then / ^ = (p. From this and (*), we have

\\ap\U = \\il-Ar'^'aM\= A+ B,

where A = ||a/(l-A)-''^^(p||, and B = \\[af, {1 -A)-''^^](p\\. Since s u p p / c B2r(z), using (*) we obtain

AS2r | aM| ( l -A) - '= /VN2r | aU | , . | | _ , .

On the other hand, since

[af, (1 -A)-*^^^] = {l-Ar''\af, (1 -A)' ' /^](l-A)- '^/^

we obtain

where we use Lemma 1.4. By (*) and Theorem 1.12, we have

B^C(r)\a\2,\\<p\U-u C(r) = C\f\2,.

Combining the estimate of A, we complete the proof. If / = -fc is odd, we have only to use

| |5p_,- | | ( l -A)-^ '^^^^/ '5f + i \\DJ{l-^)-^^^'^^^Sf 7 = 1

instead of (*). I

Finally we prove the following proposition.

Proposition 1.7. Let K be a compact set of T", and let {L(,}/=i a finite open covering of K. Take a)j{x)e C°°(T", C) with suppaj>yc: Uj so that on some open set G containing K,

J

j

Then for any s, there exists a positive constant C such that for any Se W'{J", cn with supp 5 c: X,

C-'\\Sl^i \\cojSl^C\\Sl 7 = 1

holds. C = C(s) may depend on the choice of G and co.

§2. Elliptic Partial Differential Operators on a Torus 391

Proof. Since there is a constant C with || co^S || ^ C || 51| „ the second inequality is obvious. The first inequaUty follows from the existence of positive numbers C and C such that

||5||, = ||n(x)-^n(x)5||,^c||a5||,^c'i||ci;^5||,. i

§2. Elliptic Partial Ditfereiitial Operators on a Torus

(A) Estimates by the Sobolev Norm

(a) Elliptic Operators with Constant Coefficients

We set

(A(D),(p)^= Z Z a ^ a l ) > ^ A = l , . . . , / . . (2.1) |a|^/, p = l

A(D) is called a linear partial differential operator with constant coefficients, operating on vector-valued distributions witk values in C^. When ap„#0 for some a with |a | = /, we say that A{D) is of order /. The sum of the terms with \a\ = l

A(D)= I Ia^«D« (2.2) \a\ = l p

is called the principal part of A{D). For any ^G Z" and w = (w^ . . . , w^) e C^ we have 9 = (exp(/^-X))WG D(T", C^). Let w'= {w'\... ,w'^) be defined as

w' = exp(-/^- x)A(D){exp(ff • x)w}. (2.3)

Then we have

^"= I K(/^)"w-.

The correspondence w^w' gives a (/x X;u)-matrix valued polynomial A(i^) in ^ of degree /. Let A{i^) = {a{i$Yp). Then

w'^-(A(/^)w)^= I I a ^ p « / ' " l r < (2.4)

namely,

A(i^) is called the characteristic polynomial of A{D).


We obtain from Ai{D) the matrix Ai(i^) of homogeneous polynomials of degree /, whose (p, A)-component is equal to E|at=/^P«('^)"- ^i(^i) is called the principal symbol of A(D).

Definition 2.1. When the principal symbol Ai(i^) is a regular (/x x^)-matrix for every s e Z", we say that A{D) is of elliptic type. Besides the supremum of positive constants satisfying

(lk'1')"'s5|^|'(s|w1^y'' (2.5)

is called the constant of ellipticity ofA{D), where we put w" = {Ai{i^)w)^.

Lemma 2.1. Suppose that Ai{D) is an elliptic differential operator of order I which consists of only its principal part, and let SQ be its constant of ellipticity. Then for any seU and any (p e W^'^\J", C^), the following inequality holds:

||A,(D)(^||?+5?||(p||^^2-^5?||<p||?^,. (2.6)

Proof Let

(P''=1<PI exp(i^- x), p = l,...,fi,

be the Fourier expansion of cp = (cp^,..., (p^) e 1^^(T", C^). Then, using (2.2), we have

{A,(D)cpr=l I I a : « ( / ^ ) > ? e x p ( i ^ . x )

= i' 1 iAiU)<P^)' exp(/^• x), A = 1 , . . . , M.

Hence, by the definition of the constant of ellipticity, we have

\\MD)<p\\i=ii\{A,(i)<p,r\\i+\er A e

^ A

Therefore we have

|A; (D)^ | | ?+5? | | ^ | | ?^5?I I |<^^r ( l + | ^ r r ( l + l ^ n ^ A

: 2 0o||<p||5+/-

Thus we have proved Lemma 2.L


In (2.1) put l = 2m. Then by Proposition 1.3 we find that for any (p e W"^(T",C^), A{D)(pe U^""^(T",C^). Therefore by Theorem 1.11 (2°), we can define a Hermitian form on 1^"'(T", C^) by putting

{AiD)^,il,)= I ma',^D-<p',ij;') (2.7) \a\^2m A p

for (p,ilje \y"^(T", C^). Especially for cp, if; e C°^iJ\ C^), we have

{AiD)cp,ij;) = {A{D)<p,ik) (2.8)

where ( , ) denotes the inner product of L^(T", C^). Since

(A(D) ,.A) = I 1 laU\ D>-(x).A'W dx,

we can rewrite it by partial integration as

iA{D)<p,<p) = l I IbU, A \a\,\l3\^m p

D>^(x)D^(AW^x. (2.9)

Here there appear no derivatives of order greater than m in <p nor in il/. Of course such an expression is not determined uniquely by A(D). We call {A(D)(p, ijj) or {A{D)(p, if/) the bilinear form associated to A{D). This is a continuous bilinear form on W'"{J",C^), i.e. a bilinear map form W'^ir, C^) X W^(T", C^) to C.

For W = ( W \ . . . , > V ^ ) G C ^ and w ' e C ^ set

^ =. - -w - (^'^•"w\ . . . , e'^-^w^),

where ^€Z". We define a bilinear form as follows:

(A(D) e' - w, e' -"wO = (e-'^-M(D) e'^-X >v')

= Z ( A ( / ^ ) w ) V \ (2.10) A

This is the Hermitian form on C^ associated to the matrix of characteristic polynomial of A{D). The sum of the terms of the highest order is equal to

( - l ) " X ( A 2 . ( f ) w ) V ^ (2.11) X

which is the Hermitian form associated to the matrix of the principal symbol ofA(D).

Definition 2.2. If the real part of the Hermitian form (2.11) associated to


the matrix of the principal symbol of the linear partial differential operator (2.1) is a positive definite form, we say that the differential operator A(D) is of strongly elliptic type. The maximum of such positive numbers 8 that

( - 1 ) - Re Z (A2.(^)W)^M;^ ^ 8^1^^ I \w'\' (2.12)

is called the constant of strong ellipticity.

Lemma 2.2. If a strongly elliptic partial differential operator with constant coefficients AjmiD) of order 2m contains no terms of order ^2m —1, then we get, for (pe 1^^"(T",C^),

Rc{A2rniD)<p,cp)^8l\\cpf^2-'"8l\\cp\\l,,, (2.13)

where 8o is the constant of strong ellipticity.

Proof For cp = ((p\ ...,(p^)e ^^'"(T", C^), let

(p^=Z(pêxp(i^-x), A = l , 2 , . . . , / x

be the Fourier expansion of (p. Using (2.10), we have

Re(A2^(D)(^, ip) = i-ir Re 1 1 iA{^)<p,rcp', i A

since A{D) consists only of the principal part. Therefore we have

RîA2m(D)<p,<p) + 8o\\<pf^8lll(l + \^D\<p',\' A e

î-dlMl This completes the proof.

(b) Elliptic Linear Partial Differential Operators with Variable Coefficients

Let l[7 be a domain of T". We define a differential operator A{x, D) with C°° coefficients defined on an open neighbourhood of [(7] as follows: For ( P G C ° ° ( C / , C ^ ) ,

(A(x,D)(p)^(x) = Z I < ( x ) D > - ( x ) , (2.14) *' pa


where apa(x) is a complex-valued C°° function defined on an open neighbourhood of [ U], and for some index a with |a | = /, ap^M does not vanish identically. / is called the order of A(x, D). The part of the highest order A/(x, D) of A(x, D) is given by

{A,{x,D)<p{x)Y=Y. I a^„(x)D>-(x) . (2.15)

We call Ai{x, D) the principal part of A(x, D). Let w = (w\ . . . , w^) G C^,/(x) a real-valued C°° function defined on (7,

and t 3. positive number, and put

Substituting this for (p, we see that

e-''^^"U(x, D)(e''-^("V) (2.16)

is a polynomial of degree / in t. The term of degree / in t comes from the principal part, and its coefficient is given by

(A(x,Uwr=l I < ( x ) ( f ^ J « w ^ (2.17)

where ^ = ( ^ i , . . . , ^J) is given by

df(x) = e.dx' + '"-hr.dx\ (2.18)

which is a cotangent vector of t/ at x. Namely, A/(x, ^ x) is a (/A X ^)-matrix which is determined by giving a point (pc, df(x)) on the cotangent bundle of C/. This ()Lt X/x)-matrix-valued C°° function A/(x, f ) defined on the cotangent bundle T* U is called the principal symbol ofA(x, D). If we take ^- X as a function/(x), the value of the principal symbol is equal to

(A,(x,ff)w)^=I Z < ( x ) ( f ^ ) w - (2.19) P | a | = /

and this is the part of

e - ' "%(x , D)(e'"-^w) (2.20)

of order / with respect to .

Definition 2.3. Suppose a linear partial differential operator A(x, D) of order /, is defined on a neighbourhood of [[7]. We say that A(x, D) is of elliptic type, if there is a positive number 8 such that for any point (x, ^x) T^U


with x 7 0 and for w 6 C^, the principal symbol satisfies the inequality

l\Mx,i^)wr\'^S'\i^fl\w'\\ (2.21) A A

The maximum of such 8 is called the constant of ellipticity.

In other words, A(x, D) is of elliptic type if and only if for every point Zoe[U], the differential operator A(zo, D) derived from A(x, D) by replacing the coefficients ap^(x) of A(x, d) by ap«(zo) is of elliptic type in the sense of Definition 2.1.

We want to show the corresponding fact to Lemma 2.1 concerning elliptic differential operators with variable coefficients. We introduce the following quantity: for /c = 0 , 1 , 2 , . . . ,

M f c = I Z I maxjDâ^«(x)|. (2.22)

Also we use the following notation:

Ct{ L/, C^) = {(p G C^(T", C^) I (p(x) = 0 for any x e U},

W'oi U, C^) = {the closure of C^( L/, C^) in W'(J\ C^)}.

Theorem 2.1 (Local Version of the a priori Estimate). Let A{x, D) he an elliptic linear partial differential operator defined on a neighbourhood of [ L ] and let 8Q be its constant of ellipticity. Then for sel. there is a positive constant C = C(s, 6, M,,|) such that for any cp e W'o^\U,C'')

| |<p||,^,^C(||A(x,D)(^||, + ||(p||J. (2.23)

Proof In order to prove this theorem, it suffices to show (2.23) for any cp e CîU, C^). We choose a sufficiently small e such that

2enVMi<2~^-^/^6o.

Cover [L' ] with open balls Bi,..., Bj of radius s. Furthermore we choose real-valued C°° functions OJJ(X)J = 1,...,J, defined on T" such that supp (Oj c: Bj and that Y.j (^jM = 1 on [ U]. For any (p e C^( U, C^), we set (pj(x) = (Oj(x)(p(x). Since

J

\\<P\\s^l=\\l(Oj<p\\s+l=\\l(Pj\\s+l^l \\<p\\s+h J J J

there exists some 7 such that the following inequality holds:

\\cpjlî^J-'\\cpl^, (2.24)


Let Pj be the centre of the ball Bj. We write the principal part Ai(x, D) of A(x, D) as in (2.15), and let Ai(pj, D) be the partial differential operator with constant coefficients obtained from Ai(x, D) by replacing its coefficients ^paM by their values a^podPj) at ;?,. This is an elliptic differential operator with 8Q as the constant of ellipticity. Hence (2.6) holds. Therefore we obtain the following:

||A,(p,-,D)(p,-||, + 6oi|(p,L^2-^/Xlk,L+/. . (2.25)


{{Ai{x,D)-A,{pj,D))<pjY{x)= Z Z « ( x ) - a ^ « ( / 7 , ) ) D > ; ( x ) ,

by Lemma L5 in §1, there exists a positive constant C ( E ) such that

\\l{a'pAx)-a';,^{pj))D-<p%^2eM,\\<pj\U^^

Therefore by the choice of e,

\\{Mx,D)-A,{pj,D))cp,\l

£nV(2eM,||<p,.||,+,+ C(e)M2|,|||<p,.||,+,_,)

s2-^-'/^5o||.p,||,+, + nVC(e)M2|,|||.p,.||,+,_,. (2.26)

Further, since A{x, D) — Ai{x, D) is a differential operator of order S / - 1 , there is a positive constant Ci depending only on s, I, /x and n such that

\\iA,(x, D)-A{x, D))cpjl^C,M^4<p\U,_,.

Combining this with (2.26), we have

\\A,{pj, D)<pl^\\Aix, D)cpj + {A,{x, D)-A{x, D))cpj

-{A,{x,D)-A,{pj,D))<pjl

S||A(x, D)(iPy||, + CiM|,|||(p;||,+,_,

+ n'fiC{e)M2f4<Pjl^,., + 2-'-'^'do\\iPj\U,. (2.27)

Since the commutator [A{x, D), co] is a differential operator of order S / - 1 , there is a positive constant C2 such that

\\[A(x, D), w>, . | | , g C2M|,|||.p||,+,_i.


Hence

\\A(x, D)<pjl^\\<ojA{x, D)<PI + \\[A{X, D), a>,.]<pL

s||ft>,.A(x,D)(p||, + C2M|,|||<p||,+,_,. (2.28)

Since there exists a positive constant C3 depending only on s, I, n, fi such that

\\o>jAix,D)(plsC3\\Aix,D)<p\l,

by (2.25), (2.27), (2.28) and the above inequality, we obtain

2" '^ '5O| |<P_, | | .+ ,SC3| |A(X,D)<PL + 5O||<PJ||, + C,M|,| | |<P,.| | ,+,_,

+ C2M|,||i^||,+,_i + nVC(e)M2|,|||(p,||,+,_i

+ 2-^-'/^5o|k,||.,,.

By transposing the last term of the right-hand side, we obtain

3 ' 2-'-^^'8o\\<Pjl^i^C,\\Aix, D)<PI + SolWjl

+ CiM| , | II ^ -II ,+^_i + C2M|,| II (p II,+/_!

-\-n',jLC{e)M2i4(Pj\\s+i-i, (2.29)

Since we can take a positive constant C4 such that

||(p^-||,^C4||<p||„ ||(Py||5_,/-i^C4||<p||5+/-l

we get by (2.24) and (2.29),

3'2-'-'^'8oJ-'\\<p\Ui^CMix,D)<pl-hC,8oM^ (2.30)

where we put C5= ClC4M|5l^-C2M|^l + nV^(^)^^2|5|• Taking a suitable tin Proposition 1.1, we obtain a positive constant Cg depending only on s, /, n and /JL such that

C5||(p||,^,_i^2-^-^/^5o/-^||<p||,^,+ Q||(p||,

holds. Therefore by this and (2.30), we obtain

2-^-^/^5o/-^||<p||,,,^C3||A(x,D)^||, + (C45o+Q) |kL ,

which completes the proof. I


Put / = 2m and let w = (w\ . . . , w' ) and w' = (w' \ . . . , w'^) be vectors in £^. For each (x, ^^) e T* U, we can define a Hermitian form in w, w'

( - l ) ' " I ( A 2 . ( x , ^ J w ) V ^ A

associated to the (/x x^)-matrix A2m(^, i^x) for the principal symbol of A{x,D),

Definition 2.4. We say that A(x, D) is of strongly elliptic type if the above Hermitian form associated to the principal symbol satisfies the following condition: There is a positive number 8 such that for any (x, f^) G T* (7, ^^ T 0 and for any weC^,

( - 1 ) - R e S (A2^(x, ^Jwrw' S S^l^.p- I |w^p. (2.31) A A

The supremum of such 8 is called the constant of strong ellipticity.

Theorem 2.2 (Local Version of Garding's Inequality). Let U be a domain of T". Suppose that a linear partial differential operator depending on a neighbourhood of [ V] is of strongly elliptic type and that 8 is its constant of strong ellipticity. Then there exist positive constants 8^ and 82 depending only on 8, m, fjL and M^ such that for any (p e W^{ U, C^)

Re(Aix,D)<p,cp)-^S^<pf^8l\\<p\l (2.32)

holds.

Proof For any (p,ilfe CT{ U, C^),

(A(x,D)(p,(A)= I I I aU^)D-cp'{x)ilfHx)dx. \a\^2m p,A J T "

By integrating the terms with | a | ^ m + l by parts (|Q:|-m)-times, we can rewrite this into the equality containing no partial derivatives of order ^ m + 1 as follows:

(A(x,D)vp,(A)= I 1 I b',^^{x)D-<p'{x)D^il^'{x)dx, (2.33)

where bp^p{x) is a linear combination of derivatives of order at most m of ^p'a'(x). The sum of the terms with |a | = |/3| = m is given by

(A2^(x,D)(p,(A)= Z I I fc^«^(x)D>^(x)DVW^x (2.34) | a | = |/3| = m p,A J t"

^ - 2 m ^ ^ ^ ^ n W ^ v - ^ .>' ' /(^)w . , ' ' / ( ^ ) w ' t-*oo


Let A2m{x, ^x) be the matrix of the values of the principal symbol of A(x, D) at (x, f^) E T* t/. For w, w' G C , we have

( - I ) ' " Z (A2.(x, ^Jw)^w'^ = Z Z ^:«^(x)^r^>v"w'\ (2.35)

In fact, let / ( x ) be a real-valued C°° function defined on U and g(x) any complex-valued C°° function v^ith support in U, Put ^^ = df{x) with <i/(^) ^ 0. Substituting

(p{x) = g(x) e' ^ ^ w, (A(x) = e ' ' w'

into (2.33), by the definition of A2m(x, ^x) we obtain:

I = lim r^'"(A(jc, D)g(x) e"^^ 'w, e"-'* 'w')

l ^ " |a|Sm.|;8|Sm p,A J T "

= I I [ fe^.^(x)(i^J«(-i|J''w''w'^g(x)^x.

|a | = |^| = m p,A JT"

Since g(x) is arbitrary,

( - l ) " " I ( A 2 . U ^ x ) w ) V ^ = I I C ^ ( x ) i r ^ w ' ' w ' ^ A |a | = |^| = mp,A

holds at each x. Since every non-zero element of T* U is written in the form of df{x), we obtain (2.35).

For any e > 0, we can cover [ L ] by a finite number of open balls Bi,..., Bj of radius e. Let Pu • - • ^PJ be the centres of Bi,..., Bj, respectively. We may assume that s is so small that the inequality

en^'^M^^,<2~'^-^8^ (2.36)

holds. Take real-valued C°° functions a>y(x),7 = 1 , . . . , / on T" with supp (Oj cz Bj such that

l(Oj{xy=l, xe[Ul (2.37)

For any cpeC^iU^C^), put (pj(x) = (Oj(x)(p(x). Then supp (pja Bj, Now denote by A2m(Pp D) the partial differential operator with constant coefficients obtained from (2.34) by replacing bp^pix) by its value at x = Pj,


i.e. we put

(A^^iPj, D)cp(x)r = 1 1 hU,{Pj)D-^'^'{x). (2.38)

Then we see by the hypothesis of strong ellipticity and (2.35), that A2miPj, ^ ) is a strong elHptic differential operator with the constant of strong elHpticity 8. By Lemma 2.2, we get

Re(A2.(;7„D)(p„<p,) + 5^lk,f^2—^5^| | (^ , | |L (2.39)

By the way, we have

Re(A2„(x, D)(pj, (pj) g Re{A2„{pj, D)<pj, <pj)

- \iA2^(x, D)(pj, ipj) - {A^^iPi, D)<pj, <pj)\. (2.40)

We shall take an estimate of the second term of the right-hand side.

( A 2 „ ( x , D)(pj, <pj) - {A2„,(pj, D)<pj, <pj)

= I I [ {bUi^) - b^„^(p,))D>j ' (x)DV;(^) dx. | a | = |/3| = m p,A J T "

Since |x -p^ | g g for x supp (pj, we have

P

Hence we obtain the following estimate:

| ( A 2 ^ ( X , D)ipj, CPj)-{A2m{Pj, D)<pj, 9 , - ) N e n ' ^ M ^ + l l k . l l m .

By this inequality and by (2.36), (2.39) and (2.40), we obtain

Rt{A2rn(x,D)<pj,cpj)^2---'8^cpj\\l-8^<pj\\\ (2.41)

Hence by (2.37),

iA{x,D)cp,<p)= I II | a |Sm p,X J | 0 | s m

= l{A{x,D)<pj,<pj) J

+ Z I Z [ b',^^{x)[wj, D'']<pfix)D''<p'{x) dx \a\^m p,k j J

+ I Z Z I b'^^,(x)D-cp}(x)[coj, D^]<p'(x) dx. \a\^m p,\ j J


By Lemma 1.4 there is a positive number Cj depending only on m, /JL, n, and 8 such that

Re(A(x,D)(p,(p)^IRe(A(x,D)(p,-,(p,-)-CiM^||(p|U||(p|U_i. (2.42) j

On the other hand,

Re(A(x, D)<pj, <pj)^RQ{A2m{x, D)<pj, <pj)~\(A(x, D)cpj, <pj)

-{A2m(x,D)(Pj,(Pj)\

Making the difference of (2.33) and (2.34), we have

((A(x, D)(pj, (pj)-{A2mix, D)(pj, (pj)

= T l \ b^«^(x)D>^(x)DVW^:^ \a\^m p,\ J T"

where the summation X' is taken over all the terms either with \a\^m~l or with |j8| m - 1 . There exists a positive number C2 such that

| (A(X, D)(Pj, (Pj)-{A2m{x, D)(Pj, (pj)\^C2Mm\\(Pj\\m\Wj\\m-l-

Combining the inequality with (2.41), we can find a positive constant C3 depending only on m, JJL, and 8 such that

Rc{A(x, D)<pj, cpj) g l—'d^ <pjIIi - S i ^ , f - C3M„ II.p,. |U_iII<p,II„.

By the above and (2.42), we have

Re(A(x, D)<p, <p)^2-'"-'8'l hjWi-S'iyjf J J

-C^Mml, \\(Pj\\m\\(pj\\m-l j

-C,Mj<p\U\cp\\^_,. (2.43)

There exists a positive constant C4 such that

||(;p^|U^C4||<p|U, ||(p^-|U_i^C4||<p|U-i.

On the other hand Xj ll<^7ir= ll<Pll ^^^ ^Y Lemma 1.7 there is a positive number C5 such that

§2. Elliptic Partial Differential Operators on a Torus 4 0 3

Therefore putting C^= C^JCl+ Q , we get by (2.43)

R^(A{x,D)cp,cp)^2—'Cfj-'8^cp\\l-8'\\<pr

-QM^||(p|U||(p|U_i. (2.44)

By (1.34) in Proposition 1.1, we have

By choosing a sufficiently small t in (1.33), we can find a positive number C7 such that

MJ<p\\^^,îC,MJ-'2-'"--'CfJ-'8'Ml+Q\Wf.

Therefore by (2.44) and this inequality, we obtain

Re(A(x ,D)(p , (p )^2—^C5-V-^5^ |k f . - (C , + 5^)||(pf. (2.45)

Hence (2.32) holds for any (peCoiU.C^)- Since Co(U,C^) is dense in WîU.C'^) and both sides of (2.32) are continuous in (peWîU,C''), (2.30) holds for every (p e W^{ U, £^). I

(B) Estimates by the Holder Norm

(a) The Case of Constant Coefficients

Set 0 < ^ < 1 . We say that a vector-valued function (p(x) = (<p^{x),..., (p'îx)) e C^(T", C^) is of class C^-"' if

^ V | D V ( X ) - D V ( X O | _ ^ I Z sup ——, <oo.

The totality of such functions (p is denoted by C^^^(T",C) . For (pe C^^\J\ C^) we define the norm by

. . ^ V V | D V ( X ) - D V ( X O | kk+« = k l f c+Z I sup , ne • (2.46)

A = l \a\ = k | x - x ' l ^ l |X —X j

^fc+0^ jn^ C^) is then a Banach space. The same statements as in Lemma 2.1 can be obtained by replacing || \\s-i

and II \\s by | Ifc+e and | Ik+i+e, respectively. The proof for the general case is, however, somewhat long. So we restrict ourselves to the most simple case. We assume that the order of the elliptic differential operator is 2 and


that A2(D) is of diagonal type, namely, we assume that A2iD) satisfies

{A2(D)<p)'{x)= i a^jD.Djcp'ix), A = l , . . . , / x . (2.47)

Furthermore, we suppose that a^ is constant with a^ = aj; for /, j = 1, 2 , . . . , n and that the operator is strongly elliptic. In other words, there is a positive constant 8 such that, for any ^ G R "

- R e i a^^,^j d^l fi. A = 1 , . . . , At. (2.48)

By the assumption that A2(D) is of diagonal type, the proof is reduced to the case /x = l. In what follows, putting /it = 1, we shall consider the strongly elliptic partial differential equation A{D) acting on C^'^^^^(T", C)

A{D)<p{x) = I a'^D,Dj<p(x). (2.49) u

Let A = (a'^) be an {n x n)-matrix with a' = a^\ Let B = (a^) be its inverse matrix.

We introduce the quadratic form on R" by

Q(x) = la,jx'x\ (2.50) u

Define the constant C(n) by

C{n)(n-2) \ Qizy^'^' da,(z) = 1, (2.51) J|z!=i

where dai(z) stands for the surface element on the unit sphere \z\ = 1. Set

E(jc) = C(n)(?(x)^'-"^/l

Then

AE(x ) = - ( n - 2 ) C ( n ) ( ? ( x ) - " / ^ ( l a , x ^ ) , (2.52)

and

DiDjEix) = (n-2)C(n)Q(x)-"^^

x(-a .^ + n ( ? ( x ) - ^ ( l a , . , x ^ ) ( l a y ) y (2.53)


From these equalities, we can easily see that

A{D)E{x) = 0

for X G R" with X9^0. Let p ( 0 be a C°° function if t such that p(r) = 1 for \t\<i and that p(t)^0 for | ^ |> | . Put ^{x) = p{\x\). If we put

g{x) = ax)Eix),

then we find that g(x) = 0 for |x| ^ | , and that

Djg{x) = ax)DjEix) + Djax)Eix), (2.54)

D,Djg{x) - ax)D,DjEix) + co,j(x). (2.55)

Here (Oij{x) denotes a C°° function with

SUppWyC:{x | | ^ |A: |^ |} .

Thus there is a C°° function w such that for x G R" with X9^0,

A(D)g(x) = w(x) (2.56) and that

SUpp W C l { x | 5 ^ | x | ^ | } .

We extend the functions g(x), cuy(x) and w{x) to the periodic C°° functions on R", and identify them with those on T^ We denote such function on T" by the same notation g{x), Wy(x) and w(x). g{x) is called a parametrix for the differential operator A/(D). In the general case, it is rather difficult to construct a parametrix.

Lemma 2.3. Setf{x) = A(D)u{x) for u e C^(T", C^). Then we have

u(x) = -{ JT"

g{x-y)f(y)d:y-\- w(x-y)u{y)dy. (2.57)

Proof. Set 0 < ^ < 1. Put a , = {>; G T" I ly - x| > s} for a fixed x. By means of Green's formula,

A(Dy)u{y)g(x-y)dy- u{y)A(Dy)g{x--y)d-y n,

{Av'Dy)u(y)g{x-y)da{y)

\ {Ap-Dy)gix-y)u{y)da{y), (2.58)


where Dy stands for the differentiation with respect to y, and (Av Dy) = X a'-'V,((9/^>'') with ^ = ( ^ 1 , . . . , ^„) the outer unit normal vector to dO.e'do'^y) denotes the surface element of afl^, and A{Dy)g{x-y) = w{x-y). Since \{Av' D)M(>^)[^ C|M|I and \g{x-y)\^CQ(x-y)^^-''^^^ on aHe, we have

lim e^O J . lim {Av D)u{y)g{x~y) dcT{y) = 0.

On the other hanc}, since

x-y

\\x-yy""\x-y\)' \x-y\

by (2.52), we get

(AvDy)g{x-y) = in-2)C{n)Qix-y)-''^'\x-y\

for x,y with 0<\x-y\<l, Jience we have by (2.51)

lim {Av D)g{x-y)u(y) daiy) = u{x).

Therefore putting e -» 0 in (2.58), we obtain

u{x) = - \ g{x-y)f{y)dy-h\ w{x-y)u{y) dy. I JT" JT"

Lemma 2.4. As for the integral transformation Hf{x) = jj^ h{x-y)f{y) dy, we get the following.

(1°) Ifhis integrahle and f is continuous, then Hfis continuous and

Hf{x)=[ h{y)f{x-y)dy.

Moreover the following estimate holds:

\mo^N{h)\f]o,

where we put

N{h)=\ \h{x)\dx.


(2°) If for j = 1, 2 , . . . , n, Djh is integrable, then Hf{x) is of class C \ Set Ci =\-jn \h{x)\ dx + Y^jljn \Djh{x)\ dx, then we have

l«/l. = C,\f\o.

(3°) Put fcso. If Wh is integrable for any a with \a\Sk then Hfe C'^iJ", C) and

\Hf],SC\f]o

where C = l^„^^,NiD''h).

The proof is easy, hence it is omitted. By Lemma 2.3 we write

M(X) ^ Gfix) + Wu{x), (2.59)

where

G/(x) = - | g{x-y)fiy)dy (2.60)

and

WM(A:)= wix-y)u(y)dy. (2.61) JT"

By Lemma 2.4, for any integer /c 0, we have

\Wu\,^CMo (2.62)

where C , = I | ^ , ^ , N ( P " w ) .

Since Djg{x) is integrable for any j , we obtain from Lemma 2.3,

DjGf(x) = - \ Djgix-y)f{y)dy=\ ^g{x-y)f(y) dy.

L e t / G C ^ T " , C ) . Thqn, by integration by parts, we obtain

^Gf(x) = -l , JT"

Hence, f o r / e C\J", C), we have, for J,J = 1 , . . . ,« ,

D,D,.G/(x) = - [ D,g{x-y)^f{y)dy.

f (^-> ' )—/( j ' )^J ' -ay,-


Since {dldyj)f(,y) = {dldyj){f{y)-f{x)), we obtain

D,DjGf{x) = - I D,g{x-y) ^{fiy)-f{x)) dy. JT" dyj

¥ui QL^={y eJ"\\x- y\> e). Then

D^DjGfix) = - l im I D , g ( x - y ) ^ { f { y ) -f{x)) dy. =-o Jn. dyj

By integration by parts, we get

DPjGfix) = lim i •^D,g{x-y){f{y) -f{x)) dy «-o Jn^ dyj

•l imf i ^ ^ A g ( x - > ' ) ( / ( } ' ) - / W ) d c r ( g ) . (2.63)

By the way for 6 with 0<d<\, there exists a positive number C such that

k'-yl \x~y\

\D,g{x-y)\ \f{y) -f{x)\ S C\x - yr-^'^e.

Therefore the second term of the right-hand side of (2.63) vanishes. Consequently, we have

D,DjGf(x) = I ^D,g{x-y)ifiy) -f{x)) dy

= - [ DjD,g(x-yKfiy)-f{x))dy. (2.64)

The following fact is important for this integral transformation.

Lemma 2.5. Suppose that 0 < e ^ i Then

DjDig{z)dcr,{z) = 0, f,j = l , 2 , . . . , n , (2.65) ^1^1 = 8

where dcr^iz) denotes the surface element on the sphere \z\ = e.

Proof. Suppose 77(0 is a C°° function of teU with v(t)^0 such that 7^(0 = 1 on \t\<2~^ and that 7/(r) = 0 on | r | > i Then we get

o o > M = D,g{z)Djrj{\z\) dz = \im D,giz)Djrji\z\) dz, IT" ^^0 J |z |se


since D,T/(|Z|) vanishes on |z| <2"^. Besides, since g{z) = E{z) on |z| < 2 " \

M^= D,g{z)DM\z\)dz }\z\>e-I

DjD,g(z)v{\z\) dz- f-A-g(z)T7(|z|) dcT^z) J\z\^e J\z\ = e\Z\

V{t)t-' dt DjDiE{z) da,(z)

-vis) I ^D,E{z)dcT,iz). J\z\ = l\z\

M, + T){E) f-D,E(z)da,(z)

V{t)r^ dt DjDiE(z) da.iz).

Putting e ^ 0, we see that the left-hand side converges to a finite value. Since

[' Vi Jo

Hence

](t)t ^ dt = oo^ Jo

we must have

DjDiE(z)da,(z) = 0. J|z| = l

Thus we see that, for any s with 0 < |e| < | ,

DjD,giz)dcr,{z) = 8-'p{8) DjD,E(z)da,(z) = 0, J\z\ = e J|z| = l

holds, which completes the proof of the lemma. I

Lemma 2.6. Suppose that 0 < ^ < 1. I f / e C^(T", C), then for ij = 1, 2 , . . . , n DjDiGFe C^(T", C), and there exists a positive number C such that

\DjD,GJ]e ^ CN,{g)\J]e, iJ = 1, 2 , . . . , n, (2.66)

where we put

N,{g)= I sup |zr |D"g(z) |+ I sup |zr^^|D«g(z)|. (2.67) \cc\^2 0 < | z | ^ l |a | = 3 0 < | z | ^ l


Proof. For 1 > p > 0, we set

/ ( x , p ) = I Dp,g{x-y){f{y)-f{x)) dy.

Denoting by o-{n) the area of the unit sphere, we have

/(x, p) ^ N,{g)\J]e [ I z r - dz = e-'a(n)N,(g)\f]ep'. (2.68)

Since g{z) = 0 on |z| 1, for p > 1, we have

I{x,p) = Iix,l) = DjD,Gf{x).

Hence by (2.68)

|D,AG/lo^ ^"V(n)N3(g) | / | , . (2.69)

Next suppose \x-x'\ = E<JO. We shall estimate DjDiGf{x)-DjDiGf{x'). First we write

DjD,Gf{x) = [ D,-Ag(x -}^)(/(y) -fix)) dy,

D,D,G/(x')= I DjD,g{x'-y)ifiy)-f{x')) ay + Iix',5s). J\y-x'\>5e

By Lemma 2.5, we can rewrite DjDiGf{x') as follows:

DjD,Gf(x')= I D ,Ag(x ' - j ) ( / (> ; ) - / (x ) ) tfK + /(x ' ,58)

DjD,g{x'-y)dy. \x'-y\>l/2

}\y-x'\>5e

+ {fix)-fix'))

Hence

DjD,Gfix) - DjD^Gfix') = J, + J^ + J,- lix', 5e), (2.70)

where

/ , = I {DjD^gix -y)- DjD^gix' -y)}ifiy) -fix)) dy, J |> ' -x ' |>5e

J\y-x'\<5e J2=\ DjD^ix-y)ifiy)-fix))dy, (2.71)

l\y-x'\<

J, = Mifix)~fix')),M = DjD^giz) dz. | z |> l /2


If \y-x'\ ^ 5 |x-x ' \ , then the line segment connecting y-x with y-x' does not pass through singular points to g{z). Thus, by the mean value theorem, we see that there is a t with 0<t<l such that

DjD,g{x-y)-DjD,g(x'-y)= J {x^-x'')Dj,DjD,g{tx-^{l-t)x'-y). k=i

\y-x'\> 5s = 5\x-x'\ implies | x - } ; | > 4 | x - x ' | , hence

\tx-^{l-t)x'-y\^\x-y\-\x-x'\^2~^\x-y\.

Therefore we have

\DjD,g(x-y)-DjD,g(x^-y)\^2"'-'N,(g)8\x-yr-\

Since £ < 10~\ we see that \x-y\^l-^ s<l for y with DjDig(x-y)-DjDig{x'-y) 7^0. Hence

| / i l ^ f 2"^'N,{g)8\x-yr'\Ae\x-y\'dy J\y-x\>4e

S 2"" V(n)N3(g)m« • e(l - 0)- '(4e)«- ' (2.72)

since \f{x)-f(y)\^\f\e\x-yf there. On the other hand, since |> ' -x ' |<5e implies that | j ' - x | < 6 e , we have

U,l^ \DjD,g{x-y}\\f{y)-f{x)\dy \y—x\<6e

SN,ig)\f]e\ \zr"dz=e-'N,{g)\f\e(6er. (2.73) J|z|<6e

Applying (2.68), (2.72), (2.73) and \J,\S\M\\f[ee' to (2.70), we get

\DjD,Gfix) - DjD,Gf{x')\ S C2N,(g)\f\ee'

for e < 10~\ In the case of e g 10^', since

|D,.D,G/(x) - DjD,Gf{x')\ S 20|D,A.Gyio

we finally see that there exists a positive constant C3 such that for |x — x'| ^ 1,

|D,D,G/(x) - DjD,Gf{x')\ g CN,(gM,\x - x f

holds. By this inequality and (2.69), we obtain the lemma. I


Lemma 2.7. LetA2{D) be a second-order strongly elliptic partial linear differential operator with constant coefficients of diagonal type which operates on vector-valued functions with values in C^. Suppose that its lower terms are equal to zero. Assume that 0<0<1. For any integer /cÔ, there exists a positive constant C depending only on 6, k, n, fi, and the coefficients ofA2iD) such that, for any u e C^^^^'{J\ C^),

Proof In case k = 0, the lemma follows from Lemmas 2.3, 2.4 and 2.6. For general /c, since we have

A 2 ( D ) D " M ( X ) = L>"A2(D)M(X)

for \a\ = k, by the estimate for the case of /c = 0, we have

\D-u\2ê^C{\D-A2(D)u\e^\D-uU).

On the other hand, since for any positive integer /c, there is a positive constant Ci such that

C^'M^ê^ I \D"u\e^CMkê, \a\^k

the lemma is proved for general k. I

Remark. In the general elliptic case, we have to construct a (yiix^)-matrix-valued function E{x) corresponding to that for the above case such that A{D)E(x) = 0. But is takes a long procedure. The other treatment is quite similar to the above.

(b) The Case of Variable Coefficients

Lemma 2.8. Suppose that 0<r<s. Then there are positive constants Ci and C2 such that, for any (p e C^(T", C^) and for t > 0, the following interpolation inequalities hold:

Proof We have only to prove the case /x = L (1°) The case 0 < r < 5 < L We set

A = sup \x-x'\~''\(p{x)-(p(x')\ \x-x'\^l


for (p e C^(T", C). There exist points x and x' with \x-x'\ ^ 1 such that

\x-x'\-''$p{x)-(p(x')\>2-'A.

Hence we obtain

\x - x t S 2A-\\<pix)\ + \<pix')$ S 4A- 'k |o. (2.74)


\x-x'r\<pix)-<pix')\s\x-x'r'\<p\s, we have

hence we obtain A ^ 4^'~'^^'2'^'$p\'/'\(p\l-'^\ Combining this with the trivial inequality |< |o |<p|5' 19lo~' ^^ we obtain

\<p\r^4'-^'r^'\cpV/'\<p\l-'^'. (2.75)

(2°) The case 0 < r < s = l.By virtue of the mean value theorem, we have

|x - x'\-'\(p(x) - (p{x')\ \x- xf'-'^^'-M,.

By this and (2.74), we see that

2-'A^(4A-'\<p\oy-'\ip\v

Therefore we obtain A^2''4^~''\(p\[\(p\l'~\ Thus we have

\cpl^r4'-'\cp\[\<p\'o-\ (2.76)

(3°) The case r 1< s < 2. Put 5 = 1 + ^. There exist a point XQ and j such that

M = 'ZmdLx\Dj(p{x)\^2n\Dj(p(xo)\. J xeT"

Let y be the point obtained from XQ by the translation in the direction of x by r > 0. By the mean value theorem, we find ^ between XQ and y such that

cpiy)-(pixo) = tDj<p{^).

This leads to the following estimation:

\Dj<pU)\^t-\\<p{y)\-h\<p{xo)$^2ncp\,. (2.77)


Since .p 6 C'-'^CT", C),

\Dj<pixo) - Dj<p{e\ S \xo-i\'>\Dj<p\e.

Hence

\Dj<p{Xo)\^2t-'\<p\o+t'\Dj<p\e^2r'\<p\o+t'\<pU^e

and therefore MS2n(2<~*|^lo+ t^\(p\i+e)- Taking the minimal value of the right-hand side with respect to t, we see that there exists a positive Cg such that

So we can choose a positive constant C'e^l such that

l<p l .§c i l<p |y>l r ' ' ^ (2.78)

(4°) The case r = l<s = 2. For <peC\J",C) we have \Dj{<pixo)-Dj(pi^)\^\xo-^\\(p\2- Hence, combining this with (2.77), we obtain

\Dj<p{xo)\^tW\2 + 2r'W\o^4\<p\l^^\<p\l''.

Hence we have M § 2 n • 4|(p|2^ |(pjy^. Thus we get

|<p | ,S8n |^m«pi r . (2.79)

By (1°), (2°), (3°) and (4°), we have proved the lemma for r g 1, s g 2 . The general case shall be proved by applying them repeatedly as follows: Suppose 0 < fl < 1. For some positive constants Cj, C2, C3 and C4, we have

\Dj<p\e^C,\Djcp\t\Dj<p\y

^c,i\DM\i'r'lDMo'''^'Y\Dj<p\l-'

gc3iD,,pi?ir'"(kUi'r''Vio*^''"^^)''""''*.

Hence we see that by choosing a suitable positive constant C5, we have

I^K,,sc,kl<VV''*^"''Vir<'"'" ' "'". I (2.80)

Lemma 2.9.

( r ) Suppose that 0<0<l. For a,/G C^(T",C),

| a / | , ^ | aUmo + |a|o|/1..


(2°) Let A{x, D) be a linear partial differential operator of order I which operates on C^-valued functions. A(x, D) is given as follows:

{Aix,D)cpnx)=l I aUx)D''<p''ix). A |a |g /

For 0 with 0<6<1, and for any integer /cÔ, we define Mj^ by (2.22) and put

Mfc+ = Mfc+X X I sup ^ ;7^ . A,p |a |^/ |/3|^fc |x -x ' |< l \X — X\

Then for any integer /c 0 there exists a positive constant C depending only on /, /c, 6, n and fx such that

| A ( x , D)(p\k+e ^ CMk+e\<P\i+k+e'

The proof is easy. We leave it to the reader.

Lemma 2.10. Let z he an arbitrary point of T". Let Br{z) = {x6T"|S^. (x^-z^)^<r^} be a ball of radius r<\. Let 0 < ^ < 1 , and k a non-negative integer. Then^ for any (p e C^CT", C^) with supp cp <= B^(z), we have

\<p\,^r'\<p\j,ê^ (2.81)

Proof Since cp is equal to zero outside of Bîz), for any a with \a\ = k and any x e Br(z), we have

| D > ( x ) | ^ r ^ | D > | , .

Hence we get (2.81), because \a\^k-1 implies \D"(p(x)\^ r\D"(p\i. I

Lemma 2.11. Let A be an operator of multiplication, i,e.,

{A<p)\x) = i a',{x)<p''{x). p = i

Suppose 0 < ^ < 1 and let k be a non-negative integer. Suppose that a^s C'^+'CT", C) and that

a^(z) = 0, A,p = l , 2 , . . . , ^ .

Then, there exists a positive constant C depending on k, 0 and fi such that

\A<p\i,ê^Cr'Kkê\(p\k+e


for all (p e C^'^^(J\ C^) with supp (p c B,(z), where we set K, = S , ^ \aXfor any sÔ.

Proof. There exists a positive constant Ci such that

From (2.81) we get

| A ( p | f c ^ Q r X k k + . . (2.82)

On the other hand, for a multi-index a with |a | = /c, we have

Hence there is a positive constant C2 such that

| D " A ( P U ^ I max |a^(x) | |D>|e + X,|D>|o+C2i^.+,|(^k_i^,. A,p l^-zNr

On the other hand, since, for |x — z| < r,

holds, we have

\D-A<p\e ^ /x V X p D > | , + Ke\<p\, + C2K,ê\<p\k-rê^

Applying Lemma 2.10, we see that there exists a positive constant C3 such that

\D"A<p\e^C,r'Kj,ê\<p\kê>

Combining (2.82) with this, we obtain Lemma 2.11. I

Let L be a domain of T". Suppose that a second-order linear partial differential operator A(x, D) with C°° coefficients defined on an open neighbourhood of [L^] operates on <pG C°°(T", C^) in the following way:

{Aix,D)<pnx) = l I aUx)D-cp'(x), (2.83)

We say that this operator is of diagonal type in the principal part, if ap^(x) = 0 for A 7 p and \a\ = 2.


Theorem 2.3. Let U be a domain in J". Suppose that the second-order linear partial differential operator A{x, D) with C^ coefficients defined on [ U] is of diagonal type in the principal part and strongly elliptic. Let 8 be its constant of strong ellipticity on [U\ Suppose 0<6<\. Then for any integer kÔ, there exists a positive constant C depending on /i, /x, k, 6, 8 and Mj^+0 such that for any cp e C^-^^^^{J\ C^) with supp (paU,

\cp\k+2ê ^ C(\A(x, D)9U+, + |(p|o).

Proof This is verified in an almost similar way to Theorem 2.1. We have only to use the norms | 1 + and | \k+2+e ii place of || || and || \\s+b respectively. We cover [U] with open balls B^, B2,..., Bj of radius s. Let Cj be a positive constant in Lemma 2.11. For any ze[U], let €2(2) be a constant in Lemma 2.7 for the elliptic linear partial differential operator obtained from the principal part of A(x, D) by replacing coefficients by their values at z. Take e so that for any ze{U\ CiC2{z)e^Mk+e<2~^. Choose C°° functions cOj{z), j = 1,2,... ,J, on T" such that supp coj c= Bj and that I^. (Oj(x) = 1 on [ U]. Put <pj = ojjcp for any cp e C^^^^'(T", C^). Then we have

Mk I'Pj J k+2+e

We see that for some /, the following inequality holds:

\<Pj\j,^2ê^J~'\<p\k^2ê- (2.84)

Let Pj be the centre of Bj, and denote by A2(a, D) the principal part of A{x,D):

{A2{x,D)ipY= i a\,k{x)D,D^cp\x).

We denote by A2iPj, D) the partial differential operator with constant coefficients obtained from A2{x, D) by replacing a\iu{x) by their values at x=pj:

{A2{pj,D)<pY{x)= i a\,^{pj)D,D^cp'{x). i,/c = l

Applying Lemma 2.7 to their operator, we can take a positive constant C2{pj) depending on n, /JL, k, 6, 8 and MQ such that

\<Pj\2^k-.e^C2(pj)i\A2{pj,D)<pj\j,ê^\<Pj\k)- (2.85)


Since

{{A2{x,D)-A,{pj,D))<pnx)= i {aUx)-aUpj))D,D,<p'(x), i,k = l

by Lemma 2.11, we get a positive constant Q such that

\A2{x, D)-A2(pj, D))<pj\^ê^C,8'Mj,ê\(Pj\k^2ê^ (2.86)

Ci depends only on n, /JL, k, and 0. From (2.85) and (2.86) we obtain

\<Pj\2+k+e^ C2{pj)(\A2(x, D)(pj\k+0-\-\(pj\k + CiC2(Pj)s^Mj,+e\<pM^^

Since for a sufficiently small e

C,C2{Pj)e'M^ê<2-\

we have

2+fc+0 = 2C2(/?;)(|A2(x, D)(pj\^^0-¥\(pj\j^). (2.87)

In the same way as in the proof of Theorem 2.1, we see that there is a positive constant C^ such that

Hence

|(p,-|2+fc+e^2C2(p,)(|A(x,D)(^,|fc+,4-|(p^4) + 2C2(;7,)C3|(p,|/c+i+..

Since [A(x, D ) , coj] is a first-order differential operator, there exists a positive constant C4 such that

|[A(x, D ] , co ](p|fc+0 ^ C4Mk+0\(p\k+i+e'

By the way, there is a positive constant C5 such that

\(Pj\k ^ Csl^lfc and \(pj\k+i+e ^ d(p|fc+i+^.

Therefore

J~^\(p\k+2+e ^2C2(Pj){\A(x, D)<p|fc+0 + C4Mfc+^l9U+i+e + Q k k )

+ 2C2(/7,)C3C5kU+i^,. (2.88)

By Lemma 2.8, we can find a positive constant C^ such that

2C2(/?,)(C4M;,+ , + C3C5)|9k+l + . + 2 C 2 ( / ? , ) C 5 | ^ k ^ 2 - V - ^ | ( p | , ^ 2 + e + Q k l 0 .

§3. Function Space of Sections of a Vector Bundle 419

Therefore by (2.86)

2-'j-'\<p\j,^2ê^2C2{pj)\A{x,D)cp\^ê-^Ce\(p\o-

This completes the proof. I

§3. Function Space of Sections of a Vector Bundle

Lemma 3.1. Let U and V be domains in T" which are diffeomorphic to each other, and ^: U ^ V the diffeomorphism between them. Let K be an arbitrary compact set in U. Put K' = (K). Then for any integer I, there exists a positive constant C depending on /, 0 and K such that for fe C°^{V,C^) with supp f^K\

C-II/IINII/-CDIINCII/II?. (3.1)

Proof For / ^ O , the lemma follows immediately from Proposition 1.2. In the case of / < 0, we set l = -m. Let Co be a constant with which (3.1) holds for / = m. Since supp/ci K\ f can be extended to a C°° function T" which is identically zero outside of V. Choose a C°° function x with support in V which is equal to 1 on K'. Then for any </> e C°'(T", C^),

{L<p) = {xL<p) = {Lx<p)-

Since there is a positive constant Ci such that ^x^ II m = QII <P || m, by Theorem 1.11, we have

| | / | | _ ^ ^ S U p - — ; — ^ C i SUp-T — \W\\m cp lU^llm

^ C i sup . (3.2) ^ IIÂIIm

Here (A runs over CîV, C'')\{0}. Let J{x) be the Jacobian of ^ and put 4f{x) = J{x){ilj'^){x). Then {f il/) = {f'^,ji). From Proposition 1.7, we obtain a positive constant C2 such that ||i^||^ = C2||«A'^lU- By (3.1) for l=m, we have \\iP'^\\m^jQ>Mm- Since for (/.e C^(y, C^)\{0}, iê c?^(i/ ,c^)\{0},

^^Pi r7¥~=^^ô)^^P 11/ rT il ^ W Q ) C 2 S U p — - T ^ IIÂlIm ^ I I ^ A - ^ ^ l l m ^ \m\m

^^0C2\\f' n-m> (3.3)

4 2 0 Appendix. Elliptic Partial Differential Operators on a Manifold

Hence by (3.2) and (3.3), we have the first inequality in (3.1):

| | / | | _ ^ C I V Q C 2 | | / - ^ | | _ .

Using O"^ instead of O, we obtain also the second inequality in (3.1). I

By use of Proposition 1.5, we can prove (3.1) for any real /. But we do not need this fact here, hence we omit the proof.

By using Lemma 3.1, the Sobolev space of sections of a vector bundle can be defined. Let X be an n-dimensional compact manifold, and X = VJj=i Xj an open covering of X by a system of sufficiently small coordinate neighbourhoods. Set xy. p-^ Xj(p) = (X](p),..., Xj(p)) be C°° local coordinates on Xj. This maps Xj diffeomorphically onto the open unit ball of R". Since the interior of the unit ball is diffeomorphic to a domain in an n-dimensional torus T", we may consider Xj as the diffeomorphism of Xj onto a domain Vj in T".

Let (B, X, t) be a complex vector bundle over X. The following discussion is also true for real vector bundles. Let /JL be the dimension of fibre of B. We may assume that 7r~^(Xj) = Xj xC^ on each coordinate neighbourhood Xj. Let i/ be a section of B over Xj. Then ijj can be identified with a C^-valued function il^jiXj) defined on the domain Vj in T", and it is written as follows:

iPjiXj) = (il^jiXj),..., il^^iXj)), Xj G Vj. (3.4)

Suppose that i/ is defined on X^, too. Then we have a C^-valued function M^k), with XfcG VJ,:

Mxk) = (il^i(xk),..., ^«Xfc)), Xfc G Vk. (3.4)'

For peXjnXk with Xj = Xj(p) and x^^Xkip), we have the transition relation:

il^Hxj(p))=lb^,,(p)rk(Xk(p)), (3.5) p

where b%p{p) is a C°° function of ;? G XJ n X^. We denote by C°°(X, B) the set of all C°° sections of B over X. Choose C°° functions Xj{x) on X such that

Y.xMf^h supvXj^Xj. (3.6)

For any C°° section ijj of B, define the product Xj^ by XJ^{X) = XJ{X)IIJ{X).

Then we can consider it as a C°°-valued function on Vj, since supp Xj^ ^ ^y Moreover its support is contained in a compact subset of Vj. Therefore for an integer /, we can define the norm ||A}<AIL/ in W\Vj,C^). The suffix j


means that Xj is identified with V, by Xj. Using this, we define the /-norm 11/ on X as follows:

'•=UxM\li=UxrxJ'x}{xj)\\l,. (3.7)

This norm depends on the choice of {Xj]j. Instead of {XjYj take {(OjYj such that

X (Oj{xf = 1, SUpp 0)j C= Xj. j

Then the norm X . || coyiAlll/ is equivalent to (3.7). To show this fact, it suffices to see that there exists a positive constant C independent of iff such that, for any «Ae C°°(X, J5),

C^'lWxjcil^hi^l ||a;,(A||,,/^CZ ||;t..A|lM-

For any k, we get, by (3.6),

\^Mj,i = Z Xk(Ojil/ J,l k

(3.8)

(3.9)

We denote (xl(^j)ixj \y)) = ai^kj)(y),y^ Vj- Using the representation (3.4) of i/f on Vj, we have

\\XkO)jO)\\jj = \\a(^kj)^j\\jj. (3.10)

The support of xl^^j is contained in Xj n X^ so that we can apply Lemma 3.1 to ^ = Xj-Xk\ For zeVk we put a^kjy ^iz) = d(^kj)iz)il/j'^ = Xj{z) respectively. Then by Lemma 3.1 we can take a positive Q such that

Putting ij/jiz) = (ij/jiz),..., iAf (z)), we have, by the transition formula (3.5),

^ '(^)=I^;fcp(^r(^))^?(2:). p

For bfkp is a C°° function, there is a positive C2 such that

||a(fc,-)(A7llM= ll (/c/) /c|lM- (3-12)

Since a(k,)(z) = a(fc,) • ^(z) = (xio)j){Xk\z)), we have

\\a(^kj)^k\\k,i = \\Xk(Ojil/\\k,i' (3.13)


Since Xk^j is a C°° function, there is a positive C3 such that

\\xlcoj<pUj^C,\\xkHW (3.14)

By (3.10)-(3.14), we get

\\xla^jil^\y^C,C2C,\\xM,j.

By (3.9) and the above inequality,

\\cojilj\\j,i^C,C2C,l\\xicH\,,i-k

Taking the summation over j , we obtain

l\\cojilj\\jj^C,C2C,Jl\\xkil^U,i J k

which shows the right-half of (3.8). By the way, the roles of {cOj} and {xk} are symrrietrical. Therefore the left-half of (3.8) also holds.

Hereafter, choosing {Xj} once and for all, we fix the norm as in (3.7). Clearly || <p || fc' || ( || fc, for k' < k.

By C°°(X, B) we denote the totality of C°° sections of B.

Proposition 3.1. Let /, /' and I" be three integers with I" < V <l Then

( r ) For any t>0 and any ifj e C°°(X, B), the following inequality holds

ll'/'li''^7rf'"'"'"'""ll'^li'

(2°) Foranyil,eC°°iX,B),

Y-n/a-v^^o-''mi-n_ (3.16)

Proof. For any 7, by Lemma 1.1, we have

for any ^>0 and any il/e C°°(X, B). Summing these with respect to 7, we obtain (3.15). Taking the minimum with respect to r, we get (3.16). I

§3. Function Spdce of Sections of a Vector Bundle 423

A linear operator A: C°°(X, B)-» C°°(X, B) is said to be a multiplication operator, if it is represented in the form of (3.4) as

{M)^{xj{p))=laUxjip))iljfixjip)\ peXj, (3.17) p

where a^p{Xj{p)) are C°° functions and for fixed/)eXj the ( t X/x)- matrix (aj^p(xy(p))^A=i is a linear transformatin of the fibre Tt~^{p) at /?. Therefore A is considered as a section of the vector bundle Hom c(^) = B®B'^ where B* is the dual vector bundle of B, We put

M = Z I I sup |D^^.aj^,(x,)|. j p,\ m^lxjeVj

Lemma 3.2. Let I be an integer. Then for a multiplication operator A there exists a positive C such that for any (p e C°°(X, B),

||A(p||,^CM|„||(p||;. (3.18)

C depends on n, /, /JL, but is independent ofcp and representations of A.

Proof Take {xj} as in (3.6). By (3.17) we have Xj^ = ^Xp

P

Since a^p are C°° functions, we can apply to A the case m = 0 in Proposition 1.4 and we see that there is a positive C such that

\\XjM\\j,i = ^^\i\\\xj<phi'

Squaring both sides, and summing with respect to j , we get the desired inequality by taking the square roots of both sides. I

Corollary. In the identity (3.17), if there is such a positive 8 that

inf min |det(a|p(x,))| 8, (3.19) j XjeVj

then, for each /, we can take a positive C depending on /, 8, /JL, n and M^^, such that

\m,^C\\Ail,\\, (3.20)

for any <p e C°°(X, B).

Proof. This follows from the fact that the inverse mapping A"' of A is also a multiplication operator. I


A linear map A: C°°(X, B)-> C°°(X, B) is said to be a linear partial differential operator of order m, if it has the form

(Ail^)^{xjip))= I laf,îxjip))Dîl;^{xjip)) (3.21) | Q f | ^ m p

in the local representation (3.4), where afp^ are C°° functions and for a multi-index a = (a^, a2,..., a^) Df means

» ; = D r i - - - « a = ( ^ ) " ' - - - ( ^ ) - . (3.22)

Note that the meaning of Dj above is somewhat different from the one given in §1. We put for any non-negative integer /

M, = Z I I I sup \Dfaf,îxj)\. (3.23) j a p,\ iPl^lxjeVj

Lemma 3.3. Let A be a linear partial differential operator which operates on C°°(X, B). Then there exists a positive C such that for any (p e C°°(X, B) the following inequality holds.

||A(^||,^CM|,| 11( 11; , (3.24)

where C is determined by /, m, n and /n and independent of representations of A.

Proof Take Xj ^s in (3.6). Fix j , and let a) , /c = 1 , . . . , / , be C°° function on X with supp cofcc: Xfc and Xk ^fc(^)^= 1 such that (Oj{x) = 1 identically on a neighbourhood of the support of Xj- Such {cok} do exist. Since coj is identically equal to 1 on the support of Xp Xj^^ — Xj^^j^- Hence

\\XjA(p Wjj = WxjAcojcp Wjj ^ \\Axj(p y + WlXj, A]o)j(p\\jj. (3.25)

By Proposition 1.4 and Lemma 1.4, we have a positive Q such that

\\^Xj(p\\j,i = CiMm || -<p|| ;/+ , (3.26)

and

||[^^-, A]a)j(p\\jj^CiMîi \\a)j(p\\jj+rn-i' (3.27)

By the choice of {(o^}, (3.8) holds. That is, there is a positive C2J such that

Z \\(Oj,(p\\kj+rn-l=C2jY. \\Xk<p\\Kl+m-l-k k


By (3.25), (3.26), and (3.27),

IIA'J^^^IL/^ CiM\J\\Xj(p\\j,l+m-^C2j I \\Xk<p\\k,l+m-l] •

Summing with respect to j and putting C3 = Xj Q j , we have

Z \\Xj^^\\jJ=CiMi^\T WXjiphl+m-^CsY^ \\Xk(p\\k,l+m-lj

^C,M^i^(l + C,)l\\xk<p\\kMm. k

From this inequality, it follows

j k

That is,

This proves (3.24). I

For two C°° sections (p and J/ of a vector bundle B, the inner product {(p, il/)i of order / is defined as

(<P,^)i = l(Xj^,Xj^h^ (3.28) j

Here the inner product of the right-hand side is understood to be the one of the Sobolev space, for, the supports of Xj<P and Xj^ being in Xj, Xj<P and Xjif/ can be identified by the coordinate function Xj with the vector-valued functions on VJ. The suffix j in (3.28) indicates that Xj is identified with VJ. This inner product is positive definite, and

\\(l>\\l = {<P,cp),. (3.29)

The completion of C°°(X, B) with respect to the norm || \\i is called the Sobolev space of order /, denoted by W\X,B). The inner product (3.28) extends canonically to the one on W\X, B), for which we use the same notation in (3.28). W\X, B) becomes a Hilbert space with respect to this inner product. Clearly this inner product depends on the choice of {Xj}-But the norms determined by {Xj} are all equivalent. Therefore, as a vector space, W\X, B) is independent of the choice oi{Xj}, and so is its topological structure.


By Lemma 3.3, a linear partial differential operator A can be uniquely extended to the continuous map from W\X, B) to W^'^îX, B). In particular, a multiplication operator is a continuous map from W\X, B) to W\X, B). For S 6 W\X, B), the support of S denoted by supp S is defined as the smallest compact set Ka X such that the product aS vanishes for any C°° function a(x) whose support is contained in X\K. If S is an arbitrary element of W\X, B) and xM is a C°° function, then

supp X' S'^ supp X ^supp S. (3.30)

Suppose that the support of x is contained in a coordinate neighbourhood Xi and take a family of C°° functions {OJJ} on X with supp o; c: Xj and X . 6jfy(x) = 1 such that coi(x) = 1 on supp x- Since Se W\X, B), we have a sequence {(pk}°k=î in C°°(X, B) which converges to S in W^(X, B). Since the norms are equivalent if we take {(DJ) instead of {Xj}, there is a positive Ci such that for any /c, /c'

llAr<Pk-A^<Pfc'lli,/ = lki(Ar<;pfc-Ar<PfcOllM^Qll;^<Pfc-Ar<Pfcl^

By Lemma 3.2, we have a positive C2 such that

\\x^k-X<Pk'\\i^C2\\(Pk-(Pk'\\i'

Hence for any /c, k'

\\x<Pk-X<Pk'\\i,i^C^C2\\(Pk-(pw\\i-

Therefore regarding x^k as a vector-valued function, we see that {x^k\ converges in W\yi,C^). Letting Si be its limit, we identify x^ with 5,. We can choose x arbitrarily so that we may consider an element S of W\X, B) as a section of B which can be locally identified with an element of W\Vi, C^). Of course when the support of x is contained in X^nX^, x^ can be idejitified both with S^ e W\ V , C^) and with Sj e W\ V;, C"), while between S^ and 5,, there is a relation which extends (3.5). Conversely, we can prove that the totality of section of J5 which can be considered locally as an element of W\V,C^) which satisfies this relation coincides with W\X, B). But the detail of the proof is so messy that we omit it. In case /Ô, however, it is not difficult, because the convergence of x<Pk in W'{ Vi, C^) is stronger than that in L\ V;, C^).

In the above, we take Xt as x and let 5, be the element of W^Vt, C^) corresponding to Xi^- Since


putting /c->oo, we get

\\s,\\l,sclcl\\s\\^sclcl^\\Sj\\l,. j

Thus there exists a constant C such that, for any Se W\X, B),

C-'l\\Sj\\li^\\S\\^^Cl\\S\\l (3.31) j j

Theorem 3.1 (Rellich's Theorem). Let V < I. Then W\X^ B) is contained in ^(x , B), ^d the inclusion map W\X, B) -^ W^'(X, B) is a compact linear map.

Proof. It suffices to show that the inclusion map is compact. Let {5^}^=i be a sequence in W\X, B) such that there is a positive constant R with

\\S%<R.

By the above argument we take SfeW\Vi,C^) corresponding to j = 1 , . . . , /, then by (3.31) we have

^ L W^j \\j,l=J< » j

in other words, for each j , {5}}^=i is a bounded sequence in Wl)(Vj,C^). Thus since it is a bounded sequence in W^ (T", C^), we can choose a subsequence convergent in W^'(J'',C^) (Theorem 1.10). We may assume that these subsequences have the same set of indices for all / We write them as {Sf^}. We can choose a sufficiently large KQ such that for any K, K'> KQ, and for any 7,

\\Sf-Sn\j,v<J-'C-'e

holds, where C is the same as in (3.31). Therefore, by (3.31), for K, K'> KQ, we have

||5^-S'^l|,<e.

Thus {S^}K forms a Cauchy sequence in W^'(X, B), converges in W^'{X, B). Therefore the map W\X, B)-^ W\X, B) is compact. I

Let t/ be a section of B over X, and ilfj{Xj) the vector-valued function (3.4) of Xj on Vj, corresponding to the restriction of ifj to Xj. ij/ is called a C^^^ section of B if for each j , ^j{Xj) is a C^^^ function of Xje V), where 0 ^ (9 < 1. The set of all such ifj is denoted by C^^^X, B). We take {Xj] as in (3.6) and define the norm of C^"'^(X, B) by

\^\u^e=l\xMj.k^e. (3.32)


where | 1 ; + means the norm of Xj^ as an element of C^^^{Vj, C^). For a different choice of {; }, (3.32) defines an equivalent norm, because a similar estimate to (3.8) holds for | \ji+o.

Theorem 3.2 (Sobolev's Imbedding Theorem). Let I and k be non-negative integers, and suppose l> n/2 + k.JTienforfe W\X, B), there exists a unique feC^{X,B) such that f(p)=f{p) on X except for a set of measure 0. Furthermore, there exists a positive constant C depending only on k and I such that

| / | f c^C| | / | | , (Sobolev's inequality) (3.33)

holds.

Proof Both W\X, B) and C^(X, B) are imbedded continuously into W^(X, 5 ) , and C°°(X, B) is dense in W\X, B). Therefore it suffices to show the equality (3.33) for fe C°°(X, B). In this case f{p) =f(p) for all peX.

By (1.23) in the proof of Theorem 1.3, there is a positive constant Q such that

\Xjf\j,,SCj\\x/h,, (3.34)

Putting max C, = C, and summing (3.34) with respect to j , we have

I/U^CI;||A;/IL,^CV7||/||,, J

which proves (3.33). I

Assume that X is oriented. By a volume element of X, we mean a C°° differential n-form v(dx) which is positive at every point of X. If a subset iV of X is of measure 0 with respect to one volume element, so is it with respect to another one, hence, the notion of the sets of measure 0 does not depend on the choice of a volume element.

By a metric g on a vector bundle B, we mean a C°° section of the vector bundle B*®B* such that the following condition is satisfied: Let pe V), and let {p, ^), {p,C') be two points on the fibre 7r~^(p) with fibre coordinates {, = U],..., ^n, and ^; = Uj\ . . . , Cn respectively. Then the Hermitian form on Tr'^ip) defined by

gM.n=lgi.,Mp))iH^' (3.35) A,P

is positive definite.


Let (p,il/eC°^(X,B). In terms of fibre coordinates over Xj, we write ^iip) = {<p]ix,{p)\..., cp^Xiip))), and (A/(P) = (rtiXiip)),..., ^ rU( /? ) ) ) . We define the inner product of (p and if/ with respect to the volume element v{dx) = Vi{Xi(p)) dx = Vi{Xi{p)) dx]- • • A dxt and the metric g by

(<P, *A) = I gpi(p(p). ^{p))v{dx)

= I giKp{Xi{p))ip''i{x,{p))ri{Xi{p))v,{x,{p)) dx, (3.36) J X KP

This has an intrinsic meaning. With this inner product, C°°(X, B) becomes a pre-Hilbert space. The completion with respect to the norm

lkll = (^,<p)''' (3.37)

is a Hilbert space, which we denote by L^(X, B). This is the same as W^(X, B) as a topological vector space, but with a different inner product.

Proposition 3.2. We fix a metric g and a volume element v(dx). The inner product ( , ) defined on C^{X, B) is uniquely extended to a continuous sesquilinear map of W~\X, B) x W\X, B) to C for any integer /, which we denote by the same ( , ). This is non-degenerate.

Proof Take {Xj} as in (3.6). Then for (p,il/e C°°(X, B), we have

i<P,^) = l{Xj<P,Xjil^)^ (3.38) j

Since for each j ,

(Xj^,Xj^)= I gjxp(Xj(p))Xj(pf(^j(P))Xj^j(Xj(p))vj(xj(p)) dXj(p) J X KP

(3.39)

by Theorem 1.11 in §1, for any integer /, there is a positive constant C,j such that

\{Xj<f>, xM g Qj\\xj<Pjl,-i \\xM\j,i-

Summing with respect to j , and using Schwartz' inequality we have

\{q>,il>)\^C,\\<p\\.,Uh, (3.40)

where we put C; = max^ Cy. Thus ( , ) is continuous with respect to the relative topology of C^iX, B) x C°°{X, B) in W-\X, B) x W\X, B). Since

4 3 0 Appendix. Elliptic Partial Difierential Operators on a Manifold

C°°(X, B) X C°°(X, B) is dense in W-'(X, B) x W'(X, B),( , ) is extended uniquely to a sesquilinear map of W~'{X, B)xW'{X,B).

Let 5 e W"'(X, B). Then xjS is identified with S, e WQ'(Vj, B). Define r = ( T ] , . . . , r j ' ) b y

T"; = I Vjixj)gj,,{x)Sl p = 1 , . . . , At. (3.41) A

Then by (3.38) and (3.39), we have

(S.<p)=l(Tj,Xj<Pjh (3.42) j

where ( , ) is the natural bilinear map of Wo\Vj, C'') x WliVj^C"} to C on Vj. The left-hand side is iiidependent of the choice of{xj}. Suppose that Se W-\X, B) satisfies (5, ( ) = 0 for any cp e C°°(X, B). For an arbitrary a e C°°(X) with supp a c: X^, there is a set {Xj}j=i satisfying (3.6) such that Xk(x)=l on supp a. Since (3.42) holds for such {xj}, we have for any <peC''(X,B)

0 = (5, aip) = (5fc, aAffc<p)k = ( 'S' , ;tfc(;p)fc,

hence aS^ = 0 as an element of Wo\ V , C^), hence {aS)k = 0. Since aXj = 0 for7 7 /c, we have aSj = 0. Thus (aS)j = 0 for all j , hence a5 = 0 as an element of W~^(X, B). Let {(x)j}j be a partition of unity subordinate to the open covering X = U ; y Namely, supp (Oj c Xj, and Z ^j = 1- Then by the above consideration, we have (DJS = 0,7 = 1 , . . . , /. Therefore we obtain

j

Thus 5 = 0, which implies the non-degeneracy of ( , ). I

§4. Elliptic Linear Partial Differential Operators

(A) Estimates with Respect to the Sobolev Norm

(a) A priori Estimate

Let X be a compact manifold of dimension n, B a complex vector bundle over X with /x-dimensional fibres, and TT: B^X its projection. We use the same notation as in §3.

Let A(/7, D): C°°(X, B)->C°°(X, B) be a linear partial difierential operator of order /. In terms of the local coordinates xy. Xj-^ VJ on the open

§4. Elliptic Linear Partial Differential Operators 431

set Xj, a section cp e C°°(X, B) of B is written as

<Pj{Xj(p)) = (cp'Mip)),..., cl>îxj{p))),

We write the operation of A{p, D) as

{A{p,D)cp)^{xj{p))= I YâU{xj{p))DJ<p',{xj{p)), (4.1)

where the a^pai^jip)) are C°° functions of Xj{p). The operator Ai{p,D) consisting of the terms with |a | = /, given by

{A,{p.D)^)^{xj{p))= I la%^{xj{p))DJ<p^{xj{p)) (4.2) \a\ = l p

is called the principal part of A{p, D). Let f{p) be a real-valued C°° function on X, and suppose that ^p = df(p)7^0 at /?. Then r ^ e'^^^pÂ{p, P)e"'^^^^V(p)) is a polynomial of degree / in t, whose coefficient of the terni of degree / is given by

iA(p, iQ<p)^{xj{p)) = 1 1 a^,Ax,{p))i'r, ' <p^(xjip)), (4.3) \a\ = l p

where we put ^p = df{p) = ^j^ dx] + • • • + ^ „ dxj. For any non-zero element ^p in the cotangent space r*X, there is an

fip) with fp = df{p). For any element w in the fibre Bp at /? G X, there is a section (p such that (^(p) = w. Let ^pGT*X be given by ^pj = ^ji <ix] + • • • + f „ 4^J and let w e Bj be given by Wj = ( w ] , . . . , w"). We define the linear map BpSw-^w'eBp by

wj^-(A,(p,fp)w),^= I Iaj;,«(x,(;7))^;w;, A = 1 , . . . , M (4.4) \cx\ = l p

for 0 # p G T'pX, which we call r/ie principal symbol of A(/7, D).

Definition 4.1. A(/7, D) is said to be elliptic if for any/? e X, and any fp e T^X with p 7 0, the principal symbol Ai{p, ^p) is an isomorphism of Bp = 7r~^(/?). In this case, there exists a positive constant 6 such that for any p e r * X with fp 7 0, and w e Bp, the inequaHty

l[iAm{p,$,)w)^f^8'l\wf\\ (4.5) A A

The supremum of such 8 is called the constant of ellipticity ofA{p, D).


We introduce the following constants as in the preceding section. For /c = 0 , l , . . . , put

Mfc = I I Z I sup \Dfa^,^{xj)\. (4.6) j a p,\ |j8|gfc XjVj

Theorem 4.1 (L^ a priori Estimate). Let A(p, D) be an elliptic linear partial differential operator of order I For any integer k, there is a positive constant C depending only on w, /, k, /x, the constant d of ellipticity of A{p, D) and M|fc|, such that for any u e W^^^{X^ B), the inequality

| | M | U ^ , ^ C ( | | A ( / ? , D ) M | U + | | I / | U ) (4.7)

holds.

Proof Since both sides of (4.7) are continuous with respect to the topology of W^-^'iX, B), and C°'(X, B) is dense in W^-^^X, B), it suffices to prove (4.7) for any u e C°°(X, B). Take functions {Xj}j=i as in (3.6) in the preceding section. For any u e C°°(X, B), there is a j such that

r^û\\,^,^\\Xju\yî. (4.8)

Put Uj = XjU. Take another family of functions {wi}f=i on X such that supp a)j<^ Xj and that X, îip)^^ 1 ^^ ^' Moreover we assume that Wj is identically equal to 1 in some neighbourhood of supp Xj- Then we have

\\A(p, D)uU^\\XjA{p, D)u\y

= \\XjAip,D)coju\y (4.9)

= | | A ( A D)uj\y-\\[A(p, D), iOj]coju\y.

Since [A{p, D),Xj] is a partial differential operator of order ( / - I ) , by Theorem 1.4, there is a positive constant Q such that

\\[A(p, D), Xj](oM\j,k^ C^M\k\\\(Oju\\j^k+i-i, (4.10)

while there is a positive constant C2 by (3.8) such that

||^yw|b,k+/-i^ZlkiW||,;fc+/_i^C2||M|U+/-i. (4.11) i

Since the support of Uj = XjU is contained in the coordinate neighbourhood Xj, Uj can be considered via the local coordinates Xj as an element of C°°(T", C^) with support in Vj. Thus we can apply Theorem 2.1 to find a positive constant C3 depending on k, /, n, /JL, d and M|fc| such that

||A(/7,D)M,-||,.,;,^C3||M,-||,-,;,+ /-||M,-Lfc. (4.12)


From the inequality || w, 11 , ^ || w |U, and (4.8), (4.9), (4.10), (4.11) and (4.12), we obtain

||A(AI^)w|U^C3J^/'i|t^lU-^/-||w|U-C2QM|„||u|U^;_,. (4.13)

By Proposition 3.1, there exists a positive constant C4 such that

hence from this and (4.13), we obtain

||A(p,D)i/|U + (C4+l)||M|U^2-^C3/^/^i|w|U,z. I

(b) Garding's Inequality

As stated at the end of §3, we consider the Hilbert space L^(X, B) defined by the volume element v(dx) of X and the metric g on the vector bundle B. We denote its inner product by ( , ).

Lemma 4.1. Let A{p, D) he a linear partial differential operator of order (m-\-l). Then there exists a positive constant C determined by m, n and JJL such that for any cp, il/ ^ C°°(X, B),

\iAip, D)ip, 4f)\^CM,MJ^\\, (4.14)

holds.

Proof. Take {xj} as in (3.6). Then we have

(Aip, D)<p,4f) = I {A{p, D)<p, xj^). (4.15) J

Fix7, and choose a family of C°° function {ifj^Yk^i with supp co^^ Xp, and X o)\{x) = 1 so that (Oj is identically equal to 1 in some neighbourhood of supp Xj' Then we have

(A(/7, D)(p, x]^) = (A(p, D)a>j<p, xj^)-

We may assume that A(p, D) has the form (4.1) with m + / instead of /. Then the right-hand side of the above equality is written as

+ / p,A,o- J V (A{p,D)co,<p,xj^)= I Z gjK-MiPm.M(p))

xDf{coj<pJ{xj{p)mxj>l^])(.Xjip))vjixj{p))dXjip).


Integrating the terms with |a | ^ m +1 by parts {\a\-m) times, we have

(A(p,D)coj<p,xjil^)= I Z [ bj,,^,(xjip))DJ(coj<pf{xj{p)) \cx\^m p,\,cr J Vj

xDf{xjiPf)ixj{p))vj(xj(p)) dxjip),^

where the bj^pa/iiXjip)) are Hnear combinations of partial derivatives of Xjgj\p(Xj(p))aj^^{Xj(p))vjixj{p)) of order at most /.

Therefore there exists a positive constant Q such that

By (3.8), there is a positive constant C2J depending on 7 such that

\(A(p,D)<p,x]^)\^C2jMi\\<p\U\il;l.

Summing with respect to 7, and putting X C2J = C3, we obtain

l\{A{p,D)cp,xjH^C,M,\\<pU\iljl. j

(4.14) follows from this and (4.15). I

Let A(p, D) be a linear partial differential operator of order 2m. For peX, and ^p e TpX with ^p 9^ 0, the principal symbol A2mip, ^p) of A{p, D) is a linear map Bp -^ Bp. Therefore, by virtue of the metric gp on Bp, we can define a Hermitian form on Bp x Bp associated with this linear map by putting

gpiA2m{p,Qw,w') (4.16)

for we Bp and w'e Bp. For p e Xj, if we write A{p, D) in the form of (4.1) with / = 2m, then putting ^p = ^ji dx] + ••• + !;„ dx], iv = ( w j , . . . , w"), w' = (wj\ . . . , wj"), we have

cr,\,p |a | = 2m

Definition 4.2. A linear partial differential operator A(p, D) of order 2m is said to be strongly elliptic if there exists a positive constant 8 such that for any peX, any ^p e T*X with ^p 9^ 0, and any weBp with w 7 0,

( - 1 ) - Re gp{A2mip. ^p)w, w) ^ 8'gp(w, w)|^,|^- (4.18)

holds. The supremum of such 5 is called the constant of strong ellipticity of Aip, D).


Theorem 4.2 (Garding's Inequality). Let A(p, D) he a strongly elliptic partial differential operator of order 2m. Then there are positive constants 8i, 82 depending on m, n, c, the constant 8 of strong ellipticity ofA(p, D) and M^ such that for any cp e C°°(X, B),

Re(A(p,D)<p,cp) + 8,{<p,cp)^82Ml (4.19)

holds.

Proof Take {Xj}j=i as in (3.6). For simplicity we write A for A(p, D). Since lLjXj{sY=^. we have

{Acp, (p) = Y. (A(p, Xj<p) = Z {xA<P, Xj(p) j j

= E (^Xj(p. Xj<p)-^1 (XjlXj. ^']<p, <p)-j j

Since A}[A}, ^ ] is a linear partial differential operator of order ( 2 m - 1 ) whose coefficients are linear combinations of those of A{p, D), by Lemma 4.1, there exists a positive constant Cj depending on n, m, /x and M^ such that

\{Xj[Xj,A]<p,<p)\^C,\\cp\\mMm-l-

Since the support of Xj^ is contained in Xj, Xj<P can be considered as a C^-valued C°° function on T" with support in VJ. Therefore by Theorem 2.2, there are positive constants C2, C3 such that

Re {Axjcp, Xj<P) + C2\\xj<p\\lo^ C,\\xj<p\\lm^

From the above inequalities we obtain

R^{A<p,<p)^C,I^\\Xj<p\\lr,,-C,l\\xM\lo-JC^\\<p\U<p\\,„., j j

= C3||<p||i-Q||<p||^-/C,||.p|Ui|<p|U_,. (4.20)

By Proposition 1.1, there is a positive number C4 such that

Consequently by (4.20) above, we obtain

Re{Acp,<p)m^2C4<p\\l-iC2+C,)Ml.

Since there is a positive constant C5 such that for any <p e C°°(X, B),


we obtain

Rc(Acp,cp)^'2Cs\\cp\\l-Cs(C2+Cd{<P,<p),

which proves (4.19). I

(B) A priori Estimate with Respect to the Holder Norm

Lemma 4.2. Let 0<r<s. Then there exist positive constants Cj, C2 such that forany<peC\X,B),

\<pl^CM'/'\cp\l"^\

and that for any t>0

\<pl^t'^'\cpl+C2t-''''-'^\cp\0.

This follows immediately from Lemma 2.8. I

Definition 4.3. Let A{p, D) be a second-order linear partial differential operator acting on sections of a vector bundle B over X. The principal part of A{p^ D) is said to he of diagonal type if we can choose a system of local coordinates {{Xj, X/)}/=i such that when we write A(/?, D) in terms of these coordinates as

{A{p,D)cj>)^{xj{p))= E Za;,Jx,(p))Z>;<(x,-(i7)), (4.21)

then the following conditions is satisfied:

«ia(^j(;?)) = 0 for |a | = 2 and A 7 p, P^Xj. (4.22)

Remark. If in some choice of local coordinates, the principal part of A(/7, D) is of diagonal type, and it is written as

{A2{p,D)ip)^{xj{p))= I ajAxj{p))D^cpf(xj(p)), (4.23) l«H2

then the principal part is of diagonal type in another choice of coordinates.

Theorem 4.3. Let A(p, D) he a strongly elliptic partial differential operator of order 2 with C°° coefficients acting on sections of a vector bundle B over X whose principal part is of diagonal type. Let 8 be its constant of strong ellipticity, and put 0< 6 <1. Then for any integer kÔ, there exists a positive constant C depending only on 0, k, n, /x, 8 andMj^-ê such that for any u e C^^^'^^(X, J5),

lwL -2+a ^ C{\A{p, D)u\kê + Nlo)


holds, where we put

Mfc+ = Mfc + X S I sup ^ — -e . j \a\^2 \p\^k \x-x'\^l \X — X\

Proof. We may assume that the system of local coordinates {^}/==i is so chosen that the principal part is already diagonal. Take {Xj}j=i as in (3.6). For u e C°°(X, B), by the definition of the norm (3.32), there is 3.j such that

J~^\u\k+2+e ^ \xMj,k+2+e' (4.24)

Put XjU = Uj- Choose a family of C°^ function {<w,}f=i on X with supp (Ot c: X, and Xi (îip)^= 1 such that (Oj(p)= 1 on some neighbourhood of supp;^/. Then we have

\A(p, D)u\kê ^ \XjMp, D)u\j^j,+e = \XjMp. D)(OjUy+e

^ \A{p, D)uj\j,,,ê - |[A( A D), xMAu^ê^ (4-25)

Since [A(/7, D), ; ] is a first-order partial differential operator, by Lemma 2.8 there is a positive constant Cj depending only on fc, ^, n, /x such that

|[A(/7, D), A}]w,wb,/c+0 CiMfc_, |w^M|,;fc+n. , (4.26)

while there is a positive constant C2 such that

|w^M| ;fc+l + 0 C2|M|fc+i + 0. (4 .27)

Since the support of w is contained in X , M, can be considered, via x , an element of c^''^''^(T^ C") with support in V;. Hence by Theorem 2.3, there is a positive constant C3 depending only on /c, ^, w, /x, 5 and Mk+g such that

C3|w,b;fc+2+a - I W J U O ^ | A ( A D)M,^,fc+,. (4 .28)

From thQ trivial inequality |M,|,;O^|W|O and (4.24), (4.25), (4.26), (4.27), (4.28), we obtain

|A(AD)wU^,^C3J~'|M|k+2+.-|w|o-CiC2Mfcâ|wk+i+a. (4.29)

By Lemma 4.2, there is a positive constant C4 such that

CiC2Mfc+^|M|fc+l + ^ ^ 2 ~ ' C 3 / ~ ' | M | f c + 2 + ^ + C 4 | M | o .


Consequently from this and (4.29), we obtain

which proves the theorem. |

Remark. Let A(x, D) be an elliptic operator of order / with C^ coefficients, and 8 its constant of ellipticity. Then if 0 < ^ < 1 , for any integer fcÔ, there is a positive constant C depending on n, /, /JL, k, 6, 5, Mk+e such that

\u\j,^jê ^ C(|A(x, D)M|fc+, + |M|O).

We omit the proof.

§5. The Existence of Weak Solutions of a Strongly Elliptic Partial Differential Equation

As was stated at the end of §3, we fix once for all a volume element v(dx) of X and a metric g on the vector bundle B, Then we can define L^{X, B) with respect to ( , ).

Proposition 5.1. For a linear partial differential operator A{p, D) of order /, there exists a unique linear partial differential operator A{p, D)"^ such that for any (pîjje C°^(X, B), the following equality holds:

(A(A D)cp, (A) = (<p, A(p, or^). (5.1)

Moreover A{p, D)"^ is also of order I.

Proof Let {^j}/=i be a C°° partition of unity subordinate to the open covering \Jj=.^Xj of X Then we have

{A{p, D)<p, .A) = 1 (A(p, D)(Oj<p, lij). J

If A(p, D) is written in the form (4.1) on Xj, we have

( A ( A D)COJ<P, (A) = I I [ gj.,{xj{p))a)^^(xj{p)) \a\^l p,\,(T J Yj

xDf{coj<pJixjip)Mîxjip))vj(xjip)) 4xj{p). (5.2)

§5. Weak Solutions of a St;rongly Elliptic Partial Differential Equation 439

Since cojipj vanishes near the boundary of V), by partial integration, we obtain

v,k J V.

x4>]-{x^{p))vj{xj{p)) dxj{p), (5.3)

where we put

^MAP)) A,cr,p|a|S/

xD;(g,.,,(x,-(/7))<«(x,-(/7))i;,.(x,-(p))iA;(x,(/?))), (5.4)

and {gj'îxjip))) is the inverse of the matrix {gjxp{Xj{p))). Therefore if we put tA' = Zj ^j^'j-> then A(/7, D)"^: ifj-îfj' is the required differential operator. The uniqueness if obvious. I

Definition 5.1. A(p, D)"^ is called the formal adjoint ofA{p, D). If A{p, D) is of order /, so is A(/?, D)"^.

Proposition 5.2. Let peX and ^^ T"^^ with ^^ T 0. If we write the principal symbol of the differential operator A{p, DY CLS Ai{p, ^x)" , ^^^^ It is the adjoint of the linear map ofBp determined by the principal symbol Ai{ p^ ^^) ofA(p, D) with respect to the metric gp.

Proof Let f{p) be a real-valued C°° function on X with df{p) = p. Then for any <p, /x G C°° (X , B ) , we have

lim t-\<p, e~''Â{p, D)ê 'V) = lim t-\e-''Â{p, D)e''^(p, lA), (5.5)

which proves the assertion. I

We want to solve the partial differential equation

A{p,D)u{p)=f(p). (5.6)

Definition 5.2. L e t / e W'îX, B). u e W'îX, B) is said to be a weak solution of the equation (5.6) if for any (p e CîX, B),

iu,Aip,Dr<p) = {fcp) (5.7)

holds, where A(p, D)"^ is the formal adjoint of A{p, D), and ( , ) denotes the extended one in Proposition 3.2.

Proposition 5,3.Let A(p, D) be a partial differential operator of order I. Then


by Lemma 3.3, it is extended to a continuous map ofW^{X, B) to W^~^{X, B). If u is a weak solution of (5.6), in this sense, we have

Aip,D)u=f (5.8)

in W^-'(X, B).

Proof Let {(p„}^=i be a sequence in C°°(X, B) converging to u in W^(X, B). Then {A(/?, D)<p„}^=i converges to A(p,D)u in W^-\X,B). For an arbitrary cp e C°°(X, B), by Proposition 3.2, we have

(A{p, D)u, <p) = lim (A(/7, D)<p„, cp) = lim ((^„, A(/7, D)^(^)

= (M,A(AI) ) "^<^) = ( F , ( P ) .

Hence A(/7, D ) M = / in \y'^-'(X, B), I

In particular, i f / e C^(X, B), M is a weak solution of (5.6), and ue C\X, B), then by Proposition 5.1, for any cp e C°°(X, J3), we have

{Aip,D)u,<p) = (fcp\

hence A(p, D)u = / in the sense of L^(X, B). But since both sides of this equality are continuous, we obtain

A{p,D)u(p)=f{p)

for all p&X, namely, we obtain a solution of (5.6) in the usual sense. In order to show the existence of a weak solution, the following Lax-

Milgram theorem is useful.

Theorem 5.1. Let ^ be a complex Hilbert space, ( , ) its inner product, and I I its norm. Suppose that B{x,y) is a Hermitian form on ^ satisfying the following condition: There exist positive constants Ci ^ C2 such that for any x,yeX,

\B(x,y)\^C2\x\\y\, (5.9)

ReB(x ,x )^Ci | x | ^ (5.10)

Then for any continuous conjugate linef{x) on S€, there exists a unique element Fsof^with

B{Fs,z)=f(z), (5.11)

Proof As will be shown later, there exists a continuous linear isomorphism

§5. Weak Solutions of a Strongly Elliptic Partial Differential Equation 441

B of ^ onto ^ such that for any x,ye^

B{x,y) = (Bx,y) (5.12)

holds. Assuming this for a moment, we see that by the Riesz representation theorem, there exists an element ye^ such that for any ze^

{y,z)=f{z).

Hence it suffices to put FQ = B~^y. We shall prove the existence of B satisfying the above condition. Fix an

arbitrary xe^. The associated linear map ^sy -^ B{x,y)eCis continuous by (5.9), hence, by the Riesz representation theorem, there exists a z{x) e ^ such that B(x, y) = (z(x), y) for any y. The map x -> z(x) is linear. Therefore there is a linear map B with z(x) = Bx. By (5.9) and (5.10), we have

CAx\^ — ^\Bx\ = sup—:——^ C2|x|. (5.13) 11 y^o \y\

Hence Bx is continuous. If Bx = 0, by the left inequality of (5.13) we have X = 0, hence B is injective. Furthermore the range of B is closed. In fact, if {BXn}n converges to y, then by the left inequality of (5.13), {x„}„ is also a Cauchy sequence, hence, converges to some z. Since B is continuous, Bz = y. Moreover the range of B is dense. For, if z is orthogonal to the range of B, we have

0 = \{Bz, z)\ = |B(z, z)| Re B(z, z) ^ C,\z\^,

which implies z = 0. Thus the range of B is dense. Consequently the range of B coincides with the whole ^ , and B is bijective. The continuity of B~^ is also clear. I

Let A(/7, D) be a strongly elliptic linear partial differential operator of order 2m. For a sufficiently large A > 0, we shall prove the existence of a weak solution of

(A{p,D) + XI)u = w. (5.14)

For this, take {Xj}j=i ^^ i^ (3.6). Then with the inner product (3.28), W"(X, B) becomes a Hilbert space. Put

(<p, {A(p, Dr^Dilf) = B{cp, (A) (5.15)

for (p,il/e C°°(X, B), where the inner product in the left-hand side denotes that in L^(X, B).


Proposition 5.4. Let 8i, 82 be the positive constants given in Theorem 4.2. By (5.15), B((p, if/) defined on C°°(X, B) extends uniquely by continuity to a continuous Hermitian form on W"(X, B), which we denote by the same notation B((p, ijj). Then there exists a positive constant Cj determined by n, m, jx, and M^ such that 1/ Re A > 81, for any <p,il/e WîX, B), the following inequalities hold:

\B{<p,ilf)\^C2\\cp\U\ilf\U (5.16)

RQB{<p,il;)^82Ml. (5.17)

Proof If (p.il/eCîX^B), since B(^, (/f) = ((A(/?, D) + A)(p, f^), we have (5.16) by Lemma 4.1. In other words, B((p, if/) is continuous with respect to the relative topology of C°°(X, B) in WîX, B). Since C°°(A:, B) is dense in W"(X, B), B{(p, ifj) extends uniquely to a continuous Hermitian form on W""(X,B). (5.16) still holds for the extended B{(p,ip). By Garding's inequality, (5.17) holds for cp e C°°(X, B). Hence, by continuity, (5.17) holds for all cpeW'îX^B). I

Theorem 5.2. Let A(p, D) be a strongly elliptic linear partial differential operator of order 2m, and Sj, ^2 the positive constants given in Theorem 4.2. /f Re A > 81, for any w e L^(X, B), there exists a weak solution of the equation

Aip,D)u-^Xu = w (5.18)

contained in W"^(X, B). Moreover the weak solution of this equation contained in ^"{X, B) is unique.

Proof Let A{p, D)"^ be the formal adjoint of A{p, D), and B{(p, \fj) the Hermitian form on ^ ' " (X , B) in Proposition 5.4. Since W^^X, B) is imbedded in L^(X, B), putting

f{(p) = {w,(p)

for any <p e W'îX, B), we obtain a conjugate linear form on WîX, B). Take W'îX, B) as ^ in Theorem 5.1, and let B{(p, iff) and f{(p) be the Hermitian form and the conjugate linear form for it respectively. Then by Theorem 5.1, there exists a w e WîX, B) such that for any (pWîX, B), the following holds:

B{u,cp) = {w,<p), (5.19)

where the left-hand side means B in the extended sense. We may assume that there are ifjk e C°°(X, B), /c = 1, 2 , . . . , such that {il/k}k converges to u in W'îX, B). Then {(Afc}fc converges to u also in L^{X, B). Therefore, for

§6. Regularity of Weak Solutions of Elliptic Linear Partial Differential Equations 443

(p G C°°(X, B), we can write

B{u, cp) = lim B((Afc, <p) = lim ((A , (Aip, D^ + Dcp) fc-^oo fc->oo

= (M,(A(p ,Dr + A)9). (5.20)

Hence, w is a weak solution of (5.18). The uniqueness follows from the following inequality by putting w = 0.

0 = |(w, M)| Re B{u, u) ^ 62IIM 12

§6. Regularity of Weak Solution of Elliptic Linear Partial Differential Equations

Theorem 6.1. Let A(p, D) be an elliptic linear partial differential operator of order I with C°° coefficients. Suppose that for some integers s and k, fe W'~^~^{X, B). Then for a weak solution u e W'{X, B) of the equation

A(p,D)u=f (6.1)

there exists a positive constant C such that

^C{\\fU,.,+ \\ul) (6.2)

holds, where C depends on /, n, k, /x, the constant of ellipticity 8 ofA(p, D) and M|5|, hut is independent of f and u.

Proof First we prove the case k=l. Take C°° functions cOj, j = 1,..., J on X with cOj(p)^0 and supp coj c: Xj so that

lcoj(p)^l, peX. (6.3) j

Put

Uj = COjU, fj = (Ojf j = l , . . . , / .

Then in W'-\X, B), we have

A(p, D)uj =fjHA{p, D), coj]u, 7 = 1 , . . . , / ,

while fj = (Ojfe W'~^^\X, B). Since [A(p, D), o)j] is a differential operator of order ( / - I ) , we have {A{p, D), o)j\ue W'~^'^\X, B). Furthermore, putting gj =fj-h[Aip, D), (Oj]u, we have gje W'-'^\X, B), and

A{p,D)uj = gj. (6.4)


Since the supports of Uj and gj are contained in Xj, Uj and gj can be identified with C^-valued distributions Uj e W'o( Vj, C^) and g, e W'o~'''\ V,, C^) on the open set Vj in T" respectively. The supports of these Uj and gj are contained in a compact set K. Writing w, = (M], . . . , Uj) and gj = {gj,..., g") in terms of their components, we have

gj= I i:a^,Axj)Dy^ = {Aj{p,D)uj)\ xj = xj{p). (6.5)

Since X is a compact set in V), there is a positive number /IQ such that if 0<\h\<ho, the translation of K in the direction of xf by h is contained in Vj for any /c = 1 , . . . , n. Let Tjk be the translation in the direction of Xj by h, and ^lk = h~\Tlk-I). Then we obtain

Aj;,g,. = A,-(x,-, D){^l^Uj) + Vj,u. /c - 1 , . . . , n, (6.6)

where we put

Aj;.g,= I i(Aj;,aj^,j(x)D;TJ;,u;. (6.7)

By Theorem 1.3, there exists a positive constant Cj independent of h such that

||i; ;fclb,5-z^ Qlkj;fcM;-|| > = CI||M,-||^;,. (6.8)

For 0 < |/i| < /zo, we have ^luUj e Woi V;, C^). Since AJJ w, satisfies (6.6), by Theorem 2.1, there exists a positive constant C2 independent of h such that

||Aj;,ii,-||,;,^C2(||5j;,g||,-,_,+||i;,;,||,;,_,+||Aj;fc^ (6.9)

On the other hand by Theorem 1.13, there is a positive constant C3 independent of h such that for 0<\h\< ho, the following inequalities hold:

\\^j,kgj\\j,s-i^ C4gj\\j^,_i+i, k=l,...,n,

\\^J,kUj\\j,s-l^Cs\\Uj\\j,s-M.

Using these inequalities and (6.8), we have the following estimate: For 0<|/2 |< ho, and for any k = l,... ,n,

||AJ;fclI,.||,;,^C2C3(||g,-||,;._/+l+||lJ,-||,;._/+l)+QC2||^

Since the right-hand side does not depend on h, by Theorem 1.13(3°), we see that UJE W'O''\VJ,C^). Therefore we obtain UJE W'^\X,B). By the construction of Uj and (6.3), we have w =X Wy G W^^iX, B). Therefore by (6.1), using Theorem 4.1, we see that there exists a positive constant C4

§7. Elliptic Operators in the Hilbert Space L (X, B) 445

such that

| |w| | ,+l^C4( | |g | | ._z+i+| |ML-/) .

Thus the theorem is proved for k=l. We proceed by induction on k>l. Suppose that the assertion is true for k^j. We want to show the case A:=j + 1. In this case we have fe W'-^-^'^'iX, B), and ueW'(X, B). By induction hypothesis, we have ue W^^^{X, B). Taking s' = s+j instead of 5, we see that (6.1) holds f o r / e W''-^'^\X, B), and ue W''{X, B). Since we have proved the theorem for /c = l, we have ueW^'^^{X,B) = W^^^^^{X, B). By Theorem 4.1, there is a positive constant Cs such that the following inequality holds:

Since this implies (6.2), we showed the case k=j + \, which completes the induction. I

Corollary. Let A{p, D) be an elliptic linear partial differential operator of order /, and suppose that for fe C°°(X, B), u e W'(X, B) is a weak solution of

A{p,D)u=f

Then there exists v e C°°(X, B) such that in L^{X, B)

u-v = 0 holds.

Proof S ince /e C°°(X, B), for any integer A:> 0, we h a v e / e \V'"' '^(X, B), hence, by the above theorem, we have ue W'^^'iX^B). If A:>[n/2]+l, by Sobolev's imbedding theorem (Theorem 3.2), there is a i;e ^,^-fc_c„/2]-i^^^ B) such that in W^(X, B)

u = V

holds. Thus, u = v in L^(X, B). Since k is arbitrary, we have ve C^(X,B). I

§7. Elliptic Operators in the Hilbert Space L^(X, B)

Let A{p, D) be an elliptic linear partial differential operator of order / with C°° coefficients. A{p, D) extends uniquely to a continuous linear map of W\X, B) to W'~\X, B). By restricting the domain of A(;7, D) appropriately, we treat A{p, D) as a closed operator of the Hilbert space L^(X, B).


(a) Closed Extension of Elliptic Partial Differential Operators

Definition 7.1. We define a linear operator A in the Hilbert space L^{X, B) as follows. The doniain D(A) of A is given by

D(A) = {ue L\X, B) \ A{p, P)ue L\X, B)}, (7.1)

and for u e D(A), we put Au = A{p, D)u.

Theorem 7.1.

( r ) D{A) = W\X, B), and the topology of D(A) defined by the graph norm coincides with that of W\X, B).

(2°) A is a closed operator. (3°) C°°(X, B) is dense in D{A) with respect to the graph norm, that is,

for u e D{A), there is a sequence {<Pk}t=i ^ C°°(X, B) such that (pj^ converges to u in L^{X, B), and that A{p, D)(pk converges to Au.

Proof ( r ) Since L\X, B) = W^{X, B), by Theorem 6.1, if M G D(A), then ue W\X, B). Therefore by the a priori estimate in Theorem 4.1, there are positive constants Q , C2 such that for any uD(A),

| | M | | , ^ Q ( | | A I / | | + | | I / | | ) ^C2 | |M | | / , (7.2)

while the middle part is just the graph norm of u. Thus (1°) is proved. (2°) Since D(A) endowed with the topology defined by the graph norm

coincides with W^{X, B), this space is complete. Therefore the graph of A is a complete subspace of the direct product L^(X, B) xL^(X, B), hence closed.

(3°) is clear from the density of C°°(X, B) in W\X, B). I

Since the formal adjoint A{p, D)"^ of A(/7, D) is also a partial differential operator of order /, A{p, D)"^ has the unique extension to a continuous map of W\X,B) to W'-\X,B). By Proposition 5.2, Aip^D)"" is also elliptic.

Theorem 7.2. Let A* be the adjoint of A in the sense of the operator on the Hilbert space L^{X,B). Then the domain DiA"^) is given by D{A*) = W'{X, B), and for v e D(A*), we have

A*i; = A(;?,D)^i;. (7.3)

In particular, if A(p, D) is a formally self-adjoint elliptic partial differential operator, A is self-adjoint.

§7. Elliptic Operators in the Hilbert Space L^(X, B) 447

Proof. V e D(A*) if and only if there exists an fe L^iX, B) such that for any u e D(A),

{v,Au) = {f,u). (7.4)

Since C°°(X, B) c D{A), for any <p e C^{X, B) we have

(v,A{p,D)<p) = (fcp). (7.5)

This implies that u is a weak solution of the equation

Aip,Drv=f.

Since feL^(X,B), by Theorem 6.1, we have ve W^{X,B),. Conversely, suppose that v e W\X, B) and / = A(p, D)^v. Then fe L\X, B), and for any (p e C°°(X, B), (7.5) holds. Since both sides of (7.5) are continuous in (p with respect to the graph norm, and C°^{X, B) is dense in D{A) with respect to the graph norm, (7.4) holds for every u e D(A). Thus the theorem is proved. I

Lemma 7.1. Let A(p, D) be an elliptic linear partial differential operator of order I. We define the operator A as in Definition 7.1. Then

(1°) ker A is a closed subspace of L?{X, B). (2°) For any integer /c 0, there exists a positive constant Q such that

for any u e W^'^\X, B) n (ker A)^, the following estimate holds:

||M||,^fc^Cfc||Aw|U. (7.6)

Similarly if 0 < ^ < 1, for any integer k^O, there exists a positive constant Cj, such that for u e C^^^^\X, B) n (ker A)"",

\u\i+k+e = Ck\Au\j,+e (7.7) holds.

Proof ( r ) If {M„}^=I C: ker A, and if {M„} converges to u in L^(X, B), then AM„ = 0 converges to 0, and, since A is a closed operator, u G D{A) and Au = 0. This proves (1°).

(2°) Suppose that for some /c, (7.6) does not hold for any Q . Then for any positive integer 7, there exists a WyG W^^^{X., B)n(ker A)" such that

||M,-||/+fc = l and \\AUj\\k^r\ (7.8)

Since the sequence {Uj}j is bounded in W^^^{X, B), by Theorem 3.1, {Uj}j is relatively compact in W^(X, B). Taking a subsequence, if necessary, we


may assume that {Uj}j converges to u in W^(X, B), while {AM,} converges to 0 in W^{X, B). In L^(X, B) also, {Uj}j and {Au^}j converges to u and 0 respectively. Since A is a closed operator on L^(X, B), we have u e D(A) n ker A On the other hand, since {M;}CI (ker A)-^, and (ker A)" is a closed subspace of L^(X, B), we have u e (ker A)" . Hence M = 0 in L^(X, B). Since we have already known that u e W'^iX, B), we see that M = 0 in W^{X, B). Since {uj}j converges to u in W^{X, B),

lim||M,-|U = 0. (7.9)

Using the a priori estimate in Theorem 4.1 for {«/}<= W^'^'^{X, B), we see that there is a positive constant C such that

||M,.|U^,^C(||AM,-|U + ||M,.|U), 7 = 1 , 2 , . . . .

Taking the limits of both sides for j ^ o o , we have 1 ^ 0 by (7.8) and (7.9), which is a contradiction. Hence (7.6) is true. (7.7) is proved similarly. I

Theorem 7.3. Let A(p, D) he an elliptic partial differential operator of order I. We define the operator A and its adjoint A* as in Definition 7.1. Then we have the following,

(1°) Both ker A and ker A* are finite-dimensional subspaces of C°°(X,B).

(2°) The range R{A) of A and the range R{A^) o/A* are closed subspaces ofL\X,B).

{T) R{A) = (ker A*)-', JR(A*) = (ker A)^.

Proof ( r ) By the Corollary to Theorem 6.1, we have ker A c C°°(X, B). ker A is a Banach space as a subspace of L^{X,B). Let Q = {wGker A | | |w | | ^ l} be its unit ball. Since Qc: C°°(X, B), by Theorem 4.1, we see that if M G Q , | |W| | / ^ C||M|| ^ C. Thus Q is bounded in W\X,B). Therefore by Theorem 3.1, it is compact in L^{X, B), hence in ker A. Consequently ker A is a locally compact space, hence finite dimensional.

(2°) Suppose that a sequence {fj}t=i in R{A) converges t o / i n L^(X, B). By Lemma 7.1, for any j a n d / , we have the following inequality:

\\uj-urh^C,\\fj-ffl

Since {fjj is a Cauchy sequence, {Uj}j becomes a Cauchy sequence in W\X,B), hence converges to some ue W^(X,B) in W^(X,B). On the other hand, AM„ =/„ converges to / in L^(X, B), hence we have Au =f Thus R{A) is a closed set. The closedness of RiA"^) can be proved similarly.

(3°) Let ueR(A)-^. Since for any D G D ( A ) , (AU, M) = 0, we have ue D{A*), and A * M = 0 . Conversely if A*M = 0, clearly M 1 J R ( A ) , hence we


have ker A* = R{A)^. Since R(A) is closed by (2°), we obtain (ker A*)"- = R(A). (ker A)^ = RiA"^) is proved similarly. I

Lemma 7.2. Let P and Q be the orthogonal projections to ker A and ker A* m L^(X, B) respectively. Then for any integer /cÔ, there exists a positive constant Cj^ such that for any u e L^(X, B), the following estimates hold.

\\PuUâ\\ul | | Q M | U ^ C , | | I / | | . (7.10)

\Pu\j,êâ\\ul \Qu\j,êâ\\u\\. (7.11)

Proof. Let (fu - - -, (PL be an orthonormal basis of ker A. Then Pu = Hj (w, (Pj)<Pj' Therefore, taking into account the fact that cpj e C°°(X, B), and putting Cfc = CLj \\(pj\\iy^^, we obtain

\\Pu\\,^l\(u,<pj)\\\cpj\\,^C,\\u\\. j

The other estimates are obtained similarly. I

Definition 7.2. Let D(A) n (ker A)^ = ^ . Since A is a bijection of ^ onto R{A), let G be its inverse. G is a bijection of R{A) onto ^. We define

G = G ( 1 + Q),

and call G the Green operator of A. G is a linear map of L^(X, B) to ^ which coincides with G on R{A) and vanishes on ker A*.

Theorem 7.4. T/ie Green operator G has the following properties:

( r ) G /5 defined on L^(X, B), and its range R{G) is given by R{G) = W\X, B) n (ker A)^. For any u e L\X, B), AGu = {I- Q)u, and

for any ve W\X,B), GAv = {I-P)v. (2°) For any integer /c ^ 0, there is a positive constant C^ such that for

any u e W^(X, B), Gu e W^^\X, B), and

||Gw|U^,^Cfc||M|U (7.12)

holds. If 0<6<l, for any integer /cÔ, there exists a positive constant Cj, such that for any u e C^^^(X, B), Gu e C^^^^^X, B), and

\Gu\j,^jêâ\u\kê^ (7.13)

Proof ( r ) is clear.


(2°) For ueW\X,B), by Lemma 7.2, we have | | ( 1 - Q ) M | U ^

(Cfc + l)||w||fc. Let CJc be a positive constant in (7.6) of Lemma 7.L Then

||GW|U^, = ||G(1 + Q ) M | U ^ , ^ C U C , + 1)||M|U,

which proves (7.12). (7.13) is proved similarly. I

Corollary. Let A(p, D) he a formally self-adjoint elliptic partial differential operator, P the orthogonal projection to ker A, and G the Green operator of Alfue C°°(X, B), then Gu e C°°(X, JB), and the following equality holds

u = Pu + AGu.

By this the orthogonal decomposition

^ ( X , B) = ker AeAC° ' (X, B)

is given, where ACîX, B) denotes the image of CîX, B) by A{p, D).

Proof Let A be the operator on the Hilbert space L^{X, B) defined from A{p, D) as in Definition 7.1. Since A{p, D) is formally self-adjoint, A is self-adjoint (Theorem 7.2). Therefore, since ker A = ker A*, the orthogonal projection P coincides with the orthogonal projection Q to ker A*. By Theorem 7.4, 1, for any u e C°°(X, B) c W\X, B),

u = Pu + AGu

holds. Furthermore by Theorem 7.4(2°), GueC°°{X,B), hence AGue ACîX, B). Moreover since A is self-adjoint, by Theorem 7.3 (3°), ker A and ACîX, B) are orthogonal to each other. I

(b) The Spectrum of a Strongly Elliptic Partial Differential Operator

Let A(p, D) be a strongly elliptic linear partial differential operator of order 2m. We define the operator A as in Definition 7.1. Let 8 be the constant of strong ellipticity of A(p, D), and take 5i, 2 as in Theorem 4.1. If Re A > 5i, then for any (p e W^^(X, B), we have

||(A + A)(p| |^(ReA-5i)| |(^| | . (7.14)

For, indeed, by Theorem 4.2, we have

||(A + A)(^||||<^||^Re(A(p,(^) + ReA((p , (^ )^ (ReA-5 i ) | k f .

Furthermore there exists a positive constant C depending only on n, m, /A,


8, and M^ such that if Re A > 8„ for any (p e W^'îX, B),

\\cphm^C\\(A + X)cp\\ (7.15)

holds. In fact, it suffices to use A{p, D) +A for A(p, D) in (4.7), and make an estimate of ||< ||o by (7.14).

Theorem 7.5. Let A{p, D) he a strongly elliptic partial differential operator. / / R e A>5i , thenA-^X is a linear isomorphism of W^"^(X, B) ontoL^(X,B).

Proof. It is clear that A + A is a continuous injection of W^'"(X, JB) to L^(X, B). For any fe L^{X, B), there is a weak solution of the equation

(A + A) i /= /

in W ( X , B) (Theorem 5.2). By Theorem 6.1, u e W^'îX, B). Hence A + A is surjective. Therefore by (7.15) its inverse is also continuous. I

Fix iji with iJi^Si, and put

G^ = (A + M)-^ (7.16)

Since W^'îX, B) c: L\X, B), G^ can be considered as a map of L\X, B) to V'{X,B). Then the following theorem follows from Rellich's theorem.

Theorem 7.6. G^ is a compact liner map of L^{X, B) to L^{X, B). I

For an arbitrary complex number A, consider the equation

(A-A)<p=/ (7.17)

Multiplying by G^ from the left, and putting ^ = A + t, we have

(I-^G,)cp = GJ.

Putting G^f=g, we obtain

{I-^G^)<p = g. (7.18)

We consider (7.18) instead of (7.17).

{A-\) = {A-hfji){I-^GJ = (I-CGJ{A + fi)

holds. Since (A + /x) is a bijection of \y^^(X, B) onto L\X, B), ( A - A ) is bijective if and only if I-^G^ is bijective. Thus we have the following theorem.


Theorem 7.7. A complex number A is contained in the spectrum of A if and only if ^~^ = (A — /JL)~^ is contained in the spectrum of G^. I

Since G^ is a compact operator on L^(X, B), its spectrum is rather simple. In particular, except for 0, it has only the point spectrum whose only accumulation point is 0, and the generalized eigenspace belonging to each eigenvalue is finite dimensional.

Theorem 7.8. Let A{p, D) he a strongly elliptic linear partial differential operator. Then the spectrum of A is contained in the half space Re A > — 5i, and it consists only of the point spectrum which has no finite accumulation point Furthermore the generalized eigenspace belonging to each eigen value is finite dimensional. I

§8. C^ Difierentiability of (p(0

In this section we shall prove that the vector (0, l)-form (p{t) constructed in Chapter 5, §3 is C°°.

Let S be the disk of radius r in C" with the origin as its centre, namely, we put 5 = { = (^1, . . . ,^m)|kl^ = Z]li l^jl^^ ^^}- Let M be a compact complex manifold of dimension /i, and X its underlying C°° differentiable manifold. The vector (0, l)-form (p{t) on M parametrized hy teS satisfies the following quasi-linear partial differential equation.

(- 2 7-rT^+n)<p(0-M<p(0,<p(0] = o, (s.i) \ x=\dt dt /

where [ , ] denotes the Poisson bracket. (p(t) is holomorphic in t, and C^^^ {k^2,l>0>0) with respect to p e X. Moreover we may assume that there exists a positive constant K such that

where we put

\cp(t)\^^e^KAit), (8.2)

A ( 0 = — I —ih + '-'^tJ^ (8.3) 1 6 c ^ = 1 /JL

with positive constants b, c. The aim of this section is to show that (p{t) is C°° on X x{r G C" 11 r| < 2"V}

provided that r is a sufliciently small positive number. We cover M by coordinate neighbourhoods XjJ = \,... ,J. Let Zj =

(zj,..., Zj) be the local complex coordinates on Xj with zf = xj' + v-Ixj"^". The vector-valued (0, l)-form (p(t) is represented on Xj in terms of these

§8. C°° Differentiability of <pit) 453

local coordinates as

(p{t)=l<pHz,t)d,= l cpUz,t)dz-d,. (8.4) A X,a

Let 2ga0 dz" dz^ be the Hermitian metric tensor, and (g^") = (ga^)~^ Further we denote the covariant differentiation by V. Then if we put

e[cp,cp]=lÛz''d,, (8.5)

we have

cr,p,p

-"^.(pU.cp'^-cp^V^d^cp^^). (8.6)

Since the problem is local, we can use the results from the theory of elliptic partial differential equations. Choose a C°° function (Oj(p) on X with supp (Oj c: Xj so that for any peX,

i cOj(p)^l- (8.7)

Next, for each / = 1, 2 , . . . , we choose a function rj\t) as follows.

77^(0^1 on | r | ^ ( 2 " ' + 2"'~')r, (8.8)

v\t)^0 on \t\î2-'-^2-')r.

We assume that the r]^{t) are C°°. Put

co'j(p,t) = coj(p)v'v(t)^ (8.9)

Furthermore, we choose a C°° function ;t'j(/^) with supp;^^c:A} which is identically equal to 1 on some neighbourhood of the support of coj. We put

x'j{P,t) = Xj(p)v'it). (8.10)

Since rjît) is identically equal to 1 on a neighbourhood of the support of '^^^^(0, Xj is identically equal to 1 on some neighbourhood of the support of w]^\

First we shall prove that rjîp is c^^^^^. (o^(p satisfies the equation


where we put

Fo = Wje[<p,<p]~{(ojnxj(P~Oco]x]<p}^ (8.12)

Here we use the fact that Xj is idetitically equal to 1 in a neighbourhood of the support of coj.

The support of o)](p{p, t) is contained in Xj x S. In terms of local coordinates, we have

<^j<p{pj)=lLo)]{z,t)(p\z,t)d^ A

= I o^'M t)<pU^, t) dz" d,. (8.13)

We introduce real coordinates x", a = 1, 2 , . . . , 2n + 2m, by putting

z^(;?) = x^ + V^x^'"", / = ! , . . . , n ,

By these 2n + 2m real coordinates, XjXS is identified with an open set Uj of a (2n + 2m)-dimensional torus j^^+^m

A complex-valued function/(p, t) defined on X x 5 can be considered as a function on J^"^^'" if supp/c= Xj x 5. In this case we can consider the difference quotient A^/ in the direction of x", a = 1 , . . . , 2n + 2m. Since the support of the vector-valued (0, l)-form co^cp is also contained in Xj xS, we define its difference quotient A co]<;p as

A'^co'jcpiz, t)=l A'Aco'jCp',){z, t) dz' d„ (8.14) A,p

Since (o](p^ is a complex-valued function whose support is contained in Xj X 5, the right-hand side is well defined.

(8.11) can be considered as the equation on j^n+im HQ^icc^ by taking the difference quotient of both sides of this, we obtain the following equation.

where we put

/ y ^' \ .=,3t^5f^

]^Ala>]ct> = F„ + D A^w](A = F„ (8.15)

+ (nA^-A^n)cuj.p. (8.16)

Since ( -Zr=i d^/St'' dt'' + D ) is an elliptic linear partial differential operator whose principal part is of diagonal type, by Theorem 2.3, we obtain the

§8. C°° Differentiability of (p{t) 455

following a priori estimate

|Aâ;j(p|,^, ^ C,(|FiU_2+, + |Aô>j^|o), (8.17)

where Q is a positive constant, which may depend on k. In order to make an estimate of the right-hand side of this inequality, we use the following two lemmata.

Lemma 8.1. Let u and v be complex-valued C^^^ functions defined on T' with k ^ 0 and 1> 6>0. Then the product uv is C " , and there exists a positive constant Bj^ depending only on k and /, but independent ofu and v such that

\uv\^ê^B^ I (IwUall^ls + ltÛiÛ.). (8.18) r+s = k

The proof is easy, hence we omit it. I

In order to make ah estimate of the norm of A«wJ(p, the following lemma is useful. This corresponds to Theorem 1.13.

Lemma 8.2. Let Z j , . . . , x' be coordinate functions on the l-dimensional torus J^ and A^ the difference quotient operator in the direction ofx" by h. Then for any integer kÔ and any 0 with 1> 6>0, we have the following:

(i) For Se C^"^^(T , C'^), if\h\>0, the difference quotient A«5 is again an element of C'^'-'iT^C^.

(ii) IfSe C^^^(J\ C^), then for any a = \,...J, and any h with 0 < | / j | < 1, the following estimate holds:

|A^5kâ^ 151,^1-,,. (8.19)

(iii) IfSe C^^\t\ C^), andforanya = 1 , . . . , I and any h with Q<\h\ < 1, there exists a positive constant M such that

|A^5 | , ^ , ^M, (8.20)

thenSeC'''^'-'\j\C^).

This lemma is an analogy of Theorem 1.13 with the Sobolev norm replaced by the Holder norm. The proof of Lemma 8.2 is, however, very easy unlike that of Theorem 1.13, hence we omit it.

We shall make an estimate of || Fj || .20. The second term of the right-hand side of (8.12) is a linear combination of partial derivatives of order at most 1 of x]^' Also the third term of (8.16) does not contain the difference quotient of second-order partial derivatives of (o](p since they are cancelled.


Therefore, only the first term of F^

'2^'^co]e[cp,<p] = ^2^'^co]e[x'jcp,x]<p] (8.21)

contains the difference quotients of second-order partial derivatives of a)](p or Xj(p' Using the local representation (8.5) and (8.6), we can write (8.21) as

T,cr,p,v,\

-xW"^^. dAlicoh'i)} X d r a, + • . •, (8.22)

where we omit the terms involving no difference quotients of second-order partial derivatives of (o](p or x]^-

By Lemma 8.1, there exist positive numbers L, L' such that

\\^Wjeix]^.x]<P^k-2-.e

^L Y. UiV;-|o|Aâ),V,|,^, + L>,VUêl'A,VU... (8.23)

By Lemma 8.1 and Lemma 8.2, the C^'^'^^ norm of the terms of Fj other than (8.22) can be estimated by a positive multiple of \o}](p\k+e\Xj<P\k+e- On the other hand, applying (8.2) to the estimation of the first term lAr]< lo of (8.23), we obtain

\Fi\k-2ê ^ MoA(r)\A^â>]cp\j,ê + Mt,\co]cp\j,ê\x'j<p\kê, (8.24)

where MQ and M^ are positive constants. By (8.17) and (8.24), we obtain

+ CkM,,\w](p\^ê\x](p\kê' (8.25)

We choose a sufficiently small r so that

MoCfcA(r)<2-^ (8.26)

holds. Multiplying both sides of (8.25) by 2, transposing the first term of the right-hand side, and using (8.26), we obtain

|A^cc>,VU+,^2CfcKVK + 2QMi|co,V|,+,|(A](pL^,. (8.27)

§8. C°° Differentiability of <p{t) 457

Since cp is C^^^, the right-hand side is bounded independently of h. This is true for any a = l,... ,2n + 2m. Therefore by (iii) of Lemma 8.2, (o](p is proved to be c^^^^^. Since this is true for any j = 1 , . . . , /, summing with respect to j , we see from (8.7) that r]^(p is a c^"^^^^ vector (0, l)-form.

Next we shall prove that h^(p is c^^^"^^, using the same r determined by (8.26).

Replacing co] by co] and Xj t)y x] in (8.15), and differentiating with respect to x^, we obtain

(^-l^j^+nytD,w]cp = F„ (8.28)

where we put D^ = d/dx^, and

F2 = DpF,-h{D^^^D^co^cp-DpD^^^(o]cp}. (8.29)

By (8.28), we obtain the following a priori estimate:

\A^^Dpw]<p\j,ê^CMF2\k-2ê + \âD^co]cp\o), (8.30)

where Q is the same as in (8.17). |A^Dâ;j(p|o is estimated by \(o](p\2. We want to obtain an estimate of 1 21 -2+0- Since the second term of the right-hand side of (8.29) does not contain the difference quotients of partial derivatives of order 3 of (o](p, the only term of F2 which involves difference quotients of the partial derivatives of order 3 of (o](p or x]<P is

'2âD,co]0[x]cp,x]<p] = -'2 Z g'''{x]<P^V^dA'{D,co]cp',)

T,cr,p,v,\

-XhT^cr dAaiDp(0](p',)} J r a , + • • • , ( 8 . 3 1 )

where we omit the terms not involving difference quotients of partial derivatives of order 3 of a)](p or x]<P' Comparing (8.31) with (8.22), by the same argument as for (8.24) we obtain

\F2h-2ê^MoA{r)\AtDpCo]<p\^ê-^M^^,\xj<p\j,^, +0X\(Oj<p\k+i+e, (8.32)

where M^+j may be different from M^, but MQ is the same as in (8.24). By (8.32) and (8.30), we obtain

\^tDpCo]cp\^ê CkMoA{r)\^'^DpCo]<p\k+e

+ Cj,Mk^,\x'j<P\k-îê\co]cp\j,+i+e + M'^+,\co](p\2. (8.33)


Multiplying both sides of (8.33) by 2, transposing the first term of the right-hand side, and using (8.26), we obtain

\^^Dpa)](p\j,+0 ^ 2CkMk+i\x^(p\k+i+e\o)](p\k+i+e

-^2M^,^,\co]<p\2. (8.34)

Since rj^ip is c^^^'^^, the right-hand side is bounded independently of h. Since this is true for any a = 1 , . . . , 2n + 2m, by (iii) of Lemma 8.2, D^co](p is C^^^^^. Since this is true for any j8 = 1 , . . . , In + lm, we see that (o](p is C^^^^\ Summing with respect to j , we see from (8.7) that 17 V is C "" " ^ vector (0, l)-form. Note that in the above we need not replace r satisfying (8.26) by a smaller one.

Similarly we can prove that, for any / = 1, 2 , . . . , r]^^^^(p is C " " , where we may choose r independent of /. Since rj^^^^{t) is identically equal to 1 on | r |<2"V which is independent of/, cp is C°° on X xlreC^Url <2"V}.

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Index

Abelian variety 49 admissible fibre coordinate 325 affine algebraic manifold 40 algebraic curve 40 algebraic subset 40 algebraic surface 40 analytic continuation 9, 10 analytic function 10 analytic hypersurface 19 analytic subset 33 a priori estimate 275, 396, 430, 432,

436 arithmetic genus 220 associate 15 automorphism 43

base space 97, 184 Betti number 175 biholomorphic 26 biholomorphic map 26 biholomorphically equivalent 32 Bott's theorem 244 bracket 212 bundle

canonical 164, 178 complex line 165, 166, 167 dual 102 tangent 96 trivial 99

canonical basis 358, 361 canonical bundle 164, 178

o fP" 178 Cauchy-Riemann equation 7 cell 169 Chern class 166

first 180 closed differential form 83 coboundary 115

cochain, ^-cochain 115 cocycle, ^-cocycle 116 codimension 234 cohomology 109 cohomology class 120 cohomology group 115

a-cohomology group 140 cf-cohomology group 139 with coefficients in a sheaf 116, 120 with coefficients in F 144

compact complex manifold 39 complete 228 complex analytic family 59

complete 228 effectively parametrized 215 induced by a holomorphic map 206 of hypersurfaces 234 trivial 61

complex dimension 29 complex Laplace-Beltrami

operator 154 complex Lie group 48 complex line 29 complex line bundle 165, 166, 167 complex manifold 28, 29

compact 39 complex projective space 29 complex structure 28, 29, 38 complex torus 48

number of moduli of 238 complex vector bundle 101 component

contravariant 153 CO variant 153 of a tangent vector 96

composite function 12 constant of ellipticity 392, 396, 431 constant of strong ellipticity 394, 399,

434 contravariant component 153 contravariant tensor field 107

462 Index

contravariant vector 106 convergent power series 14 coordinate multi-interval 127 coordinate polydisk 32 coordinate transformation 29 coprime 15 covariant component 153 CO variant tensor field 106 covariant vector 106 covering manifold 45 covering transformation group 45

a-closed form 88 ^-closed vector (0, ^)-form 266 a-cohomology group 140 5-cohomology group with coefficients

in F 144 <i-cohomology group 139 deformation 61

infinitesimal 182, 188, 190, 198 de Rham's theorem 134, 139 determining set 2 diagonal type 404, 416, 436 diffeomorphism 38 difference quotient operator 367 differentiable family

locally trivial 193 of compact complex

manifolds 182, 183, 184, 345 of complex vector bundles 324 of holomorphic vector bundles 346 of linear operators 325 of strongly elliptic partial differential

operators 325 system of local coordinates of 185 trivial 193

differentiable manifold 37, 38 differentiable structure 37, 38 differential form 76

closed 83 exact 83 harmonic 144, 152, 156, 157 on complex manifolds 85

differential operator 62 dimension 33 directed set 119 discrete subgroup 48 distinguished polynomial 13 displacement 236

infinitesimal 236 distribution 363, 364

vector-valued 368 divisor 167

Dolbeault's lemma Dolbeault's theorem dual bundle 102 dual form 150 dual space 102

134, 139, 140

effective 215 effectively parametrized 215 eigenfunction 323 eigenvalue 323 elliptic curve 47

complex analytic family of 68 elliptic modular function 69 elliptic operator with constant

coefficients 391 elliptic partial differential

equation 443 elliptic partial differential

operator 363, 391, 430 strongly 320

elliptic surface 319 elliptic type 392, 395, 431 equicontinuous 276 equivalent 99, 109, 192 exact commutative diagram 133 exact differential form 83 exact sequence 123, 126 exterior differential 78

fibre 97 fibre coordinate 98

admissible 325 fine resolution 137 fine sheaf 134 fixed point 44 form

a-closed 101 dual 150 harmonic 156

with coefficients in F 160 holomorphic p-form 88

with coefficients in F 143 1-form 77 (p, g)-form 86

with coefficients in F 143 r-form 77 (0, ^)-form with coefficients in

F 142 formal adjoint 152, 439 formally self-adjoint 156, 321 Fourier series 373

Index 463

Fourier series expansion 368 Friedrichs' inequality 330 fundamental domain 47

Jacobian 24 Jacobian matrix 24

Garding's inequality 399, 433, 435 geometric genus 220 germ 109 Green operator 275, 315, 323 group of automorphisms 43

Kahler form 174 Kahler manifold 174 Kahler metric 174 kernel 123 Kronecker product 104 Kuranishi family 318

harmonic differential form 144, 152, 156, 157

harmonic form 156 with coefficients in F 160

harmonic function 9 harmonic part 315 harmonic vector (0, <3f)-form 274 hermitian metric 144

on the fibres 158 Holder norm 274, 303, 403, 436 holomorphic 2 holomorphic function 1, 2, 30 holomorphic map 23, 30 holomorphic p-form 88

with coefficients in F 143 holomorphic vector bundle 101 holomorphic vector field 67, 69 holomorphically equivalent 32, 61 homogeneous coordinate, of P" 29 homomorphism, of sheaves 123 Hopf manifold 49, 55 Hopf surface 69

deformation of 208 hypersurface 40

inclusion 124 infinitesimal deformation 182,188,190

along d/dt 198 infinitesimal displacement 236 inner product 102, 147, 425 integrability condition 269 integral domain 15 intersection number 54 irreducible 15 irreducible equation 20 irreducible factor 15 irreducible factorization 15 irregularity 220 isomorphism, of sheaves 124

Laplacian 9 Lax-Milgram theorem 440 Leibnitz' formula 367 length, of a multi-index 363 line at infinity 57 linear partial differential operator 424 local C°° function 109 local C°° vector (0, ^)-form 256 local complex coordinate 28 local coordinate 29

with centre q 32 local equation 33 local homeomorphism 110 locally constant function 112 locally finite open covering 32 locally trivial 193

majorant 277 manifold

affine algebraic 40 compact complex 39 complex 28, 29 differentiable 37, 38 projective algebraic 40 topological 37, 38

map biholomorphic 26 holomorphic 23, 30

meromorphic function 35 minimal equation 22 mixed tensor field 107 modification 53 monoidal transformation 319 multiple 15 multiplication operator 423

Noether's formula 220 non-singular prime divisor 167 non-homogeneous coordinate,

o fP" 30

464 Index

norm 148 number of moduli

314 215, 227, 228, 305,

obstruction 209, 255 first 214 Kh 255

1-form 77 orbit 43 order 321 orientable 80 orientation 80, 86

parameter 184 parameter space 60, 184 parametrix 405 partial derivative, of a

distribution 366 period matrix 48 Poincare's lemma 83 point at infinity 57 polydisk 2 power series ring 14, 15 (p, ^)-form 86

with coefficients in F 143 principal part 154, 395, 431 principal symbol 392, 395, 431 projection 97, 112 projective algebraic manifold 40 projective space 174, 216

complex 29 homogeneous coordinate of 29 non-homogeneous coordinate of 30

properly discontinuous 44

^-cochain 115 ^-cocycle 115 quadratic transformation 57 quartic surface 307 quotient sheaf 125 quotient space 43

r-form 77 rank 97 rational function 42 real vector bundle 101 reducible 15 refinement 117 region of convergence 7

regular 17 relatively prime 15, 19, 35 Rellich's theorem 427 restriction 114, 171, 172 Riemann matrix 49 Riemann-Roch formula 227 Riemann-Roch-Hirzebruch

theorem 220, 221 Riemannian metric 321

sheaf 109, 112 fine 134 of germs of C°° functions 111 of germs of C°° sections of F 112 of germs of C°° (0, ^)-forms with

coefficients in F 142 of germs of holomorphic

functions 111 of germs of locally constant

functions 112 quotient 125

simplicial decomposition 168 singular point 33 smooth 33 Sobolev norm 372, 430 Sobolev space 376, 425 Sobolev's imbedding theorem 369,

370, 428 Sobolev's inequality 330, 428 space of moduli 233 spectrum, of a strongly elliptic

operator 450 stalk 112 strongly elliptic 156, 322, 394, 399,

434 strongly elliptic partial differential

operator 320 structure theorem of distributions 375 subbundle 107 submanifold 33, 34, 39 subsheaf 123 support 82, 134, 367 surgery 52, 53 system of local complex

coordinates 28, 29 system of local coordinates, of a

differentiable family 185 system of local C°° coordinates 37, 38

tangent bundle 96 tangent space 94, 96

Index 465

tangent vector 62, 95, 96 tensor field 105

contravariant 107 covariant 106 mixed 107

tensor product 103, 104 theorem of completeness topological manifold 37, transition function 98 translation 367 trivial 61, 99, 193

230, 284 38

holomorphic 101 rank of 97 real 101

vector field 63 holomorphic 67, 69

vector-valued distribution 368 vector (0, ^)-form

5-closed 266 harmonic 274 local C^ 256

volume element 146

unique factorization domain 15 unit 15 upper semicontinuous 200, 326, 351,

352

weak solution 438, 439 Weierstrass P-function 46 Weierstrass preparation theorem Whitney sum 108

13

vector bundle 94, 97 complex 101 (0, q)-foTm with coefficients in F 142

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