JJ Rodriguez, Local Zeta Functions and Pseudo-Differential Equations Over P-Adic Fields

8/12/2019 JJ Rodriguez, Local Zeta Functions and Pseudo-Differential Equations Over P-Adic Fields

1/57

p


2/57

p


3/57


4/57

p

p

Qnp

p

M

p

p


5/57

L


6/57

p

p

p p

p n

p

p

Qp p Zp p pZp Zp Zp/pZp Fp Qp x Qp v v(p) = 1 |x|p = p

v(x)

ac(x) =xpv(x) Qp Zp p1Zp


7/57

S(Qnp ) C Qnp

S(Qnp ) Qnp x y Q

np x y=

ni=1 xiyi

f =f(x) Qp

[x1

, . . . , xn

]

p f(, ) |f|p

f(, )= F1

|f|pF

,

F :S(Qnp )S(Qnp )

Qnp

(x y)(x) dnx

Qnp dnx Qnp

Zn

p

1 f(, ) L2(Qnp ) f(, ) p

f(, )u= g;

E u= E g

Du

(x) =F1x

||pFxu

, >0, u S(Qp).

p 1

2i

dudx

=

R

e2ix

Fxu

() d, N.

p

u(x, t)

t + a

0(x, t)(Du)(x, t) +

n

k=1 ak(x, t)(Dku)(x, t)+ b(x, t)u(x, t) =f(x, t), x Qp, 0< t T ,

u(x, 0) =(x),

0< 1 < < n < D

ak(x, t)f(x, t)


8/57

ak(x, t) k= 0, 1, . . . , n b(x, t)

t

M (x) Qp

|(x)| C(1 + |x|p);

t M t C t

0 < 1 M t

u(x, t) =

t0

Qp

(x,t,, )f(, ) d d+

Qp

(x,t,, 0)(0) d,

(x,t,, ) x Qp 0 < t T

(x,t,, ) =Z(x , t , , ) +W(x,t,, ).

Z(x,t,y,) =

Qp

(x) exp(a0(y, )||p ) d,

|W(x,t,, )| x t

n

p

n n

u(x, t)t + (f(, )u) (x, t) = 0, x Qnp , n 1, t (0, T]u(x, 0) =(x), (f(, ))

(f(, )) (x, t) =F1x

|f()|pFx(x, t)

, x Qnp ,


9/57

> 0 f() Qp[1, . . . , n] d

B(Qnp ) : Qnp C ||||L 0

p

u(x, t)

t + a(DTu)(x, t) =f(x, t), x Q

np , t (0, T],

u(x, 0) =(x),


10/57

Qp

Qp n

|f()|p =

max1in

|i|pd

, Qnp ,

d f

(A) (x) =a0(x, t)(DT)(x) +

n

k=1 ak(x, t)(DkT )(x) +b(x, t)(x), >1 0< 1 < . . . < n < ak(x, t) b(x, t)

g(x) Qp[x1, . . . , xn] p |g|sp

g

|g|s

p, = Qnp\g1(0) (x)|g(x)|sp dnx, (x)> 0, S(Qnp ), g

g |g|sp


11/57

|g|sK, ps

S(Qnp ) (ac(g))|g|sp Z

p

pm pm

Z(s,g,) =(ac(g))|g|sp,

p

E(, g) =

Qnp

(x)(g(x)) dnx;

|E(, g)|= O(||0+p ),

0 {Re(s)| s Z(s,g,)}

|f|sp

T S(Qnp ) > 0 |f|

pT = 1

E=F1TS

(Qnp ) E p

p


12/57

p

p

p p p Qp Q p | |p

|x|p =

0 x= 0

p x Q x= pab a b p

= (x) p x ord (x) | |p p Qp

p

|x|p 0 |x|p = 0 x= 0

|xy|p= |x|p|y|p

|x + y|p (|x|p, |y|p)

Zp ={x Qp | |x|p 1} p pZp= {x Qp | |x|p< 1} Zp/pZp p Fp


13/57

P

p x= 0

x= p

j=0xjp

j ,

= ord (x) Z xj {0, 1, . . . , p 1} x Qp {x}p

{x}p =

0, x= 0 0

p||1j=0

xjpj, 0 (x + x

) = (x) x Bnl(x)

: Qnp C C

S(Qnp ) S(Qnp ) l 0

(x + x) = (x) x Bnl x Qnp

l= l ()


14/57

P

S(Qnp ) S(Qnp )

S(Qnp )

= (1, . . . , n) x= (x1, . . . , xn) Qnp x:=

ni=1 ixi

S(Qnp

)

(F)() =

Qnp

( x)() dnx Qnp

( x) =n

i=1(ixi) = exp

2in

i=1 {ixi}p

(xj) =

exp

2in

i=1 {xj}p

Qp

S(Qnp )

r r Z Br(0) := (prZp)

n

S

r

:= {

rN

|

|Br(0)= (0)

}.

L:= L(Qnp ) ={ S |Qnp

(x) dnx= 0} W :=W(Qnp )

C r r Z

S L + W

W = prn

Qnp

(x) dnx

r W L = W L S

L W W {F()| L}

p

f() Qp[1, . . . , n] f(D, ) >0 |f()|p

(f(D, )) =F1

|f|p F

,

S f(D, ) L2(Qnp )

(D) (x) =F1x ||p Fx

, >0, S(Qp).

p

1

2i

ddx

=

R

e2ix

Fx

() dn, N.


15/57

P

p (n)p (s)

(n)p (s) =1 psn

1 ps s C, s= 0.

n + 2ilnpZ s= 0

(n)p (s)(n)p (n s) = 1, s / {0} {n+

2i

lnpZ}.

Rs

Rs(x) =||x||snp

(n)p (s)

, (s)> 0, s /n+2i

lnpZ, x Qnp .

C

Rs(x), (x)= 1 pn

1 psn(0) +

1 ps

1 psn

||x||p>1

||x||snp (x) dnx

+ 1 ps

1 psn ||x||p1||x||snp ((x) (0)) d

nx,

n+ 2ilnpZ (s)> 0

Rs(x), (x)= 1 ps

1 psn

Qnp

(x)||x||snp dnx, s /n+

2i

lnpZ,

Rs(x), (x)= 1 ps

1 psn

Qnp

((x) (0))||x||snp dnx.

s= 0

R0(x), (x):= lims0

Rs(x), (x)= (0),

R0(x) = (x) Rs S(Qnp ) s

C\

n+ 2ilnpZ


16/57

P

||x||sp (s) > 0

||x||sp, (x) =

1 pn

1 psn

(0) + ||x||p>1 ||x||sp(x) d

nx

+

||x||p1

||x||sp((x) (0)) dnx, S.

||x||sp n

S(Qnp ) (F Rs) (x)

||x||sp s /n+ 2ilnpZ

DT >0

(DT)(x) =F1x

||||p Fx

S(Qnp )

(DT) (x) = (R ) (x) =

1 p

1 pn

Qnp

||y||np ((x y) (x)) dny.

(x) ||x||p1

||x||np |(x)| dnx 0 |f()|p

(f(D, )) =F1

|f|p F

,

S f(D, ) L2(Qnp )


17/57

P

f() Qp[1, . . . , n] f() d (i) f() d (ii)f() = 0 = 0

f() Qp[1, . . . , n]

d C0(f) C1(f)

C0(f)||||dp |f()|p C1(f)||||

dp, Q

np .

f() cf() c Qp Zp

f() Qp[1, . . . , n] d A Qnp 0 /A m = m(A, f) |f()|p p

m A

A Li=1Bi Bi =zi+ (pmZp)

n |f()|p =|f(zi)|p

Bi

f() Zp[1, . . . , n] d

|f|p f(D, ) d

g(x) Qp[x] x = (x1, . . . , xn) g(x)

|g|sp, =

Qnpg1(0)

|g(x)|sp(x) dnx,

s C (s) > 0 S dnx Qnp

g(x)

|g|sp |g|sp, p

s S

EE{(NE, nE)}

EE(1 pnENEs)|g|sp S |g|sp

S |g|sp, Z(s, g) Z(s, g) =Z0(s, g)


18/57

P

f(x) Zp[x] x= (x1, . . . , xn) d

Z(s, f) = L(ps)

1 pdsn,

L(ps) ps s= n/d Z(s, f)

A = {x Znp | (xi) d, i = 1, . . . , n} A = {x Znp | (xi) 0

|f|sp, F= Z(s, f)i

cipdris, (s)> 0.

1 pds

1 pnRds+n, p

nriri= 1 pdsn

1 pn ||x||dsp , p

nriri= pdris,

ri Z (s)> 0

|f|s

p, F=

1 pds

1 pn Z(s, f)Rds+n, F,

(s)> 0 Z(s, f) Rds+n C

C W


20/57

u(x, t)


np , t (0, T],

u(x, 0) =(x),

a > 0 >0 DT p

Z(x, t) =

Qnp

(x )eat||||p dn.

Z(x, t) = (1 pn)||x||np k=0 qkneat(qk||x||1p ) ||x||np eat(p||x||1p ) ; Z(x, t) =

m=1

(1)m

m!1pm

1pmn(at)m||x||mnp x= 0;

Z(x, t) 0 x Qnp t (0, T].


21/57

Z(x, t)

Z(x, t) =

k=||||p=pk(x )eat||||

p dn,

||||p=pk

(x ) dn=

pkn(1 pn), ||x||p p

k

pknpn, xp = pk+1

0, ||x||p > pk+1,

Z(x, t) = (1 pn)||x||np

k=0

pkneat(pk||x||1p ) ||x||np eat(p||x||

1p ) .

Z(x, t)

ex

Z(x, t) =

m=1

(1)m

m!

1 pm

1 pmn(at)m||x||mnp x= 0.

l(x) Bnl(0) Fl =

pnll

Z(x, t) =

l=

eatpl

||||p=pl

(x ) dn

=

l=

eatpl

(pn(l)l(x) pn(l1)l+1(x))

=

l=

pnl(eatpl

eatp(l+1)

)l(x) 0.

Z(x, t) C t(t1/ + ||x||p)n

t >0 x Qnp


22/57

l pl1 t1/ pl

Z(x, t)

Qnp

eat||||p dn

Qnp

eap(l1)||||p dn=

Qnp

ea||p(l1)||p dn

=p(l1)n Qnp

ea||||p d= C0()pnpln C1tn/.

||x||p t1/

Z(x, t) ||x||np

m=1

Cm2m!

(t||x||p )m C3t||x||

np .

||x||p t1/

Z(x, t) C3t||x||np 2

+nC3t(t1/ + ||x||p)

n.

||x||p < t1/

Z(x, t) C1tn/ 2+nC1t(t

1/ + ||x||p)n.

Z(x, t) x L1(Q

np ) L2(Q

np )

Qnp

Z(x, t)dnx= 1.

M

M >0 (x) Qnp

|(x)| C() (1 + ||x||p).

t M

t

C

l ()

t

M <

limt0+

Qnp

Z(x , t)() dn= (x).


23/57

Qnp

Z(x , t)() dn (x)

=

Qnp

Z(x , t) (() (x)) dn

Qnp

Z(x , t)|() (x)| dn

C

Qnp

t(t1/ + ||x ||p)n|() (x)| dn:= I(x, t) .

M < I(x, t)

I(x, t) =C

||x||p>pt(t1/ + || x||p)

n|() (x)| dn

I1(x, t) +I2(x, t),

I1(x, t) :=C1t

||x||p>p

1 + ||||p(t1/ + ||x ||p)+n

dn,

I2(x, t) :=C t|(x)|

||x||p>p

(t1/ + || x||p)n dn.

>0 t >0 I2(x, t) C2t|(x)|,

<

I1(x, t) C1t

C3+

||||p>p

||x ||p||||p+n

dn

C1t

C3+

pxp

||x ||p||||p+n

dn

=

C1tC4(x) + ||||p>xp 1||||p+n dn =C5(x) t.

limt0+

Qnp

Z(x , t)() dn (x)

limt0+ C6(x) t= 0.


24/57

Z(x, t) 0 x Qnp t (0, T];

Qnp

Z(x, t) dnx= 1 t >0

SQnp

lim(x,t)(x0,0)

Qnp

Z(x , t) () dn= (x0)

Z(x, t+ t) =Qnp

Z(x y, t) Z(y, t) dny t t >0

eat||||p L1

Qnp

Qnp Z(x y, t1)Z(y, t2) dny= F1 (F(Z(y, t1) Z(y, t2)))

=F1

eat1||||p eat2||||

p

=Z(x, t1+ t2).

b >0 0 < x Qnp

I(b, x) =

Qnp

(b+ ||x ||p)n ||||pd

n C b

1 + ||x||p

,

C b x

m pm1 b pm

(b+ ||x ||p)n

pm1 + ||x ||p

n,

I(b, x) I(pm1, x) =

Qnp

pm1|p+ ||x ||p

n||||pd

n

=p(m1)(n)Qnp

1 + ||pm1x pm1||p

n||||pd

n

=p

(m1)() Qnp 1 + ||pm1x ||pn ||||pdn=p(m1)()I(1, pm1x).

pm1x= y ||y||p=pl

I(1, y) =I1(y) +I2(y) +I3(y),


25/57

I1(y) =l1

k=||||p=pk

(1 + ||y ||p)n ||||pd

n,

I2(y) =||||p=pl

(1 + ||y ||p)n ||||pd

n,

I3(y) =

k=l+1

||||p=pk

(1 + ||y ||p)n ||||pd

n.

I1(y) C0(1 + ||y||p)n||y||+np ;

I2(y) C1||y||p

I2(y) C2

I(1, y) C3(1 + ||y||p)

I(b, x) C3p(m1)()(1 +p(1m)||x||p )

C3pm(1 + ||x||p) C b

1 + ||x||p

.

I1(y) =l1

k=||||p=pk(1 + ||y ||p)

n ||||pdn,

= (1 pn)(1 + ||y||p)n

l1k=

p(+n)k

C0(1 + ||y||p)n||y||+np ,

C0=(1 pn)pn

1 pn .

y Qp |

y|p= p

l =||y||p

I2(y) = ||||p=pl

(1 + ||y ||p)n

||||pdn

=||y||p

||||p=pl

1 + |y|p||y1y y1||pn dn

=||y||p

||||p=1

||y||1p + ||u ||p

ndn u=y1y.


26/57

Am= { Qnp | ||||p = 1 ||u ||p = p

m}, m N,

I {1, 2, . . . , n}

Am,I={ Am| |ui i|p= pm i I |ui i|p < pm i /I},

u= (u1, . . . , un) = (1, . . . , n) Qnp ||||p = ||u||p= 1

Am

IAm,I

vol(Am,I) (pm(1 p1))|I|(pm1)n|I|,

|I| I

vol(Am)n

|I|=0

n

|I|

(pm(1 p1))|I|(pm1)n|I| =pmn,

I2(y) =||y||p

m=0

Am

||y||1p + ||u ||p

ndn

||y||p

m=0

||y||1p +p

mn

pmn

=||y||p1 pn

m=0

||||p=pm

||y||1p + ||||p

ndn

=||y||p1 pn

||||p1

||y||1p + ||||p

n

dn

C1||y||p

Qnp

||y||1p + ||||pn dn=C1||y||

p

Qnp

||y||1p + ||y||

1p ||y||pn dn

=C1||y||+np

Qnp

(1 + ||y||p)n dn=C1||y||

p

Qnp

(1 + ||||p)n dC1||y||

p .

I3(y) =

k=l+1

||||p=pk

(1 + ||||p)n ||||pd

n,

Qnp

(1 + ||||p)n ||||pd

n= C.


27/57

>0

||x||p = 1

(n)p ()

Qnp

||y||np ((x y) 1) dny

x Qnp

0<

(DTZ)(x, t) =

Qnp

(x )||||peat||||p dn.

Z(x y, t) =Z(x, t) ||y||< ||x|| (DTZ)(x, t)

(DTZ)(x, t) = 1

(n)p ()

Qnp

||y||np (Z(x y, t) Z(x, t)) dny

= 1

(n)p ()

||y||p||x||p

||y||np (Z(x y, t) Z(x, t)) dny.

(DTZ)(x, t)

1

(n)p ()

||y||p||x||p

Ct||y||2np + Z(x, t)||y||

np

dny

< .

(DTZ)(x, t)

Z(m)(x, t) :=

||||ppm

(x )eat||||p dn.

Z(m)(x, t) x m (DTZ)(x, t)

(DTZ(m))(x, t) =

1

(n)p () Qnp

||y||np (Z(m)(x y, t) Z(m)(x, t)) dny

= 1

(n)p ()

||y||p>pm

||y||np (Z(m)(x y, t) Z(m)(x, t)) dny

= 1

(n)p ()

||y||p>pm

||y||np

||||ppm

eat||||p (x )((y ) 1) dn dny


28/57

=

||||ppm

eat||||

(x )

1

(n)

p () ||y||p>pm ||y||np ((y ) 1)d

ny dn.

||y||p pm

(y ) = 1 ||||p pm

(DTZ(m))(x, t)

||||ppmeat||||

p (x )

1

(n)p ()

Qnp

||y||np ((y ) 1) dny

dn

=

||||ppm

eat||||

(x )||||pdn.

(DTZ)(x, t) =Qnp

eat||||p (x )||||pd

n.

Z

t(x, t) =a

Qnp

(x )||||p eat||||p dn;

Z

t(x, t) =a(DTZ)(x, t) 0< .

Zt (x, t) Ct1/ + ||x||pn ;(DTZ)(x, t) Ct1/ + ||x||pn .

Qnp

(DTZ)(x, t) dnx= 0.


29/57

p

(x), f(x, t) M

0 <

u(x, t)


np , t (0, T],

u(x, 0) =(x),

a >0 >0 M

u(x, t) =

QnpZ(x , t)() dn+

t

0

Qnp

Z(x , t )f(, ) dn

d.

u1(x, t) :=

Qnp

Z(x , t)() dn,

u2(x, t) :=

t0

Qnp

Z(x , t )f(, ) dn

d .

u(x, t) M t u(x, t)

u1(x, t) M t l N (+ y) =() yp p

l

y = u1(x, t) u1(x, t) |u1(x, t)| C(1 + x)

u1(x, t) M

t

u2(x, t) x |u2(x, t)| C T(1 + x)

u(x, t) =u1(x, t) + u2(x, t) M

t

limt0+ u(x, t) =(x) limt0+ u1(x, t) =(x) lim

t0+ u

2(x, t) = 0 |u

2(x, t)| C t (1 + x) t T

u1(x, t), u2(x, t) t

u1t

(x, t) =

Qnp

Z

t(x , t)() dn.


30/57

u2t (x, t) = t0 Qnp Zt (x , t )f(, ) f(x, ) dn d+ f(x, t).

u2,h(x, t) :=

th0

Qnp

Z(x , t )f(, ) dn

d ,

h uh(x, t+ t

) uh(x, t)

t

th0

Qnp

Z(x , t+ t ) Z(x , t )

t f(, ) dn

d

+ th+tth

Qnp

Z(x , t+ t ) Z(x , t )

t f(, ) dn

d

+1

t

th+tth

Qnp

Z(x , t )f(, ) dn

d .

t 0+ th0

Qnp

Z

t(x , t )f(, ) dn

d ,

Qnp

Z(x , t+ t ) Z(x , t )

f(, ) dn,

Qnp

Z(x , t )f(, ) dn,

Qnp

Z(x , h)f(, t h) dn.

u2,ht

(x, t) =

th0

Qnp

Z

t(x , t )f(, ) dn

d

+

Qnp

Z(x , h)f(, t h) dn.


31/57

u2,ht

(x, t) =

th0

Qnp

Z

t(x , t )(f(, ) f(x, )) dn

d

+ th0

Qnp

Z

t(x , t )f(x, ) dn

d

+

Qnp

Z(x , h)(f(, t h) f(, t)) dn

+

Qnp

Z(x , h)f(, t) dn.

t = f

Qnp | p pm f h f(x, t) h 0+

u2t

(x, t) =

t0

Qnp

Z

t(x , t )(f(, ) f(x, t)) dn

d+ f(x, t).

u

t(x, t) =

Qnp

Z

t(x , t)() dn

+

t0

Qnp

Z

t(x , t )(f(, ) f(x, t)) dn

d+ f(x, t).

DT 0 < u(x, t) (DTu)(x, t) > u(x, t) M

<

(DTu1)(x, t) = Qnp

(DTZ)(x , t)() dn.

Z(x, t) := (DTZ)(x, t)

Z,l(x, t) := 1

(n)p ()

||y||p>pl

||y||np

Z(x y, t) Z(x, t)

dny.


32/57

1

(n)p ()

||y||p>pl

||y||np

u1(x y, t) u1(x, t)

dny

= 1

(n)p ()

||y||p>pl

||y||np

Qnp

Z(x y , t) Z(x , t)

() dn

dny

=

Qnp

()

1

(n)p ()

||y||p>pl

||y||np

Z(x y , t) Z(x , t)

dny

dn

=

Qnp

Z,l(x , t)() dn.

m

||x||ppm

Z,l(x , t)() dn+ ||x||p m Z,l(x, t) =Z(x, t)

1

(n)p ()

||y||p>pl

||y||np

u1(x y, t) u1(x, t)

dy=

||x||ppm

Z(x , t)() dn

+ ||x||p m liml

liml

||x||p


33/57

=

||x||p liml

<

(DTu2)(x, t) =

t0

Qnp

(DTZ)(x , t )f(, ) dn

d .

u2,h(x, t) := th

0 Qnp

Z(x y, t )f(y, ) dny d.

1

(n)p ()

||y||p>pl

||y||np

u2,h(x y, t) u2,h(x, t)

dny

= 1

(n)p ()

||y||p>pl

||y||np

th

0 Qnp Z(x y , t ) Z(x , t )

f(, ) dn

d

dny

=

th0

Qnp

Z,l(x , t )f(, ) dn

d, Z,l(x, t)

Z,l(x, t) =

Qnp

(x )Pl()eat||||p dn,

Pl() =

1

(n)p () ||y||p>pl

||y||n

p (y ) 1 dny.

|Pl()| 2||||p

|(n)p ()|

||u||p>1

||u||np dnu= C||||p ,


34/57

|Z,l(x, t)| C.

||x ||pp(l1)

Z,l(x , t ) =Z(x , t )

th0

Qnp

Z,l(x , t )f(, ) dn

d

=

th0

||x||pp(l1)

Z(x , t )f(, ) dn

d

+

th0

||x||ppl

(DTZ)(x , t ) (f(, ) f(x, )) dn

d ,

l f(, ) u2,h M h h 0+

(DTu2)(x, t) =

t0

Qnp

(DTZ)(x , t ) (f(, ) f(x, )) dn

d.

(DTu)(x, t) =Qnp

(DTZ)(x , t)() dn

+

t0

Qnp

(DTZ)(x , t )(f(, ) f(x, )) dn

d;


35/57

a(DTu)(x, t) =

Qnp

Z

t(x , t)() dn

t

0 QnpZ

t

(x , t )(f(, ) f(x, )) dn d , 0<

u(x, t) M t u(x, t) u(x, t)

g(x) Zp[x1, . . . , xn] g(x) Fp[x1, . . . , xn] p g(x) p f(x) Zp[x1, . . . , xn] f(0) = 0 d f(x)= 0 f(x) p

x Fnp |f(x) = 0

={0} ,

f(x) Qp x Qnp |f(x) = 0

={0} .

f p f Qp

I {1, . . . , n} fI(x) fI(x) f(x)

TI :=

x Znp |xi= 0 i I

,

TI := x Fnp |xi= 0 i I . f(x) Zp[x1, . . . , xn] f(0) = 0 d Zp f(x)

p I {1, . . . , n} fI(x) p


36/57

f(x) =x2 y2 Zp \Zp

2

Zp

2

:=

x Zp |x = y

2 y Zp

.

f(x) p

p

n n= 1 1 n k k 2 g (x1, . . . , xk) p d Zp

l Fp l 2 f(x1, . . . , xk+1) = g (x1, . . . , xk)l xldk+1

f(x1, . . . , xk+1) p

f(x) Zp[x1, . . . , xn] f(0) = 0 d Zp f(x) p

|f(x)|p= xdp x Q

np

A:=

(z1, . . . , zn) Znp | |zi|p= 1 i

f(x)

Qp supzA

|f(z)|p

xdp |f(x)|p

infzA

|f(z)|p

xdp

|f|p|A1.

I {1, . . . , n}

AI={x A | |xi|= 1 i I} .

IAI A I {1, . . . , n}

|f|p |AI1 I

I={1, . . . , r} 1 r n x AI xi Z

p i= 1, . . . , r xi pZp i= r+1, . . . , n f(x) =fI(x)= 0

f p |f|p|AI1


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Z(x, t)

( (t) f) (x) =

Qnp

Z(x , t) f() dn.

Z(x, t)> 0 (t) f(x) 1 (t)


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p

u(x, t)

t + a0(x, t)(D

Tu)(x, t) +

nk=1 ak(x, t)(D

kT u)(x, t)

+b(x, t)u(x, t) =f(x, t), x Qnp , t (0, T],

u(x, 0) =(x).

> 1 0 < 1 < . . . < n < a0(x, t) a1(x, t), . . . , an(x, t)

b(x, t) DT

M ( 0) (x) Qnp

|(x)| C(1 + ||x||

p). M

n= 1

Z(x,t,y,)

u(x, t)

t + a0(y, )(D

Tu)(x, t) = 0, x Q

np , t (0, T],

u(x, 0) =(x),

y Qnp > 0


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P

Z(x,t,y,) =

Qnp

(x )ea0(y,)t||||p dn.

Z(x,t,y,)

Z(x,t,y,) C t(t1/+||x||p)n, Zt (x,t,y,) C(t1/+||x||p)n, (DTZ)(x,t,y,) C(t1/+||x||p)n,

y

Qnp

Z(x,t,y,) dnx= 1,

Z

t(x,t,y,) =a0(y, )

Qnp

(x )||||p ea0(y,)t||||p dn,

(DTZ)(x,t,y,) =

Qnp

(x )||||pea0(y,)t||||

p dn,

Qnp

(DTZ)(x,t,y,) dnx= 0.

ak(x, t) k= 0, 1, . . . , n b(x, t) 0 < 1 a1(x, t) = = an(x, t) = 0 0 <

n= 1

u(x, t) f(x, t) = 0 u M t u(x, 0) = 0 u(x, t) = 0 x Qnp t (0, T]


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P

u(x,t,) := t

Qnp

Z(x y, t ,y ,)f(y, ) dny d,

< t f(x, t) x Qnp (x, t) Qnp (0, T]

|f(x, t)| C t(1 + ||x||p),

0


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P

|R(x,t,, )| Cn+1

k=1(t )1/ + ||x ||p

kn,

C

J(x,,t , ) =

t

(t )/( )/Qnp

(t )1/ + ||x ||p

nb1 ( )1/ + || ||p

nb2dn

d,

0 < t x, Qnp b1, b2> 0 + b1< + b2<

J(x,,t , )

C

(t )

(++b2) B(1

, 1

+ b2

)

(t )1/ + ||x ||pnb1+

C

(t )

(++b1) B(1

+ b1

, 1

)

(t )1/ + ||x ||pnb2 ,

C b1 b2 B(z1, z2)

|Rm(x,t,, )|

C Mm(t )(m1)/((/))m

(m/)

n+1k=1

(t )1/ + ||x ||p

kn, C

m m = 1 m

|Rm+1(x,t,, )| t

Qnp |R(x,t,,)| |Rm(,,, )| dn d

=C Mm((/))m

(m/)

n+1k,l=1

t

( )(m1)/Qnp

(t )1/ + ||x ||p

kn( )1/ + || ||pln dnd.


43/57

P

Ik,l(x,,t , ) =

t

( )(m1)/

Qnp

(t )1/ + ||x ||pkn( )1/ + || ||pln dnd.

Ik,l(x,,t , )

C B( k

,

m +

)(t )(m+k)/

(t )1/ + ||x ||p

ln+ CB(1,

m+ l

)(t )(m+l)/

(t )1/ + ||x ||pkn.

, >0

B(x + , y+ ) B(x, y)

B( k

,

m+

) B(

,m

),

B(1,m+ l

) B(

,m

),

(t )(m+k)/ C(t )(m+11)/.

|Rm+1(x,t,, )|

C Mm+1(t )m/ ((/))m+1

((m+ 1)/)

n+1k=1

(t )1/ + ||x ||p

kn.

(x,t,, ) =

m=1

Rm(x,t,, ),

|(x,t,, )| Cn+1

k=1(t )1/ + ||x ||pkn

n 0 < 1 a1(x, t), . . . , an(x, t) 0 <


44/57

P

u(x, t)

t + a0(x, t)(D

Tu)(x, t) +

nk=1 ak(x, t)(D

kT u)(x, t)

+b(x, t)u(x, t) =f(x, t), x Qn

p , t (0, T],u(x, 0) =(x),

u(x, t) =

t0

Qnp

(x,t,, )f(, ) dn d+

Qnp

(x,t,, 0)() dn,

Qnp [0, T] t M t (x,t,, ) x, Q

np

0 < t T

(x,t,, ) =Z(x , t , , ) +W(x,t,, ),

|W(x,t,, )| C{(t )n+[(t )1/ + ||x ||p]n

+ (t )n+1k=1

[(t )1/ + ||x ||p]kn}.

u1(x, t) u2(x, t)

u1(x, t) = t0

Qnp

Z(x , t , , )f(, ) dn d

+

t0

Qnp

Z(x , t ,,)F(, ) dnd,

u2(x, t) =

Qnp

Z(x ,t,, 0)() dn

+

t

0 QnpZ(x , t ,,)G(, ) dnd,

F(, ) =

0

Qnp

(,,, )f(, ) dnd,

G(, ) =

Qnp

(,,, 0)() dn.


45/57

P

|F(, )| C, |G(, )| C n+1/,

Qn

p

(0, T]

F

G

N ai b Zi Z || q

N

(x + ,t,+, ) =(x,t,, ),

F(+ , ) =F(, ), G(+, ) =G(, ).

u1(x, t) u2(x, t)

u(x, t)

u(x, t) (x) t 0 u2(x, t) (x) t 0

u2(x, t) =

Qnp

[Z(x ,t,, 0) Z(x ,t,x, 0)]() dn

+

Qnp

Z(x ,t,x, 0)[() (x)] dn+ (x).

Z

{| ||x ||ppN}.

t 0

(x,t,, )

(x,t,, )

b(x, t) = 0


46/57

DTu= v, v S, >0.

E S

E v

E c E+ c

E

DT((E+ c) v) =DT((E v) + (c v))=u+ DT(c v) =u,

u c v DT


47/57

E(x) =

1 p

1 pn||x||np =n

1 pn

pn lnpln(||x||p) = n

E FE

||x||p FE= 1,

||x||sp =mZ

cm(s+ )m

cm S

m S ||x||sp ||x||

s+p =||x||

p ||x||

sp

s=

||x||p cm= 0

m n 0< < n

||x||p1||x||np d

nx


48/57

E, = 1 p

1 pn (Qp)n ||x||np (x) dnx

= 1 p

1 pn

Qnp

||x||np (x) dnx.

(x) = (x) S F(

)

F

F()

=

= n

c0

||x||sp, F()= mZcm, F()(s + n)

m,

||x||sp, F()= 1 ps

1 psn

Qnp

||x||snp (x) dnx

= (1 ps)

Qnp

pv(x)(s+n)

1 psn(x) dnx,

x= (x1, . . . , xn) v(x) := 1in

v(xi) ||x||p= pv(x)

(1 ps)pv(x)(s+n)

1 psn =

1 pn

lnp (s + n)

1

+(1 p1)v(x) lnp lnpp +

(p1)2p lnp

lnp + O((s + n)),

En, = c0, =

Qnp

1 pnpn ln(p)

ln(||x||p) +pn 3

2pn

(x) dnx.

F() S

L

v L >1 DTu= v =n L 1/= 1/+ /n ||u||LC||v||L


49/57

u= Iv, (Iv) (x) = 1

()

Qnp

v(y)||x y||n dny.

f(D, ) d E f(D, )u= v >0 v S

E(x) =

L(p)(1 pd)

(1 pn)(1 pdn)||x||dnp W =n/d

L(pn/d

)(1 pn

)(1 pn)(pn lnp)

ln(||x||p) W = n/d,

L(ps) Z(s, f)

E FE

|f|p FE= 1 S.

|f|

p , =

(1 pd)L(p)

(1 pn)(1 pdn)R

d+n,

W s C W

=n/d

|F(E )(x)| C()||x||dp |F()(x)|,

x Qnp W

S l

||||2Hl =

Qnp

[ (1, ||||p)]2l|F()()|2 dn.


50/57

S ||||Hl l Hl :=Hl(Qnp )

Hl S

f(x) = 0 ||x||p 1||x||p ||x||p > 1, > n

||f||2Hl =

||||p1

(1 pn)(1 ||||np pn)(1 pn)

p||||np

2 dn. ||f||2

Hl n/2 Hl

Hl F() L1 Qnp

|F()()| dn=

Qnp

( (1, ||||p))l

( (1, ||||p))l|F()()| dn,

Qnp

( (1, ||||p))2l dn,

l > n/2 Qnp

|F()()| dn C||||Hl .

>0 l 0 f(D, ) :Hl+d Hl

S f(D, )

||f(D, )||2Hl = Qnp [ (1, ||||p)]2l|f()|2|F()()|2 dnC1

Qnp

[ (1, ||||p)]2(l+)|F()()|2 dn= C1||||

2Hl+ .

S Hl+d


51/57

I() :=

||||p1

||||pdn.

I() = 1 pn

1 pnn, >n.

I() =

||||p1

||||2lp|F(vW)()|2 dn

.


52/57

vW() = prnCr() r > 0

F(vW)() =C r()

||uW||2Hl+d C(,d,n)C2p2rn ||||p1 ||||

2dp d

n

+ ||vW||2Hl+d

C(,d,n)

C1(,d,n) + ||vW||

2Hl

,

2d >n uWHl+d

F(uL) =F(vL)|f|p ,

f

|F(uL)()| C||||d|F(vL)()| .

||uL||2Hl+d

||||p1

||||2dp |F(vL)()|2 dn

+

||||p>1

||||2lp|F(vL)()|2 dn.

||vL||2Hl

0< < n/2d ||||p1

||||2dp |()|2 dn C||||L2 ,

S

||uL||2Hl+d C||F(vL)||L2+ ||vL||

2Hl .

v Hl+d f(D, )u= v v S u f(D, )u =v

f(D, )(u u) = 0, |f|p F(u u) = 0,

F(uu)() = 0 = 0 f (x)(uu)() = 0 (u u)() = 0 u u u() =u() Qnp


53/57

f(D, )

S l

||||2Hl :=

Qnp

||||2lp|F()()|2 dn.

S | | | |Hl l Hl :=

Hl(Qnp ) Hl Hl l 0 ||||Hl ||||Hl

> 0 l 0 f(D, ) : Hl+d Hl

Ll Wl L W | | | |Hl

Hl0:= Ll + Wl Hl.

f(D, ) >0 d l f(D, ) : Hl+d Wl

v Wl {vn} Wl

v vn W

{un}

Hl+d

f(D, )un =vn

{un} Hl+d

||un um||2Hl+d C

Qnp

||||2(l+d)p ||||2dp |F(vn vm)()|

2 dn

C||vn vm||2Hl ,

u Hl+d un u f(D, ) f(D, )u= v

f(D, ) >0 d l f(D, ) : Hl+d Ll

v Ll {vn} Ll

v vn L {un} H

l+d f(D, )un = vn

{un} H

l+d

|F(un)()| C||||d|F(vn)()|,


54/57

||un um||2Hl+d C

Qnp

||||2(l+d)p ||||2dp |F(vn vm)()|

2 dn

C||vn vm||2Hl ,

u Hl+d un u f(D, ) f(D, )u= v


55/57

p p

p

p

p p

p


56/57

p

p

p

p

p

p

p

p

p

p


57/57

p

p

p

JJ Rodriguez, Local Zeta Functions and Pseudo-Differential Equations Over P-Adic Fields

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Transcript of JJ Rodriguez, Local Zeta Functions and Pseudo-Differential Equations Over P-Adic Fields