Dahlberg B.E.J., Kenig C.E. Harmonic Analysis and Partial Differential Equations (1996)(en)(138s)

7/30/2019 Dahlberg B.E.J., Kenig C.E. Harmonic Analysis and Partial Differential Equations (1996)(en)(138s)

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H A R M O N I C A N A L Y S I S A N D

P A R T I A L D I F F E R E N T I A L E Q U A T I O N S

B J R N E . J . D A H L B E R G

C A R L O S E . K E N I G

I S S N 0 3 4 7 - 2 8 0 9

D E P A R T M E N T O F M A T H E M A T I C S

C H A L M E R S U N I V E R S I T Y O F T E C H N O L O G Y

A N D T H E U N I V E R S I T Y O F G T E B O R G

G T E B O R G 1 9 8 5 / 1 9 9 6


2/144


3/144

F O R E W O R D

T h e s e l e c t u r e n o t e s a r e b a s e d o n a c o u r s e I g a v e r s t a t U n i v e r s i t y o f T e x a s , A u s t i n

d u r i n g t h e a c a d e m i c y e a r 1 9 8 3 - 1 9 8 4 a n d a t U n i v e r s i t y o f G t e b o r g i n t h e f a l l o f 1 9 8 4 .

M y p u r p o s e i n t h o s e l e c t u r e s w a s t o p r e s e n t s o m e o f t h e r e q u i r e d b a c k g r o u n d i n o r d e r

t o p r e s e n t t h e r e c e n t r e s u l t s o n t h e s o l v a b i l i t y o f b o u n d a r y v a l u e p r o b l e m s i n d o m a i n s

w i t h b a d b o u n d a r i e s . T h e s e n o t e s c o n c e n t r a t e o n t h e b o u n d a r y v a l u e p r o b l e m s f o r t h e

L a p l a c e o p e r a t o r f o r a c o m p l e t e s u r v e y o f r e s u l t s , w e r e f e r t o t h e s u r v e y a r t i c l e b y C a r l o s

K e n i g I a m v e r y g r a t e f u l f o r t h i s k i n d p e r m i s s i o n t o i n c l u d e i t h e r e . I t i s a l s o m y p l e a s u r e

t o a c k n o w l e d g e m y g r a t i t u d e t o P e t e r K u m l i n f o r e x c e l l e n t w o r k i n p r e p a r i n g t h e s e n o t e s

f o r p u b l i c a t i o n .

J a n u a r y 1 9 8 5

B j r n E . J . D a h l b e r g

i


4/144

i i


5/144

C o n t e n t s

0 I n t r o d u c t i o n 1

1 D i r i c h l e t P r o b l e m f o r L i p s c h i t z D o m a i n . T h e S e t u p 1 1

2 P r o o f s o f T h e o r e m 1 . 1 a n d T h e o r e m 1 . 2 1 9

3 P r o o f o f T h e o r e m 1 . 6 2 5

4 P r o o f o f T h e o r e m 1 . 3 3 7

5 P r o o f o f T h e o r e m 1 . 4 4 7

6 D i r i c h l e t P r o b l e m f o r L i p s c h i t z d o m a i n s . T h e n a l a r g u m e n t s f o r t h e

L

2

- t h e o r y 5 1

7 E x i s t e n c e o f s o l u t i o n s t o D i r i c h l e t a n d N e u m a n n p r o b l e m s f o r L i p s c h i t z

d o m a i n s . T h e o p t i m a l L

p

- r e s u l t s 5 7

I n d e x : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 4

A p p e n d i x 1 C . E . K e n i g : R e c e n t P r o g r e s s o n B o u n d a r y V a l u e P r o b l e m s o n

L i p s c h i t z D o m a i n s : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 7

A p p e n d i x 2 B . E . D a h l b e r g / C . E . K e n i g : H a r d y s p a c e s a n d t h e N e u m a n n

P r o b l e m i n L

p

f o r L a p l a c e ' s e q u a t i o n i n L i p s c h i t z d o m a i n s : : : : : : : : : 1 0 7

i i i


6/144

i v


7/144

C h a p t e r 0

I n t r o d u c t i o n

I n t h i s c o u r s e w e w i l l s t u d y b o u n d a r y v a l u e p r o b l e m s ( B V P : s ) f o r l i n e a r e l l i p t i c P D E : s

w i t h c o n s t a n t c o e c i e n t s i n L i p s c h i t z - d o m a i n s , i . e . , d o m a i n s w h e r e t h e b o u n d a r y @

l o c a l l y i s g i v e n b y t h e g r a p h o f L i p s c h i t z f u n c t i o n . W e r e c a l l t h a t a f u n c t i o n ' i s L i p s c h i t z

i f t h e r e e x i s t s a c o n s t a n t M


8/144

D i r i c h l e t p r o b l e m

u = 0 i n

u = f o n @

N e u m a n n p r o b l e m

u = 0 i n

@ u

@ n

= f o n @

t h e c l a m p e d p l a t e p r o b l e m

8


9/144

a n d t h a t

( ) s u p

y > 0

k u ( y ) k

p

k f k

p

T h u s w i t h X = L

p

( R

n

) a n d Y = f u : u h a r m o n i c i n R

n + 1

+

a n d u s a t i s e s ( ) g w e h a v e t h e

i m p l i c a t i o n

f 2 X ) u 2 Y

H o w e v e r , w e c a n a l s o r e v e r s e t h e i m p l i c a t i o n s i n c e a h a r m o n i c f u n c t i o n u w h i c h s a t i s e s

( ) h a s n o n - t a n g e n t i a l l i m i t s a . e . o n @ R

n + 1

+

, t h e l i m i t - f u n c t i o n u

0

= u ( 0 ) 2 L

p

( R

n

) a n d

u ( x y ) = p

y

u

0

( x )

S k e t c h o f a p r o o f . A s s u m e u h a r m o n i c f u n c t i o n i n R

n + 1

+

t h a t s a t i s e s ( ) . T h e s e m i g r o u p

p r o p e r t i e s o f f p

y

g

y 0

i m p l i e s

u ( x y + ) = p

y

u

( x ) > 0 y > 0

w h e r e u

( x ) = u ( x )

( ) ) u

n

* v i n L

p

( R

n

) a s

n

# 0

) p

y

u

n

( x ) ! p

y

v ( x ) a s

n

# 0 y > 0

B u t p

y

u

n

( x ) = u ( x y +

n

) a n d t h u s

u ( x y ) = p

y

v ( x ) w h e r e v 2 L

p

( R

n

)

F o r t h e p r o o f o f t h e e x i s t e n c e o f n o n - t a n g e n t i a l l i m i t s o f p

y

u

0

w e r e f e r t o e . g . S t e i n / W e i s s

2 ] .

T h e n o t i o n o f s o l u t i o n o f t h e D i r i c h l e t p r o b l e m a n d a n y o t h e r p r o b l e m , i s s o u n d o n l y

i f w e h a v e s u c h a m a t c h i n g b e t w e e n t h e b o u n d a r y v a l u e f o f u a n d t h e s o l u t i o n u i t s e l f ,

i . e . , w e s h o u l d n o t a c c e p t c o n c e p t s o f s o l u t i o n w h i c h a r e s o w e a k s u c h t h a t t h e r e v e r s e d

i m p l i c a t i o n i s i m p o s s i b l e .

N o w a s s u m e t h a t i s a b o u n d e d ( c o n n e c t e d ) d o m a i n i n R

n

n 3 w i t h C

2

b o u n d a r y . ( T o

a v o i d t e c h n i c a l i t i e s , w e h a v e a s s u m e d n 6= 2 ) . C o n s i d e r t h e D i r i c h l e t p r o b l e m

( D )

u = 0 i n

u

@

= f 2 C ( @ )

L e t r d e n o t e ( ; 1 ) ( t h e f u n d a m e n t a l s o l u t i o n ) o f t h e L a p l a c e o p e r a t o r i n R

n

, t h a t i s ,

r ( x ) = c

n

1

x

n ; 2

c

n

= ;

1

( 2 ; n ) !

n

;

1

2 ; n

;

( n = 2 )

2

n = 2

3


10/144

a n d s e t

R ( x y ) = r ( x ; y )

F o r f 2 C ( @ ) w e d e n e

D f ( P ) =

Z

@

@

@ n

Q

R ( P Q ) f ( Q ) d ( Q ) P =2 @

S f ( P ) =

Z

@

R ( P Q ) f ( Q ) d ( Q ) P =2 @

T h u s D f a n d S f d e n o t e t h e d o u b l e l a y e r p o t e n t i a l a n d s i n g l e l a y e r p o t e n t i a l r e s p . H e r e d

i s t h e s u r f a c e m e a s u r e o n @ a n d

@

@ n

Q

i s t h e d i r e c t i o n a l d e r i v a t i v e a l o n g t h e u n i t o u t w a r d

n o r m a l f o r @ a t Q . I t i s i m m e d i a t e t h a t

D f ( P ) = 0 P 2 R

n

n @

a n d D f w i l l b e o u r c a n d i d a t e f o r s o l u t i o n o f ( D ) . I t r e m a i n s t o s t u d y t h e b e h a v i o u r o f D f

a t @

P a r t o f t h a t s t o r y i s

L e m m a 1 . I f f 2 C ( @ ) , t h e n

1 ) D f 2 C (

)

2 ) D f 2 C ( { )

M o r e p r e c i s e l y : D f c a n b e e x t e n d e d a s a c o n t i n u o u s f u n c t i o n f r o m i n s i d e t o

a n d f r o m

o u t s i d e t o { . L e t D

+

f a n d D

;

f d e n o t e t h e r e s t r i c t i o n s o f t h e s e f u n c t i o n s t o @ r e s p .

S e t K ( P Q ) =

@

@ n

Q

R ( P Q ) f o r P 6= Q P Q 2 @ . W e n o t e t h a t

i ) K 2 C ( @ @ n f ( P P ) : P 2 @ g )

i i ) K ( P Q )

C

P ; Q

n 2

f o r P Q 2 @ a n d s o m e C


11/144

S i n c e ' i s a C

2

f u n c t i o n , w e h a v e t h a t

' ( x ) = ' ( y ) + h x ; y r ' ( y ) i + e ( x y ) w h e r e e ( x y ) = O ( x ; y

2

)

H e n c e

K ( P Q ) C

h P ; Q ( r ' ( y ) ; 1 ) i

P ; Q

n

C

e ( x y )

P ; Q

n

C

P ; Q

n ; 2

T h i s e s t i m a t e i s u n i f o r m i n P a n d Q s i n c e @ c o m p a c t .

F o r f 2 C ( @ ) d e n e

T f ( P ) =

Z

@

K ( P Q ) f ( Q ) d ( Q ) P 2 @

W e c a n n o w f o r m u l a t e

L e m m a 2 ( j u m p r e l a t i o n f o r D )

1 ) D

+

=

1

2

I + T

2 ) D

;

= ;

1

2

I + T

a n d

L e m m a 3 . T : C ( @ ) ! C ( @ ) i s c o m p a c t .

S k e t c h o f p r o o f o f L e m m a 3 . D e n e t h e o p e r a t o r s T

n

b y

T

n

f ( P ) =

Z

@

K

n

( P Q ) f ( Q ) d ( Q ) P 2 @

f o r f 2 C ( @ ) , w h e r e

K

n

( P Q ) = s i g n ( K ( P Q ) ) m i n ( n K ( P Q ) ) n 2 Z

+

T h u s K

n

i s c o n t i n u o u s o n @ @ a n d A r z e l a - A s c o l i ' s t h e o r e m i m p l i e s t h a t T

n

i s a c o m p a c t

o p e r a t o r o n C ( @ ) . F u r t h e r m o r e s i n c e k T

n

k s u p

Q 2 @

k K

n

( Q ) k

1

C


12/144

P r o o f o f L e m m a 1 a n d 2 . S o m e b a s i c f a c t s :

1 )

Z

@

@

@ n

Q

R ( P Q ) d ( Q ) = 1 , i f P 2

P r o o f : A p p l y G r e e n ' s f o r m u l a t o t h e h a r m o n i c f u n c t i o n R ( ; Q ) i n n B

( P ) f o r

> 0 s m a l l , w h e r e B

( P ) = f x 2 R

n

: P ; x g

2 )

Z

@

@

@ n

Q

R ( P Q ) d ( Q ) = 0 , i f P =2

P r o o f : E x e r c i s e .

3 )

Z

@

K ( P Q ) d ( Q ) =

1

2

, i f P 2 @


L e t P 2 @ . W e w a n t t o s h o w t h a t

D f ( Q ) !

1

2

f ( P ) + T f ( P ) a s 3 Q ! P

A : A s s u m e P =2 s u p p f : E a s y .

B : A s s u m e f ( P ) = 0 : W e n e e d .

4 ) 9 C > 0 :

Z

@

@

@ n

Q

R ( P Q )

d ( Q ) < C f o r a l l P =2 @


4 ) i m p l i e s t h e e s t i m a t e

k D f k

L

1

( R

n

n @ )

C k f k

L

1

( @ )

C h o o s e f f

k

g C ( @ ) w i t h P =2 s u p p f

k

s u c h t h a t

k f ; f

k

k

L

1

( @ )

! 0 a s k ! 1

T b o u n d e d o p e r a t o r i m p l i e s T f

k

( P ) ! T f ( P ) a s k ! 1 . H e n c e

D f ( Q ) ; T f ( P ) C k ( f ; f

k

) k

L

1

( R

n

n @ )

+ D f

k

( Q ) ; T f

k

( P ) +

+ T f

k

( P ) ; T f ( P ) ! 0 a s k ! 1 a n d 3 Q ! P

C : E n o u g h t o c h e c k f 1

T h e r e s u l t f o l l o w s f r o m b a s i c f a c t s 1 ) a n d 3 ) . H e n c e w e h a v e p r o v e d L e m m a 1 a n d 2 p a r t

1 ) . P a r t 2 ) f o l l o w s a n a l o g o u s l y .

6


13/144

W e n o w r e t u r n t o t h e s i n g l e l a y e r p o t e n t i a l a n d o b s e r v e t h a t S f i s h a r m o n i c i n R

n

n @

a n d c o n t i n u o u s i n R

n

i f f 2 C ( @ ) . N e x t w e w a n t t o c o m p a r e t h e n o r m a l d e r i v a t i v e o f

S f w i t h D f a t @ . S i n c e @ i s C

2

w e h a v e f o l l o w i n g r e s u l t :

F o r " > 0 s m a l l e n o u g h

; " " @ 3 ( t P ) ! P + t n

p

2 V

i s a d i e o m o r p h i s m , w h e r e n

p

i s t h e o u t w a r d u n i t n o r m a l o f @ a t P , a n d V i s a n e i g h -

b o r h o o d o f @ . F o r P 2 @ a n d t 2 ; " " s e t

D S f ( P + t n

p

) =

Z

@

@

@ n

p

R ( P + t n

p

Q ) f ( Q ) d ( Q )

T h e c l o s e r e l a t i o n s b e t w e e n D f a n d D S f i s f o r m u l a t e d i n

L e m m a 4 . I f f 2 C ( @ ) t h e n

1 ) D S f 2 C ( V \ )

2 ) D S f 2 C ( V \ { )

( C o m p a r e L e m m a 1 ) .

L e t D

+

S f b e t h e r e s t r i c t i o n t o @ o f t h e f u n c t i o n D S f e x t e n d e d t o V \ f r o m i n s i d e

a n d D

;

S f t h e r e s t r i c t i o n t o @ o f t h e f u n c t i o n D S f e x t e n d e d t o V \ { f r o m o u t s i d e .

B u t R ( P Q ) = R ( Q P ) s o w i t h R

n

( P Q ) = K ( Q P ) , w h i c h i s t h e r e a l - v a l u e d k e r n e l i n

T

f ( P ) =

Z

@

K

( P Q ) f ( Q ) d ( Q ) P 2 @

w e h a v e t h a t T

i s t h e a d j o i n t o p e r a t o r o f T

L e m m a 5 ( j u m p r e l a t i o n s f o r D S ) 1 ) D

+

S = ;

1

2

I + T

2 ) D

;

S =

1

2

I + T

P r o o f o f L e m m a 4 a n d 5 . L e t f 2 C ( @ ) a n d d e n e

w

f

( P ) =

D f ( P ) + D S f ( P ) P 2 V n @

T f ( P ) + T

f ( P ) P 2 @

C l a i m : w

f

2 C ( V )

P r o o f : w

f

c o n t i n u o u s o n V n @ a n d o n @ . H e n c e i t i s e n o u g h t o s h o w t a h t

w

f

( P + t n

p

) ! w

f

( P ) u n i f o r m l y f o r P 2 @ a s t ! 0

7


14/144

A s s u m e

0

2 C ( @ ) s u c h t h a t 0

0

1

0

= 1 i n a n e i g h b o r h o o d o f P a n d

s u p p

0

B

( P )

D e c o m p o s e f a s

f = f

1

+ f

2

0

f + ( 1 ;

0

) f

A : w

f

2

( P + t n

p

) ! w

f

2

( P ) a s t ! 0 . E a s y

B : A s s u m e t 6= 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1

P

t

= P + t n

P

P

t


15/144

T h e r e f o r e

T f ( P ) + T

f ( P ) = D

+

f ( P ) + D

+

S f ( P ) = D

;

f ( P ) + D

;

S f ( P ) P 2 @

T h e j u m p r e l a t i o n s f o r D S f o l l o w .

W e n o w g i v e t h e n a l a r g u m e n t f o r t h e e x i s t e n c e o f a s o l u t i o n o f t h e D i r i c h l e t p r o b l e m i n

a n d t h a t i s

D

+

: C ( @ ) ! C ( @ )

i s o n t o .

S i n c e D

+

=

1

2

I + T , w h e r e T i s c o m p a c t , F r e d h o l m ' s A l t e r n a t i v e t h e o r e m c a n b e a p p l i e d .

H e n c e ,

1

2

I + T = D

+

o n t o

i

1

2

I + T

= D

;

S 1 ; 1

T o p r o v e D

;

S i s 1 ; 1 i s e a s y :

A s s u m e D

;

S f = 0 f o r s o m e f 2 C ( @ ) . S e t v = S f . T h e n

i ) v h a r m o n i c i n {

i i ) v ( P ) = O ( P

2 ; n

) a s P ! 1

i i i )

@ v

@ n

@

= 0

G r e e n ' s f o r m u l a i m p l i e s

Z

{

r v

2

=

Z

{

v v +

Z

@

v

@ v

@ n

d = 0

T h u s v = 0 i n {

. B u t v 2 C ( R

n

) a n d v = 0 i n

M a x i m u m p r i n c i p l e ) v = 0 i n R

n

) f = 0

R e m a r k : T h e p r o o f a b o v e i s v a l i d f o r d o m a i n s w i t h C

1 +

b o u n d a r i e s w h e r e > 0 , b u t

n o t f o r d o m a i n s w i t h b o u n d a r i e s w i t h l e s s r e g u l a r i t y .

R e m a r k : W e o b s e r v e t h a t t h e m e t h o d i s n o n - c o n s t r u c t i v e a s a c o n s e q u e n c e o f t h e s o f t

a r g u m e n t s ( i . e . , c o m p a c t n e s s a r g u m e n t s ) w e h a v e u s e d . H e n c e i t i s n o t p o s s i b l e t o s o l v e

t h e D i r i c h l e t p r o b l e m f o r , s a y L i p s c h i t z - d o m a i n s b y a p p r o x i m a t i n g w i t h C

2

d o m a i n s

k

, s o l v e s o m e D i r i c h l e t p r o b l e m s f o r t h e s e a n d o b t a i n a n a p p r o x i m a t i o n o f a s o l u t i o n f o r

, s i n c e w e d o n o t h a v e a n y e s t i m a t e s o f t h e i n v e r s e s o f t h e D

+

: s

9


16/144

R e f e r e n c e s

1 ] F o l l a n d , G : I n t r o d u c t i o n t o p a r t i a l d i e r e n t i a l e q u a t i o n s . M a t h . N o t e s 1 7 , P r i n c e t o n

U . P .

2 ] S t e i n , E . M . / W e i s s , G : I n t r o d u c t i o n t o F o u r i e r A n a l y s i s o n E u c l i d e a n S p a c e s . P r i n c e -

t o n U . P .

1 0


17/144

C h a p t e r 1

D i r i c h l e t P r o b l e m f o r L i p s c h i t z

D o m a i n . T h e S e t u p

A f u n c t i o n ' : R

n

! R s u c h t h a t

' ( x ) ; ' ( y ) M x ; y f o r a l l x y 2 R

n

i s c a l l e d L i p s c h i t z f u n c t i o n . A b o u n d e d d o m a i n R

n + 1

i s c a l l e d L i p s c h i t z d o m a i n i f @

c a n b e c o v e r e d b y n i t e l y m a n y r i g h t c i r c u l a r c y l i n d e r s L w h o s e b a s e s a r e a t a p o s i t i v e

d i s t a n c e f r o m @ s u c h t h a t t o e a c h c y l i d e r L t h e r e i s a L i p s c h i t z f u n c t i o n ' : R

n

! R a n d

a c o o r d i n a t e s y s t e m ( x y ) x 2 R

n

y 2 R s u c h t h a t t h e y - a x i s i s p a r a l l e l t o t h e a x i s o f

s y m m e t r y o f L a n d L \ = L \ f ( x y ) : y > ' ( x ) g a n d L \ @ = L \ f ( x y ) : y = ' ( x ) g

A d o m a i n D R

n + 1

i s c a l l e d s p e c i a l L i p s c h i t z d o m a i n i f t h e r e i s a L i p s c h i t z f u n c t i o n

' : R

n

! R s u c h t h a t D = f ( x y ) : y > ' ( x ) g a n d @ D = f ( x y ) : y = ' ( x ) g . I n t h i s a n d

a l l p r o c e e d i n g c h a p t e r s w e r e s e r v e t h e n o t a t i o n f o r b o u n d e d L i p s c h i t z d o m a i n s a n d D

f o r s p e c i a l L i p s c h i t z d o m a i n s r e s p e c t i v e l y . W i t h a c o n e ; w e m e a n a c i r c u l a r c o n e w h i c h

i s o p e n . A c o n e ; w i t h v e r t e x a t a p o i n t P 2 @ C , w h e r e C R

n + 1

i s a d o m a i n , i s c a l l e d

a n o n t a n g e n t i a l c o n e i f t h e r e i s a c o n e ;

0

a n d a > 0 s u c h t h a t

6= (

; \ B

( P ) ) n f P g ;

0

\ B

( P ) C

B

r

( Q ) i s o u r s t a n d a r d n o t a t i o n f o r t h e b a l l f x 2 R

n

: x ; Q r g . W e s a y t h a t a f u n c t i o n

u d e n e d i n a d o m a i n C h a s n o n t a n g e n t i a l l i m i t L a t a p o i n t P 2 @ C i f

u ( Q ) ! L a s Q ! P Q 2 ;

f o r a l l n o n t a n g e n t i a l c o n e s ; w i t h v e r t i c e s a t P . F i n a l l y w e d e n e t h e n o n t a n g e n t i a l

m a x i m a l f u n c t i o n M

u f o r > 1 a n d f u n c t i o n u d e n e d i n L i p s c h i t z d o m a i n b y

M

u ( P ) = s u p f u ( Q ) : P ; Q < d i s t ( Q @ ) Q 2 g P 2 @

1 1


18/144

O n e o f t h e m a i n r e s u l t s i n t h i s c o u r s e w i l l b e t h e e x i s t e n c e o f a s o l u t i o n t o t h e D i r i c h l e t

p r o b l e m

u = 0 i n R

n + 1

u

@

= f 2 L

2

( @ )

w h e r e i s a b o u n d e d L i p s c h i t z d o m a i n . B y t h i s w e m e a n t h a t t h e r e e x i s t s a h a r m o n i c

f u n c t i o n u i n w h i c h c o n v e r g e s n o n t a n g e n t i a l l y t o f a l m o s t e v e r y w h e r e w i t h r e s p e c t t o

t h e s u r f a c e m e a s u r e d ( @ ) a n d t h a t t h e m a x i m a l f u n c t i o n M

u 2 L

2

( @ ) f o r > 1 . T h e

s t a r t i n g p o i n t f o r o u r e n t e r p r i s e o f p r o v i n g t h e e x i s t e n c e o f a s o l u t i o n t o D i r i c h l e t p r o b l e m

f o r t h e L i p s c h i t z d o m a i n i s t h e d o u b l e l a y e r p o t e n t i a l

D g ( P ) =

Z

@

@

@ n

Q

R ( P Q ) g ( Q ) d ( Q ) P 2

w h e r e R ( P Q ) i s t h e f u n d a m e n t a l s o l u t i o n f o r L a p l a c e e q u a t i o n i n R

n + 1

( m u l t i p l i e d w i t h

; 1 ) a n d g 2 L

2

( @ ) . S i n c e D g i s h a r m o n i c i n , w e a r e d o n e i f w e c a n s h o w t h a t

f o r s o m e c h o i c e o f g w e h a v e t h e r i g h t b e h a v i o u r o f D g a t @ . H o w e v e r , t h i s i s n o t

e a s y s i n c e f o r K ( P Q ) =

@

@ n

Q

R ( P Q ) P Q 2 @ P 6= Q , w e o n l y h a v e t h e e s t i m a t e

K ( P Q )

C

P ; Q

n 1

w h i c h c a n n o t b e i m p r o v e d i n g e n e r a l . T h u s w e h a v e t o r e l y o n t h e

c a n c e l l a t i o n p r o p e r t i e s o f K ( P Q ) , a n d t h e o p e r a t o r T w h i c h a p p e a r e d i n C h a p t e r 0 c a n

o n l y b e d e n e d a s a p r i n c i p a l v a l u e o p e r a t o r . B e f o r e w e s t u d y t h e c a s e w i t h a g e n e r a l

b o u n d e d L i p s c h i t z d o m a i n w e t r e a t t h e c a s e w i t h a s p e c i a l L i p s c h i t z d o m a i n D . F r o m

t h i s w e o b t a i n t h e r e s u l t f o r u s i n g s t a n d a r d p a t c h i n g t e c h n i q u e s ( s e e A p p e n d i x 2 ) .

C o n s i d e r

D g ( P ) =

Z

@ D

@

@ n

Q

R ( P Q ) g ( Q ) d ( Q ) P 2 D

w h e r e D = f ( x y ) : y > ' ( x ) g f o r a L i p s c h i t z f u n c t i o n ' : R

n

! R . W e r e m a r k t h a t

' L i p s c h i t z f u n c t i o n i m p l i e s t h a t '

0

e x i s t s a . e . s o t h e d e n i t i o n o f D g m a k e s s e n s e a n d

@

@ n

Q

R ( P Q ) = C

n

h n

Q

P ; Q i

P ; Q

n + 1

, w i t h n

Q

=

( r ' ( x ) ; 1 )

p

r ' ( x )

2

+ 1

f o r Q = ( x ' ( x ) ) , e x i s t s a . e .

d ( @ D ) . T o s t a t e t h e r s t p r o p o s i t i o n , w e n e e d s o m e m o r e n o t a t i o n : F o r e v e r y m e a s u r e

a n d e a c h - m e a s u r a b l e f u n c t i o n g a n d m e a s u r a b l e s e t A w i t h ( A ) 6= 0 w e l e t

R

A

g d ,

d e n o t e t h e m e a n v a l u e

1

( A )

Z

A

g d . F u r t h e r m o r e f o r g 2 L

1

o c

( @ D ) w e d e n e t h e m a x i m a l

f u n c t i o n M g b y

M g ( P ) = s u p

r > 0

Z

@ D \ B

r

( P )

g ( Q ) d ( Q ) P 2 @ D

T h e f o l l o w i n g r e s u l t i s c r u c i a l .

1 2


19/144

P r o p o s i t i o n 1 . 1 . L e t D = f ( x y ) : y > ' ( x ) g w h e r e ' : R

n

! R i s a L i p s c h i t z f u n c t i o n

w i t h k '

0

k

1

= A . L e t P = ( x y ) 2 D a n d P

2 ( x ' ( x ) ) 2 @ D a n d s e t = y ; ' ( x )

A s s u m e g 2 L

p

( @ D ) f o r s o m e p w h e r e 1 < p 0 s u c h t h a t

f x : M g ( x ) > g C

k g k

1

f o r a l l g 2 L

1

5 ) M i s b o u n d e d i n L

p

1 < p 0 s u c h t h a t

k M g k

p

C

p

k g k

p

f o r a l l g 2 L

p

H e r e 3 ) i s t r i v i a l , 4 ) c a n b e p r o v e n b y a c o v e r i n g l e m m a a r g u m e n t a n d 5 ) f o l l o w s f r o m 3 ) ,

4 ) a n d M a r c i n k i e w i c z ' i n t e r p o l a t i o n t h e o r e m ( s e e S t e i n 1 ] ) . F o r l a t e r r e f e r e n c e w e s t a t e

1 3


20/144

M a r c i n k i e w i c z ' i n t e r p o l a t i o n t h e o r e m . L e t 1 p < q 1 a n d l e t T b e a s u b a d -

d i t i v e o p e r a t o r d e n e d o n L

p

+ L

q

. A s s u m e T i s a w e a k ( p p ) o p e r a t o r a n d a w e a k ( q q )

o p e r a t o r . T h e n T i s b u n d e d o n L

r

w h e r e p < r < q . A n o p e r a t o r T i s a w e a k ( p p )

o p e r a t o r i f t h e r e e x i s t s a c o n s t a n t C > 0 s u c h t h a t

f x : T g ( x ) > g C

;

k g k

p

p

f o r a l l g 2 L

p

a n d > 0

H e n c e , i f T i s b o u n d e d o n L

p

, t h e n T i s a w e a k ( p p ) o p e r a t o r , b u t t h e c o n v e r s e i s n o t t r u e

i n g e n e r a l .

P r o o f o f P r o p o s i t i o n 1 . 1 .

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

P

D

P

D g ( P ) ; T

g ( P

) C

Z

@ D \ f P

; Q > g

h n

Q

P ; Q i

P ; Q

n + 1

;

h n

Q

P

; Q i

P

; Q

n + 1

g ( Q ) d

+ C

Z

@ D n f P

; Q > g

h n

Q

P ; Q i

P ; Q

n + 1

g ( Q ) d ( Q )

C

Z

@ D \ f P

; Q > g

( + Q ; P

)

n + 1

g ( Q ) d ( Q ) +

+ C

Z

@ D n f P

; Q > g

1

n

g ( Q ) d ( Q )

w h e r e w e h a v e a p p l i e d t h e m e a n v a l u e t h e o r e m t o t h e r s t i n t e g r a l . T h e s e c o n d i n t e g r a l i s

C M g ( P

) a n d t h e r s t i n t e g r a l c a n a l s o b e e s t i m a t e d f r o m a b o v e w i t h t h e s a m e b o u n d

a c c o r d i n g t o

L e m m a 1 . 1 . L e t 0 b e a r a d i a l d e c r e a s i n g f u n c t i o n d e n e d i n R

n

. A s s u m e f 2 L

1

+ L

1

a n d s e t m f ( x ) = s u p

r > 0

R

B

r

( x )

f ( x ) d x f o r x 2 R

n

. T h e n f ( x ) B m f ( x ) f o r a l l x 2 R

n

w h e r e B =

R

( x ) d x

1 4


21/144

I f w e t a k e t h i s l e m m a f o r g r a n t e d f o r a m o m e n t a n d s e t

( x ) =

( x + )

n + 1

t h e r s t i n t e g r a l a b o v e i s b o u n d e d f r o m a b o v e b y C M g ( P

) a n d w e a r e d o n e .

P r o o f o f L e m m a 1 . 1 . I t i s e n o u g h t o p r o v e t h e l e m m a f o r 0 f 2 C

1

0

2 C

1

0

a n d x = 0

S e t S

n

= @ B

1

( 0 ) a n d A ( r ) =

R

B

r

( 0 )

f ( x ) d x

W e o b t a i n

f ( 0 ) =

Z

R

n

( x ) f ( x ) d x =

Z

1

0

( r ) r

n ; 1

Z

S

n

f ( r w ) d ( w ) d r =

=

Z

1

0

( r ) A

0

( r ) d r = ;

Z

1

0

0

( r ) A ( r ) d r ;

Z

1

0

0

( r ) B

r

( 0 ) d r m f ( 0 )

S e t f 1 i n t h e c a l c u l a t i o n s a b o v e a n d w e g e t ;

R

1

0

0

( r ) B

r

( 0 ) d r = B . T h e l e m m a i s

p r o v e n .

I f w e d e n e t h e o p e r a t o r T

b y

T

g ( P

) = s u p

> 0

T

g ( P

) P

2 @ D

f o r g 2 L

p

( @ D ) , t h e n

D g ( P ) C ( T

g ( P

) + M g ( P

) )

f o r a l l P = ( x y ) 2 D a n d P

= ( x ' ( x ) ) 2 @ D . T h u s i f w e c a n p r o v e t h a t T

i s b o u n d e d

o n L

p

( @ D ) , t h e n D g ( P

) 0 a n d T

. T h i s c a l l s f o r s o m e d e n i t i o n s . L e t S ( R

n

) d e n o t e

t h e S c h w a r t z c l a s s ( i . e . , t h e s p a c e o f a l l C

1

- f u n c t i o n s i n R

n

w h i c h t o g e t h e r w i t h a l l t h e i r

d e r i v a t i v e s d i e o u t f a s t e r t h a n a n y p o w e r o f x a t i n n i t y ) w i t h t h e u s u a l t o p o l o g y .

T i s c a l l e d a s i n g u l a r i n t e g r a l o p e r a t o r ( S I O ) i f T : S ( R

n

) ; ! S ( R

n

)

i s l i n e a r a n d

c o n t i n u o u s a n d t h e r e e x i s t s a k e r n e l K s u c h t h a t f o r a l l ' , 2 C

1

0

( R

n

) w i t h s u p p ' \

s u p p =

h T ' i =

Z Z

K ( x y ) ' ( y ) ( x ) d y d x

1 5


22/144

w h e r e h i i s t h e u s u a l S ; S

p a r i n g . W e o b s e r v e t h a t K d o e s n o t d e t e r m i n e T u n i q u e l y .

C o n s i d e r f o r i n s t a n c e T f = f

0

f o r w h i c h K = 0 i s a k e r n e l .

W e s a y t h a t a k e r n e l K i s o f C a l d e r n - Z y g m u n d t y p e ( C Z - t y p e ) i f

1 ) K ( x y )

C

x ; y

n

2 ) r

x

K ( x y ) + r

y

K ( x y )

C

x ; y

n + 1

3 ) K ( x y ) = ; K ( y x )

T h e o p e r a t o r - k e r n e l s , w e w i l l s t u d y , w i l l b e o f t h e f o r m

K

i

( x y ) =

( ( x ' ( x ) ) ; ( y ' ( y ) ) )

i

( x ' ( x ) ) ; ( y ' ( y ) )

n + 1

i = 1 2 : : : n + 1

w h e r e ( a )

i

d e n o t e s t h e i - t h c o m p o n e n t o f a 2 R

n + 1

W e o b s e r v e t h a t t h e s e k e r n e l s a r e o f C Z - t y p e a n d a d o p t t h e c o n v e n t i o n t h a t w h e n e v e r

w e d i s c u s s k e r n e l s K , t h e y a r e a s s u m e d t o b e o f C Z - t y p e u n l e s s w e e x p l i c i t l y s t a t e t h e

c o n v e r s e . S t a r t i n g w i t h a k e r n e l K , w e c a n f o r m a w e l l - d e n e d S I O w i t h K a s t h e k e r n e l

n a m e l y t h e p r i n c i p a l v a l u e o p e r a t o r ( P V O ) T . N o t e t h a t f o r ' 2 S ( R

n

) )

Z Z

x ; y > "

K ( x y ) ' ( y ) ( x ) d y d x =

1

2

Z Z

x ; y > "

K ( x y ) ( ' ( y ) ( x ) ; ' ( x ) ( y ) ) d y d x

s i n c e K ( x y ) = ; K ( y x ) a n d t h u s

l i m

" ! 0

Z Z

x ; y > "

K ( x y ) ' ( y ) ( x ) d y d x

e x i s t s s i n c e ' ( y ) ( x ) ; ' ( x ) ( y ) = O ( x ; y ) a n d ' 2 S ( R

n

) d e c a y f a s t e n o u g h a t

i n n i t y .

H e n c e T : S ( R

n

) 3 ' ! T ' 2 S ( R

n

)

w h e r e

h T ' i = l i m

" ! 0

Z Z

x ; y > "

K ( x y ) ' ( y ) ( x ) d y d x

i s a S I O .

W e l e a v e t h e p r o o f o f c o n t i n u i t y o f T a s a n e x e r c i s e . F r o m n o w o n w e a s s u m e t h a t a l l

o p e r a t o r s T a r e P V O w i t h k e r n e l s K o f C Z - t y p e .

T o a c h i e v e o u r g o a l t o e s t a b l i s h t h e e x i s t e n c e o f a s o l u t i o n t o t h e D i r i c h l e t p r o b l e m f o r

L i p s c h i t z d o m a i n s , w e w i l l p r o v e t h e f o l l o w i n g s e q u e n c e o f t h e o r e m s

T h e o r e m 1 . 1 . I f T b o u n d e d o n L

2

, t h e n T i s a w e a k ( 1 1 ) o p e r a t o r .

T h i s i m p l i e s

1 6


23/144

T h e o r e m 1 . 1 ' . I f T b o u n d e d o n L

2

, t h e n T b o u n d e d o n L

p

f o r 1 < p


24/144

1 8


25/144

C h a p t e r 2

P r o o f s o f T h e o r e m 1 . 1 a n d T h e o r e m

1 . 2

W e r e c a l l t h a t T i s a P V O w i t h k e r n e l K o f C Z - t y p e . I n t h i s c h a p t e r w e g i v e a p r o o f o f

t h e f o l l o w i n g r e s u l t o f C a l d e r n - Z y g m u n d 1 ] .

T h e o r e m 1 . 1 : I f T b o u n d e d o n L

2

, t h e n T i s a w e a k ( 1 1 ) o p e r a t o r .

T h e f o l l o w i n g b o u n d o n T

i s d u e t o C o t l a r 3 ] .

T h e o r e m 1 . 2 : I f T b o u n d e d o n L

2

, t h e n T

i s b o u n d e d o n L

p

f o r 1 < p 0

s u c h t h a t

f x 2 R

n

: T f ( x ) > g C

k f k

1

f o r a l l f 2 L

1

a n d > 0

b y s p l i t t i n g f i n a g o o d p a r t g , w h i c h i s a L

2

- f u n c t i o n a n d a b a d p a r t b . T h i s i s d o n e w i t h

t h e f o l l o w i n g l e m m a

L e m m a ( C a l d e r n - Z y g m u n d d e c o m p o s i t i o n ) . L e t f 2 L

1

( R

n

) a n d > 0 . T h e n

t h e r e e x i s t c u b e s Q

j

j = 1 2 : : : s u c h t h a t

1 ) Q

j

\ Q

k

= 0 f o r j 6= k

2 ) f ( x ) a . e . f o r x 2 R

n

n

1

j = 1

Q

j

1 9


26/144

3 )

R

Q

j

f ( x ) d x g j j f x 2 R

n

: T g ( x ) >

2

g + f x 2 R

n

: T b ( x ) >

2

g

f x 2 R

n

: T g ( x ) >

2

g +

1

j = 1

2 Q

j

+ f x 2 R

n

n

1

j = 1

2 Q

j

: T b ( x ) >

2

g

w h e r e 2 Q

j

i s t h e c u b e w i t h t h e s a m e c e n t e r a s Q

j

, w h i c h w e d e n o t e y

j

, w i t h s i d e s p a r a l l e l

w i t h Q

j

a n d w i t h d o u b l e d s i d e l e n g t h s c o m p a r e d w i t h Q

j

. H e r e

f x 2 R

n

: T g ( x ) >

2

g

4

2

k T g k

2

C

2

k g k

2

C

1

k f k

1

a n d

1

j = 1

2 Q

j

2

n

1

X

j = 1

Q

j

2

n

k f k

1

s o i t r e m a i n s t o e s t i m a t e f x 2 R

n

n

1

j = 1

2 Q

j

: T b ( x ) >

2

g a n d t h i s i s t h e p o i n t w h e r e

w e u s e t h e p r o p e r t i e s o f t h e k e r n e l K . S e t

b

j

( x ) =

b ( x ) x 2 Q

j

0 o t h e r w i s e

F o r , x =2 2 Q

j

w e h a v e

T b

j

( x ) =

Z

Q

j

( K ( x y ) ; K ( x y

j

) ) b

j

( y ) d y

2 0


27/144

a n d t h u s

T b

j

( x ) C

Z

Q

j

y ; y

j

x ; y

n + 1

b

j

( y ) d y

w h e r e w e u s e d

Z

Q

j

b

j

( y ) d y = 0 . I n t e g r a t i n g t h e s e i n e q u a l i t i e s g i v e s

Z

R

n

n

1

j = 1

2 Q

j

T b ( x ) d x

1

X

j = 1

Z

R

n

n 2 Q j

T b

j

( x ) d x

C

1

X

j = 1

Z

Q

j

b

j

( y ) d y = C k b k

1

C k f k

1

a n d c o n s e q u e n t l y

f x 2 R

n

n

1

j = 1

2 Q j : T b ( x ) >

2

g

2

k T b k

L

1

( R

n

n

1

j = 1

2 Q j )

C

k f k

1

T h e p r o o f i s d o n e .

C o r o l l a r y : I f T i s b o u n d e d o n L

2

, t h e n T i s b o u n d e d o n L

p

f o r 1 < p


28/144

a n d f

2

( x ) = f ( x ) ; f

1

( x ) . T h u s T

"

f ( 0 ) = T f

2

( 0 ) . T h e s t r a t e g y i s t o p r o v e t h a t f o r x "

"

y

n + 1

f ( y ) d y C

Z

m i n

;

"

y

n + 1

"

; n

f ( y ) d y C M f ( 0 )

f o r x


29/144

R e f e r e n c e s

1 ] A . P . C a l d e r o n / A . Z y g m u n d : O n t h e e x i s t e n c e o f c e r t a i n s i n g u l a r i n t e g r a l s , A c t a

M a t h . 8 8 ( 1 9 5 2 ) p p . 8 5 - 1 3 9 .

2 ] E . M . S t e i n : S i n g u l a r i n t e g r a l s a n d d i e r e n t i a b i l i t y p r o p e r t i e s o f f u n c t i o n s . P r i n c e t o n

U n i v e r s i t y P r e s s 1 9 7 0 .

3 ] M . C o t l a r : S o m e g e n e r a l i z a t i o n s o f t h e H a r d y - L i t t l e w o o d m a x i m a l t h e o r e m R e v .

M a t . C u y a n a 1 ( 1 9 5 5 ) p p . 8 5 - 1 0 4 .

2 3


30/144

2 4


31/144

C h a p t e r 3

P r o o f o f T h e o r e m 1 . 6

W e b e g i n t h i s s e c t i o n b y i n t r o d u c i n g s o m e o f t h e t o o l s w e n e e d t o p r o v e t h e L

2

- b o u n d e d n e s s

o f T

B M O

p

( R

n

) : F o r f 2 L

1

o c

( R

n

) w e s e t

k f k

p

= s u p

Q c u b e

;

Z

Q

f ( x ) ; f

Q

p

d x

1

p

1 p


32/144

a n d w i t h r a t i o n a l s i d e l e n g t h s . S e t E =

j

@ Q

j

. F o r e a c h p a i r ( x

1

x

2

) 2 ( R

n

n E ) ( R

n

n E )

c h o o s e a c u b e Q 2 f Q

j

g s u c h t h a t x

1

x

2

2 Q . S e t f

1

= f

2 Q

a n d f

2

= f ; f

1

. D e n e

F ( x

1

x

2

) = T f

1

( x

1

) ; T f

1

( x

2

) +

Z

R

n

( K ( x

1

y ) ; K ( x

2

y ) ) f

2

( y ) d y

W e n o t e t h a t F i s d e n e d a . e . a n d t h a t F i s i n d e p e n d e n t o f Q ( a s l o n g a s x

1

x

2

2 Q )

C h e c k i t ! F u r t h e r m o r e , f o r a . e . x

1

2 R

n

a n d x

2

2 R

n

F ( x x

1

) ; F ( x x

2

) i s a c o n s t a n t

( r e g a r d e d a s a f u n c t i o n o f x ) . W e n o w d e n e T f a s t h e c l a s s x ! F ( x x

1

) f o r a . e . x

1

2 R

n

I t r e m a i n s t o s h o w t h a t T : L

1

! B M O i s b o u n d e d . I t i s e n o u g h t o s h o w t h a t

Z

Q

F ( x x

Q

) ; T f

1

( x

Q

) d x C k f k

L

1

f 2 L

1

( R

n

)

f o r a l l c u b e s Q 2 f Q

j

g

B u t

Z

Q

T f

1

( x ) d x (

Z

Q

T f

1

( x )

2

d x )

1 = 2

C (

Z

Q

f

1

( x )

2

d x )

1 = 2

C k f k

1

s i n c e

Z

R

n

( K ( x y ) ; K ( x

Q

y ) ) f

2

( y ) d y

Z

R

n

n 2 Q

K ( x y ) ; K ( x

Q

y ) f ( y ) d y

C

Z

R

n

n 2 Q

x ; x

Q

y ; x

Q

n + 1

d y k f k

1

C k f k

1

f o r x 2 Q

T h e p r o p o s i t i o n f o l l o w s .

F a c t 2 : J o h n - N i r e n b e r g i n e q u a l i t y

T h e o r e m : L e t ' 2 B M O ( R

n

) . T h e n t h e r e e x i s t s c o n s t a n t s , C > 0 > 0 , d e p e n d i n g

o n l y o n n , s u c h t h a t

f x 2 Q : ' ( x ) ; '

Q

> g C Q e x p

;

;

k ' k

f o r a l l > 0 a n d c u b e s Q

S k e t c h o f a p r o o f . I t i s e n o u g h t o s h o w t h a t

s u p

Q c u b e

Z

Q

e x p

;

k ' k

' ( x ) ; '

Q

d x C


33/144

A s s u m e k ' k

= 1 a n d ' 2 L

1

. S i n c e t h e c o n s t a n t s C a n d w i l l b e i n d e p e n d e n t o f k ' k

1

,

t h e r e s u l t f o l l o w s f o r a g e n e r a l ' . F i x a c u b e Q . C o n s i d e r a l l c u b e s Q

j

i n t h e d y a d i c m e s h

o f Q a n d c h o o s e a t > 1 . L e t

~

Q

j

d e n o t e t h o s e d y a d i c c u b e s w h i c h a r e m a x i m a l w i t h r e s p e c t

t o i n c l u s i o n s a t i s f y i n g

Z

~

Q

j

' ( x ) ; '

Q

d x > t

a n d

' ( x ) ; '

Q

t a . e . f o r x 2 Q n

1

j = 1

~

Q

j

C l e a r l y

~

Q

j

Q a n d

1

j = 1

~

Q

j

1

t

k ' ; '

Q

k

L

1

( Q )

1

t

Q

T h e m a x i m a l i t y o f

~

Q

j

i m p l i e s t h a t

Z

Q

j

' ( x ) ; '

Q

d x t

w h e r e

Q

j

i s t h e m i n i m a l c u b e i n t h e d y a d i c m e s h o f Q w i t h r e s p e c t t o i n c l u s i o n f o r w h i c h

~

Q

j

6

Q

j

. F u r t h e r m o r e

Q

Q

j

~

Q

j

'

~

Q

j

; '

Q

'

~

Q

j

; '

Q

j

+ '

Q

j

; '

Q

Z

~

Q

j

' ( x ) ; '

Q

j

d x + t

2

n

Z

Q

j

' ( x ) ; '

Q

j

d x + t

( 2

n

+ 1 ) t

S e t X ( Q ) = s u p

Q

j

2 d y a d i c m e s h o f Q

R

Q

j

e x p ( ' ( x ) ; '

Q

j

) d x w h i c h i s


34/144

T a k e s u p r e m u m o v e r a l l c u b e s Q . T h u s

s u p

Q c u b e

X ( Q ) 1 ;

1

t

e x p ( t ( 2

n

+ 1 ) ) ] e

t

w h i c h i m p l i e s s u p

Q c u b e

X ( Q ) C i f > 0 s m a l l e n o u g h . T h e p r o o f i s d o n e .

R e m a r k : I t i s a n e a s y c o n s e q u e n c e o f J o h n - N i r e n b e r g ' s i n e q u a l i t y t h a t t h e n o r m s k k

p

a n d k k

k k

1

a r e e q u i v a l e n t f o r 1 < p 0 s u c h t h a t x

p

C e x p ( x ) f o r

x > 0 . C h o o s e =

2

a n d a p p l y t h e i n e q u a l i t y a b o v e . H e n c e

k

'

k ' k

k

p

C s u p

Q c u b e

Z

Q

e x p

;

2

' ( x ) ; '

Q

k ' k

d x =

= C s u p

Q c u b e

1

Q

Z

1

0

e x p

;

2

t

d

;

f x 2 Q :

' ( x ) ; '

Q

k ' k

> t g

C s u p

Q c u b e

1

Q

Z

1

0

e x p

;

2

t

C Q e x p ( ; t ) ( ; ) d t = C

a n d k ' k

p

C k ' k

. T h e i n e q u a l i t y k ' k

k ' k

p

f o l l o w s f r o m H l d e r ' s i n e q u a l i t y .

F a c t 3 : C o n n e c t i o n b e t w e e n B M O a n d C a r l e s o n m e a s u r e s .

C a r l e s o n m e a s u r e s o r i g i n a l l y a p p e a r e d a s a n s w e r s t o t h e f o l l o w i n g q u e s t i o n .

Q u e s t i o n : W h i c h p o s i t i v e m e a s u r e s o n R

n + 1

+

h a v e t h e p r o p e r t y

Z Z

R

n + 1

+

P

y

f ( x )

2

d ( x y ) C ( ) k f k

2

2

f o r a l l f 2 L

2

( R

n

)

w h e r e P

y

f ( x ) = p

y

f ( x ) w i t h t h e P o i s s o n k e r n e l p

y

( x ) = c

n

y

( x

2

+ y

2

)

n + 1

2

?

T o o b t a i n a n e c e s s a r y c o n d i t i o n o n c o n s i d e r f =

Q

, i . e . , f i s t h e c h a r a c t e r i s t i c f u n c t i o n

f o r a c u b e Q R

n

. W e i m m e d i a t e l y o b s e r v e t h a t P

y

f ( x ) C > 0 f o r f ( x y ) : x 2

1

2

Q 0


35/144


36/144

R e m a r k : T h e o r e m 3 . 1 i s a l s o v a l i d f o r a l l o p e r a t o r s o f t h e f o r m

P

t

f ( x ) = '

t

f ( x )

w h e r e ' i s a s m o o t h f u n c t i o n w h i c h d e c a y s a t i n n i t y a n d s u c h t h a t ' ( x ) ( x ) f o r s o m e

r a d i a l f u n c t i o n 2 L

1

( R

n

) '

t

( x ) d e n o t e s

1

t

n

'

;

x

t

. W e l e a v e t h e p r o o f o f t h i s r e m a r k a s

a n e x e r c i s e .

W e n o w i n t r o d u c e t w o f a m i l i e s o f o p e r a t o r s d e n o t e d P

t

a n d Q

t

o f w h i c h t h e r s t i s a n

a p p r o x i m a t i o n o f t h e i d e n t i t y a n d t h e s e c o n d i s a n a p p r o x i m a t i o n o f t h e z e r o o p e r a -

t o r . L e t ' b e s m o o t h f u n c t i o n s t h a t d e c a y a t i n n i t y s u c h t h a t

R

R

n

' ( x ) d x = 1 a n d

R

R

n

( x ) d x = 0 . D e n e P

t

a n d Q

t

b y

d

P

t

f ( ) = ^' ( t )

^

f ( )

d

Q

t

f ( ) =

^

( t )

^

f ( )

f o r n i c e f u n c t i o n s f i n R

n

. W e i m m e d i a t e l y o b s e r v e

L e m m a 3 . 2 . I f f 2 L

2

( R

n

) , t h e n

Z

1

0

k Q

t

f k

2

2

d t

t

C ( ) k f k

2

2

P r o o f . A p p l y P l a n c h e r e l ' s f o r m u l a .

T h e o r e m 3 . 2 . I f f 2 B M O ( R

n

) , t h e n

d ( x t ) = Q

t

f ( x )

2

d x d t

t

i s a C a r l e s o n m e a s u r e w i t h C a r l e s o n n o r m C ( ) k f k

2

T o c a r r y t h r o u g h t h e a r g u m e n t i n t h e p r o o f o f t h i s t h e o r e m , w e n e e d a l e m m a .

L e m m a 3 . 3 . I f f 2 B M O ( R

n

) a n d Q

0

i s t h e u n i t c u b e ( c e n t e r e d a t 0 ) , t h e n

Z

R

n

f ( x ) ; f

Q

0

1 + x

n + 1

d x C k f k

w h e r e C o n l y d e p e n d s o n n

P r o o f . F o r a > 0 l e t a Q d e n o t e t h e c u b e w i t h s i d e s p a r a l l e l w i t h t h e s i d e s o f Q a n d o f

l e n g t h s a t i m e s t h e s i d e l e n g t h s o f Q a n d w i t h t h e s a m e c e n t e r a s Q . W e o b s e r v e t h a t f o r

e v e r y c u b e Q R

n

f

Q

; f

2 Q

1

Q

Z

Q

f ( x ) ; f

2 Q

d x

2

n

2 Q

Z

2 Q

f ( x ) ; f

2 Q

d x 2

n

k f k

3 0


37/144

S e t Q

j

= 2

j

Q

0

f o r j 2 N a n d a s s u m e k f k

= 1 . H e r e

f

Q

j + 1

; f

Q

j

2

n

f o r j 2 N

w h i c h i m p l i e s

f

Q

j + 1

; f

Q

0

( j + 1 ) 2

n

f o r j 2 N

H e n c e

Z

R

n

f ( x ) ; f

Q

0

1 + x

n + 1

d x =

1

X

j = 0

Z

Q

j + 1

n Q

j

f ( x ) ; f

Q

0

1 + x

n + 1

d x +

+

Z

Q

0

f ( x ) ; f

Q

0

1 + x

n + 1

d x

1

X

j = 0

;

Z

Q

j + 1

f ( x ) ; f

Q

j + 1

2

j ( n + 1 )

d x +

+

Z

Q

j + 1

f

Q

j + 1

; f

Q

0

2

j ( n + 1 )

d x

+ 1

1

X

j = 0

( 2

n ; j

+ ( j + 1 ) 2

2 n ; j

) + 1 = C

w h i c h c o m p l e t e s t h e p r o o f .

P r o o f o f T h e o r e m 3 . 2 . Q

t

f ( x ) i s a w e l l - d e n e d f u n c t i o n i n R

n + 1

+

s i n c e Q

t

1 = 0 . W e w a n t

t o p r o v e t h a t f o r e a c h c u b e Q R

n

( )

Z Z

~

Q

Q

t

f ( x )

2

d x d t

t

C ( ) k f k

2

Q

I t i s e n o u g h t o c o n s i d e r Q = u n i t c u b e Q

0

s i n c e B M O i s s c a l e - a n d t r a n s l a t i o n i n v a r i a n t ,

i . e . , k f k

= k f

s

t

k

w h e r e f

s

t

( x ) = f ( t ( x ; s ) ) a n d

d t

t

i s s c a l e i n v a r i a n t . F u r t h e r m o r e w e m a y

a s s u m e f

2 Q

0

= 0 s i n c e Q

t

1 = 0 . T h u s w e h a v e t o p r o v e ( ) f o r Q = Q

0

a n d a l l f 2 B M O

w i t h f

2 Q

0

= 0 . S e t f

1

= f

2 Q

0

a n d f

2

= f ; f

1

T h e n Q

t

f = Q

t

f

1

+ Q

t

f

2

a n d w e o b t a i n

Z Z

Q

0

Q

t

f

1

2

d x d t

t

Z Z

R

n + 1

+

Q

t

f

1

2

d x d t

t

C ( ) k f

1

k

2

2

C ( ) k f k

2

f r o m l e m m a 3 . 2 a n d f o r ( x t ) 2

~

Q

0

w e o b t a i n

Q

t

f

2

( x )

Z

R

n

n 2 Q

0

1

t

n

;

x ; z

t

f

2

( z ) d z

C ( )

Z

R

n

n 2 Q

0

t

t

n + 1

+ x ; z

n + 1

f

2

( z ) d z

C ( ) t

Z

R

n

f ( z )

1 + z

n + 1

d z C ( ) t k f k

a c c o r d i n g t o l e m m a 3 . 3 . T h i s c o m p l e t e s t h e p r o o f .

W e h a v e n o w p r e p a r e d t h e t o o l s w e n e e d t o p r o v e t h e t h e o r e m o f D a v i d a n d J o u r n .

3 1


38/144

T h e o r e m 1 . 6 : I f T i s a P V O w i t h C Z t y p e k e r n e l K , t h e n

T i s b o u n d e d o n L

2

i T 1 2 B M O

H e r e t h e o n l y i f - p a r t f o l l o w s f r o m P r o p o s i t i o n 3 . 1 . T h e i f - p a r t i s t h e h a r d p a r t . T h e

p r o o f w e p r e s e n t i s d u e t o C o i f m a n / M e y e r 1 ] . C h o o s e ' 2 C

1

0

( R

n

) s u c h t h a t ' r a d i a l

w i t h s u p p o r t i n t h e u n i t b a l l B

1

( 0 ) a n d s u c h t h a t

^' ( ) = 1 + O (

4

) a s ! 0

D e n e P

t

a s a b o v e b y

d

P

t

f ( ) = ^' ( t )

^

f ( ) . A n a l o g o u s l y d e n e Q

t

b y

d

Q

t

f ( ) = ( t )

2

' ( t ) f ( )

a n d R

t

b y

d

R

t

f ( ) = ( )

; 1

^'

0

( t )

^

f ( ) . H e n c e P

t

Q

t

a n d R

t

c o m m u t e s a n d

d

d t

P

2

t

=

2

t

R

t

Q

t

T h i s i m p l i e s

1

2

d

d t

P

2

t

T P

2

t

=

1

t

( R

t

Q

t

T P

2

t

+ P

2

t

T R

t

Q

t

)

T h e i d e a i s a s f o l l o w s : W e w a n t t o s h o w

h

1

T

2

i C k

1

k

2

k

2

k

2

f o r a l l

1

2

2 S ( R

n

)

W e n o t e t h a t h

1

P

2

t

T P

2

t

2

i ! 0 a s t ! 1 a n d h e n c e i t i s e n o u g h t o p r o v e

Z

1

0

d

d t

h

1

P

2

t

T P

2

t

2

i d t C k

1

k

2

k

2

k

2

f o r a l l

1

2

2 S ( R

n

)

s i n c e P

0

i s t h e i d e n t i t y o p e r a t o r .

F u r t h e r m o r e , i t i s e n o u g h t o p r o v e

Z

1

0

h

1

R

t

Q

t

T P

2

t

2

i

d t

t

C k

1

k

2

k

2

k

2

f o r a l l

1

2

2 S ( R

n

)

s i n c e P

t

Q

t

R

t

a r e s e l f a d j o i n t o p e r a t o r s a n d T

= ; T . W e n e e d t h e f o l l o w i n g e s t i m a t e .

L e m m a 3 . 4 . L e t ' 2 C

1

0

( R

n

) w i t h s u p p o r t i n t h e u n i t b a l l B

1

( 0 ) a n d a s s u m e

Z

R

n

( x ) d x = 0

T h e n

h

x

t

T '

y

t

i C p

t

( x ; y )

w h e r e

x

t

( z ) =

1

t

n

;

z ; x

t

'

y

t

( z ) =

1

t

n

'

;

z ; y

t

a n d p

t

i s t h e P o i s s o n k e r n e l .

3 2


39/144

P r o o f . T h e a r g u m e n t c o n s i s t s o f a s t r a i g h t f o r w a r d c a l c u l a t i o n w h e r e w e u s e t h e C Z t y p e

p r o p e r t i e s o f t h e k e r n e l K

2 h

x

t

T '

y

t

i = l i m

" # 0

Z Z

; > "

K ( ) (

x

t

( ) '

y

t

( ) ;

x

t

( ) '

y

t

( ) ) d d

= l i m

" # 0

Z Z

; > "

K ( t + y t + y )

;

x y

t

1

( ) ' ( ) ;

x y

t

1

( ) ' ( ) ) d d :

H e n c e , i t i s e n o u g h t o p r o v e t h e l e m m a f o r y = 0

A s s u m e x


40/144

H e r e

I =

Z

1

0

k

^'

0

( t )

t

^

1

( ) k

2

2

d t

t

C k ^

1

k

2

2

= C k

1

k

2

2

T o c o p e w i t h I I , w e r e w i r t e Q

t

T P

2

t

2

= L

t

P

t

2

a s

( ( L

t

P

t

)

2

) ( x ) = L

t

P

t

2

; P

t

2

( x ) ( x ) + P

t

2

( x ) L

t

1 ( x ) =

= L

t

P

t

2

; P

t

2

( x ) ( x ) + P

t

2

( x ) Q

t

T 1 ( x )

B u t T 1 2 B M O i m p l i e s Q

t

T 1

2

d x d t

t

i s a C a r l e s o n m e a s u r e a c c o r d i n g t o T h e o r e m 3 . 2 a n d

h e n c e

Z

1

0

k P

t

2

( x ) Q

t

T 1 k

2

2

d t

t

=

Z Z

R

n + 1

+

P

t

2

( x )

2

Q

t

T 1

2

d x d t

t

C k

2

k

2

2

w h e r e w e h a v e u s e d t h e r e m a r k t o T h e o r e m 3 . 1 . F u r t h e r m o r e u s i n g J e n s e n ' s i n e q u a l i t y

A ( x t ) L

t

( P

t

2

; P

t

2

( x ) ) ( x )

2

=

=

Z

R

n

1

t

( x y ) ( P

t

2

( y ) ; P

2

2

( x ) ) d y

2

C

Z

R

n

p

t

( x ; y ) P

t

2

( y ) ; P

t

2

( x )

2

d y

a n d t h u s

Z

1

0

k L

t

P

t

2

; P

t

2

( x ) ( x ) k

2

2

d t

t

=

Z

1

0

Z

R

n

A ( x t )

d x d t

t

C

Z

1

0

Z

R

n

Z

R

n

p

t

( x ; y ) P

t

2

( x ) ; P

t

2

( y )

2

d y d x

d t

t

=

= C

Z

1

0

Z

R

n

Z

R

n

p

t

( x ) P

t

2

( x + y ) ; P

t

2

( y )

2

d x d y

d t

t

= C

Z

1

0

Z

R

n

Z

R

n

p

t

( x )

\

( P

t

2

( + x ) ; P

t

2

( ) ) ( )

2

d d x

d t

t

=

= C

Z

1

0

Z

R

n

Z

R

n

p

t

( x ) ^' ( t )

2

1 ; e

i h x i 2

^

2

( )

2

d d x

d t

t

B u t

Z

R

n

p

t

( x ) 1 ; e

i h x i 2

d x = 2 ; 2 e

; t

T h u s

Z

1

0

k L

t

P

t

2

; P

t

2

( x ) ( x ) k

2

2

d t

t

C

Z

R

n

Z

1

0

2 ( 1 ; e

; t

) ^' ( t )

2

d t

t

^

2

( )

2

d

C

Z

R

n

;

Z

1

0

( 1 ; e

; t

)

d t

t

+

Z

1

1

^' ( t )

2

d t

t

^

2

( )

2

d

C k

2

k

2

2

T h i s c o m p l e t e s t h e p r o o f .

3 4


41/144

R e f e r e n c e s

1 ] R . R . C o i f m a n / Y . M e y e r : p e r s o n a l c o m m u n i c a t i o n .

3 5


42/144

3 6


43/144

C h a p t e r 4

P r o o f o f T h e o r e m 1 . 3

I n t h i s s e c t i o n w e p r o v e

T h e o r e m 1 . 3 : I f ' : R ! R L i p s c h i t z f u n c t i o n a n d K

'

( x y ) =

1

x ; y + i ( ' ( x ) ; ' ( y ) )

,

t h e n t h e c o r r e s p o n d i n g o p e r a t o r T

'

i s b o u n d e d o n L

2

a n d k T

'

k C ( k '

0

k

1

)

W e i m m e d i a t e l y o b s e r v e t h a t t h e k e r n e l K

'

i s o f C Z t y p e a n d t h u s t h e L

2

b o u n d e d n e s s

o f t h e P V O T

'

c a n b e p r o v e d u s i n g T h e o r e m 1 . 6 , i . e . , i t i s e n o u g h t o p r o v e T

'

1 2 B M O .

H o w e v e r , t h i s i s n o t e a s y .

W e g i v e t h e p r o o f i n t w o s t e p s .

A : T h e r e e x i s t s a n "

0

> 0 s u c h t h a t i f k '

0

k

1

"

0

t h e n k T

'

k C ( k '

0

k

1

)

B : R e m o v a l o f t h e c o n s t r a i n t k '

0

k

1

"

0

P a r t A w a s p r o v e d b y C a l d e r n 1 ] . P a r t B w a s p r o v e d b y C o i f m a n / M c I n t o s h / M e y e r 2 ] .

T h e p r o o f w e p r e s e n t i s d u e t o D a v i d 3 ] .

P r o o f o f A : A s s u m e ' 2 C

1

0

( R

n

) a n d d e n e

K

N

( x y ) =

( ' ( x ) ; ' ( y ) )

N

( x ; y )

N + 1

N = 0 1 2 : : : :

T

N

c o r r e s p o n d i n g P V O w i t h k e r n e l K

N

W e r e m a r k t h a t t h e T

N

' s a r e c a l l e d c o m m u t a t o r s a n d a r i s e n a t u r a l l y w h e n o n e t r i e s t o

c o n s t r u c t a c a l c u l u s o f s i n g u l a r i n t e g r a l o p e r a t o r s t o h a n d l e d i e r e n t i a l e q u a t i o n s w i t h n o n -

s m o o t h c o e c i e n t s . W e r e f e r t o C a l d e r n 4 ] f o r a n e x t e n s i v e d i s c u s s i o n o f c o m m u t a t o r s

a n d P D E ' s . S i n c e T =

P

1

N = 0

( ; i )

N

T

N

, i t i s e n o u g h t o p r o v e t h a t

k T

N

k C

N + 1

k '

0

k

N

1

N = 0 1 2 : : : :

3 7


44/144

W e a l s o n o t e t h a t i t i s e n o u g h t o p r o v e t h a t t h e r e e x i s t s a n "

1

> 0 s u c h t h a t i f k '

0

k

1

"

1

,

t h e n

k T

N

k C

N

N = 1 2 : : : :

s i n c e T

0

i s t h e H i l b e r t t r a n s f o r m a n d t h i s o p e r a t o r i s L

2

b o u n d e d . T h e r e a r e m a n y p r o o f s

o f t h i s f a c t a n d o n e p r o o f i s s u p p l i e d b y T h e o r e m 1 . 6 . T o p r o v e t h a t T

N

N = 1 2 : : : a r e

L

2

b o u n d e d w e m a k e t h e f o l l o w i n g o b s e r v a t i o n .

L e m m a 4 . 1 . I f ' 2 C

1

0

, t h e n

T

N + 1

( 1 ) = T

N

( '

0

) N = 0 1 : : : :

P r o o f . T h e l e m m a i s a c o n s e q u e n c e o f t h e i d e n t i t y

d

d y

;

' ( x ) ; ' ( y )

x ; y

N + 1

= ( N + 1 )

( ' ( x ) ; ' ( y ) )

N + 1

( x ; y )

N + 2

; ( N + 1 )

( ' ( x ) ; ' ( y ) )

N

( x ; y )

N + 1

'

0

( y )

a n d

T

N + 1

1 ( x ) = l i m

" # 0

Z

x ; y > "

( ' ( x ) ; ' ( y ) )

N + 1

( x ; y )

N + 2

d y =

= l i m

" # 0

1

N + 1

;

' ( x ) ; ' ( x ; " )

"

;

' ( x ) ; ' ( x + " )

; "

+ T

N

'

0

( x ) = T

N

'

0

( x )

A r e c u r s i o n a r g u m e n t u s i n g P r o p o s i t i o n 3 . 1 , L e m m a 4 . 1 a n d t h e f a c t t h a t ' L i p s c h i t z

f u n c t i o n i m p l i e s '

0

2 L

1

s h o w s t h a t T

N

N = 0 1 2 : : : a r e L

2

b o u n d e d . W h a t r e m a i n s

t o b e d o n e i s t o s h o w t h a t

k T

N

k C

N

N = 1 2 : : :

f o r s o m e c h o i c e o f C > 0 . H e r e , o f c o u r s e , k k d e n o t e s t h e n o r m k k

L

2

! L

2

. T o c o n c l u d e t h e

p r o o f o f A w e n o t e t h a t

1 : I f K i s a k e r n e l o f C Z t y p e s u c h t h a t

K ( x y ) + ( r

x

K ( x y ) + r

y

K ( x y ) ) x ; y

C

1

x ; y

n

a n d i f k T 1 k

C

2

, t h e n

k T k D

n

( C

1

+ C

2

)

f o r s o m e c o n s t a n t D

n

w h i c h o n l y d e p e n d s o n d i m e n s i o n n

3 8


45/144

2 : U n d e r t h e s a m e a s s u m p t i o n s a s i n 1

k T k

L

1

! B M O

D

n

( k T k + C

1

)

T h i s f o l l o w s f r o m t h e p r o o f s o f P r o p o s i t i o n 3 . 1 a n d T h e o r e m 1 . 6 . H e n c e

k T

N + 1

k C

1

+ C

2

k T

N

k N = 0 1 2 : : :

f o r s o m e c o n s t a n t s C

1

C

2

i n d e p e n d e n t o f N a n d

k T

N

k C

N

N = 1 2 : : :

f o r s o m e c o n s t a n t C > 0 f o l l o w s . T h e p r o o f o f A i s c o m p l e t e d .

P r o o f o f B : W e s t a r t w i t h t h r e e l e m m a s .

L e m m a 4 . 2 . T h e r e e x i s t s a n "

0

> 0 s u c h t h a t i f

' ( x ) = A x + ( x ) x 2 R

w h e r e A 2 R a n d : R ! R L i p s c h i t z f u n c t i o n w i t h k

0

k

1

"

0

t h e n

k T

'

k C

0

f o r s o m e c o n s t a n t C

0

> 0 w h i c h i s i n d e p e n d e n t o f A .

P r o o f . R e p e a t i n g t h e a r g u m e n t a b o v e , w e s e e t h a t i f h : R ! C L i p s c h i t z f u n c t i o n w i t h

k h

0

k


46/144

R e m a r k :

~

L 2 L ; M L + M

P r o o f . W i t h o u t l o s s o f g e n e r a l i t y w e c a n a s s u m e t h a t I = 0 1 M = 1 L = ;

4

5

U

f x 2 I : '

0

( x ) + L 0 g h a s m e a s u r e

1

2

. C h e c k t h i s !

H e n c e ;

1

5

'

0

9

5

D e n e ~' : R ! R b y

~' ( x ) =

8

0 s u c h t h a t x x + h 2 0 1 . T h u s 0 ~'

0

9

5

a n d ~' s a t i s e s ( i i ) w i t h

~

L =

; 9

1 0

R e m a i n s t o c h e c k t h a t ( i ) i s s a t i s e d . S e t

E = f x 2 I : ' ( x ) = ~' ( x ) g

T h e n 0 2 E a n d E c l o s e d i m p l i e s t h a t

I n E =

k

I

k

w h e r e t h e c o m p o n e n t s I

k

a r e o f t h e f o r m a

k

b

k

o r a

k

1 w h e r e 0 < a

k

< b

k

1 . B u t ~'

i s c o n s t a n t o n e a c h i n t e r v a l I

k

a n d t h u s

' ( a

k

) = ~' ( a

k

) = ~' ( b

k

) = ' ( b

k

) i f I

k

= a

k

b

k

' ( a

k

) = ~' ( a

k

) = ~' ( 1 ) ' ( 1 ) i f I

k

= a

k

1

H e n c e

Z

I

k

'

0

( x ) d x 0 f o r e a c h k a n d w e o b t a i n

Z

'

0

( x ) d x 0

F i n a l l y

0

Z

'

0

( x ) d x =

Z

\ U

'

0

( x ) d x +

Z

\ ( I n U )

'

0

( x ) d x

4

5

\ U ;

1

5

\ ( I n U )

i m p l i e s

\ U

1

4

\ ( I n U )

4 0


47/144

a n d t h i s g i v e s u s

= \ U + \ ( I n U )

5

4

\ ( I n U )

5

8

H e n c e

E

3

8

a n d t h e p r o o f i s d o n e .

L e m m a 4 . 4 ( J o h n 6 ] ) . A s s u m e f : R ! R m e a s u r a b l e . A s s u m e t h e r e e x i s t s a n > 0

a n d a c o n t i n u o u s f u n c t i o n C : ! R , w h e r e = f ( a b ) 2 R

2

: a < b g , s u c h t h a t

f x 2 I : f ( x ) ; C ( I ) < g >

1

3

I f o r e a c h i n t e r v a l I = ( a b ) . T h e n f 2 B M O ( R ) a n d

k f k

C w h e r e C i s i n d e p e n d e n t o f f a n d t h e f u n c t i o n C

P r o o f . I t i s e n o u g h t o p r o v e t h e l e m m a f o r = 1 . T h e a r g u m e n t s a r e s i m i l a r t o t h o s e o n e

u s e s t o p r o v e J o h n - N i r e n b e r g ' s i n e q u a l i t y o n c e w e h a v e p r o v e d t h e f o l l o w i n g

C l a i m : I f I J R a r e i n t e r v a l s s u c h t h a t J = 2 I , t h e n C ( I ) ; C ( J ) 1 5

P r o o f o f c l a i m : L e t d e n o t e t h e i n t e r v a l w i t h e n d p o i n t s C ( I ) a n d C ( J ) a n d a s s u m e t o

o b t a i n a c o n t r a d i c t i o n t h a t > 1 5 . T h e n t h e r e e x i s t s p o i n t s z

k

2 k = 1 2 : : : 6 s u c h

t h a t z

k

2 a n d m i n

k 6= 1

z

k

; z

1

> 2 . S e t M

k

= f x 2 J : f ( x ) ; z

k

P

6

k = 1

M

k

S i n c e C : ! R i s c o n t i n u o u s , t h e r e e x i s t i n t e r v a l s I

k

k = 1 2 : : : 6 s u c h t h a t I I

k

J

a n d C ( I

k

) = z

k

. H e n c e

J >

6

X

k = 1

M

k

6

X

k = 1

M

k

\ I

k

6

1

3

I = 2 I

T h i s c o n t r a d i c t s J = 2 I a n d c l a i m i s p r o v e n .

W e n o w s h o w t h a t u n d e r t h e h y p o t h e s i s i n t h e l e m m a w i t h = 1

Z

I

f ( x ) ; C ( I ) d x C I

f o r e a c h i n t e r v a l I R . O b s e r v e t h a t w e d o n o t k n o w w h e t h e r f 2 L

1

o c

o r n o t . S i n c e t h e

a s s u m p t i o n s o n f a r e s c a l e - a n d t r a n s l a t i o n i n v a r i a n t w e c a n a s s u m e I = 0 1 . W e c a n a l s o

a s s u m e C ( I ) = 0

N o w , s e t

k

= 1 0 0 k k = 1 2 : : : a n d l e t

k

=

j

I

k

j

b e t h e u n i o n o f t h o s e i n t e r v a l s

i n t h e d y a d i c m e s h o f I w h i c h a r e m a x i m a l w i t h r e s p e c t t o i n c l u s i o n w i t h t h e p r o p e r t y

C ( I

k

j

) >

k

. W e s e e t h a t I =2

1

a n d

1

2

3

. F o r e a c h I

k

j

i n

k

t h e r e e x i s t s

4 1


48/144

a n i n t e r v a l

~

I

k

j

i n t h e d y a d i c m e s h o f I w h i c h i s m i n i m a l w i t h r e s p e c t t o i n c l u s i o n a n d w i t h

t h e p r o p e r t y I

k

j

*

~

I

k

j

. H e n c e C (

~

I

k

j

)

k

a n d t h e c l a i m a b o v e i m p l i e s

k

k

+ 2 g w h i c h i s a m e a s u r a b l e s e t a n d t a k e a p o i n t

o f d e n s i t y x

0

2 A

k

. T h e n t h e r e e x i s t s a s u c i e n t l y s m a l l i n t e r v a l J i n t h e d y a d i c m e s h o f

I s u c h t h a t x

0

2 J a n d

f x 2 J : f ( x ) >

k

+ 2 g

9 9

1 0 0

J

T h i s i m p l i e s C ( J ) >

k

a n d t h u s x

0

2 J

k

. H e n c e A

k

k

e x c e p t f o r a s e t o f m e a s u r e

0 a n d

A

k

k

B u t

k + 6

1

2

k

i m p l i e s t h a t t h e r e e x i s t c o n s t a n t s B C > 0 s u c h t h a t

A

k

B e

; C (

k

+ 2 )

k = 1 2 : : : :

C h o o s i n g a s l i g h t l y l a r g e r c o n s t a n t C w e g e t

f x 2 I : f ( x ) > g B e

; C

T h e r e m a i n i n g a r g u m e n t i s t h e s a m e a s i n t h e p r o o f o f t h e r e m a r k o n p a g e 2 9 .

F i n a l l y w e g i v e t h e a r g u m e n t f o r

k + 6

1

2

k

P r o o f . S e t E

k

= f x 2 I : f ( x ) ;

k


49/144

b y t a k i n g t h e u n i o n o v e r a l l I

k

j

I

k

0

j

0

a n d u s i n g E

k

\ I

k

j

1

3

I

k

j

. H e n c e

I

k

0

j

0

X

k

0

< k k

0

+ 6

I

k

0

j

0

\ E

k

1

3

X

k

0

< k k

0

+ 6

I

k

0

j

0

\

k

2 I

k

0

j

0

\

k

0

+ 6


k + 6

1

2

k

W e h a v e n o w p r e p a r e d a l l t h e m a c h i n e r y w e n e e d t o p r o v e p a r t B . T h e i d e a i s t o m a k e a n

i n d u c t i o n a r g u m e n t w h e r e p a r t A i s t h e b a s e a n d t h e i n d u c t i o n s t e p i s t h e f o l l o w i n g : L e t

M > 0 . I f t h e r e e x i s t s a c o n s t a n t

C = C

;

9

1 0

M

s u c h t h a t k T

~'

k C f o r a l l L i p s c h i t z f u n c t i o n s ~' w i t h o s c ~'

9

1 0

M t h e n t h e r e e x i s t s a

c o n s t a n t C = C ( M ) s u c h t h a t k T

'

k C f o r a l l L i p s c h i t z f u n c t i o n s ' w i t h o s c ' M

H e r e w e l e t o s c ' d e n o t e i n f f M : ' ( x ) + L x ; ( ' ( y ) + L y ) M x ; y f o r a l l x 6= y a n d

s o m e L 2 R g

W e r e m a r k t h a t L e m m a 4 . 2 i m p l i e s t h a t k T

'

k o n l y d e p e n d s o n o s c ' a n d n o t o n k '

0

k

1

W e

o b s e r v e t h a t i t i s e n o u g h t o p r o v e t h e i n d u c t i o n s t e p f o r L i p s c h i t z - f u n t i o n s ' w i t h ' 2 C

1

a n d i t i s e n o u g h t o p r o v e t h a t t h e r e e x i s t s a c o n s t a n t C = C ( M ) s u c h t h a t

k T

'

k

L

1

! B M O

C

f o r a l l L i p s c h i t z - f u n c t i o n s ' 2 C

1

w i t h o s c ' M . T a k e f 2 L

1

w i t h k f k

1

= 1 . L e t

I b e a n i n t e r v a l a n d l e t x

I

d e n o t e t h e c e n t e r o f I . S e t z ( x ) = x + i ' ( x ) . D e c o m p o s e

f i n f

1

a n d f

2

w h e r e f

1

= f

I

a n d f

2

= f ; f

1

. F i n a l l y s e t C

f

( I ) = T

'

f

2

( x

I

) W e

s e e t h a t C i s a c o n t i n u o u s f u n c t i o n w i t h r e s p e c t t o t h e e n d p o i n t s o f I . W i t h o u t l o s s o f

g e n e r a l i t y w e a s s u m e I = 0 1 . F r o m L e m m a 4 . 3 w e o b t a i n a L i p s c h i t z f u n c t i o n s u c h

t h a t o s c

9

1 0

M a n d E f x 2 I : ' ( x ) = ( x ) g h a s m e a s u r e

3

8

. S e t z

( x ) = x + i ( x )

O u r p u r p o s e i s t o u s e t h e c h a r a c t e r i z a t i o n o f B M O f u n c t i o n s w h i c h i s g i v e n i n L e m m a 4 . 4

b y s h o w i n g t h a t

f x 2 I : T

'

f ( x ) ; C

f

( I ) < C ( M ) g >

1

3

f o r c o n s t a n t C = C ( M ) c h o s e n l a r g e e n o u g h a n d w h i c h i s i n d e p e n d e n t o f f . I f t h i s i s d o n e ,

T h e o r e m 1 . 3 f o l l o w s . W e s t a r t w i t h

T

'

f ( x ) ; C

f

( I ) = T

'

f ( x ) ; T

'

f

2

( x

I

)

T

'

f

2

( x ) ; T

'

f

2

( x

I

) + T

'

f

1

( x ) ; T

f

1

( x ) + T

f

1

( x )

w h e r e x b e l o n g s t o a s u b s e t o f I w h i c h w e w i l l d e n e l a t e r .

4 3


50/144

F o r x 2 I ,

T

'

f

2

( x ) ; T

'

f

2

( x

I

)

Z

R n I

z ( x ) ; z (

1

2

)

( z ( x ) ; z ( y ) ) ( z (

1

2

) ; z ( y ) )

f ( y ) d y

C

0

( M )

Z

R n I

d y

x ; y

1

2

; y


Z

I

T

'

f

2

( x ) ; T

'

f

2

( x

I

) d x C

0

( M )

F o r x 2 E I ,

T

'

f

1

( x ) ; T

f

1

( x )

Z

I n E

z ( y ) ; z

( y )

( z ( x ) ; z ( y ) ) ( z

( x ) ; z

( y ) )

f ( y ) d y

B u t I n E = I

k

w h e r e t h e c o m p o n e n t s I

k

a r e i n t e r v a l s a n d

z ( y ) ; z

( y ) C

0

( M ) I

k

f o r y 2 I

k

H e n c e

T

'

f

1

( x ) ; T

f

1

( x )

X

k

Z

I

k

C

0

( M ) I

k

x ; y

2

d y x 2 E

S e t J

k

=

1 0 0 1

1 0 0 0

I

k

a n d E

= E \ { (

k

J

k

)

I

k 1

J

k 1

E

E

J

k

I

k

T h e n E n E

1

1 0 0 0

a n d

Z

E

T

'

f

1

( x ) ; T

f

1

( x ) d x C

0

( M )

X

k

Z

I

k

Z

J

k

I

k

x ; y

2

d x d y C

0

( M )

X

k

Z

I

k

d y C

0

( M )

F i n a l l y , f r o m t h e h y p o t h e s i s i n t h e i n d u c t i o n s t e p

Z

I

T

f

1

( x ) d x

;

Z

I

T

f

1

( x )

2

d x

1 = 2

C

;

9

1 0

M

B u t

f x 2 I : T

'

f

2

( x ) ; T

'

f

2

( x

I

)

C

0

( M )

3

g

1

1 0 0 0

f x 2 E

: T

'

f

1

( x ) ; T

f

1

( x )

C

0

( M )

3

g

1

1 0 0 0

f x 2 I : T

f

1

( x )

C

0

( M )

3

g

1

1 0 0 0

4 4


51/144

i f C

0

( M ) l a r g e e n o u g h . H e n c e f x 2 E

: T

'

f ( x ) ; T f

2

( x

I

) < C

0

( M ) g j j E ;

4

1 0 0 0

>

1

3

i f C

0

( M ) l a r g e e n o u g h a n d w h e r e C

0

( M ) i s i n d e p e n d e n t o f f . ( N o t e t h a t w e h a v e a s s u m e d

k f k

1

= 1 . ) L e m m a 4 . 4 c o n c l u d e s t h a t

k T

'

k C C

0

( M ) = C ( M )

a n d t h e i n d u c t i o n s t e p i s p r o v e d .

R e f e r e n c e s

1 : A .

Dahlberg B.E.J., Kenig C.E. Harmonic Analysis and Partial Differential Equations (1996)(en)(138s)

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Transcript of Dahlberg B.E.J., Kenig C.E. Harmonic Analysis and Partial Differential Equations (1996)(en)(138s)