Harmonic analysis and partial differential equations

HARMONIC ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS BJ RN E. J. DAHLBERG CARLOS E. KENIG ISSN 0347-2809

DEPARTMENT OF MATHEMATICS CHALMERS UNIVERSITY OF TECHNOLOGY AND THE UNIVERSITY OF G TEBORG G TEBORG 1985/1996

FOREWORD These lecture notes are based on a course I gave rst at University of Texas, Austin during the academic year 1983 - 1984 and at University of Gteborg in the fall of 1984. My purpose in those lectures was to present some of the required background in order to present the recent results on the solvability of boundary value problems in domains with bad boundaries. These notes concentrate on the boundary value problems for the Laplace operator for a complete survey of results, we refer to the survey article by Carlos Kenig I am very grateful for this kind permission to include it here. It is also my pleasure to acknowledge my gratitude to Peter Kumlin for excellent work in preparing these notes for publication. January 1985 Bjrn E. J. Dahlberg

i

ii

Contents 0 Introduction

1

1 Dirichlet Problem for Lipschitz Domain. The Setup

11

2 Proofs of Theorem 1.1 and Theorem 1.2

19

3 Proof of Theorem 1.6

25

4 Proof of Theorem 1.3

37

5 Proof of Theorem 1.4

47

6 Dirichlet Problem for Lipschitz domains. The nal arguments for the L2-theory 51 7 Existence of solutions to Dirichlet and Neumann problems for Lipschitz domains. The optimal Lp-results 57

Index : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 64 Appendix 1 C. E. Kenig: Recent Progress on Boundary Value Problems on Lipschitz Domains : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 67 Appendix 2 B. E. Dahlberg/C. E. Kenig: Hardy spaces and the Neumann Problem in Lp for Laplace's equation in Lipschitz domains : : : : : : : : : 107

iii

iv

Chapter 0 Introduction In this course we will study boundary value problems (BVP:s) for linear elliptic PDE:s with constant coecients in Lipschitz-domains , i.e., domains where the boundary @ locally is given by the graph of Lipschitz function. We recall that a function ' is Lipschitz if there exists a constant M < 1 such that j'(x) ; '(z)j  M jx ; zj for all x and z. y = '(x)

x

To solve the BVP:s we will reformulate the problems in terms of integral equations. It therefore becomes necessary to study singular integral operators of Caldern-Zygmund type, which we prove to be Lp-bounded for 1 < p < 1 and invertible. The Lp-boundedness is a consequence of the Lp-boundedness of the Cauchy integral (Coifman, McIntosh and Meyer) Z f (w) Tf (z) = w ; z dw ; where ; is a Lipschitz-curve (method of rotation). The invertability will be proved by a new set of ideas recently developed by Dahlberg, Kenig and Verchota. Among the BVP:s which can be solved by this technique are the 1

Dirichlet problem u = 0 in u = f on @ Neumann problem u = 0 in @u = f on @ @n the clamped plate problem

8 < u = 0 in : @uu == fg on @ @n

and BVP's for systems e.g. the elasticity problem u + rdiv u = 0 in (ru + ruT )n = g on @ and Stoke's equation 8 < u = rp in : div uu == f0 on @ where u = (u1 u2 u3) in R3.

Fredholm theory for Dirichlet problem for domain with C 2 boundary We start with an example. Example (Dirichlet problem for a halfspace). If the function f 2 Lp (Rn) 1 < p < 1, it is well known that u(x y) = py  f (x) (x y) 2 Rn++1 = Rn  R+  where

;



; n+1 y 2  Py (x) = +1 2  2 (jxj2 + y2) +1 denotes the Poisson kernel, is a solution of u = 0 in Rn++1 u = f on @ Rn++1 = Rn n

n

2

and that ()

sup ku( y)kp  kf kp: y>0

Thus with X = Lp(Rn) and Y = fu : u harmonic in Rn++1 and u satises ()g we have the implication

f 2 X ) u 2 Y:

However, we can also reverse the implication since a harmonic function u which satises () has non-tangential limits a.e. on @ Rn++1, the limit-function u0 = u( 0) 2 Lp(Rn) and u(x y) = py  u0(x). Sketch of a proof. Assume u harmonic function in Rn++1 that satises (). The semigroup properties of fpy gy 0 implies

u(x y + ) = py  u(x)  > 0 y > 0

where u(x) = u(x ).

() ) u * v in Lp(Rn) ) py  u (x) ! py  v(x)

as n # 0 as n # 0 y > 0:

n

n

But py  u (x) = u(x y + n ) and thus n

u(x y) = py  v(x) where v 2 Lp(Rn):

For the proof of the existence of non-tangential limits of py  u0 we refer to e.g. Stein/Weiss 2]. The notion of solution of the Dirichlet problem and any other problem, is sound only if we have such a matching between the boundary value f of u and the solution u itself, i.e., we should not accept concepts of solution which are so weak such that the reversed implication is impossible. Now assume that is a bounded (connected) domain in Rn n 3 with C 2 boundary. (To avoid technicalities, we have assumed n 6= 2). Consider the Dirichlet problem (D)

u = 0 in uj@= f 2 C (@ )

Let r denote (;1)  (the fundamental solution) of the Laplace operator in Rn, that is,

r(x) = cn jxj1n;2  cn = ; (2 ;1n)!

(n=2)

1 ; ; 2 ; n 2n=2 n

3

and set

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

For f 2 C (@ ) we dene

Z

@ R(P Q)f (Q)d (Q) Z@ @nQ Sf (P ) = R(P Q)f (Q)d (Q)

Df (P ) =

@

P 2= @ P 2= @

Thus Df and Sf denote the double layer potential and single layer potential resp. Here d

is the surface measure on @ and @n@ is the directional derivative along the unit outward normal for @ at Q. It is immediate that Q

Df (P ) = 0 P 2 Rn n @

and Df will be our candidate for solution of (D). It remains to study the behaviour of Df at @ . Part of that story is Lemma 1. If f 2 C (@ ), then 1) Df 2 C ( ) 2) Df 2 C ({ ). More precisely: Df can be extended as a continuous function from inside to and from outside to { . Let D+ f and D; f denote the restrictions of these functions to @ resp. Set K (P Q) = @n@ R(P Q) for P 6= Q P Q 2 @ . We note that Q

i) K 2 C (@  @ n f(P P ) : P 2 @ g) ii) jK (P Q)j  jP ;QCj ;2 for P Q 2 @ and some C < 1. n

ii) is a consequence of the regularity of the boundary and can be seen as follows: Assume @ given by the graph of the C 2-function '. Set P = (x '(x) and Q = (y '(y)). Then K (P Q) = !1 hPjP;;QQnjnQi where h i is the inner product in `2(Rn) and n nQ = p(r'(y) ;2 1) : jr'(y)j + 1 4

Since ' is a C 2 function, we have that

'(x) = '(y) + hx ; y r'(y)i + e(x y) where je(x y)j = O(jx ; yj2):

Hence

jK (P Q)j  C jhP ; QjP(;r'Q(jyn) ;1)ij  C jjPe(;xQy)jjn  jP ;CQjn;2 :

This estimate is uniform in P and Q since @ compact. For f 2 C (@ ) dene

Tf (P ) =

Z

@

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

We can now formulate

Lemma 2 (jump relation for D). 1) D+ = 21 I + T 2) D; = ; 12 I + T and

Lemma 3. T : C (@ ) ! C (@ ) is compact. Sketch of proof of Lemma 3. Dene the operators Tn by

Tnf (P ) = for f 2 C (@ ), where

Z

@

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

Kn (P Q) = sign (K (P Q))  min(n jK (P Q)j) n 2 Z+ :

Thus Kn is continuous on @ @ and Arzela-Ascoli's theorem implies that Tn is a compact operator on C (@ ). Furthermore since kTnk  supQ2@ kKn ( Q)k1  C < 1, where C is independent of n we see that

Tn ! T

in the space B = fbounded linear operators on C (@ )g. But the compact operators in B form a closed subspace in B , and hence T is compact. 5

Proof of Lemma 1 and 2. Some basic facts:

Z

@ R(P Q)d (Q) = 1, if P 2 @  @nQ Proof: Apply Green's formula to the harmonic function R(; Q) in

> 0 small, where B (P ) = fx 2 Rn : jP ; xj  g. Z @ 2) R(P Q)d (Q) = 0, if P 2= . @n @ Q Proof: Exercise. Z 3) K (P Q)d (Q) = 21 , if P 2 @ @ Proof: Exercise. Let P 2 @ . We want to show that Df (Q) ! 21 f (P ) + Tf (P ) as 3 Q ! P A: Assume P 2= supp f : Easy. B: Assume f (P ) = 0: We need. Z  @  4) 9C > 0 :  @n R(P Q)d (Q) < C for all P 2= @ Q @ Proof: Exercise. 4) implies the estimate

1)

n B (P ) for

kDf kL1 (R n@)  C kf kL1 (@): Choose ffk g  C (@ ) with P 2= supp fk such that kf ; fk kL1(@) ! 0 as k ! 1: T bounded operator implies Tfk(P ) ! Tf (P ) as k ! 1. Hence jDf (Q) ; Tf (P )j  C k(f ; fk)kL1 (R n@) + jDfk (Q) ; Tfk(P )j + + jTfk(P ) ; Tf (P )j ! 0 as k ! 1 and 3 Q ! P: C: Enough to check f 1. n

n

The result follows from basic facts 1) and 3). Hence we have proved Lemma 1 and 2 part 1). Part 2) follows analogously. 6

We now return to the single layer potential and observe that Sf is harmonic in Rn n @ and continuous in Rn if f 2 C (@ ). Next we want to compare the normal derivative of Sf with Df at @ . Since @ is C 2 we have following result: For " > 0 small enough ] ; " "@

3 (t P ) ! P + tnp 2 V

is a dieomorphism, where np is the outward unit normal of @ at P , and V is a neighborhood of @ . For P 2 @ and t 2] ; " " set Z @ DSf (P + tnp) = @n R(P + tnp Q)f (Q)d (Q): @ p The close relations between Df and DSf is formulated in Lemma 4. If f 2 C (@ ) then 1) DSf 2 C (V \ ) 2) DSf 2 C (V \ { ) (Compare Lemma 1). Let D+ Sf be the restriction to @ of the function DSf extended to V \ from inside and D; Sf the restriction to @ of the function DSf extended to V \ { from outside. But R(P Q) = R(Q P ) so with Rn (P Q) = K (Q P ), which is the real-valued kernel in

T f (P ) =

Z

@

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

we have that T  is the adjoint operator of T . Lemma 5 (jump relations for DS ). 1) D+ S = ; 21 I + T  2) D; S = 12 I + T . Proof of Lemma 4 and 5. Let f 2 C (@ ) and dene

wf (P ) =

Df (P ) + DSf (P ) P 2 V n @  Tf (P ) + T f (P ) P 2 @ :

Claim: wf 2 C (V ). Proof: wf continuous on V n @ and on @ . Hence it is enough to show taht

wf (P + tnp) ! wf (P ) uniformly for P 2 @ as t ! 0: 7

Assume 0 2 C (@ ) such that 0  0  1 0 = 1 in a neighborhood of P and supp 0  B (P ):

Decompose f as

f = f1 + f2 0f + (1 ; 0)f:

A: wf2 (P + tnp) ! wf2 (P ) as t ! 0. Easy B: Assume t 6= 0

11111111111111 00000000000000 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 t 0, but not for domains with boundaries with less regularity. Remark: We observe that the method is non-constructive as a consequence of the soft arguments (i.e., compactness arguments) we have used. Hence it is not possible to solve the Dirichlet problem for, say Lipschitz-domains by approximating with C 2 domains k , solve some Dirichlet problems for these and obtain an approximation of a solution for , since we do not have any estimates of the inverses of the D+ :s. 9

References 1] Folland, G: Introduction to partial dierential equations. Math. Notes 17, Princeton U.P. 2] Stein, E.M./ Weiss, G: Introduction to Fourier Analysis on Euclidean Spaces. Princeton U.P.

10

Chapter 1 Dirichlet Problem for Lipschitz Domain. The Setup A function ' : Rn ! R such that

j'(x) ; '(y)j  M jx ; yj

for all x y 2 Rn

is called Lipschitz function. A bounded domain  Rn+1 is called Lipschitz domain if @ can be covered by nitely many right circular cylinders L whose bases are at a positive distance from @ such that to each cylider L there is a Lipschitz function ' : Rn ! R and a coordinate system (x y) x 2 Rn y 2 R such that the y-axis is parallel to the axis of symmetry of L and L \ = L \ f(x y) : y > '(x)g and L \ @ = L \ f(x y) : y = '(x)g. A domain D  Rn+1 is called special Lipschitz domain if there is a Lipschitz function ' : Rn ! R such that D = f(x y) : y > '(x)g and @D = f(x y) : y = '(x)g. In this and all proceeding chapters we reserve the notation for bounded Lipschitz domains and D for special Lipschitz domains respectively. With a cone ; we mean a circular cone which is open. A cone ; with vertex at a point P 2 @C , where C  Rn+1 is a domain, is called a nontangential cone if there is a cone ;0 and a > 0 such that  6= (; \ B(P )) n fP g  ;0 \ B (P )  C:

Br (Q) is our standard notation for the ball fx 2 Rn : jx ; Qj  rg. We say that a function u dened in a domain C has nontangential limit L at a point P 2 @C if u(Q) ! L as Q ! P Q 2 ;

for all nontangential cones ; with vertices at P . Finally we dene the nontangential maximal function M u for  > 1 and function u dened in Lipschitz domain by

M u(P ) = supfju(Q)j : jP ; Qj <  dist (Q @ ) Q 2 11

g

P 2@ :

One of the main results in this course will be the existence of a solution to the Dirichlet problem u = 0 in  Rn+1 uj@= f 2 L2(@ )

where is a bounded Lipschitz domain. By this we mean that there exists a harmonic function u in which converges nontangentially to f almost everywhere with respect to the surface measure d (@ ) and that the maximal function M u 2 L2(@ ) for  > 1. The starting point for our enterprise of proving the existence of a solution to Dirichlet problem for the Lipschitz domain is the double layer potential

Dg(P ) =

Z

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

where R(P Q) is the fundamental solution for Laplace equation in Rn+1 (multiplied with ;1) and g 2 L2(@ ). Since Dg is harmonic in , we are done if we can show that for some choice of g we have the right behaviour of Dg at @ . However, this is not easy since for K (P Q) = @n@ R(P Q) P Q 2 @ P 6= Q, we only have the estimate jK (P Q)j  jP ;QCj ;1 which cannot be improved in general. Thus we have to rely on the cancellation properties of K (P Q), and the operator T which appeared in Chapter 0 can only be dened as a principal value operator. Before we study the case with a general bounded Lipschitz domain we treat the case with a special Lipschitz domain D. From this we obtain the result for using standard patching techniques (see Appendix 2). Consider Z @ Dg(P ) = @D @nQ R(P Q)g(Q)d (Q) P 2 D Q

n

where D = f(x y) : y > '(x)g for a Lipschitz function ' : Rn ! R. We remark that ' Lipschitz function implies that '0 exists a.e. so the denition of Dg makes sense and @ R(P Q) = C hnQ  P ; Qi , with n = p(r'(x) ;1) for Q = (x '(x)), exists a.e. n Q @nQ jP ; Qjn+1 jr'(x)j2 + 1 d (@D). To state the rst proposition, we need some more notation: For every measure R  and each -measurable function g and measurable set A with  ( A ) = 6 0 we let A gd, Z 1 denote the mean value (A) gd. Furthermore for g 2 L1loc(@D) we dene the maximal A function Mg by

Mg(P ) = sup

Z

r>0 @D\Br (P )

jg(Q)jd (Q)

The following result is crucial. 12

P 2 @D:

Proposition 1.1. Let D = f(x y) : y > '(x)g where ' : Rn ! R is a Lipschitz function

with k'0k1 = A. Let P = (x y ) 2 D and P  2 (x '(x)) 2 @D and set  = y ; '(x). Assume g 2 Lp (@D) for some p where 1 < p < 1. Then jDg(P ) ; Tg(P )j  CMg(P ) where

T

g(P ) =

Z

@DnB
(P  )

K (P  Q)g(Q)d (Q):

The constant C depends only on the dimension a. Before we prove this proposition, we make some remarks on the maximal function M .

1) Another maximal function, which is quite similar to the Hardy-Littlewood maximal function M is M  dened by

Z

j g(Q0)jd (Q0) P 2 @D B (Q)3P @D\B (Q) We immediately observe that Mg  M g  CMg for some 

M g(P ) =

sup

r

r

for g 2 L1loc(@D). dimensional constant C , i.e., M and M are equivalent. 2) Let  denote the projection  : @D ! Rn where (x '(x)) 7! x and dene the maximal function M~ by ~ (x) = sup Mg

Z

r>0 Br (x)

jg  ;1(y)jdy x 2 Rn

for g 2 L1loc(@D). Since ' is a Lipschitz function, we see that M and M~ are equivalent. 3) M is bounded in L1 with norm 1, i.e., kMgk1  kgk1 for all g 2 L1 4) M is a weak (1 1) operator, i.e., there exists a C > 0 such that jfx : Mg(x) > gj  C kgk1 for all g 2 L1: 5) M is bounded in Lp 1 < p < 1, i.e., there exists a Cp > 0 such that kMgkp  Cpkgkp for all g 2 Lp: Here 3) is trivial, 4) can be proven by a covering lemma argument and 5) follows from 3), 4) and Marcinkiewicz' interpolation theorem (see Stein 1]). For later reference we state 13

Marcinkiewicz' interpolation theorem. Let 1  p < q  1 and let T be a subad-

ditive operator dened on Lp + Lq . Assume T is a weak (p p) operator and a weak (q q) operator. Then T is bunded on Lr where p < r < q. An operator T is a weak (p p) operator if there exists a constant C > 0 such that jfx : Tg(x)j > gj  C ; kgkp p for all g 2 Lp and  > 0: Hence, if T is bounded on Lp, then T is a weak (p p) operator, but the converse is not true in general.

11111111111111111111111111 00000000000000000000000000 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 D 00000000000000000000000000 11111111111111111111111111 P 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111
00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 P 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 Proof of Proposition 1.1.11111111111111111111111111 00000000000000000000000000




Z

   j  C @D\fjP ;Qj>g  hjnPQ; PQ;jn+1Qi ; hjnPQ;P Q;jnQ+1i   jg(Q)jd

Z  hnQ  P ; Qi  +C  n+1 jg (Q)jd (Q)  Z@DnfjP ;Qj>g jP ; Qj  C @D\fjP ;Qj>g ( + jQ ; P j)n+1 jg(Q)jd (Q) + Z 1 jg(Q)jd (Q) +C @DnfjP ;Qj>g n

jDg(P ) ; T

g(P )

where we have applied the mean value theorem to the rst integral. The second integral is  CMg(P ) and the rst integral can also be estimated from above with the same bound according to Lemma 1.1. Let  0Rbe a radial decreasing function de ned in Rn. Assume f 2 L1 +L1 and set mf (Rx) = supr>0 B (x) jf (x)jdx for x 2 Rn. Then  f (x)  Bmf (x) for all x 2 Rn where B = (x)dx.
r

14

If we take this lemma for granted for a moment and set (x) = (jxj +)n+1 

the rst integral above is bounded from above by CMg(P ) and we are done. 1 1 Proof of Lemma 1.1. It is enough R to prove the lemma for 0  f 2 C0   2 C0 and x = 0. n Set S = @B1(0) and A(r) = Br (0) f (x)dx. We obtain

  f (0) =

Z

(jxj)f (x)dx =

Z1

(r)rn;1

Z

f (rw)d (w)dr = Z1 0 0 = (r)A (r)dr = ;  (r)A(r)dr  ; 0(r)jBr (0)jdr mf (0): 0 0 R1 0 0 Set f 1 in the calculations above and we get ; 0  (r)Br (0)dr = B . The lemma is proven.

ZR1 n

0Z 1

Sn

If we dene the operator T by Tg(P ) = sup jTg(P  )j P  2 @D for g 2

Lp(@D),

>0

then

jDg(P )j  C (Tg(P ) + Mg(P )) for all P = (x y) 2 D and P  = (x '(x)) 2 @D. Thus if we can prove that T is bounded on Lp(@D), then jDg(P  )j < 1 for a.e. P  2 @D. We remark that with some additional considerations one can prove that jDg(Q)j  C (Tg(P ) + Mg(P )) for all Q in a nontangential cone ; with vertex at P  2 @D, and thus supQ2; jg(Q)j in non-tangential cones ; with vertices at P  2 @D for almost every P  2 @D. It then follows that Dg has nite nontangential limit a.e. d (@D) (see Dahlberg 2]).

The limitfunction belongs to Lp(@D). If we also can prove that the limitfunction is equal to f for some choice of g, we are done. To be successful in our approach, we have to study the operators T for  > 0 and T. This calls for some denitions. Let S (Rn) denote the Schwartz class (i.e., the space of all C 1-functions in Rn which together with all their derivatives die out faster than any power of x at innity) with the usual topology. T is called a singular integral operator (SIO) if T : S (Rn) ;! S (Rn) is linear and continuous and there exists a kernel K such that for all ',  2 C01(Rn) with supp ' \ supp  = 

hT' i =

Z Z

K (x y)'(y)(x)dydx 15

where h i is the usual S ; S  paring. We observe that K does not determine T uniquely. Consider for instance Tf = f 0 for which K = 0 is a kernel. We say that a kernel K is of Caldern-Zygmund type (CZ-type) if 1) jK (x y)j  jx ;C yjn 2) jrxK (x y)j + jry K (x y)j  jx ;Cyjn+1 3) K (x y) = ;K (y x). The operator-kernels, we will study, will be of the form y)))i i = 1 2 : : :  n + 1 Ki (x y) = j(((xx''(x(x))));;(y(y''(y()) jn+1 where (a)i denotes the i-th component of a 2 Rn+1. We observe that these kernels are of CZ -type and adopt the convention that whenever we discuss kernels K , they are assumed to be of CZ -type unless we explicitly state the converse. Starting with a kernel K , we can form a well-dened SIO with K as the kernel namely the principal value operator (PVO) T . Note that for '  2 S (Rn)) Z Z Z Z 1 K (x y)'(y)(x)dydx = 2 K (x y)('(y)(x) ; '(x)(y))dydx jx;yj>" jx;yj>" since K (x y) = ;K (y x) and thus lim

"!0

Z Z

jx;yj>"

K (x y)'(y)(x)dydx

exists since j'(y)(x) ; '(x)(y)j = O(jx ; yj) and '  2 S (Rn) decay fast enough at innity. Hence T : S (Rn) 3 ' 7! T' 2 S (Rn) where

hT' i = "lim !0

Z Z

jx;yj>"

K (x y)'(y)(x)dydx

is a SIO. We leave the proof of continuity of T as an exercise. From now on we assume that all operators T are PVO with kernels K of CZ -type. To achieve our goal to establish the existence of a solution to the Dirichlet problem for Lipschitz domains, we will prove the following sequence of theorems Theorem 1.1. If T bounded on L2, then T is a weak (1 1) operator. This implies 16

Theorem 1.1'. If T bounded on L2, then T bounded on Lp for 1 < p < 1. and

2  p  Theorem R 1.2. If T bounded on L , then T bounded on L for 1 < p < 1 where T g(x) =

sup">0 j

j

jx;yj>" K (x y )g (y )dy .

Thus it is crucial for us to be able to prove L2-boundedness of T . This is done in two steps. Theorem 1.3. If ' : R ;! R Lipschitz function and K (x y) = x;y+i('(1x);'(y)) , then the corresponding operator T is L2 bounded. Theorem 1.4. If ' : Rn ;! R Lipschitz function and y)))i i = 1 2 : : :  n + 1 Ki (x y) = j(((xx''(x(x))));;(y(y''(y()) jn+1 then the corresponding operators Ti are L2 bounded.

Finally we prove

Theorem 1.5.

Dj@D is invertible.

To prove Theorem 1.3, we will characterize those kernels K of CZ -type which correspond to L2 bounded operators T . This is done by a theorem of Daivd and Journ 3]. Theorem 1.6. T bounded on L2 i T 1 2 BMO. The denition of BMO and the theorem and its proof will be discussed in Chapter 3.

References 1] E. M. Stein: Singular integral and dierentiability properties of functions, Princeton University Press 1970. 2] B.E.J. Dahlberg: Harmonic functions in Lipschitz domains, Proceedings of Symposia in Pure Mathematics Vol XXXV, Part 1 (1979) pp. 313-322. 3] G. David and J.-L. Journ: A boundedness criterion for generalized Caldern-Zygmund operators, Preprint.

17

18

Chapter 2 Proofs of Theorem 1.1 and Theorem 1.2 We recall that T is a PVO with kernel K of CZ-type. In this chapter we give a proof of the following result of Caldern-Zygmund 1].

Theorem 1.1: If T bounded on L2, then T is a weak (1 1) operator. The following bound on T is due to Cotlar 3].

Theorem 1.2: RIf T bounded on L2, then T is bounded on Lp for 1 < p < 1 where Tf (x) = sup">0 j

j

jx;yj>" K (x y )f (y )dy ,

Proof of Theorem 1.1. The idea of the proof is the same as when T is a translationinvariant L2-bounded operator (See Stein 2]). Thus we show that there exists C > 0 such that jfx 2 Rn : jTf (x)j > gj  C kfk1 for all f 2 L1 and  > 0 by splitting f in a good part g, which is a L2-function and a bad part b. This is done with the following lemma

Lemma (Caldern-Zygmund decomposition). Let f 2 L1(Rn) and  > 0. Then there exist cubes Qj  j = 1 2 : : : such that 1) jQj \ Qk j = 0 for j 6= k 2) jf (x)j   a.e. for x 2 Rn n 1 j =1 Qj

19

R

3)   Q jf (x)jdx < 2n . The proof is based on a recursive stop-time argument and can be found e.g. Stein 2]. j

Now set

f (x) x 2 Rn n  1 Q g(x) = R jf (x)jdx x 2 Qj j=i j j = 1 2 : : : Q j

and b(x) = f (x) ; g(x). We immediately observe that j 1) supp b  1 j =1 Q R 2) Q b(x)dx = 0 j = 1 2 : : : and 3) kgk2  2n kf k1. Furthermore, jfx 2 Rn : jTf (x)j > gj  jfx 2 Rn : jTg(x)j > 2 gj + jfx 2 Rn : jTb(x)j > 2 gj  jfx 2 Rn : jTg(x)j > 2 gj + j 1j=1 2Qj j + jfx 2 Rn n 1j=12Qj : jTb(x)j > 2 gj j

where 2Qj is the cube with the same center as Qj , which we denote yj , with sides parallel with Qj and with doubled sidelengths compared with Qj . Here jfx 2 Rn : jTg(x)j > 2 gj  42 kTgk2  C2 kgk2  C 1 kf k1 and 1 n X 1 n j j=1 2Qj j  2 jQjj  2 kf k1 j =1 so it remains to estimate jfx 2 Rn n 1j=1 2Qj : jTb(x)j > 2 gj and this is the point where we use the properties of the kernel K . Set

bj (x) = For, x 2= 2Qj we have

Tbj (x) =

Z Qj

b(x) x 2 Qj 0 otherwise

(K (x y) ; K (x yj ))bj (y)dy 20

and thus

Z

where we used

jTbj (x)j  C Q jxjy;;yyjnj+1j jbj (y)jdy j

Z Qj

bj (y)dy = 0. Integrating these inequalities gives

Z Rn n 1 j =1 2Qj

C

1 Z X

jTb(x)jdx 

j =1 Qj

1 Z X j =1 Rn n2Qj

jTbj (x)jdx 

jbj (y)jdy = C kbk1  C kf k1

and consequently

jfx 2 Rn n 1j=1 2Qj : jTb(x)j > 2 gj  2 kTbkL1(R n 1=12Qj)  C kfk1 : n

j

The proof is done.




Corollary: If T is bounded on L2, then T is bounded on Lp for 1 < p < 1. Proof. Marcinkiewicz' interpolation theorem and Theorem 1.1 implies that T is bounded on Lp for 1 < p  2. But the adjoint operator T  of T is a PVO with CZ-kernel K (x y) = K (y x) and T  is bounded on L2. An application of Marcinkiewicz' interpolation theorem and Theorem 1.1 to T  gives T  bounded on Lp for 1 < p  2. Hence T = (T ) is bounded on Lp for 2  p < 1 by duality and we are done. Proof of Theorem 1.2. The proof is an easy consequence of Cotlar's inequality, i.e., if T bounded on L2 then

(2.1) Tf  C (T )(Mf + MTF ) where M is the Harcy-Littlewood maximal function since M is bounded on Lp and T is bounded on Lp according to the corollary above. Remains to show Cotlar's inequality: It is enough to prove (2.1) for x = 0. Fix an " > 0. We will show that (2.2) T"f (0)  C (Mf (0) + MTF (0)) where C is independent of ". Set " f1(x) = f0 (x) jjxxjj
3 g E2 = fx 2 B : jTf1(x)j > 3 g: "

Here B = E1  E2 and thus We see that and

1  jjEB1jj + jjEB2jj :

Z 3 jE1j   B jTf (x)jdx  3jBj MTf (0)

jE2j  C kf1k1  C jBj Mf (0)

where we have used that T is a weak (1 1) operator. Thus   C (Mf (0) + MTf (0)) and it only remains to show (2.3), which is a straightforward calculation:

Z

jTf2(x) ; Tf2(0)j  jyj>" jK (x y) ; K (0 y)jjf (y)jdy  Z Z "  C jyj>" jyjn+1 jf (y)jdy  C min ; jyj"n+1  ";njf (y)jdy  CMf (0)

for jxj < 2" where we have applied Lemma 1.2. This completes the proof. 22

References 1] A. P. Calderon/A. Zygmund: On the existence of certain singular integrals, Acta Math. 88 (1952) pp. 85-139. 2] E. M. Stein: Singular integrals and dierentiability properties of functions. Princeton University Press 1970. 3] M. Cotlar: Some generalizations of the Hardy-Littlewood maximal theorem Rev. Mat. Cuyana 1(1955) pp. 85-104.

23

24

Chapter 3 Proof of Theorem 1.6 We begin this section by introducing some of the tools we need to prove the L2-boundedness of T . BMOp(Rn): For f 2 L1loc(Rn) we set

;Z

R

1 kf kp  = Qsup j f (x) ; fQjpdx  1  p < 1 cube Q p

where fQ = Q f (x)dx and dene the space BMOp to consist of those functions f such that kf kp  < 1. Thus (BMOp kkp ) becomes a semi-normed vectorspace with semi-norm vanishing on the constant functions. The letters BMO stand for bounded mean oscillation. Examples of BMO-functions: 1) log jxj 2 BMOp(Rn) 2) L1(Rn)  BMOp(Rn) Z 3) log jx ; yjd(y) 2 BMOp(Rn) for nite measures . To be able to work with this space, we only need to know three basic facts.

R

Fact 1: Let f 2 L1loc. If for all cubes Q, there exist constants CQ such that ( Q jf (x) ; CQjpdx)1=p  `, then f 2 BMOp and kf kp   2`.

Proof. Exercise.

This fact can be used to prove following proposition. Proposition 3.1. If T bounded on L2, then T : L1 ! BMO. Proof. The rst step in the proof is to give a denition of the function Tf where f 2 L1 . We therefore introduce fQj g = the set of all cubes Q with centers with rational coordinates

25

and with rational sidelengths. Set E = j @Qj . For each pair (x1 x2) 2 (Rn n E )  (Rn n E ) choose a cube Q 2 fQj g such that x1 x2 2 Q. Set f1 = f  2Q and f2 = f ; f1. Dene

F (x1 x2) = Tf1(x1) ; Tf1(x2) +

Z

Rn

(K (x1 y) ; K (x2 y))f2(y)dy:

We note that F is dened a.e. and that F is independent of Q (as long as x1 x2 2 Q). Check it! Furthermore, for a.e. x1 2 Rn and x2 2 Rn F (x x1) ; F (x x2) is a constant (regarded as a function of x). We now dene Tf as the class x ! F (x x1) for a.e. x1 2 Rn. It remains to show that T : L1 ! BMO is bounded. It is enough to show that

Z

jF (x xQ) ; Tf1(xQ)jdx  C kf kL1  Q for all cubes Q 2 fQj g. But

Z Q

since

Z

jTf1(x)jdx  ( Q jTf1 j

(x) 2dx)1=2

Z

f 2 L1(Rn)

Z

 C ( Q jf1(x)j2dx)1=2  C kf k1

Z

j R (K (x y) ; K (xQ y))f2(y)dyj  j R n2Q jK (x y) ; K (xQ y)jjf (y)jdy  Z  C R n2Q jyjx;;xQxjQnj+1 dykf k1  C kf k1 for x 2 Q: n

n

n

The proposition follows.

Fact 2: John-Nirenberg inequality Theorem: Let ' 2 BMO (Rn). Then there exists constants, C > 0  > 0, depending only on n, such that

 jfx 2 Q : j'(x) ; 'Qj > gj  C jQj exp ; ; k 'k

for all  > 0 and cubes Q.

Sketch of a proof. It is enough to show that

sup

Q cube

Z

;  exp k'k j'(x) ; 'Qj dx  C < 1:  Q 26

Assume k'k = 1 and ' 2 L1 . Since the constants C and  will be independent of k'k1, the result follows for a general '. Fix a cube Q. Consider all cubes Qj in the dyadic mesh of Q and choose a t > 1. Let Q~ j denote those dyadic cubes which are maximal with respect to inclusion satisfying

Z

Q~ j

and

j'(x) ; 'Qjdx > t

j'(x) ; 'Qj  t a.e. for x 2 Q n 1j=1Q~ j :

Clearly Q~ j  Q and

j 1j=1 Q~ j j  1t k' ; 'QkL1(Q)  1t jQj:

The maximality of Q~ j implies that

Z Q j

j'(x) ; 'Qjdx  t

where Q j is the minimal cube in the dyadic mesh of Q with respect to inclusion for which Q~ j 6 Q j . Furthermore

Q~ j Qj

j'Q~Z ; 'Qj  j'Q~ ; 'Q j + j'Q ; 'Qj   Q~ j'(x) ; 'Q jdx + t  Z n  2 Q j'(x) ; 'Q jdx + t   (2n + 1)t: j

j

j

j

j

j

Q

j

j

R

Set X ( Q) = supQ 2dyadicmeshofQ Q exp(j'(x) ; 'Q j)dx which is < 1 since ' 2 L1 . From the properties of Q~ j it follows that Z Z 1 exp(j'(x) ; 'Qj)dx  jQj etdx + 1 ~ Q Qn =1 Q Z 1 X + jQ1 j jQ~ j j exp(j'(x) ; 'Q~ j)dx exp(t(2n + 1))  Q~ j =1  et + 1t exp(t(2n + 1))X ( Q) j

j

j

j

j

j

27

j

Take supremum over all cubes Q. Thus sup X ( Q)1 ; 1t exp(t(2n + 1))]  et

Q cube

which implies supQ cube X ( Q)  C if  > 0 small enough. The proof is done.

Remark: It is an easy consequence of John-Nirenberg's inequality that the norms kkp  and kk kk1  are equivalent for 1 < p < 1.

Proof. For every 1 < p < 1 and  > 0 there exists a C > 0 such that xp  C exp(x) for x > 0. Choose  = 2 and apply the inequality above. Hence

Z ;  ' k k'k kp   C Qsup exp 2 j'(xk)';k 'Qj dx =  Q Z cube 1 ;  ;  1 = C sup jQj exp 2 t d jfx 2 Q : j'(xk)';k 'Qj > tgj  Q cube  Z0 1 ;   1  C Qsup exp 2 t C jQj exp(;t)  (;)dt = C cube jQj 0

and k'kp   C k'k. The inequality k'k  k'kp  follows from Hlder's inequality.

Fact 3: Connection between BMO and Carleson measures. Carleson measures originally appeared as answers to the following question.

Question: Which positive measures  on Rn++1 have the property Z Z

R++1 n

jPy f (x)j2d(x y)  C ()kf k22

for all f 2 L2(Rn)

where Py f (x) = py  f (x) with the Poisson kernel py (x) = cn (jxj2+yy2 )

+1 ? 2

n

To obtain a necessary condition on  consider f = Q, i.e., f is the characteristic function for a cube Q  Rn. We immediately observe that Py f (x) C > 0 for f(x y) : x 2 12 Q 0 < y < `(Q)g where `(Q) = side length of Q. Set Q~ = f( ) 2 Rn++1 :  2 Q 0 <  < `(Q)g. Hence (C ) (Q~ )  C jQj for all cubes Q  Rn is a necessary condition on . We call a positive measure  a Carleson measure if  satises (C) and inf fC : (Q~ )  C jQj for all cubes Qg is called the Carleson norm for . 28

Lemma 3.1. Let  be a continuous function in Rn++1 and set

u(x) = supfju( )j : jx ; j < g x 2 Rn:

Let  be a Carleson measure. Then (f(x y) 2 Rn++1 : ju(x y)j > g)  C jfx 2 R : u(x) > gj for all  > 0, where C only depends on n and the Carleson norm of . Proof. The lemma is a consequence of following geometric fact: For every covergin fQj g of countably many cubes there is a subcovering fQ0j g such that Qj = Q0j and each x 2 Qj belongs to at most 2n of the Q0j 's. We leave the proof of this fact as an exercise. For  > 0 set E = f(x y) 2 Rn++1 : ju(x y)j > g and for each (x y) 2 E dene Q~ (x y) = f( ) 2 Rn++1 : k ; xk < y 0 <  < yg Q(x y) = f 2 Rn : k ; xk < yg where kxk = maxi=1 ::: n jxij. Select a covering of E consisting of countably many cubes Q~ (x y) by a compactness argument. It is obvious that u() >  for all  2 Q(x y) and hence (E )  X (Q~ (x y)) = (Q~X (x y)0)   (Q~ (x y)0)  C jQ(x y)0j   2nC jf 2 Rn : u() > gj:

We can now answer the question posed above by Theorem 3.1. If  is a Carleson measure on Rn++1, then

Z Z

p d(x y )  C (p  n)kf kp  j P f ( x ) j y p R +1 n

+

1 < p  1:

Proof. Let py (x) denote the Poisson kernel. If jx ; xj < y, then py ( x ; t)  Cpy (x ; t) for all t 2 Rn where C is independent of x x and y. Furthermore Py f (x) py  f (x)  CMf (x) and thus supfjPy f ( x)j : jx ; xj < yg  CMf (x): Lemma 3.1 implies Theorem 3.1 since M is bounded on Lp for 1 < p  1.

29

Remark: Theorem 3.1 is also valid for all operators of the form

Pt f (x) = 't  f (x) where ' is a smooth function which decays at innity and such that j'(x)j  (x) for some ;  radial function  2 L1(Rn). 't(x) denotes t1n ' xt . We leave the proof of this remark as an exercise. We now introduce two families of operators denoted Pt and Qt of which the rst is an approximation of the identity and the second is an approximationR of the zero operator. R Let '  be smooth functions that decay at innity such that R '(x)dx = 1 and R  (x)dx = 0. Dene Pt and Qt by Pd t f ( ) = '^(t )f^( ) Qd t f ( ) = ^(t )f^( ) for nice functions f in Rn . We immediately observe Lemma 3.2. If f 2 L2(Rn), then Z1 k Qtf k22 dtt  C ()kf k22: 0 n

n

Proof. Apply Plancherel's formula. Theorem 3.2. If f 2 BMO (Rn), then

d(x t) = jQtf (x)j2 dxdt t

is a Carleson measure with Carleson norm

 C ()kf k2.

To carry through the argument in the proof of this theorem, we need a lemma. Lemma 3.3. If f 2 BMO (Rn) and Q0 is the unit cube (centered at 0), then Z jf (x) ; f j Q0 dx  C kf k n +1 R 1 + jxj where C only depends on n. n

Proof. For a > 0 let aQ denote the cube with sides parallel with the sides of Q and of lengths a times the sidelengths of Q and with the same center as Q. We observe that for every cube Q  Rn Z n Z 1 2 jfQ ; f2Qj  jQj Q jf (x) ; f2Qjdx  j2Qj 2Q jf (x) ; f2Qjdx  2nkf k:

30

Set Qj = 2j Q0 for j 2 N and assume kf k = 1. Here jfQ +1 ; fQ j  2n for j 2 N which implies jfQ +1 ; fQ0 j  (j + 1)2n for j 2 N: Hence Z jf (x) ; f j 1 Z X jf (x) ; fQ0 j dx + Q0 dx = n +1 n+1 R 1 + jxj j =0 Q +1 nQ 1 + jxj Z jf (x) ; f j 1 ;Z X jf (x) ; fQ +1 j dx + Q0 + dx  n+1 2j(n+1) Q0 1 + jxj j =0 Q +1 Z jf ; f j  1 X Q +1 Q0 n;j + (j + 1)22n;j ) + 1 = C dx + 1  (2 + j ( n +1) 2 Q +1 j =0 which completes the proof. j

j

j

n

j

j

j

j

j

j

Proof of Theorem 3.2. Qtf (x) is a well-dened function in Rn++1 since Qt1 = 0. We want to prove that for each cube Q  Rn

()

Z Z

2 j Qtf (x)j2 dxdt t  C ()kf kjQj: Q~

It is enough to consider Q = unit cube Q0 since BMO is scale- and translation invariant, i.e., kf k = kftsk where fts(x) = f (t(x ; s)) and dtt is scale invariant. Furthermore we may assume f2Q0 = 0 since Qt1 = 0. Thus we have to prove () for Q = Q0 and all f 2 BMO with f2Q0 = 0. Set f1 = f2Q0 and f2 = f ; f1. Then Qtf = Qtf1 + Qtf2 and we obtain Z Z Z Z dxdt 2 2 dxdt  j Qtf1j t  j Q f j t 1 t Q0 R++1 2 2  C ()kf1k2  C ()kf k from lemma 3.2 and for (x t) 2 Q~ 0 we obtain Z jQtf2(x)j  R n2Q0 t1n j; x ;t z jjf2(z)jdz  Z  C () R n2Q0 tn+1 + jxt ; zjn+1 jf2(z)jdz  Z jf (z)j  C ()t R 1 + jzjn+1 dz  C ()tkf k according to lemma 3.3. This completes the proof. n

n

n

n

We have now prepared the tools we need to prove the theorem of David and Journ. 31

Theorem 1.6: If T is a PVO with CZ type kernel K , then

T is bounded on L2 i T 1 2 BMO: Here the only if-part follows from Proposition 3.1. The if-part is the hard part. The proof we present is due to Coifman/Meyer 1]. Choose ' 2 C01(Rn) such that ' radial with support in the unit ball B1(0) and such that '^(jj) = 1 + O(jj4) as jj ! 0: d Dene Pt as above by Pd t f ( ) = '^(tj j)f^( ). Analogously dene Qt by Q tf ( ) = (tj j)2'(tj j)f ( ) and Rt by Rd t f ( ) = (j j);1 '^0(tj j)f^( ). Hence Pt  Qt and Rt commutes and d P 2 = 2R Q : dt t t t t This implies 1 d P 2TP 2 = 1 (R Q TP 2 + P 2TR Q ): 2 dt t t t t t t t t t The idea is as follows: We want to show jh1 T2ij  C k1k2k2k2 for all 1 2 2 S (Rn): We note that h1 Pt2TPt22i ! 0 as t ! 1 and hence it is enough to prove Z1d j 0 dt h1 Pt2TPt22idtj  C k1k2k2k2 for all 1 2 2 S (Rn) since P0 is the identity operator. Furthermore, it is enough to prove Z1 j 0 h1 RtQtTPt22i dtt j  C k1k2k2k2 for all 1 2 2 S (Rn) since Pt  Qt Rt are selfadjoint operators and T  = ;T . We need the following estimate. Lemma 3.4. Let '  2 C01(Rn) with support in the unit ball B1(0) and assume Z (x)dx = 0: Rn

Then

jhtx T'yt ij  Cpt(x ; y)

;  ;  where tx(z) = t1n  z;t x  'yt (z) = t1n ' z;t y and pt is the Poisson kernel. 32

Proof. The argument consists of a straightforward calculation where we use the CZ type properties of the kernel K .

2h

tx  T'yt

= lim "#0

i = lim "#0

Z Z

Z Z

K ( )(tx()'yt () ; tx()'yt ())dd j ; j>" ; ; ; K (t + y t + y) 1 ()'() ; 1 ()'())dd: x y t

j ; j>"

Hence, it is enough to prove the lemma for y = 0. Assume jxj < 10t: Then we obtain jhtx T'ij  t1n ; ; jCxj 2 1+ t Assume jxj 10t: Then we obtain

x y t

+1 2

n

= Cpt(x):

Z Z

jh Z Z ij = j (K ( ) ; K (x ))tx()'t()dd   jK (t t) ; K (x t)j1 ()'()dd   C jxjtn+1  C (t2 + jxt j2) +12 = Cpt(x) tx T'

x t

n

which concludes the proof of the lemma. Now set Lt = QtTPt where

Ltf (x) =

Z Rn

1t(x y)f (y)dy

with j1t(x y)j  Cpt(x ; y) according to Lemma 3.4. We recall that it is enough to show that Z1 j 0 h1 RtQtTPt22i dtt j  C k1k2k2k2 for all 1 2 2 S (Rn): Hence it is enough to show Z1 j 0 h1 RtQtTPt22i dtt j  C (k1k22 + k2k22) for all 1 2 2 S (Rn): But Z1 Z1 dt 2 j 0Z h1 RtQtTPt 2Zi t j  0 jhRt1 QtTPt22ij dtt  1 1  krt1k22 dtt + kQtTPt22k22 dtt I + II: 0

0

33

Here

Z 1 '^0(tjj) 2 dt 2 2 I= k  ^ 1 ( )k2  C k^1k2 = C k1k2 : t j  j t 0 To cope with II, we rewirte QtTPt22 = LtPt2 as ((LtPt)2)(x) = LtPt2 ; Pt2(x)](x) + Pt2(x)Lt1(x) = = LtPt2 ; Pt 2(x)](x) + Pt 2(x)QtT 1(x) But T 1 2 BMO implies jQtT 1j2 dxdt t is a Carleson measure according to Theorem 3.2 and hence Z Z Z1 dt 2jQ T 1j2 dxdt  C k k2 2 j P  ( x ) j k Pt2(x)QtT 1k2 t = t 2 t 2 2 t R++1 0 n

where we have used the remark to Theorem 3.1. Furthermore using Jensen's inequality A(xZt) jLt(Pt 2 ; Pt 2(x))(x)j2 = Z 2 = j 1t(x y)(Pt2(y) ; P2 2(x))dyj  C pt (x ; y)jPt2(y) ; Pt2(x)j2dy and thus

But Thus

Rn

Rn

Z1

Z 1Z

A(x t) dxdt k LtPt2 ; Pt 2(x)](x)k = t  0 R 0 Z 1Z Z  C 0 R R pt(x ; y)jPt2(x) ; Pt2(y)j2dydx dtt = Z 1Z Z =C pt (x)jPt2(x + y) ; Pt 2(y)j2dxdy dtt Z0 1 ZR ZR =C pt (x)j(Pt2( + x) ; Pt2())()j2 ddx dtt = Z0 1 ZR ZR =C pt (x)j'^(tjj)j2j1 ; eihx ij2j^2()j2 ddx dtt : 0 R R 2 dt 2t

n

n

n

n

n

n

n

n

Z Rn

n

\

pt(x)j1 ; eihx ij2dx = 2 ; 2e;j jt:

Z1

k LtPt2 ; Pt 2(x)](x)k22 dtt  0 Z Z 1  C R 0 2(1 ; e;j jt)j'^(tjj)j2 dtt j^2()j2d Z ; Z j1j Z1 dt ;j j t  C R 0 (1 ; e ) t + 1 j'^(tjj)j2 dtt j^2()j2d  jj  C k2k22: n



n



This completes the proof.




34

References 1] R. R. Coifman/ Y. Meyer: personal communication.

35

36

Chapter 4 Proof of Theorem 1.3 In this section we prove

Theorem 1.3: If ' : R ! R Lipschitz function and K' (x y) = x ; y + i('1(x) ; '(y)) ,

then the corresponding operator T' is bounded on L2 and kT'k  C (k'0k1). We immediately observe that the kernel K' is of CZ type and thus the L2 boundedness of the PVO T' can be proved using Theorem 1.6, i.e., it is enough to prove T'1 2 BMO. However, this is not easy. We give the proof in two steps. A: There exists an "0 > 0 such that if k'0k1  "0 then kT'k  C (k'0k1). B: Removal of the constraint k'0k1  "0. Part A was proved by Caldern 1]. Part B was proved by Coifman/McIntosh/Meyer 2]. The proof we present is due to David 3]. Proof of A: Assume ' 2 C01(Rn) and dene

N KN (x y) = ('((xx);;y')N(y+1)) N = 0 1 2 : : : : TN corresponding PVO with kernel KN . We remark that the TN 's are called commutators and arise naturally when one tries to construct a calculus of singular integral operators to handle dierential equations with nonsmooth coecients. WePrefer to Caldern 4] for an extensive discussion of commutators N and PDE's. Since T = 1 N =0 (;i) TN , it is enough to prove that

kTN k  C N +1k'0kN1

37

N = 0 1 2 : : : :

We also note that it is enough to prove that there exists an "1 > 0 such that if k'0k1  "1, then

kTN k  C N

N = 1 2 : : : : since T0 is the Hilbert transform and this operator is L2 bounded. There are many proofs of this fact and one proof is supplied by Theorem 1.6. To prove that TN N = 1 2 : : : are L2 bounded we make the following observation. Lemma 4.1. If ' 2 C01, then TN +1(1) = TN ('0) N = 0 1 : : : : Proof. The lemma is a consequence of the identity

and

d ; '(x) ; '(y) N +1 = (N + 1) ('(x) ; '(y))N +1 ; (N + 1) ('(x) ; '(y))N '0(y) dy x ; y (x ; y)N +2 (x ; y)N +1

Z

('(x) ; '(y))N +1 dy = TN +11(x) = lim "#0 jx;yj>" (x ; y)N +2 1 ; '(x) ; '(x ; ") ; '(x) ; '(x + ")  + T '0(x) = T '0(x): = lim N N "#0 N + 1 " ;"

A recursion argument using Proposition 3.1, Lemma 4.1 and the fact that ' Lipschitz function implies '0 2 L1 shows that TN  N = 0 1 2 : : : are L2 bounded. What remains to be done is to show that

kTN k  C N N = 1 2 : : :  for some choice of C > 0. Here, of course, kk denotes the norm kkL2 !L2 . To conclude the proof of A we note that

1: If K is a kernel of CZ type such that

jK (x y)j + (jrxK (x y)j + jry K (x y)j)jx ; yj  jx ;C1yjn and if kT 1k  C2, then kT k  Dn(C1 + C2) for some constant Dn which only depends on dimension n. 38

2: Under the same assumptions as in 1

kT kL1!BMO  Dn (kT k + C1)

This follows from the proofs of Proposition 3.1 and Theorem 1.6. Hence

kTN +1k  C1 + C2kTN k

N = 0 1 2 : : :

for some constants C1 C2 independent of N and

kTN k  C N

N = 1 2 : : : for some constant C > 0 follows. The proof of A is completed. Proof of B: We start with three lemmas. Lemma 4.2. There exists an "0 > 0 such that if '(x) = Ax + (x) x 2 R where A 2 R and  : R ! R Lipschitz function with k0k1  "0 then kT'k  C0 for some constant C0 > 0 which is independent of A. Proof. Repeating the argument above, we see that if h : R ! C Lipschitz function with kh0k  < 1 then kThk  C where the constant C is independent of h. Furthermore consider T' with kernel K' (x y) = x ; y + i(Ax + 1(x) ; Ay ; (y)) = 1 +1 iA  x ; y + i(h1(x) ; h(y)) (x) . Then the rst observation gives the desired result. where h(x) = 1+ iA Lemma 4.3 (David 3]). Assume ' : R ! R Lipschitz function and L 2 R such that

j'(x) + Lx ; ('(y) + Ly)j  M jx ; yj

x y 2 R:

Let I  R be an interval. Then there exists a Lipschtz function '~ : R ! R and a L~ 2 R such that (i) jfx 2 I : '(x) = '~(x)gj 83 jI j (ii) j'~(x) + L~ x ; ('~(y ) + L~ y )j  109 M jx ; y j x y 2 R.

39

Remark: L~ 2 L ; M L + M ]. Proof. Without loss of generality we can assume that I = 0 1] M = 1 L = ; 45  U fx 2 I : '0(x) + L 0g has measure 12 . Check this! Hence ; 15  '0  95 . Dene '~ : R ! R by 8 x 15. Then there exists points zk 2  k = 1 2 : : :  6 such that zk 2  and mink6=1 jzk ; z1j > 2. Set Mk = fx 2 J : jf (x) ; zkj < 1g. The sets P6 Mk are mutually disjoint. Furthermore the points zk can be chosen such that jJ j > k=1 jMk j. Since C :  ! R is continuous, there exist intervals Ik  k = 1 2 : : :  6 such that I  Ik  J and C (Ik) = zk . Hence 6 X

6 X

k=1

k=1

jJ j > jMk j jMk \ Ik j 6  31 jI j = 2jI j: This contradicts jJ j = 2jI j and claim is proven. We now show that under the hypothesis in the lemma with  = 1

Z

I

jf (x) ; C (I )jdx  C jI j

for each interval I  R. Observe that we do not know whether f 2 L1loc or not. Since the assumptions on f are scale- and translationinvariant we can assume I = 0 1]. We can also assume C (I ) = 0. Now, set k = 100k k = 1 2 : : : and let k = j Ijk be the union of those intervals in the dyadic mesh of I which are maximal with respect to inclusion with the property jC (Ijk)j > k . We see that I 2= 1 and 1  2  3  : : : . For each Ijk in k there exists 41

*

an interval I~jk in the dyadic mesh of I which is minimal with respect to inclusion and with the property Ijk I~jk . Hence jC (I~jk)j  k and the claim above implies

k < jC (Ijk)j < k + 15:

We claim that if we can prove that

j we are done.

k+6

j  12 j k j

Proof. Consider Ak = fx 2 I : jf (x)j > k + 2g which is a measurable set and take a point of density x0 2 Ak . Then there exists a suciently small interval J in the dyadic mesh of I such that x0 2 J and 99 jJ j: jfx 2 J : jf (x)j > k + 2gj 100 This implies C (J ) > k and thus x0 2 J  k . Hence Ak  k except for a set of measure 0 and

But j

jAk j  j k j:

k+6

j  12 j k j implies that there exist constants B C > 0 such that jAkj  Be;C( +2) k = 1 2 : : : : k

Choosing a slightly larger constant C we get

jfx 2 I : jf (x)j > gj  Be;C :

The remaining argument is the same as in the proof of the remark on page 29.

j  21 j k j. Proof. Set Ek = fx 2 I : jf (x) ; k j < 16g k = 1 2 : : : . Hence Ek are mutually disjoint Finally we give the argument for j

k+6

and

jEk \ Ijkj jfx 2 Ijk : jf (x) ; C (Ijk )j < 1gj 13 jIjkj: Furthermore, for k k0 we obtain jIjk00 \ Ekj 31 jIjk00 \ k j 42

by taking the union overall Ijk  Ijk00 and using jEk \ Ijk j 31 jIjkj. Hence

jIjk00 j X jIjk00 \ Ek j 31 X jIjk00 \ kj

k0 1 and k'0k1. Theorem 6.2. If f 2 L2(@D), then there exists a u such that

u = 0 in D @u @n j@D = f on @D where the boundary values are taken in the sense np  ru(Q) ! f (P ) as Q ! P nontangentialy a.e. @D and M (ru) 2 L2 (@D) with kM (ru)k2  C kf k2 where C only depends on  > 1 and k'0k1 . We recall that M u(P ) = supfju(Q)j : jP ; Qj <  dist (Q @D)g P 2 @D and that np denotes the outward unit normal at P 2 @D which is dened a.e. on @D. Corollary 6.1. Theorem 6.1 is valid if L2 replaced by Lp for 2  p  1. 51

This is a straightforward consequence of the maximum principle and interpolation between L2 and L1 , but apart from this result we discuss the Lp-theory for the Dirichlet problem and Neumann problem in Chapter 7. The proofs of Theorems 6.1 and 6.2 involves the layer potentials Z @ Df (P ) = @D @nQ r(P Q)f (Q)d (Q) P 2 D Z S f (P ) = r(P Q)f (Q)d (Q) P 2 D D

where r(P Q) = cn jP ; Qj1;n. With Q = (x '(x)) 2 @D and P = (z y) 2 D

Z y ; '(x) ; (z ; x)  r'(x) Df (z y) = cn R (jx ; zj2 + ('(x) ; y)2) +12 f (x)dx p Z S f (z y) = cn R (jx ; zj21++('jr(x')(x;)jy2)2) ;2 1 f (x)dx: n

n

n

n

From Proposition 1.1, the remark on page 17 and Theorems 1.2 and 1.4 it follows that

kM (Df )kp  C ( k'0k1)kf kp f 2 Lp for 1 < p < 1. Thus Dj@D f (P ) = lim D3Q!P Df (Q) exists and it remains to show non;tangentially that Dj@D is invertible on Lp . We now observe that D; = f(x y) 2 Rn+1 : '(x) > yg is

also a special Lipschitz domain and we have the following jump relations at the interface between D and D; .

Lemma 6.1. Let f 2 L2(@D) and let Tf (P ) = lim "#0

Z

@ r(P Q)f (Q)d (Q) P 2 @D jP ;Qj>" @nQ

Then

Dj@Df (p) = 21 f (P ) + Tf (P ) Dj@D; f (P ) = ; 12 f (P ) + Tf (P ) @ Sj f (P ) = ; 1 f (P ) + T f (P ) @np @D 2 @ Sj f (P ) = 1 f (P ) + T f (P ) @nP @D; 2 where T  is the adjoint operator of T .

52

a.e. @D a.e. @D a.e. @D a.e. @D

We observe that it is equivalent to solve the Dirichlet problem in D and the Neumann problem in D; . The jumprelations are similar to those in the case of regular boundary in chapter 0 with the dierence that here the operator T has to be interpreted as a PVO. The main ingredient in the proof is Proposition 1.1. The idea is to approximate @D by a C 2-boundary and use the result from Chapter 0. We leave the proof as an exercise. The nal step in the proof of Theorem 6.1 and 6.2 is to prove that  21 I + T and  21 I + T  are invertible on L2. The proof of this is due to Verchota 1], who used an indentity due to Rellich 2]. See also Jerison/Kenig 3].

Lemma 6.2. Let f 2 L2(@D) and u = S f jD . Then there exists C1 C2 > 0 such that Z ; @u  Z Z ; @u  2 2 2 C1 d

 jr d

t uj d  C2 @D @n @D @D @n

where rt denotes the tangential derivative.

Remark: Let  : @D ! Rn denote the projection mapping (x '(x)) 7! x. Then rtu is r(u  ;1) lifted with ;1 to @D. Proof. Assume f has compact support. Set e = (0 1). Since u = 0 in D we get

div (jruj2e ; 2 @u ru) = 0 @y called Rellich identity. Apply the divergence theorem. Hence Z Z @u @u 2 () jruj he nid = 2 @D @y @n d : @D Since 0 < c0  he ni  1 for Lipschitz domain, we obtain Z Z  @u  2 c0 jruj d  2 jruj @n d

@D @D and Schwartz inequality implies Z Z ; @u  2 2 d

jr uj d  C @D @D @n Remains to prove the reversed inequality. @u = hru ei @y 53

where e = he nin + et on @D. Hence @u = @u he ni + hru e i t @y @n

; 2 2 which we introduce in formula () above together with jruj2 = @u @n + jrtuj and jhru etij  jrtuj. Then we obtain Z @u Z Z ; @u  2 2 jrtuj he nid = @D @n he nid + 2 @D @n hru etid : @D Thus Z ; @u  ;Z ; Z ; @u 2  12 ; Z 2 2 2 1  d

 C jr d

jr tuj d + tuj d 2 @D @n @D @D @n @D and Z ; @u  Z 2 2 d

 C jr tuj d

@D @n @D To prove that  12 I + T and  12 I + T  are invertible on L2, it is enough to prove that  21 I + T  are invertible. We rst claim that there exists a C > 0 such that k( 12 I + T )f k2 C jjf k2 f 2 L2(@D)

Proof. Assume k( 12 I + T )f k2 = "kf k2 for some f 2 L2(@D). But k 12 f + T f k2 =k @n@ Sj@D; f k2  krtSj@D; f k2 = @ Sj f k = k ; 1 f + T f k = krtSj@Df k2  k @n @D 2 2 2 since rtS f is continuous across the boundary. Now f = ( 12 f + T f ) ; (; 21 f + T f ) and k 12 f + T f k2  k ; 21 f + T f k2 implies that " above cannot be too small. The proof is completed.

54

The nal step for proving the invertability of  21 I + T  is done with a method of continuity argument. Let U s donote the operator  12 I + T  where ' is replaced by s'. We note that U s : L2 ! L2 bounded such that for 0  s  1

kU sf k2 C kf k2 C independent of s kU sf ; U tf k2  C jt ; sjkf k2 C independent of s and t

(6.1) (6.2) (6.3)

U 0 = 21 I invertible

The invertability follows easily.

Proof. Set S = fs 2 0 1] : U s invertibleg. Then that S 6=  follows from (6.3), that S is open which follows from (6.2) and that S is closed is a consequence of (6.1) and (6.2). We only indicate the closedness of S . Assume sj ! s and U (sj ) invertible. Take g 2 L2 and fj 2 L2 such that U (sj )fj = g. (6.1) implies fj * f in L2 for a subsequence. We can also assume U (s)fj * U (s)f . Claim: U (s)f = g. Take an h 2 L2. Then

jhU (s)f ; g hij  jhU (s)f ; U (s)fj  hij + jh(U (s) ; U (sj ))fj  hij ! 0

as j ! 1. Hence U (s)f = g.

Remark on uniqueness of solutions in Theorem 6.1 and 6.2: The u appearing in Theorem 6.1 is unique while the u appearing in Theorem 6.2 is uniquely dened up to an additive constant.

References 1] G. C. Verchota: Layer potentials and boundary value problems for Laplace's equation in Lispchitz domain. (to appear in J. Funct. Anal.) 2] F. Rellich: Darstellung der eigenwerte von ru + u durch ein Randintegral Math. Z. 46 (1940) pp. 635 - 646. 3] D. S. Jerison/C. E. Kenig: The Neumann problem on Lipschitz domains. Bull AMS vol. 4 (1981) pp. 203 - 207.

55

56

Chapter 7 Existence of solutions to Dirichlet and Neumann problems for Lipschitz domains. The optimal Lp-results In this chapter we give parts of the proofs of Theorem 7.1. For every Lipschitz domain D = f(x y) 2 Rn+1 : '(x) < yg for ' : Rn ! R Lipschitz function there exists an " = "(D) > 0 such that for all 2 ; " < p  1 and all f 2 Lp(@D) there exists an u such that u = 0 in D uj@D = f on @D

where the boundary values are taken non-tangentially a.e. @D and such that M u 2 Lp (@D) with kM ukp  C kf kp where C only depends on  > 1 and k'0 k1 .

Theorem 7.2. For every Lipschiatz domain D as above there exists an " = "(D) > 0 such that for all 1 < p < 2 + " and all f 2 Lp(@D) there exists an u such that

u = 0 in D @u j = f on @D @n @D

where the boundary values are taken in the sense np  ru(Q) ! f (P ) as Q ! P nontangentialy a.e. @D and such that M ru 2 Lp (@D) with kM rukp  C kf kp where C only depends on  > 1 and k'0k1 .

Remark: The ranges 2 ; " < p  1 for the Dirichlet problem and 1 < p < 2 + " for the Neumann problem are optimal. The estimate kM ruk1  C kf k1 fails even for 57

smooth regions and for each p > 2 it is possible to construct a Lipschitz domain such that kM rukp  C kf kp fails. The situation is analogous for the Dirichlet problem. See Dahlberg 1]. The extension 2 ; " < p < 2 for the Dirichlet problem and 2 < p < 2 + " for the Neumann problem is done by a real variable argument using the result for p = 2 an a good  inequality. The extension to 1 < p < 2 in the Neumann problem is shown by establishing that for f 2 Hat1 , the atomic H 1 on @D, the solution of the Neumann problem with data f satises kM ruk1  C kf kH 1 . This is done by estimating the maximal function of gradient of the L2 solutions of atoms using the regularity theory for uniformly elliptic equations in self-adjoint form. The full result then follows by interpolation. The extension to 2 < p  1 in the Dirichlet problem is a consequence of the maximumprinciple and interpolation. We begin to discuss the regularity theory for uniformly elliptic equations in self-adjoint form. Let A(x) = (aij (x)) be a n  n dimensional symmetric matrix valued function in D where the entries aij (x) are bounded real-valued measurable functions. We assume that A(x) is uniformly elliptic on D, i.e., there exists a  1 such that at

n 1 jj2  X aij (x)ij  jj2 for all  2 Rn  i j =1

n X @ a (x) @ . We call u a (weak) solution of Lu = 0 in @xj ij @xj i j =1 Z 2 D if u 2 L1 loc(D) and hAru r'idx = 0 for all ' 2 C01(D). Here L21 loc denotes D the space of functions in L2loc(D) with distributional derivatives of rst order in L2loc(D). We (supersolution) of Lu = 0 in D if u 2 L21 loc(D) and Z Z say that u is a subsolution hAru r'idx  0 ( hAru r'idx 0) for all 0  ' 2 C01(D). The main result is

Let L denote the operator

D

D

Theorem 7.3 (DeGiorgi 2], Nash 3]). If u is a solution of Lu = 0 in D, then u is Hlder continuous.

This follows from

Theorem 7.4 (Harnack's inequality). If u 0 and Lu = 0 in D and if K  D is a compact set, then

ess supK u  C ess infK u

where C = C (n  K D).

Remark: Harnack's inequality is a quantitative version of the maximum principle. Remark: For notational convenience we let min and max denote ess inf and ess sup, resp. 58

Proof of Theorem 7.3. This is done using Harnack's inequality. Assume Lu = 0 in D = fx 2 Rn : jxj < 2g. Set

M (r) = maxjxjr u m(r) = minjxjr u

r < 1:

Then M (r) ; u and u ; m(r) are solutions and 0 in fx 2 Rn : jxj  rg. Hence ;  ;  M (r) ; m r2  C (M (r) ; M r2 ; ;   ;  M 2r ; m(r)  C m 2r ; m(r) Add these inequalities and set (r) = M (r) ; m(r). We obtain ;  ; 1 (r)  2r  CC + 1 and hence

(r)  r (1) for some  > 0:

Note that we have used the same constant C in the repeated uses of Harnack's inequality. This is justied by the scale invariance properties of C . We leave it as an exercise to check this. Proof of Theorem 7.4. (The proof is due to Moser 4]). Set Q(h) = fx 2 Rn : jxj j < h2 g. By a covering argument it is enough to prove the theorem for K = Q(1) and D = Q(4). Thus we assume u 0 and Lu = 0 in Q(4). Set

; 1 Z 1=p ;Z  p '(p h) = jQ(h)j u dx updx 1=p Q(h) Q(h) for 0 < h < 4 and ;1 < p < 1. For ;1 < p < 0 we study u + " for " > 0 small and then let " tend to zero in the estimates. Since max u = plim '(p h) !1 Q(h) min u = p!;1 lim '(p h) Q(h) the theorem is equivalent to show that for some C = C ( n) we have

'(1 1)  C'(;1 1):

The proof is based on three general inequalties relating integrals of functions v = v(x) to integrals of the gradient of v. We assume n 3. 59

Inequality A (Poincare's inequality) Z

where vQ(h) =

jv ; vQ(h)j  Q(h)

R

2dx

Z

Ch2

jr vj2dx Q(h)

Q(h) vdx.

Inequality B (Sobolev's inequality). Set K = n ;n 2 . ;Z

jvj Q(h)

2Kdx K1



Z

jrvj Q(h)

C (h2

2dx +

Z

 j vj2dx Q(h)

Inequality C (John-Nirenberg's inequality). If kvk  1, then there exists  > 0 and C only depeding on n such that

Z Q(2)

evdx

Z Q(2)

e;v dx  C:

The proof will be given in two parts viz.

Proposition 7.1. If u 0, subsolution in Q(4), then

; p 2;Z p dx1=p for p > 1 max u  C u Q(1) p ; 1 Q(2)

Proposition 7.2. If u > 0, supersolution in Q(4), then ;Z

Q(3)

updx)1=p  C

; 1 2 u for 0 < p < K: Q(1) K ; p min

Since K > 1, the theorem follows from the propositions. The proof of the propositions will be done by estimating '(p h)='(p0 h0) for p > p0 and derive the desired estimates by iteration. We need the following lemma.

Lemma 7.1. If u > 0 is a subsolution in D and v = u , then for any function  2 C01(D) one has

Z D

2

jrvj

2dx

;  2 Z  C 2 ; 1 D jrj2v2dx if  > 12 :

The same assertion is true for supersolutions if  < 21 .

60

R

Proof. u subsolution in D implies that D hAru r'idx  0 for all 0  ' 2 C01 (D). Choose  2 C01(D) and  > 0 and set ' = u2. (This ' does not belong to C01(D), but we have the approximation business as an exercise to the reader). This implies

Z

hAru rui D

u;12dx +

Z

h Aru ri2udx  0: D

The uniform ellipticity of A implies 1 Z jruj2u;12dx + Z hAru ri2udx  0:  D D 1 . Then we obtain Introduce v = u where  =  + 2 Z ;1  jrvj22dx  C Z jhru rij jj ju +1 2 jju 2 jdx  2 D D 

Thus



Z 1  C  D(jrjv)(jrvj jj)dx

;  2 Z jr j2v2dx  > 12 : jr vj  c 2 ; 1 D D The argument for u supersolution and  < 12 is similar. Z

2 2dx

Now let 0 < h0 < h < 2h0 < 4 and choose  2 C01(Q(h)) such that 0    1  = 1 on Q(h0) and jrj  C h ;1 h0 . Since u solution to Lu = 0 in Q(h), u is both a subsolution and a supersolution and Lemma 7.1 implies that for  6= 12 Z ; 2  1 Z  2 u2 dx: jr u j dx  C (2 ; 1) (h ; h0)2 0 Q(h) Q(h ) Set p = 2 . Inequality B gives us Z ;  ;Z 1 p 1 2 p K u dx) K  C p ; 1 ; h 2 updx 0 Q(h ) h0 ; 1 Q(h) i.e, ; 2 ;  2 '(Kp h0)  C p ;p 1 hh0 ; 1 ; '(p h) if p > 1 and ;  2 '(Kp h0) C hh0 ; 1 ; '(p h) if p < 0: 61 p

p

p

Now for p > 1 let

We nd since

 ; Y i=1

p = K  p h = 1 + 2; h0 = h+1

 = 0 1 2 : : :

Kip   C that Kip ; 1 '(p+1  h+1 )  C =K '(p  h )

for  = 1 2 : : :

and iteration yields

;  ;  '(p+1  h+1)  C p ;p 1 2'(p h) = C p ;p 1 2'(p 2): But lim sup '(p  h ) '(+1 1) and the proof of Proposition 7.1 is completed.  !1

To prove Proposition 7.2, we rst note that

'(;1 1) C '(;q 2) q > 0

follows if we apply the same iteration technique as above to p < 0 and especially to ;q < 0 close to 0. What remains to be shown is that

'(;q 2) C'(q 2)

and

'(q 2) C'(p 3)

where 0 < p < K is the parameter that appears in the proposition. The rst inequality follows from Inequality C if q  . Proof. It is enough to show that v = log u 2 BMO (Q(2)). Take any cube Q  Q(2) and choose  2 C0(Q(3)) such that  = 1 on Q. Since u is a supersolution

Z

D

hAru r'idx 0

for all 0  ' 2 C01(D):

Choose ' = 2 u1 . We get

Z

Z 1 h Aru rui u2 dx  C h Aru ri u1 dx Q(3) Q(3) 2

62

and with Schwarz' inequality and the uniform ellipticity Z Z 1 2 2 jruj  u2 dx  C jrjjruj u1 dx:

Hence

Q(3)

and Inequality A implies

Q(3)

Z Q

jrvj  C 2dx

Z Q(3)

jrjdx

Z

j v ; vQj2dx  C Q

where C independent of Q  Q(2). The inequality '(;q 2) '(q 2) follows from Inequality C .

Finally, for 0 < p < K choose a q > 0 such that qK = p for some  2 N and q   in Ineauality C . Finitely many applicaitons of Lemma 7.1 and Inequality B regarding u as a positive supersolution gives '(q 2) C'(p 3): This concludes the proof of Proposition 7.2.
The rest of the proof the Theorem 7.1 and Theorem 7.2 can be found in Dahlberg/Kenig 5]. See Appendix. There can also be found the corresponding results for bounded Lipschitz domains using a patching technique. We nally remark that the solution of the Dirichlet problem is unique and the solution of the Neumann problem is unique to an additive constant.


References 1] Dahlberg, B. E. J.: Estimates of harmonic measure. Arch. Rat. Mech. Anal. 65 (1977) pp. 278 - 288. 2] DeGiori, E.: Sulle dierenziabilata e analiticita della estremali degli integrali multipli regotari. Mem. Acad. Sci. Torino 3 (1957) pp. 25 - 43. 3] Nash, J.: Continuity of the solutions of parabolic and elliptic equations. Amer. J. Math. 80 (1957) pp. 931 - 954. 4] Moser, J.: On Harnack's theorem for elliptic dierential operator Comm. P.A.M. 14 (1961) pp. 577 - 591. 5] Dahlberg, B. E. J. /Kenig, C. E.: Hardy spaces and the Lp Neumann problem for Laplace's equation in a Lipschitz domain. Preprint. 63

Index page

BMO : : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : 26 Caldern-Zygmund decomposition : : : :: : : : : :: : : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : : 21 Caldern-Zygmund type (CZ) kernel : : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : 18 Carleson measure : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : 30 Carleson norm : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : 30 Cauchy integral : : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : 1 characteristic function : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : 30 commutators : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : 39 cone : : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : : 12 Cotlar inequality : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : 23 Hardy-Littlewood maximal function : : : :: : : : : :: : : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : 14 John-Nirenberg's inequality :: : : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : 27 Lipschitz domain : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : 12 Lipschitz function : : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : : 12 Marcinkiewicz' interpolation theorem : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : 15 Method of rotations : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : : :: : : : : 51 Nontangential cone : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: 12 Nontangential limit : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : : :: : 12 Nontangential maximal function :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : 12 Poisson kernel : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : : 29 Principal value operator (PVO) : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : 18 Rellich identity : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : 55 Schwartz class S : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : 17 Singular integral operators (SIO) : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : 17 64

Special Lipschitz domain : : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : : 12 Solution to the Dirichlet Problem : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : : :: : : : : : 12 subsolution : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : : 60 supersolution : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : :: : : : : : :: : : : : :: : : : : : :: : : : : : 60

65

66

Appendix 1

University of Minnesota Mathematics Report 84-133

Recent Progress on Boundary Value Problems on Lipschitz Domains by Carlos E. Kenig School of Mathematics University of Minnesota Minneapolis, MN 55455 U.S.A.

Supported in part by the NSF, and the Alfred P. Sloan Foundation

Introduction In this note we will describe, and sketch the proofs of some recent developments on boundary value problems on Lipschitz domains. In 1977, B. E. J. Dahlberg was able to show the solvability of the Dirichlet problem for Laplace's equation on a Lipschitz domain D, and with L2(@D d ) data and optimal estimates. In fact, he proved that given a Lipschitz domain D, there exists " = "(D) such that this can be done for data in Lp(@D d ) 2 ; "  p  1. (See 6], 7] and 8]). Also, simple examples show that given p < 2, there exists a Lipschitz domain D where this fails in Lp(@D d ). Dahlberg's method consisted of a careful analysis of the harmonic measure. His techniques relied on positivity, Harnack's inequality and the maximum principle, and thus, they were not applicable to the Neumann problem, to systems of equations, or to higher order equations. In 1978, E. Fabes, M. Jodeit, Jr. and N.Riviere (15]) were able to utilize A. P. Calderon's theorem (1]) on the boundedness of the Cauchy integral on C 1 curves, to extend the classical method of layer potentials to C 1 domains. They were thus able to resolve the Dirichlet and Neumann problem for Laplace's equation, with Lp(@D d ) data, and optimal estimates, for C 1 domains. They relied on the Fredholm theory, exploiting the compactness of the layer potentials in the C 1 case. In 1979, D. Jerison and C. Kenig 20], 21] were able to give a simplied proof of Dahlberg's results, using an integral identity that goes back to Rellich (33]). However, the method still relied on positivity. Shortly afterwards, D. Jerison and C. Kenig, (22]) were also able to treat the Neumann problem on Lipschitz domains, with L2(@D d ) data and optimal estimates. To do so, they combined the Rellich type formulas with Dahlberg's results on the Dirichlet problem. This still relied on positivity, and dealt only with the L2 case, leaving the corresponding Lp theory open. In 1981, R. Coifman, A. McIntosh and Y. Meyer 3] established the boundedness of the Cauchy integral on any Lipschitz curve, opening the door to the applicability of the method of layer potentials to Lipschitz domains. This method is very exible, does not rely on positivity, and does not in principle dierentiate between a single equation or a system of equations. The diculty then becomes the solvability of the integral equations, since unlike in the C 1 case, the Fredholm theory is not applicable, because on a Lipschitz domain operators like the double layer potential are not compact. For the case of the Laplace equation, with L2(@D @ ) data, this diculty was overcome by G. C. Verchota (36]) in 1982, in his doctoral dissertation. He made the key observation that the Rellich identities mentioned before are the appropriate substitutes to compactness, in the case of Lipschitz domains. Thus, Verchota was able to recover the L2 results of Dahlberg 7] and of Jerison and Kenig 22], for Laplace's equation on a Lipschitz domain, but using the method of layer potentials. This paper is divided into two sections. The rst section which consists of two parts, deals with Laplace's equation on Lipschitz domains. The rst part explains the L2 results of 70

Verchota mentioned above. The second part deals with a sketch of recent joint work of B. Dahlberg and C. Kenig (1984) (9]). We were able to show that given a Lipschitz domain D  Rn, there exists " = "(D) such that one can solve the Neumann problem for Laplace's equation with data in Lp(@D @ ) 1 < p  2 + ". Easy examples show that this range of p's is optimal. Moreover, we showed that the solution can be obtained by the method of layer potentials, and that Dahlbergs solution of the Lp Dirichlet problem can also be obtained by the method of layer potentials. We also obtained endpoint estimates for the Hardy space H 1(@D d ), which generalize the results for n = 2 in 25] and 26], and for C 1 domains in 16]. The key idea in this work is that one can estimate the regularity of the so-called Neumann function for D, by using the De Giorgi-Nash regularity theory for elliptic equations with bounded measurable coecients. This, combined with the use of the so-called 'atoms' yields the desired results. The second section, which consists of three parts, deals with higher order problems. In parts 1 and 2, we treat L2 boundary value problems for systems of equations. Part 1 deals with the systems of elastostatics, whicle part 2 deals with the Stokes system of hydrostatics. The results in part 1 are joint work of B. Dahlberg, C. Kenig and G. Verchota (see 12]), while the results in part 2 are joint work of E. Fabes, C. Kenig and G. Verchota (see 17]). The results obtained had not been previously available for general Lipschitz domains, although a lot of work has been devoted to the case of piecewise linear domains. (See 27], 28] and their bibliographies). For the case of C 1 domains, our results for the systems of elastostatics had been previously obtained by A. Gutierrez (19]), using compactness and the Fredholm theory. This is, of course, not available for the case of Lipschitz domains. We are able to use once more the method of layer potentials. Invertability is shown again by means of Rellich type formulas. This works very well in the Dirichlet problem for the Stokes system (see part 2), but serious diculties occur for the systems of elastostatics (see part 1). These diculties are overcome by proving a Korn type inequality at the boundary. The proof of this inequality proceeds in three steps. One rst establishes it for the case of small Lipschitz constant. One then proves an analogous inequality for non-tangential maximal functions on any Lipschitz domain, by using the ideas of G. David (13]), on icreasing the Lipschitz constant. Finally, one can remove the non-tangential maximal function, using the results on the Dirichlet problem for the Stokes system, which are established in part 2. See parts 1 and 2 for the details. Some partial results in this direction were previously announced in 26]. The third part of Section 2 deals with the Dirichlet problem for the biharmonic equation 2 ( a fourth order elliptic equation), on an arbitrary Lipschitz domain in Rn. This sketches joint work of B. Dahlberg, C. Kenig and G. Verchota (11]). The case of C 1 domains in the plane was previsouly treated by J. Cohen and J. Gosselin 2], using layer potentials and compactness. We are able to reduce the problem, for an arbitrary Lipschitz domain in Rn, to a bilinear estimate for harmonic functions. This is a Lipschitz domain version of the paraproduct of J. M. Bony. See part 3 of Section 2 for further details. Compete proofs of the results explained in Section 1, part 2, and Section 2, will appear in future publicaitons. 71

Acknowledgements: As was mentioned before, the results in Section 1, part 2 are joint

work with B. Dahlberg, the results in Section 2, part 1, joint work with B. Dahlberg and G. Verchota, the results in Section 2, part 2, joint work with E. Fabes and G. Verchota, and the results in Seciton 2, part 3, joint work with B. Dahlberg and G. Verchota. It is a great pleasure to express my gratitutde to B. Dahlberg, E. Fabes and G. Verchota, for their contributions to our joint work. I would also like to thank A. McIntosh for pointing out the applicability of the continuity method in Seciton 1, part 1, and for pointing out to us the work of Ne!cas (30]). I also would like to thank G. David for making his unpublished result (Lemma 2.1.10 in Section 2, part 1) available to us. Part of this research was carried out while I was visiting the Center for Mathematical Analysis at the Australian National University, Princeton University and the University of Paris, Orsay. I would like to thank these institutions for their kind hospitality.

72

Section 1: Laplace's equation Part 1: The theory on a Lipschitz domain, for Laplace's equation, by the method of layer potentials A bounded Lipschitz domain D  Rn is one which is locally given by the domain above the graph of a Lipschitz function. For such a domain D, the non-tangential region of opening  at a point Q 2 @D is ;(Q) = fX 2 D : jX ; Qj < (1 + )dist (X @D)g. All the results in this paer are valid, when suitably interpreted for all bounded Lipschitz domains in Rn n 2, with the non-tangential approach regions dened above. For simplicity, in this exposition we will restrict ourselves to the case n 3 (and sometimes even to the case n = 3), and to domains D  Rn D = f(x y) : y > '(x)g, where ' : Rn;1 ! R is a Lipschitz function, with Lipschitz constant M , i.e., j'(x) ; '(x0)j  M jx ; x0j. D; = f(x y) : y < '(x)g. For xed M 0 < M0 ;e (x) = f(z y) : (y ; '(x)) > ;M 0jz ; xjg  D;, and ;i(x) = f(z y) : (y ; '(x)) > M jz ; xjg  D. Points in D will usually be denoted by X , while points on @D by Q = (x '(x)) or simply by x. Nx or NQ will denote the unit normal to @D =  at Q = (x '(x)). If u is a function dened on Rn n , and Q 2 @D u (Q) will denote lim X !Q u(X ) or lim X !Q u(X ), respectively. If u is a X 2; (Q) X 2; (Q) function dened on D N (u)(Q) = supX 2; (Q) ju(X )j. e

i

i

We wish to solve the problems u = 0 in D u = 0 in D (D)  @u j = f 2 L2 (@D d ) uj@D= f 2 L2(@D d )  (N) @N @D The results here are

Theorem 1.1.1: There exists a unique u such that N (u) 2 L2(@D d ), solving (D), where the boundary values are taken non-tangentially a.e. Moreover, the solution u has the form Z hX ; Q N i 1 Q g(Q)d (Q) u(X ) = ! n n @D jQ ; X j for some g 2 L2(@D d ).

Theorem 1.1.2. There exists a unique u tending to 0 at 1, such that N (ru)

2

solving (N) in the sense that NQ ru(X ) ! f (Q) as X ! Q non-tantentially a.e. Moreover, the solution u has the form Z ; 1 u(X ) = ! (n ; 2) jX ;1Qjn;2 g(Q)d (Q) n @D

L2(@D d ),

73

for some g 2 L2(@D d ). In order to prove the above theorems, we introduce Z hX ; Q N i 1 Q Kg(x) = ! g(Q)d (Q) n j X ; Q j n @D and Z ; 1 Sg(X ) = ! (n ; 2) jX ;1Qjn;2 g(Q)d (Q): n @D If Q = (X '(x)) X = (z y), then Z y ; '(x) ; (z ; x)  r'(x) 1 Kg(z y) = ! g(x)dx n R ;1 jx ; z j2 + '(x) ; '(z )]2] 2 n

n

p Z 1 + jr'(x)j2 ; 1 Sg(z y) = ! (n ; 2) ;1 ;2 g (x)dx: n R jx ; z j2 + '(x) ; y ]2] 2 n

n

Theorem 1.1.3. a) If g 2 Lp(@D d ) 1 < p < 1, then N (rSg) N (Kg) also belong to Lp(@D d ) and their norms are bounded by C kgkL (@D d). Z 1 '(z) ; '(x) ; (z ; x)  r'(x) g(x)dx = Kg(z) lim (b) "!0 !n jx;zj>" jx ; z j2 + '(x) ; '(z )]2]n=2 exists a.e. and kKgkL (@D d)  C kgkL (@D d) 1 < p < 1 p Z ( z ; x ' ( z ) ; ' ( x )) 1 + jr'(x)j2 g(x)dx ; 1 lim "!0 !n jz;xj>" j2 ; xj2 + '(z) ; '(x)]2]n=2 exists a.e. and in Lp(@D d ), and its Lp norm is bounded by C kgkL (@D d) 1 < p < 1. (c) (Kg) (Q) =  21 g(Q) + Kg(Q) p Z (z ; x '(z) ; '(x)) (1 + jr'(x)j2) 1 1

(rSg) (z) =  2 g(z)Nz + ! "lim !0 jz ; xj2 + "(z) ; "(x)]2]n=2 g p

p

p

p

n

jz;xj>"

Corollary 1.1.4. (Nz rSg) (z) = 21 g(z) ; K g(z), where K  is the L2(@D d ) adjoint

of K . The proof of Theorem 1.1.3 is an easy consequence of the deep results of Coifman-McIntoshMeyer (3]). 74

It is easy to see that (at least the existence part) of Theorems 1.1. and 1.1.2 will follow immediately if we can show that ( 12 I + K ) and 12 I + K ) are invertible on L2(@D d ). This is the result of G. Verchota (36]).

Theorem 1.1.5. ( 21 I + K ) ( 21 I + K ) are invertible on L2(@D d ).

In order to do so, we show that if f 2 L2(@D d ) k( 21 I + K )f kL2(@D d)  k( 12 I ; K )f kL2(@D d), where the constants of equivalence depend only on the Lipschitz constant M . Let us take this for granted, and show, for example, that 12 I + K  is invertible. To do this, note rst that if T = 21 I + K  kTf kL2 C kf kL2 , where C depends only on the Lipschitz constant M . For 0  t  1, consider the operator Tt = 21 I + Kt, where Kt is the operator corresponding to the domain dened by t'. Then, T0 = 12 I T1 = T , and @ T : Lp(Rn;1) ! Lp(Rn;1) 1 < p < 1 with bound independent of t, by the theorem of @t t Coifman-McIntosh-Meyer. Moreover, for each t, kTtf kL2 C kf kL2 , C independent of t. The invertibility of T now follows from the continuity mehtod:

Lemma 1.1.6. Suppose that Tt : L2(Rn;1 ) ! L2(Rn;1) satisfy

(a) kTtf kL2 C1kf kL2 (b) kTtf ; Tsf kL2  C2jt ; sjkf kL2  0  t s  1 (c) T0 : L2(Rn;1) ! L2(Rn;1) is invertible. Then, T1 is invertible. The proof of 1.1.6 is very simple. We are thus reduced to proving (1.1.7) k( 21 I + K )f kL2 (@D d)  k( 21 I ; K )f kL2(@D d). In order to prove (1.1.7), we will use the following formula, which goes back to Rellich 33] (see also 31], 30], 22]).

Lemma 1.1.8. Assume that u 2 Lip (D ) u = 0 in D, and u and its derivatives are suitablly small at 1. Then if en is the unit vector in the direction of the y-axis,

Z

@D

hNQ enijr j

u 2d

Z @u @u = 2 @y  @N d : @D

75

@ ru  ru, while div ; @u ru = Proof. Observe that div (enjruj2) = @y@ jruj2 = 2 @y @y @ ru  ru + @u  div ru = @ ruru. Stokes' theorem now gives the lemma. @y @y @y We will now deduce a few consequences of the Rellich identity. Recall that p 1 Nx = (;r'(x) 1)= 1 + jr'(x)j2, so that (1 + M 2 )1=2  hNx  en i  1.

Corollary 1.1.9. Let u be as in 1.1.8, and let T1(x) T2(x) : : :  Tn;1(x) be an orthogonal basis for the tangent plane to @D at (x '(x)). Let jrt

j

u(x) 2

=

Z Z ; @u  2 2 d

 C jr tuj d : @D @D @N

n;1 X j =1

jhru(x) Tj (x)ij2. Then,

Proof. Let  = en;hNx eniNx , so that  is a linear combination of T1(x) T2(x) : : :  Tn;1(x). Then,

@u = hN  e i @u + h rui: x n @y @N Also,

@u 2 + jr uj2 jruj2 = ; @N t

and so,

Z

Z ; @u 2 h Nx eni @N d + hNx  enijrtuj2d = @D Z ; @u 2 @DZ ; @u  d : = 2 hNx eni @N + 2 h rui @N @D

@D

Hence,

So,

Z

Z Z ; @u 2 @u d : 2 h Nx  eni @N d = hNx enijrtuj d ; 2 h rui @N @D @D @D

Z Z Z ; @u   Z ; @u  2 2 1=2 2 2 1 = 2 d

 C jr jr d

 tuj d + C ( tuj d ) ( @D @D @D @N @D @N

and the corollary follows. 76

Corollary 1.1.10. Let u be as in 1.1.8. Then, Z

@D

Z

;Z

jr j  2 @D

Proof.

u 2d

jrt j

u 2d

Z ; @u   c @D @N 2d :

 ; u 2d 1=2

jr j @D

Z

@u j2d 1=2, and the corollary follows. j @D @N

Corollary 1.1.11. Let u be as in 1.1.8. Then Z

jrt j  @D u 2d

Z

@u j2d : j @D @N

In order to prove 1.1.7, let u = Sg. Because of 1.1.3c, rtu is continuous across the boundary, while by 1.1.4, ; @u  1 I ; K )g: = (  @N 2 We now apply 1.1.11 in D and D; , to obtain 1.1.7. This nishes the proof of 1.1.1 and 1.1.2. We now turn our attention to L2 regularity in the Dirichlet problem.

De nition 1.1.12. f 2 Lp1() 1 < p < 1, if f (x '(x)) has a distributional gradient in Lp(Rn;1). It is easy to check that if F is any extension to Rn of f , then rxF (x '(x)) p is well dened, and belongs to Lp(). We call this krtf kL (). p

rt f .

The norm in L1() will be

Theorem 1.1.13. The single layer potential S maps L2() into L21() boundedly, and

has a bounded inverse.

Proof. The boundedness follows from 1.1.3 a). Because of the L2-Neumann theory, and

@S (f )k 2 C kf k 2 . The argument used in the proof of 1.1.11, krtS (f )kL2 () C @N L () L () 1.1.5 now proves 1.1.13.

Theorem 1.1.14. Given f 2 L21(), there exists a harmonic function u, with

kN (ru)kL2()  C krtf kL2(), and such that rtu = rtf (a.e.) non-tangentially on . u 2 is unique (modulo constants), and we can choose u = S (g), where g 2 L (). The existence part of 1.1.14 follows directly from 1.1.13. 77

Part 2: The Lp theory for Laplace's equation on a Lipschitz domain The main results in this section are:

Theorem 1.2.1. There exists " = "(M ) > 0 such that, given f 2 Lp(@D d ), 2 ; " 

p < 1, there exists a unique u harmonic in D, with N (u) 2 Lp(@D d ), such that u converges non-tangentially almost everywhere to f . Moreover, the solution u has the form Z hX ; Q N i 1 Q u(x) = ! g(Q)d (Q) n j X ; Q j n @D for some g 2 Lp(@D d ).

Theorem 1.2.2. The exists " = "(M ) > 0, such that, given f 2 Lp(@D d ) 1 < p 

2+ ", there exists a unique u harmonic in D, tending to 0 at 1, with N (ru) 2 Lp(@D d ), such that NQ ru(X ) covnergens non-tangentially a.e. to f (Q). Moreover, u has the form

Z ; 1 u(X ) = ! (n ; 2) jX ;1Qjn;2 g(Q)d (Q) n @D

for some g 2 Lp(@D d ).

Theorem 1.2.3. There exists " = "(M ) > 0 such that given f 2 Lp1() 1 < p  2 + ", there exists a harmonic funciton u, with

kN (ru)kL ()  C krtf kL () p

p

and such that rtu = rtf a.e. non-tangentially on , u is unique (modulo constants). Moreover, u has the form

Z ; 1 u(x) = ! (n ; 2) jX ;1Qjn;2 g(Q)d (Q) n @D

for some g 2 Lp(@D d ). The case p = 2 of the above theorems was discussed in Part 1. The rst part of 1.2.1 (i.e., without the representation formula), is due to B. Dahlberg (1977) (7]). Theorem 1.2.3 was rst proved by G. Verchota (1982) (36]). The representation formula in 1.2.1, Theorem 1.2.2, and the proof that we are going to present of 1.2.3 are due to B. Dahlberg and C. Kenig (1984) (9]). Just like in Seciton 1, 1.2.1, 1.2.2, and 1.2.3 follow from: 78

Theorem 1.2.4. There exists " = "(M ) > 0 such taht ( 12 I ; K ) is invertible in Lp(@D d ) 1 < p  2 + " ( 12 I + K ) is invertible in Lp(@D d ) 2 ; "  p < 1, and S : Lp(@D @ ) ! Lp1(@D d ) is invertible, 1 < p  2 + ". In order to prove Theorem 1.2.4, just as in Part 1, it is enough to show that if u = Sf , f since, then, for 1 < p  2 + ", @u krtukL (@D d)  @N L (@D d): This will be done by proving the following two theorems: p

p



@u Theorem 1.2.5. Let u = 0 in D. Then kN (ru)kL (@D d)  C @N L (@D d) 1 < p  p

2 + ".

p

Theorem 1.2.6. Let u = 0 in D. Then

kN (ru)kL (@D d)  C krtukL (@D d) 1 < p  2 + ": p

p

We rst turn our attention to the case 1 < p < 2 of Theorem 1.2.5. In order to do so, we introduce some denitions. A surface ball B in  is a set of the form (x '(x)), where x belongs to a ball in Rn;1.

De nition 1.2.7. An atom R a on  is a function supported in a surface ball B , with

kakL1  1= (B), and with

 ad

= 0. Notice that atoms are in particular L2 functions. The following interpolation theorem will be of importance to us.

Theorem 1.2.8. Let T be a linear operator such that kTf kL2()  C kf kL2(), and such

that for all atoms a, kTakL1()  C . Then, for 1 < p < 2, kTf kL ()  C kf kL (). For a proof of this theorem, see 5]. Thus, in order to establish the case 1 < p < 1 of 1.2.5, @u is an atom, then kN (ru)k 1  C . By dilation and it suces to show that if a = @N L () translation invariance we can assume that '(0) = 0 supp a  B1 = f(x '(x)) : jxj < 1g. Let B  be a large ball centred at (0 0) in Rn, which contains (x '(x)) jxj < 2. The diameter of B  depends only on M . Since kakL2()  (B1 )1=2 = C , by the L2-Neuman 1 theory, p

Z

@D\B 

N (ru)  C

Z

@D\B 

79

N (ru)2d  c:

p

R

Thus, we only have to estimate CB\@D(ru)d . We will do so by appealing to the regularity theory for divergence form elliptic equations. Consider the bi-Lipschitzian mapping  : D ! D; given by (x y) = (x '(x) ; y ; '(x)]). Dene u on D; by the formula u = u  ;1 . A simple calculation shows that, in D;  u veries (in the weak sense) the equation div (A(x y)ru) = 0, where A(x y) = J1(X )  (0)t(X )  (0)(X ), where X = ;1(x y). It is easy to see that A 2 L1(D; ), and hA(x y) i C jj2. Notice also @u  B  B  \ @D. Dene now that supp @N 1 I for (x y) 2 D B (x y) = A (x y) for (x y) 2 D;  and y) for (x y) 2 D u~(x y) = uu(x (x y) for (x y) 2 D; : @u = 0 in @D n B , it is very easy to see that u~ is a (weak) solution in Rn n Because @N B  of the divergence form elliptic equation with bounded measurable coecients, Lu~ = div B (x y)ru~ = 0. In order to estimate u, (and hence ru) at 1, we use the following theorem of J. Serrin and H. Weinberger (34]).

Theorem 1.2.9. Let u~ solve Lu~ = 0 in Rn n B , and suppose that ku~kL1(R nB) < 1.

Let g(X ) solve Lg = 0 in jX j > 1, with g(X )  jX j Then, u~(X ) = u~1 + g(X )+ v(X ), where Lv = 0 in Rn n B , and jv(X )j  C ku~kL1(R nB)  jX j2;nR; , where  > 0 C > 0 depend only on the ellipticity constants of L., Moreover,  = c B (X )ru~(X )  r(X ), where  2 C 1(Rn)  = 0 for X in 2B , and  1 for large X . Let us assume for the time being that u is bounded, and let us show that if  is as in 1.2.9, then  = 0. Pick a  as in 1.2.9. In D B (X ) = I , and so n

2;n . n

Z

D

B rur =

Z

D

Z

ru  r = "lim ru  r !0 D "

where

D" = f(x y) : j(x y)j <  y > '(x) + "g and  is large. The right-hand side equals Z Z @u @u   ; 1] @N lim   @N = "lim !0 @D "!0 @D since, by the harmonicity of u, Z @u = 0: @D @N "


"


"


80

Let

@D " 1 = f(x y) 2 @D" : y = '(x) + "g

and @D " 2 = @D" n @D " 1. Then,

Z

Z Z Z @u @u @u lim  ; 1] @N = "lim  ; 1] @N + "lim  ; 1] @N =  ; 1]a = "!0 @D !0 @D 1 !0 @D 2 @D Z Z Z = a ; a = a = 0 @D @D @D R R since  0 on supp a. Moreover D; B ru~r = D ru  r, where  =   , by our construction of B . The last term is also 0 by the same argument, and so  = 0. We now show that u (and hence u~) is bounded. We that n 4 for simplicity. Z willfassume ) d (Q), with kf k 2  C . Since kakL2()  C , we know that u(X ) = Cn jX ;(Q L () Qjn;2 @D Now, for X 2 D1 = f(x y) : y > '(x) + 1g jX ;1Qjn;2  1 + jCQjn;2 2 L2() and so u 2 L1 (D1). Let now B be any ball in Rn so that 2B  Rn n B , B is of unit size, and such thatR a xed fraction of B is contained in D1 . Since N (ru) 2 L2(), with norm 2 less than C 2B\D jr by the ;R R uj 2 C , and moreover on B \ D1 ju(x)j  C . Therefore, 2 1 = Poincare inequality 2B u~  C . But, since u~ solves Lu~ = 0 maxB u~  C 2B ju~j ) 2  C , (29]). Therefore, u~ 2 L1 (Rn n B ) ku~kL1(R nB)  C . Hence, since  = 0R ru = rv, and jv(x y)j  C=(jxj + jyj)n;2+   > 0. For R R0 = diam B , set b(R) = A N (ru)2, where AR = f(x '(x)) : R < jxj < 2Rg. For each xed R, let "


"


"


n

R

N1(ru)(x) = supfjru(z y)j : (z y) 2 ;i (x) dis ((z y) @D)  Rg N2(ru)(x) = supfjru(z y)j : (z y) 2 ;i (x) dist ((z y) @D) Rg:

In the set where the sup in N2 is taken, u is harmonic, and the distance of any point X to the boundary is comparable to jX j. Thus, bound on v, we see that R using our n ; 1+  n ; 1+  2 N2(ru)(x)  C=jX j  C=R , and so A N2(ru)  CR1;n;2 . Let now R

;  = f(x y) : '(x) < y < '(x) + CR R < jX j < r;1Rg  2 41  12 : R R By the L2-Neumann theory in   A N1(ru)2d  C @ jruj2d . Integrating in  from 1=4 to 1=2 gives Z Z Z C C 2 2 N1(ru) d  R jr uj dX  R3 u2 A 1 4 n1 2 C1 R<jX j 0 such that

Z

m2+"dx

Z

 c jf j2+" dx

for all 0 < "  "0, where f = @u . Let h = M (f 2 )1=2, where M denotes the Hardy@N Littlewood maximal operator. Let E = fx 2 Rn;1 : m(x) > g: We claim that

Z

fm> h ]



C2

jE j + C 83

Z

fm > g

m2:

Let us assume the claim, and prove the desired estimate. First, note that

Z

E



m2

Z

fm > h g

m2 +

Z

fh> g

m2



j j + C

C2 E

Z

fm> g

m2 +

Z

by the claim. Choose now and x  so that C   < 1=2. Then,

Z

E

For " > 0,

Z

m2+"

Z1

m2



Z

C2

Z

jE j + C

Z1

m2:

Z

m2d  0Z fm> g 0 Z E Z 1 1 1+ "  " ;  C"  jfm > gjd + C"  1; m2d: ="

";1

m2d

"

fh g

fh> g

m2 

0

";1



h>

0

By a well-knwon inequality (see 18] for example), jE j  Cjfm > gj. Thus,

Z

m2+"

Z1

 C" 0 jfm > gjd + C" Z Z 2+ "  C" m + C m2n": 1+"

If we now choose "0 so that

C"0 < 1=2 for " < "0

Z

m2+"

If we now use Holder's inequality with exponents

Z

m2+"

C

;Z

Z1

2; m2+" 2+"

2+" 2

Z

0

; ";1

C

Z

Z

h>

 m2 d 

m2 n" :

and 2+" " , we see that

M (f 2) 2+2

"

 2+ "

"



and the desired inequality follows from the Hardy-Littlewood maximal theorem. It remains to establish the claim. Let fQk g be a Whitney decomposition of the set E = fm > g, such that 3Qk  E , and f3Qk g has boundedR overlap. Fix k, we can assume that there exists x 2 Qk such that h(x)  , and hence, 2Q f 2  C2jQk j. For 1    2, let Qk  = Qk , and Q~ k  = f(x y) : x 2 Qk  0 < y <  length (Qk )g: Q~ k  (and (Q~ k  )) is a Lipschitz domain, uniformly in k  . Also, by construction of Qk , there exists xk with dist (xk  Qk )  length (Qk ) and such that m(xk )  . Let Ak  = @Qk  \ xk +   Bk  = @Qk  \ Rn+ n Ak   k

84

so that

@Qk  = Qk   Ak   Bk  :

Note that the height of Bk  is dominated by C length (Qk ), and that jruj   on Ak  . Let m1 be the maximal function of ru, corresponding to the domain Q~ k  (i.e., where the cones are truncated at height  `(Qk )). Then, for x 2 Qk  m(x)  m1(x) + . Also,

Z

Qk

C

m21

Z

Bk



Z

m21  (using the L2-theory on Q~ k  ) 

@ Q~ k

jr j

u 2d + c

Z Ak

jr j

u 2d + c

Z 2Qk

f2

C

Z Bk

Integrating in  between 1 and 2, we see that

Z

Qk

m21

Z `(Q ) Z C  `(Qk) 0 2Q k

k

Thus,

Z Qk

m2

jr j

u 2 + C2

 C

Z 2Qk

jQkj  C

fm > h g

m2



C2

Z 2Qk

m2 + C2jQk j:

m2 + C2jQk j:

Adding in k, we see that

Z

jruj2d + C2jQk j:

jE j + C

Z fm > g

m2

which is the claim. Note also that the same argument gives the estimate kN (ru)kp C krtukp 2 < p < 2 + ", and the Lp theory is thus completed.



Section 2. Higher order boundary value problems Part 1: The systems of elastostatics In this part we will sketch the extension of the L2 results for the Laplace equation to the systems of linear elastostatics on Lipschitz domains. These results are joint work of B. Dahlberg, C. Kenig and G. Verchota, and will be discussed in detail in a forthcoming paper (12]). Here we will describe some of the main ideas in that work. For simplicity, here we restrict our attention to domains D above the graph of a Lipschitz function ' : R2 ! R. 85

Let  > 0  0 be constants (Lame moduli). We will seek to solve the following boundary value problems, where ~u = (u1 u2 u3) (2.1.1)

~u + ( + )rdiv ~u = 0 in D ~uj@D = f~ 2 L2(@D d )

(2.1.2)

~u + ( + )rdiv ~u = 0 in D (div ~u)N + fr~u + (r~u)tgN j@D = f~ 2 L2(@D d ):

(2.1.1) corresponds to knowing the displacement vector ~u on the boundary of D, while (2.1.2) corresponds to knowing the surface stresses on the boundary of D. We seek to solve (2.1.1) and (2.1.2) by the method of layer potentials. In order to do so, we introduce the Kelvin matrix of fundamental solutions (see 27] for example), ;(X ) = (;ij (X )) where ;ij (X ) = 4A j Xijj + 4C XjXi Xj3j  and

    A = 12 1 + 2 1+   C = 12 1 ; 2 1+  :

We will also introduce the stress operator T , where

T~u = (div ~u)N + fr~u + r~utgN:

The double layer potential of a density ~g(Q) is then given by

~u(X ) = K~g(X ) =

Z

f T (Q);(X ; Q)gt~g(Q)d (Q) @D

where the operator T is applied to each column of the matrix ;. The single layer potential of a denisty ~g(Q) is

~u(X ) = S~g(X ) =

Z

@D

;(X ; Q)  ~g(Q)d (Q):

Our main results here parallel those of Section 1, Part 1. They are 86

Theorem 2.1.3. (a) There exists a unique solution of problem 2.1.1 in D, with N (~u) 2

L2(@D d ). Moreover, the solution u has the form ~u(x) = K~g(x) ~g 2 L2(@D d ). (b) There exists a unique solution of (2.1.2) in D, which is 0 at innity, with N (r~u) 2 L2(@D d ). Moreover, the solution ~u has the form ~u(X ) = S~g(X ) ~g 2 L2(@D d ). (c) If the data f~ in 2.1.1 belongs to L21(@D d ), then we can solve (2.1.1), with N (r~u) 2 L2(@D d ). The proof of Theorem 2.1.3 starts out following the pattern we used to prove 1.1.1, 1.1.2 and 1.1.14. We rst show, as in Theorem 1.1.3, that the following lemma holds:

Lemma 2.1.4. Let K~g S~g be dened as above, so that they both solve ~u + ( + )r div ~u = 0 in R3 n @D. Then,

kN (K~g)kL (@D d)  C k~gkL (@D d) kN (rS~g)kL (@D d)  C k~gkL (@D d) for 1 < p < 1: (K~g) (P ) =  12 ~g(P ) + K~g(P ) (b) ; @

A + C 

( S~ g ) n j ) (P ) =  i (P )gj (P ) ; ni (P )  nj (P )hNp  g (P )i + @Xi 2 ; Z @ + p:v: @P ;(P ; Q)~g(Q)d (Q))j  @D i R where K~g(P ) = p:v: @DfT (Q);(P ; Q)gt~g(Q)d (Q), and A C are the constants in the

(a)

p

p

p

p

denition of the fundamental solution. Thus, just as in Section 1, part 1 reduces to proving the invertibility on L2(@D d ) of  21 I + K  12 I + K , and the invertibility from L2(@D d ) onto L21(@D d ) of S. Just as before, using the jump relations, it suces to show that if ~u(X ) = S~g(X ), then kT~ukL2(@D d)  krt~ukL2(@D d): Before explaining the diculties in doing so, it is very useful to explain the stress operator T (and thus the boundary value problem 2.1.2), from the point of view of the theory of constant coecient second order elliptic systems. We go back to working on Rn, and use the summation convention. Let arsij 1  r s  m 1  i j  n be constants satisfying the ellipticity condition arsijij r s C jj2jj2 and the symmetry condition arsij = asrji. Consider vector valued functions ~u = (u1 : : :  um) on Rn satisfying the divergence form system @ ars @ us = 0 in D: @Xi ij @Xj 87

From variational considerations, the most natural boundary conditions are Dirichlet cons @~ u @u ditions (~uj@D = f~) or Neumann type conditions, @ = ni arsij @X = fr . The interpretaj tion of problem 2.1.2 in this context is that we can nd constants arsij 1  i j  3 1  r s  3, which satisfy the ellipticity condition and the ssymmetry condition, and such that @ ars @u = 0 in D, and with T~u = @ ~u. In ~u +( + )rdiv ~u = 0 in D if and only if @X ij @ i @Xj order to obtain the equivalence between the tangential derivatives and the stress operator, we need an identity of the Rellich type. Such identities are available for general constant coecient systems (see 32], 30]).

Lemma 2.1.5 (The Rellich, Payne-Weinberger, Necas identities). Suppose that @ ars @ us = 0 in D, ars = asr  ~h is a constant vector in Rn, and ~u and its derivatives ij ji @X ij @X i

j

are suitably small at 1. Then,

Z

@ur @us d

h`n` arsij @X i @Xj @D

=2

Z @D

@ur n ars @us d : hi @X ` `j @X i

j

Proof. Apply the divergence theorem to the formula

@ ;h ars ; h ars ; h ars @ur  @us  = 0 @X` ` ij i `j j i` @Xi @Xj

Remark 1: Note that if we are dealing with the case m = 1 aij = I , and we choose ~h = en, we recover the identity we used before for Laplace's equation. Remark 2: Note that if we had the stronger ellipticity assumption that arsijir js C P` t j`tj2, we would have, if @D = f(x '(x)) : ' : Rn;1 ! R kr'k1  M g, that krtukL2(@D d)  k @u@ kL2(@D d). In fact, if we take ~h = en, then we would have Z XZ @ur @us d = r 2 jr u j d  c h` n`arsij @X i @Xj @D r @D Z r s 1=2; Z @u 2 1=2 ;XZ @u @u r 2 rs jru j d @D j @ j d : = 2C hi @X  n`a`j @X d  2C i j @D r @D R PR 2 Thus, r @D jrurj2d  C @D j @u @ j d . For the opposite inequality, observe that for each r s j xed, the vector hin` ars`j ; h` n` arsij is perpendicular to N . Because of Lemma 2.1.5 Z Z r @us @u`  @us d : @u rs h` n`aij @X @X d = 2 (h`n` arsij ; hin`ars`j) @X i j i @Xj @D @D 88

Hence, and so

Z

jr j  C @D u 2d

Z

;Z

 ; u 2d 1=2

jrt j @D

Z

 jr uj2d 1=2 @D

Z Z @u 2 2 j j d  c jruj d  c jrtuj2: @ @D @D @D Remark 3: In the case in which we are interested, i.e., the case of the systems of elastostatics, @us  @ur = (div ~u)2 +  X ; @uj + @ui 2 arsij @X 2 i j @Xi @Xj i @Xj which clearly does not satisfy X arsijir js C j`tj2 ` t

since the quadratic form involves only the symmetric part of the matrix (ir ). In this case, @~u = T~u = (div ~u)N + fr~u + r~utgN . of course @

Remark 4: The inequality

kr~ukL2(@D d)  C krt~ukL2(@D d)

holds in the general case, directly from Lemma 2.1.5, by a more complicated algebraic argument. In fact, as in Remark 2, Z Z r @us @u @u`  @us d  rs h` n`aij @X @X d = 2 (h`n` arsij ; hin`ars`j) @X i j i @Xj @D @D rs rs and for xed r s j (h`n` aij ; hin`a`j ) is a tangential vector. Thus, Z r @us ;Z 1=2; Z  @u rs 2 jr jr ~uj2d 1=2: h`n` aij @X @X d  C t~uj d

i j @D @D @D rs ; 1 Consider now the matrix drs = (aij ni nj ) . This is a strictly positive matrix, since arsijij r s C jj2jj2. Moreover, ; @u  ; @u  rs @ur @us drs @ r @ s ; aij @X @X = i j m r s t @u @u rs @u @u = ; a = drtniartij @X  n`asm `k @X ij @X @X j k i j t  t @u @u @u @u s t = drsnk artk`  n ; a m amv @X`t @Xv v` @Xtv @X` = @u  n as @u ; at @u @u = = drsnk artkv @X m m` @X` t v` @Xv @X` v @u @u : = fdrsnk artkv nm asm` ; atv`g @X v @X` 89

Now, note that for t  ` xed fdrs nk artkv nm asm`;atv` g is perpendicular to N , by our denition of drs, and the symmetry of arsij:

drs nk artkv nmastm`nv ; atv`nv = artkv nk nv drsastm` nm ; atm`nm = = atrvk nv nk drs astm`nm ; atm` nm = tsastm` nm ; atm`nm = atm` nm ; atm`nm = 0:

Therefore,

Z

; @~u  ; @~u  ;Z 1=2; Z  2 h` n`drs @ r @ sd  c jr jr ~uj2d 1=2 t~uj d

@D @D @D

Now, s s ; @~u  rs @us rs @u ; ars n n n @u = ; a n n = n a @ r kjs k j @N i ijs@Xj kj k j i @Xi @u ; arsn n n @u = fn ars ; arsn n n g @us = = niarsij @X i ij ki k j i @X ki k i j @X j j j s @u = fniarsij ; arsiknk ninj g @X : j But, for i r s xed, arsij ; arsiknk nj is perpendicular to N , and so Z @ut gfas n n @u gd  h` n` drsfartkj nk nj @N i` i ` @N @D Z Z

; 1=2; 1=2 Z 2 2 C jrt~uj @

jr~uj d + jrt~uj2d : @D

@D

@D

We now choose ~h = en , so that h`n` C , and recall that (drs ) and (artkj nk nj ) are strictly positive denite matrices. We then see that Z @~u

; Z 1=2; Z 1=2 Z 2 2 2 2  j d

 C jr jr ~ u j d

+ jr j t~uj d

t~uj d : @D @D @D @D @N @~u j2 , the remark follows. Now, as jr~uj2 = jrt~uj2 + j @N

Remark 5: In order to show that R@D jrt~uj2d  C R@D jT~uj2d , it suces to show that Z

Z

jr j  c j(div ~u)I + fr~u + r~utgj2d : @D @D ~u 2d

In fact, if this inequality holds, we would clearly have that

Z

Z

jr j  C jr~u + r~utj2d

@D @D ~u 2d

90

(Korn type inequality at the boundary). The Rellich-Payne/Weinberger-Ne!cas identity is, in this case (with ~h = en), Z nnf 2 jr~u + r~utj2 + (div ~u)2gd = @D Z u  f(div ~u)N + fr~u + r~utgN gd : = 2 @~ @y @D

But then,

Z

jr j  C @D ~u 2d

;Z

 ; ~u 2d 1=2

jr j @D

Z

 j (div ~u)N + fr~u + r~utgN j2d 1=2: @D

The rest of part 1 is devoted to sketching the proof of the above inequality.

Theorem 2.1.6. Let ~u solve ~u + ( + )r div ~u = 0 in D ~u = S (~g), where ~g is nice.

Then, there exists a constant C , which depends only on the Lipschitz constant of ' so that

Z

Z

jr j  C @D

j (div ~u)I + fr~u + r~utgj2d : @D

~u 2d

The proof of the above theorem proceeds in two steps. They are:

Lemma 2.1.7. Let ~u be as in Theorem 2.1.6. Then, Z

@D

N (r

~u)2d

c

Z

@D

N ((div ~u)I + fr~u + r~utg)2d :

Lemma 2.1.8. Let ~u be as in Theorem 2.1.6. Then, Z

@D

N ((div ~u)I + fr~u + r

g

~ut )2d

C

Z

@D

j(div ~u)I + fr~u + r~utgj2d :

Lemma 2.1.7 is proved by rst doing so in the case when the Lipschitz constant is small, and then passing to the general case by using the ideas of G. David (13]). Lemma 2.1.8 is proved by observing that if ~v is any row of the matrix (div ~u)I + fr~u + r~utg, then ~v is a solution of the Stokes system 8 < ~v = rp in D ~v = 0 in D (S) : ~div vj@D = f~ 2 L2(@D d ) This is checked directly by using the system of equations ~u + ( + )rdiv ~u = 0. One then invokes the following Theorem of E. Fabes, C. Kenig and G. Verchota, whose proof will be presented in the next section. 91

Theorem 2.1.9. Given f~ 2 L2(@D d ), there exists a unique solution (~v p) to system

(S) with p tending to 0 at 1, and N (~v) 2 L2(@D d ). Moreover, kN (~v)kL2(@D d)  C kf~kL2(@D d):

We now turn to a sketch of the proof of Lemma 2.1.7. We will need the following unpublished real variable lemma of G. David (14]).

Lemma 2.1.10. Let F : R  Rn ! R be a function of two variables t 2 R x = (x1 : : :  xn) 2 Rn. Assume that for each x, the function t ! F (t x) is Lipschitz, with

Lipschitz constant less than or equal to M , and for each i 1  i  n, the function xi ! F (t x) is Lipschitz, with Lipschitz constant less than or equal to Mi, for any choice of the other variables. Given an interval I  J = I  J1  : : :  Jn, where the Ji's and I are 1-dimensional compact intervals, there exists a function G(t x) : R  Rn ! R with the following properties: (a) G(t x) F (t x) on I  J (b) If E = f(t x) 2 I  J : F (t x) = G(t x)g, then jE j 38 jI jjJ j. (c) For each i, the function G(t x1 x2 : : :  xi;1 ; xi+1 : : :  xn) is Lipschitz, with Lipschitz constant less than or equal to Mi, and one of the following statements is true: 4M , or for each x, ;4M  @G (t x)  M . Either for each x, ;M  @G ( t x )  @t 5 5 @t The proof of this lemma is the same as in the 1-dimensional case, treating x as a parameter (see 13]). Before we procedd with the proof of Lemma 2.1.7, we would like to point out that in the analogue of Lemma 2.1.7 for bounded domains, a normalization is necessary since if ~u(x) solves the systems of elastostatics so does ~u(x) + ~a + BX , where ~a is a constant vector, while B is any antisymmetric 3  3 matrix. The right-hand side of the inequality in the Lemma of course remains unchanged, while the left-hand side increases if B 'increases'. The most convenient normalization is that for some xed point X  in the domain r~u(X  ); r~u(X )t = 0. This also gives uniqueness modulo constants to problem 2.1.2 in bounded domains. We now need to introduce some denitions. Let D0  Rn+ be a xed, C 1 domain with f(x 0) : jjxjj = max jxij  1g  @D0, f(x y) : 0 < y < 1 jjxjj  1g  D0  f(x y) : 0 < y < 2 jjxjj < 2g: If ' : Rn;1 ! R is Lipschitz, with jjr'jj  M , we construct theRmapping T' : Rn+ ! Rn by T'(x y) = (x cy + y  '(x)) where  2 C01(Rn;1) is radial,  = 1, and c = c(M ) is 92

chosen so that T'(Rn+)  f(x y) : y > '(x)g, and so that T' is a bi-Lipschitzian mapping. Also, it is clear that T' is smooth for (x y) with y > 0, and T'(x 0) = (x '(x)). We will denote by A' the point T'(0 1). Lemma 2.1.7 is an easy consequence of the following result.

Lemma 2.1.11. Given M > 0 and ' with jjr'jj  M , there exists a constant C = C (M )

such that for all functions ~u in D' , which are Lipschitz in D ', which satisfy ~u + ( + )rdiv ~u = 0 in D' and r~u(A') = r~u(A')t, we have kN'(r~u)kL2(@D d)  C kN'((div ~u)I + fr~u + r~utg)kL2(@D d): Here N' is the non-tangential maximal operator corresponding to the domain D'. This lemma will be proved by a series of propositions. Before we proceed, we need to introduce one more denition. We say that proposition (M ") holds if whenever ' is such that jjr'jj  M , and there exists a constant vector ~a with jj~ajj  M so that jjr' ;~ajj  ", then for all Lipschitz functions ~u on D ' , with ~u + ( + )rdiv ~u = 0 in D' , with r~u(A') = r~ut(A') we have kN'(r~u)kL2(@D d)  C kN'((div ~u)I + fr~u + r~utgkL2(@D d) where C = C (M "). Note that if proposition (M ") holds, then the corresponding estimates automatically hold for all translates, rotates or dilates of the domains D' when ' satises the conditions in proposition (M "). In the rest of this section, a coordinate chart will be a translate, rotate or dilate of a domain D'. The bottom B' of @D' will be T'(@D0 \ (x 0) : x 2 Rn;1).

Proposition 2.1.12. Given M > 0, there exists " = "(M ) so that propostion (M ")

holds. We will not give the proof of Proposition 2.1.12 here. We will just make a few remarks about its proof. First, in this case the stronger estimate kN'(r~u)kL2(@D d)  C k(div ~u)N + fr~u + r~utgN kL2 (@D d) holds. This is because in this case, the domain D' is a small perturbation of the smooth domain Dax . For the smooth domain Dax, we can solve problem 2.1.2 by the method of layer potentials (see 27], for example). If " is small, a perturbation analysis based on the theorem of Coifman-McIntosh-Meyer (3]) shows that this is still the case. This easily gives the estimate claimed above.

Proposition 2.1.13. For all M > 0 " > 0  2 (0 0:1), if proposition (M ") holds, then propostion (1 ; M 1:1") holds. We postpone the proof of Proposition 2.1.13, and show rst how Proposition 2.1.12 and Proposition 2.1.13 yield Lemma 2.1.11. 93

Proof of Lemma 2.1.11. We will show that proposition (M ") holds for any M ". Fix M ", and choose N so large that if "(10M ) is as in Proposition 2.1.12, then (1:1)N "(10M )

N Y ". Pick now j > 0 so that (1 ; j ) = 1=10. Then, since proposition (10M "(10M )) j =1 holds, by Proposition 2.1.12, applying Proposition 2.1.13 N times we see that proposition (M ") holds. We will now sketch the proof of Proposition 2.1.13. We rst note that it suces to show that kN'(r~u)kL2(@D d)  C kN~'((div ~u)I + fr~u + r~vtgkL2(@D d) where N~' is the non-tangential maximal operator with a wider opening of the non-tangential region. This follows because of classical arguments relating non-tangential maximal functions with dierent openings (see 18]) for example). Pick now ' with jjr'jj  (1 ; )M , and such that there exists ~a with jjr' ; ~ajj  1:1" jj~ajj  (1 ; )M . We will choose N~' as follows: Since @D' n B' is smooth, it is easy to see that we can nd a nite number of coordinate charts (i.e., rotates, translates and dilates of D ), which are entirely contained in D', such that their bottoms B are contained in @D', such that T ((x 0) : jjxjj < 1=2) cover @D', and such that the 's involved satisfy jjrjj  (1 ; 2 )M and there exist ~a such that jj~ajj  (1 ; 2 )M , and jjr ; ~a jj  1:11". The non-tantential region dening N~', on T ((x 0) : jjxjj < 21 ) is dened as follows: let F  f(x 0) : jjxjj < 1=2g be a closed set. Consider the cone on Rn+  = f(x y) 2 Rn+ : bjxj < yg, where b is a small constant. Consider now the domain DF on Rn+, given by DF = x2F ((x 0) +  ). Then DF is the domain above the graph of a Lipschitz function , for which jjrjj  cb, for some absolute constant c (independent of F ). It is also easy to see that we can take now b so small, depending only on M and " such that T (DF ) is the domain above the graph of a Lipschitz function ~, with ~ , and which statises jjr~jj  (1 ; 10 )M jjr~ ; ~a jj  1:111": The non-tangential region dening N~', for Q 2 T ((x 0) : jjxjj < 1=2) is then the image under T of (x 0)+  , with b chosen as above, suitably truncated, and where Q = T ((x 0). Let now, to lighten notation, m = N'(r~u) m = N~'((div ~u)I + fr~u + r~utg). For t > 0, consider the open-set Et = fm > tg. We now produce a Whitney type decomposition of Et into a family of disjoint sets fUj g with the property that each Uj is contained in T((x 0) : jjxjj < 1:2) for a coordinate chart D , each Uj containes T (Ij ), where Ij is a cube in jjxjj < 1=2, and is contained in T(I j ), where I j is a xed multiple of Ij . Finally, we can also assume that there exists a constant 0 such that if diam (Uj )  0, there exists a point Qj in @D', with dist (Qj  Uj )  diam Uj , such that m(Qj )  t. Let now 94

 > 1 be given. We claim that there exists > 0 so small that if Ej = Uj \fm > t m  tg then (Ej )  (1 ; M ) (Uj ), where M > 0. Assume the claim for the time being. Then, Z Z1 Z1 X 2Z 1 2 2 m d = 2 t (Et)dt = 2 t (Et)dt = 2 t (Qj \ Et)dt  @D 0 0 0 j Z1 X 2Z 1 2  2 0 t (Ej)dt + 2 0 t (m > t)dt  j Z1 2Z 1 X 2   2 (1 ; M ) 0 t (Qj)dt + 2 2 0 t fm > tgdt = j Z 2Z  2 2 =   (1 ; M ) m d + 2 m 2d : @D @D '

'

'

Thus, if we choose  > 1, but so that  2  (1 ; M ) < 1, the desired result follows. It remains to establish the claim. We argue by contradiction. Suppose not, then (Ej ) > (1 ; M ) (Uj ). Let E~j = T;1(Ej ). If M is chosen suciently small, we can gurarantee that jE~j \ Ij j :99jIj j. Let now Fj = E~j \ Ij , and construct now the Lipschitz function ~  )M jjr~ ; corresponding to it, as in the denition of N~'. Thus, ~  jjr~jj  (1 ; 10 ~a jj  1:111". We now apply Lemma 2.1.10 to ~, one variable at a time, to nd a Lipschitz function f , with f ~ on Ij , such that if F j = fx 2 Ij : f = ~g, then jF j \ Fj j c (Uj ),  )M , and such that there exists ~a , with jj~a jj  (1 ;  )M so that with jjrf jj  (1 ; 10 f f 10 4 jjrf ; ~af jj  5 1:111" < ". We can also arrange the truncation of our non-tangential regions in such a way that on the appropriate rotate, translate and dilate of Df (which of course is contained in the corresponding coordinate chart associated to D , which is contained in D'),

j(div ~u)I + fr~u + r~utgj  t: To lighten the exposition, we will still denote by Df the translate, rotate and dilate of Df . Note that proposition (M ") applies to it it. We divide the sets Uj into two types. Type I are those with diam Uj 0, and type II those for which diam Uj  0. We rst deal with the Uj of type I. In this case, Df has diameter of the order of 1. Because of the solvability of problem 2.1.2 for balls, and our normalization, we see that on a ball B  D'  diam B  1 A' 2 B , we have

Z

Z

jr j  c jdiv ~uI + fr~u + r~utgj2: B B ~u 2

Joining Af to A' by a nite number of balls, and using interior regularity results for the system ~u +( + )r div ~u = 0, we see that jr~u(Af )j  C t, for some absolute constant 95

C . Then C (Uj

) 2t2



Z T (F Zj \Fj )

 C (Uj) 2t2 + C @D Z 2 2  C (Uj) t + C @D

f

f

Z

 c @D Nf2(ru)d  t ;  Nf2 r~u ; r~u(Af ) ;2 r~u (Af )] d  Nf2((div ~u)I + fr~u + r~utg)2d  m2d

f

by (M "). The last quantity is also bounded by C (Uj ) 2t2, which is a contradiction for small . Now, assume that Uj is of type II. Note that in this case there exists Qj 2 @D', with dist (Qj  Uj )  diam Uj , and such that jr~u(x)j  t for all x in the non-tantential region associated to Qj . Because of this, it is easy to see, using the arguments we used to bound jr~u(Af )j in case I, that for all X in a neighborhood of Af and also on the top part of Df , we have that jr~u(X )j  t + C t. Since for Q 2 T (F j \ Fj ) m(Q) t, and  > 1, if is small enough, we see that we must have Nf (r~u)(Q) m(Q). Hence, t ;   Nf r~u ; r~u(Af ) ;2 r~u (Af ) (Q) ( ; 1 ; C )t ( ;2 1) t

if is small and Q 2 T (F j \ Fj ). Thus, applying (M ") to Df , we see that

C ( ; Z



@Df

Z

t  T (F \F ) Nf (r~u ;  r~u(Af ) ;2 r~u (Af ) 2d  ;  ~ut(Af ) 2d  C (U ) 2t2 Nf r~u ; r~u(Af ) ;r j 2

1)2t2 (Uj )



j

j

a contradiction if is small. This nishes the proof of Proposition 2.1.13, and hence of Lemma 2.1.11.

Part 2: The Stokes system of linear hydrostatics In this part I will sketch the proof of the L2 results for the Stokes system of hydrostatics. These results are joint work of E. Fabes, C. Kenig and G. Verchota (17]). We will keep using the notation introduced in Part 1. We seek a vector valued function ~u = (u1 u2 u3) and a scalar valued function p satisfying (2.2.1)

8 < ~u = rp in D ~u = 0 in D : ~div uj@D = f~ 2 L2(@D d ) in the non-tangential sense 96

Theorem 2.2.2 (Also Theorem 2.1.9). Given f~ 2 L2(@D d ), there exists a unique

solution (~u p) to (2.2.1), with p tending to 0 at 1, and N (~u) 2 L2(@D d ). Moreover, ~u(X ) = K~g(X ), with ~g 2 L2(@D d ). (K will be dened below). In order to sketch the proof of 2.2.2, we introduce the matrix ;(X ) of fundamental solutions (see the book of Ladyzhenskaya 28]), ;(X ) = (;ij (X )), where ;ij (X ) = 81 j Xijj + 81 XjXiXj3j , and its corresponding pressure vector q(X ) = (qi(X )) where qi(X ) = 4XjXi j3 : Our solution of (2.2.2) will be given in the form of a double layer potential,

~u(X ) = K~g(X ) = ; where

Z

f H 0(Q);(X ; Q)g~g(Q)d (Q) @D

;i` (X ; Q)n (Q): (H 0(Q);(X ; Q))i` = ij q`(X ; Q)nj (Q) + @@Q j j

We will also use the single layer potential

~u(X ) = S~g(X ) =

Z @D

;(X Q)~g(Q)d (Q):

In the same way as one establishes 2.1.4,

Lemma 2.2.3. Let K~g S~g be dened as above, with ~g 2 L2(@D d ). Then, they both solve ~u = rp in D, and D; div ~u = 0 in D and D; . Also

(a) kN (K~g)kL2(@D d)  C k~gkL2 (@D d), Z 1

(b) (K~g) (P ) =  2 ~g(P ) ; p:v: fH 0(Q);(P ; Q)g~g(Q)d (Q) @D (c) kN (rS~g)kL2(@D d)  C k~gkL2(@D d) 

; @ (S~g)j (P ) =  ni(P )2gj (P ) ; ni(P )2nj (P ) hNp~g(P )ig (d) @X i Z @ ;(P Q)~g(Q)d (Q) + p:v: @P @D

(e)

i

(HS~g) (P ) =

Z 1  2~g(P ) + p:v: @DfH (P );(P ; Q)g~g(Q)d (Q), 97

where

;i` (X ; Q) ; q`(X ; Q)n (X ): (H (X );(X ; Q))i` = nj (x) @@X ij j j

For the proof of this lemma in the case of smooth domains, see 28]. The proof of Theorem 2.2.2. (at least the existence part of it), reduces to the invertibility in R L2(@D d ) of the operator 12 I +K , where K~g(P ) = ;p:v: @D fH 0(Q);(P ;Q)g~g(Q)d (Q). As in previous cases, it is enough to show (2.2.4) k( 12 I ; K )~gkL2(@D d)  k( 12 I + K )~gkL2(@D d): This is shown by using the following two integral identities.

Lemma 2.2.5. Let ~h be a constant vector in Rn , and suppose that ~u = p div ~u = 0, in D, and that ~u p and their derivatives are suitablly small at 1. Then, Z Z @us @us Z s @us @u @us d : h`n` @X  @X d = 2 @N  h` @X d ; 2 pns h` @X j j ` ` @D @D @D

Lemma 2.2.6. Let ~h p and ~u be as in 2.2.5. Then, Z

@D

h` n`

p2d

Z Z r @ui r @u @us @ur d : @u = 2 hr @N pd ; 2 hr @X @N d + 2 hr ns @X i j @Xi @D @D @D Z

The proofs of 2.2.5 and 2.2.6 are simple applicaitons of the properties of ~u p, and the divergence theorem. Choosing ~h = e3, we see that, from 2.2.6 we obtain

Corollary 2.2.7. Let ~u p be as in 2.2.6. Then , R@D p2d

depends only on M .

 c R@D jr~uj2d , where C

@~u = @~u A consequence of Corollary 2.2.7 and Lemma 2.2.5, is that if @ @N have

Corollary 2.2.8. Let ~u p be as in 2.2.5. Then, Z

Z XZ @~ u @us j2d  2 2 j j d

 jr ~ u j d

+ j n t s @D @ @D j @D @Xj where the constants of equivalence depend only on M . 98

; p  N , then we

Proof. 2.2.5 clearly implies, by Schwartz's inequality, that Z

jr j  C @D ~u 2d

Z

@~u j2d : j @D @

Moreover, arguing as in the second part of the Remark 2 after 2.1.5, we see that 2.2.5 shows that Z Z Z s @u 2 2 jr~uj d  C @D jrt~uj d + j @D pns h` @X` d j: @D By Corollary 2.2.7, the right-hand side is bounded by

C

;Z

@D

 ; ~u 2d 1=2

jr j

s 1=2 Z @u 2 j ns @X j d + c jrtuj2d

j @D @D

XZ j

2.2.8 follows now, using 2.2.7 once more. @u are continuous across @D. To prove 2.2.4, let ~u = S (~g). By d) in 2.2.3, rt~u and ns @X Using this fact, 2.2.3 e) and Corollary 2.2.8, 2.2.4 follows. In closing this part, we would like to point out another boundary value problem for the Stokes system, which is of physical siginicance, the so-called slip boundary condition s

j

(2.2.9)

8 < ~u = rp in D ~u = 0 in D : div ((r~u + r~ut)N ; p  N )j@D = f~ 2 L2(@D d ):

This problem is very similar to (2.1.2). Using the techniques introduced in Part 1, together with the observation that if ~u = rp div ~u = 0 in D, the same is true for each row ~v of the matrix r~u + r~ut ; pI ], we have obtained

Theorem 2.2.10. Given f~ 2 L2(@D d ), there exists a unique solution (~u p) to (2.2.9),

which tends to 0 at 1, and with N (r~u) 2 L2(@D d ). Moreover, ~u(X ) = S (~g)(X ), with ~g 2 L2(@D d ).

Part 3: The Dirichlet problem for the biharmonic equation on Lipschitz domains This part deals with the Dirichlet problem for 2 on an arbitrary Lipschitz domain in Rn. The results are joint work of B. Dahlberg, C. Kenig and G. Verchota (11]). We continue using the notation introduced before. 99

We seek a function u dened in D, such that

8 2 <  u = 0 in D 2 d ) : u@uj@Dj@D==f g22LL1(2@D (@D d )

(2.3.1)

@N

where the boundary values are taken non-tangentially a.e.

Theorem 2.3.2. There exists a unique u solving (2.3.1), with N (ru) 2 L2(@D d )

kN (ru)kL2(@D d)  C fkgkL2(@D d) + kf kL21(@D d)g

where C depends only on M . We will only discuss existence. By 1.1.14, we may assume f = 0 on @DR. Let G(X Y ) be the Green function for  on D. Then, since uj@D= 0, we have u(X ) = D G(X Y )u(Y )dy. @ v(Y ), where Notice that w(y) = u(y) is harmonic in D. We claim that w(Y ) = @y v is a harmonic function in D, with L2(@D d ) Dirichlet data, and that the operator @u j is an invertible map from L2(@D d ) onto L2(@D d ). This would T : vj@D! @N @D establish 2.3.2. In fact, by using the Green's potential representation, Fubini's theorem, @ G(; Y ) is the density of harmonic measure at Y 2 D, and the fact that @N

Z

Z

Z Z @ 1 2 vTvd = v(Y ) @y v(Y )dY = 2 v(x '(x)) dx C v2d : @D D R ;1 @D n

This shows that if T : L2(@D d ) ! L2(@D d ) is bounded, it will have a bounded inverse. To establish the boundedness of T , note that if h is harmonic in D, then the argument given above shows that

Z

Z @v hTvd = @y (Y )h(Y )dY: @D D

All we need therefore, is the following bilinear estimate.

Theorem 2.3.3. If v h are harmonic in D, tend to 0 at 1, then

Z @v j D @y (Y )  h(Y )dY j  C kvkL2(@D d)  khkL2(@D d): 100

Proof. This theorem is a generalization to Lipschitz domains of the fact that the para-

product of two L2 functions is in L1 (see 4]). In order to establish the inequality, because of the invertibility of the double layer potential (the representation formula in 1.1.1), we can assume that Z hY ; Q N i 1 Q h(Y ) = ! g(Q)d (Q) n ; n @D jY ; Qj 2 with kgkL2(@D d)  C khkL2(@D d): Thus, since hY ; Q NQi = Cn @ ; 1  jY ; Qjn;2 @NQ jY ; Qjn;2 it suces to show thatZ k @N@ jY ;1Qjn;2 @v @y (Y )dY kL2(@D d)  C kvkL2(@D d): Q D

In order to do so, we will obtain a respresentation formula for 1 @v (Y )dY: @ Z n ; 2 @NQ D jY ; Qj @y Fix Q 2 @D, and let B satisfy Y B (Y ; Q) = jY ;1Qjn;2 , i.e., B is the fundamental solution for 2 (for example, if n 5 B (Y ) = Cn jY j4;n). We recall the denition of the Riesz transforms vj = Rj v j = 1 : : :  n ; 1. They are harmonic functions, which, together @v = @ R v, with v satisfy the generalized Cauchy-Riemann equations (see 35]), i.e., @X @y j j

n;1 X @v @ v(Y ) =  B . @v = (using the and @y = ; @x@ Rj v. If Y = (x y), then jY ; 1Qjn;2 @y y @y j =1 j

summation convention) ; @2 @ 2 B  @ v = @ 2 B @ v ; @ 2B @v + B + @x2j @y2 @y @x2j @y @xj @y @y 2 j v @ 2B @ + @x@ @y B @R @y ; @y2  @x Rj v: j e1 e2 : : :  en;1 en

j

Let now be the standard basis of Rn, with en pointing in the direction of the y axis. Then, we can rewrite the right-hand side as n;1 2  2 ;@ 2B  : : :  ;@ 2B  X @ B  rvi + h; ; @x@ 1B@y  @x @xn;1@y j=1 @x2j 2@y n;1 2 n;1 2 X X @ B + h @x @y en rRj vi ; h @@yB2 ej  rRj vi: j j =1 j =1 101

2 2 ; ;@ 2 B  : : :  ;@ 2 B  Pn;1 @ 2B2 , ~j = @ B en ; @ B ej . Note that Let ~ = ; @x@21B@y  @x j =1 @x @x ;1 @y 2 @y @xj @y @y2 div ~j = 0, and div ~ = 0, and that Z n;1 Z @ v(Y )dY = Z h~  rvi + X 1 h~j  rRj vi = D jY ; Qjn;2 @y D D j =1 n

=

Z

@D

j

v(P )  h~ (P ) Npid (P ) +

n;1 Z X

j =1 @D

Rj v(P )  h~j (P ) Npid (p)

by the divergence theorem. This can be rewritten as Z @ @ B (P ; Q)]v(P )d (P ) + (;nj (P ) @P@ @P@ B (P ; Q) + nn (P ) @P j n j @Pj @D n;1 Z X @ @ B (P ; Q) ; n (P ) @ 2 B (P ; Q)]R v(P )d (P ): + nn(P ) @P j j @Pn2 j @Pn j =1 @D HenceZ @ 1 @ v(Y )dY = n ; 2 @NZQ D jY ; Qj @y @ @ hrB (P ; Q) N i + n (P ) @ 2 hrB (P ; Q N i]v(P )d (P ) + = ;nj (P ) @P Q n Q @Pn2 j @Pn @D n;1 Z X @ @ hrB (P ; Q) N i ; n (P ) @ 2 hrB (P ; Q) N iR v(P )d (P ): + nn(P ) @P Q j Q j @Pn2 j @Pn j =1 @D @ @ @ B (P ; Q) is the kernel But, by the theorem of Coifman-McIntosh-Meyer, 3], @P j @Pi @Qk 2 of a bounded operator in L (@D d ). Thus, Z n;1 X @ 1 @v k @NQ D jY ; Qjn;2 @y (Y )dY j  C fkvkL2(@D d) + kRj vkL2(@D d)g: j =1 Finally, we invoke a result of Dahlberg (8]), who showed that kRj vkL2(@D d)  C kvkL2(@D d): This concludes the proof of 2.3.3. As a nal comment, we would like to point out that in this exposition we have emphasized non-tangential maximal function estimates, but that optimal Sobolev space estimates also hold. For example, the solution ~u of (2.1.1) is in the Sobolev space H 1=2(D), the one of (2.1.2) in the Sobolev space H 3=2(D), and the same is true for ~u in 2.1.3 c). The solution of (2.2.1) is in H 1=2(D), while the one of (2.2.9) is in H 3=2(D). Finally, the solution u of 2.3.1 is in H 3=2(D). All of these results can be proved in a unied fashion using a variant of the proof of Lemma 2.1.11. The details will appear in a forthcoming paper of B. Dahlberg and C. Kenig 10]. 102

References 1] A. P. Calderon, Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sc. U.S.A. 74 (1977), 1324-1327. 2] J. Cohen and J. Gosselin, The Dirichlet problem for the biharmonic equation in a bounded C 1 domain in the plane, Indiana Math. J. 32 (1983), 635-685. 3] R. R. Coifman, A. McIntosh and Y. Meyer, L'integrale de Cauchy denit un operateur borne sur L2 pour les courbes Lipschitziennes, Annals of Math. 116 (1982), 361-387. 4] R. R. Coifman and Y. Meyer, Au dela des operaterus pseudodierentials, Asterisque 57 (1978). 5] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. AMS 83 (1977), 569-645. 6] B. E. J. Aahlberg, On estimates of harmonic meaure, Arch. Rational Mech. and Anal. 65 (1977), 272-288. 7] B. E. J. Dahlberg, On the Poisson integral for Lipschitz and C 1 domains, Studia Math. 66 (1979), 13-24. 8] B. E. J. Dahlberg, Weighted norm inequalities for the Lusin area integral and the nontangential maximal functions for functions harmonic in a Lipschitz domain, Studia Math. 67 (1980), 297-314. 9] B. E. J. Dahlberg and C. E. Kenig, Hardy spaces and the Lp Neumann problem for Laplace's equation in Lipschitz domain, Annals of Math. 125 (1987), 437-466. 10] B. E. J. Dahlberg and C. E. Kenig, Area integral estimates for higher order boundary value problems on Lipschitz domains, to appear. 11] B. E. J. Dahlberg, C. E. Kenig and G. C. Verchota, The Dirichlet problem for the biharmonic equation in a Lipschitz domain, Annales de l'Institut Fourier 36 (1986), 109-135. 12] B. E. J. Dahlberg, C. E. Kenig and G. C. Verchota, Boundary value problems for the systems of elastostatics on a Lipschitz domain, Duke Math. J. 57 (1988), 795-818.. 13] G. David, Operateurs integraux singuliers sur certaines courbes du plan complex, Ann. Sci. de L'Ecole Norm. Sup. 17 (1984), 157-189. 14] G. David, Personal communication, 1983. 15] E. Fabes, M. Jodeit, Jr. and N. Riviere, Potential techniques for boundary value problems on C 1 domains, Acta Math. 141, (1978), 165-186. 103

16] E. Fabes and C. E. Kenig, On the Hardy space H 1 of a C 1 domain, Ark. Mat. 19 (1981), 1-22. 17] E. Fabes, C. E. Kenig and G. C. Verchota, Boundary value problems for the Stokes system on Lipschitz domains, Duke Math. J. 57 (1988), 769-793. 18] C. Feerman and E. Stein, H p spaces of several variables, Acta Math. 129 (1972), 137-193. 19] A. Gutierrez, Boundary value problems for linear elastostatics on C 1 domains, University of Minnesota preprint, 1980. 20] D. S. Jerison and C. E. Kenig, An identity with applications to harmonic measure, Bull. AMS Vol. 2 (1980), 447-451. 21] D. S. Jerison and C. E. Kenig, The Dirichlet problem in non-smooth domains, Annals of Math., 113 (1981), 367-382. 22] D. S. Jerison and C. E. Kenig, The Neumann problem on Lipschitz domains, Bull. AMS Vol. 4 (1981), 203-207. 23] D. S. Jerison and C. E. Kenig, Boundary value problems on Lipschitz domains, MAA Studies in Mathematics, vol. 23, Studies in Partial Dierential Equations, W. Littmann, editor (1982), 1-68. 24] C. E. Kenig, Weighted H p spaces on Lipschitz domains, Amer. J. of Math. 102 (1980), 129-163. 25] C. E. Kenig, Weighted Hardy spaces on Lipschitz domains, Proceedings of Symposia in Pure Mathematics, Vol. 35, Part 1, (1979), 263-274. 26] C. E. Kenig, Boundary value problems of linear elastostatics and hydrostatics on Lipschitz domains, Seminaire Goulaouic-Meyer-Schwartz, 1983-1984, Expose no XXI, Ecole Polytechnique, Palaiseau, France. 27] V. D. Kupradze, Three dimensional problems of the mathematical theory of elasticity and thermoelasticity, North Holland, New York, 1979. 28] C. A. Ladyzhenskaya, The mathematical theory of viscous incompressible ow, Gordon and Breach, New York, 1963. 29] J. Moser, On Harnack's theorem for elliptic dierential equations, Comm. Pure Appl. Math., Vol. XIV (1961), 577-591. 30] J. Ne!cas, Les methodes directes en theorie des equations elliptiques, Academia, Prague, 1967. 104

31] L. Payne and H. Weinberger, New bounds in harmonic and biharmonic problems, J. Math. Phys. 33 (1954), 291-307. 32] L. Payne and H. Weinberger, New bounds for solutions of second order elliptic partial dierential equations, Pacif J. of Math. 8 (1958), 551-573. 33] F. Rellich, Darstellung der eigenwerte von u + u durch ein randintegral, Math. Z. 46 (1940), 635-646. 34] J. Serrin and H. Weinbrger, Isolated singularities of solutions of linear elliptic equations, Amer. J. of Math., 88 (1966), 258-272. 35] E. Stein and G. Weiss, On the theory of harmonic functions of several variables, I, Acta Math. 103 (1960), 25-62. 36] G. C. Verchota, Layer potentials and boundary value problems for Laplace's equation in Lipschitz domains, Thesis, University of Minnesota (1982), also in J. of Funcitonal Analysis, 59 (1984), 572-611.

105

106

Appendix 2

107

108

Hardy spaces and the Neumann Problem in Lp for Laplace's equation in Lipschitz domains By Bjrn E. J. Dahlbergand Carlos E. Kenig

1. Introduction The purpose of this paper is to give optimal results for the solvability of the Neumann problem in Lipschitz domains with data in Lp. We also obtain corresponding end point results for Hardy spaces. Our main theorem asserts that if D  Rn n 3, is a bounded Lipschitz domain with connected boundary, thenR there exists " = "(D) > 0 such that, for all f 2 Lp(@D d ), with 1 < p < 2 + ", and @D fd = 0, there is a unique (modulo constants) harmonic function u in D with

kM (ru)kL (d)  Cp(D)kf kL (d)

(1.1)

p

p

and

@u = f on @D: @n Here is the surface measure on @D, and M (ru), the non-tangential maximal function of ru, the gradient of u, is dened, for Q 2 @D by (for example) (1.2)

M (F )(Q) = supfjF (X )j : X 2 D jX ; Qj < 2 dist (X @D)g:

(1.3)

It is known (see 4]) that if v is harmonic in a Lipschitz domain and M (v) < 1 a.e. on @D, then v has non-tangential limits a.e. on @D. Here almost everywhere is taken with respect to the surface measure on @D, and the existence of the non-tangential limit means that Xlim v(X ) exists and is nite for all  > 0, where !Q X 2;(Q)

;(Q) = fX 2 D : jX ; Qj < (1 + ) dist (X @D)g:

Consequently, if (1.1) holds, then ru has non-tangential limits a.e. on @D, and the meaning of the generlized normal derivative @u=@n in (1.2) is the limit of hru(X ) n(Q)i as X ! Q 2 @D non-tangentially. Here hA B i denotes the inner product in Rn, and n(Q) the unit normal to @D at Q. As is well known, n(Q) exists for a.e. Q, since D is a Lipschitz domain. Both authors were supported in part by the N.S.F.

109

The result was rst established for the case p = 2 by Jerison and Kenig (13]) and for the case of C 1 domains by Fabes, Jodeit and Riviere (7]). Our extension of the Lipschitz domain case to the range 1 < p < 2 + " is done by two dierent methods. The extension to 2 < p < 2+ " is done by a real variable argument, using the result for p = 2, and a variant of the 'good ' inequalities. The extension to 1 < p < 2 is accomplished by establishing that for f 2 Hat1 (@D), the atomic H 1 space on @D, there exists a unique (modulo constants) solution u of the Neumann problem with data f which satises

kM (ru)kL1(d)  C (D)kf kH 1 (@D): at

This is proved by estimating the non-tangential maximal functions of gradients of the L2-solutions with data atoms. We do this in turn by using the regularity theory for uniformly elliptic operators in selfadjoint form. The full result then follows by interpolation. Combining our estimates for atoms with the techniques in 8], we are able to obtain a generalization of the Stein-Weiss theory of Hardy spaces, valid for Lipschitz domains in Rn. This generalizes the results for C 1 domains in 8], and some of the two dimensional results in 14]. The range 1 < p < 2 + " is optimal for the Neumann problem. The estimate (1.1) fails for p = 1 even for smooth regions (Hardy space results are the appropriate analogue). Moreover, for each p0 > 2 it is possible to construct a Lipschitz domain D, for which (1.1) fails for p = p0 . (See for example 16] for the relevant examples.) The situation is similar to the case of the Dirichlet problem (see Dahlberg, 5]), where one has for the solution u of the problem u = 0 in D u = f on @D, (1.4)

kM (u)kL (d)  cp(D)kf kL (d) p

p

whenever 2 ; " < p  1 for an " = "(D) > 0. The relationship between the results (1.1) and (1.3) can be best understood by the use of the method of integral equations. For f on @D, let Z 1 D(f )(X ) = !n(2 ; n) @D f (Q) @n@ Q (jX ; Qj2;n)d (Q) be the double layer potential of f , and let Z 1 S (f )(X ) = ! (n ; 2) f (Q)jX ; Qj2;nd (Q) n @D be the single layer potential of f . A consequence of the boundedness of the Cauchy integral on Lipschitz curves (see Coifman, McIntosh and Meyer 2]), is that M (D(f )) and M (rS (f )) take Lp(d ) into Lp(d ) 1 < p < 1. In the classical case when the domain D is smooth, one can use the Fredholm theory to see that the operators D(f )j@D and (@=@n)S (f )j@D are invertible on Lp(d ) 1 < p < 1. 110

This is the result that was extended to C 1 domains by Fabes, Jodeit and Riviere 7], using the work of A. P. Calden 1.A]. However, this compactness argument does not extend to the case of Lipschitz domains (see for example 16] for simple couterexamples). The invertibility of the layer potentials for general Lipschitz domains was established in L2(d ) by Verchota. As a consequence of (1.1) and the analogous estimate (1.5)

kM (ru)kL (d)  Cp(D)krT f kL (d) p

p

1 < p < 2 + "

" = "(D), where u is the solution of the Dirichlet problem, and rT denotes the tangential gradient, we are able to establish that the operator f ! D(f )j@D is one-to-one and onto on Lp(d ) 2 ; " < p < 1, and that the operator f ! (@=@n)S (f )j@D is one-to-one and onto Lp0(d ) 1 < p < 2 + ", where Z

 p p L0(d ) = f 2 L (d ) : fd = 0 : @D

This is again the optimal range of p's in both cases. At this point we would like to point out that the case p = 2 of (1.5) is due to Jerison and Kenig 13], while the general case is due to Verchota 25]. Here we will also give a new proof of (1.5), analogous to our proof of (1.1). We will also present endpoint results on the invertibility of D and (@=@n)S on BMO and Hat1 respectively. For a more complete description of these results, we refer to the body of the paper. Capitial letters X Y Z will denote points of a xed domain D  Rn , while P Q will be reserved for points in @D. Lower case letters x y z are reserved for points in Rn;1, while the letters s t will be reserved for real numbers. As was mentioned before, in the sequel we assume that n 3. The results remain valid when n = 2 with the obvious modications.

2. The Neumann problem on graphs We begin by treating the case when

D = f(x y) 2 Rn : y > '(x) x 2 Rn;1g

where ' : Rn;1 ! R is Lipschitz continous i.e., j'(x) ; '(x0)j  mjx ; x0j. We start out by reviewing the Neumann problem with data f 2 L2(),  = @D. Let us a priori assume that f is bounded and has compact support. Let Z 1 u(X ) = S (f )(X ) = ! (n ; 2) f (Q)jX ; Qj2;nd (Q) n  111

be the single layer potential of f . Since u is harmonic in D, we have the identity ;  div jruj2e ; 2 @u (2.1) r u = 0 @y where e = (0 1). The results in 2] show that kM (ru)kL2()  C kf kL2(): Since ru(X ) = O(jX j1;n ) as X ! 1, it follows from (2.1), the estimate above, and the divergence theorem applied in the domains D " = f(x y) : y > '(x) + " jxj2 + y2 < 2g, that Z ; @u  @u d = 0 (2.2) jr uj2he ni ; 2 @n @y  where the derivatives on  are taken as the non-tangential limits from D, of the corresponding expressions in D. This is the Rellich (21]) identity on . To exploit (2.2), let T1(x) T2(x) : : :  Tn;1(x) be an orthonormal basis for the tangent plane to  at (x '(x)). The Ti(x) exist for a.e. x. Let

jrT

j

u(x '(x)) 2 =

n;1 X j =1

jhru(x '(x)) Ti(x)ij2:

A well-known argument (see 15], Corollary 2.1.11 for example) shows that there are constants c1 c2, that depend only on the Lipschitz constant m of ' such that Z  @u  Z Z  @u  2 2 C1  @n  d  jrT uj d  C2  @n 2d : (2.3)    Note that in the above estimate the values of the derivatives of u are taken as the limits from above the graph, but the same estimate can be obtained by taking limits from below the graph. Let now Tf and T;f denote the normal derivatives of Sf as a function in D and Rn n D respectivly. We then have (see 15]) (2.4) kTf kL2() ckf kL2() where c = c(m). To establish (2.4), we recall the classical jump relation T + T; = I , where I is the identity operator (again, see 15] for a proof of this in our case). To prove (2.4) we only need to remark that jrT uj2 is continuous a.e. across  (see 15]). Thus, (2.4) follows by application of (2.3) in D and Rn n D , together with the jump relation. The boundedness of T (2]) together with (2.4) immediately show that T is one-to-one, with closed range. 112

To see that the range of T is all of L2(), we let U : L2(Rn;1) ! L2(Rn;1) be the pullback of T , i.e., Ug = T (g  F ;1)  F , where F : Rn;1 !  is given by F (x) = (x '(x)). Letting Us denote the operator corresponding to the graph of x ! s'(x) 0  s  1, we see easily, using the results in 2], that kUs ; Us1 k  C js ; s1j 0  s s1  1. (See 15] for the details.) Since U0 = 21 I , the continuity method shows that U1, and hence T , is onto (see 15], Lemma 2.1.6).

Theorem 2.5. Given any f 2 L2() there is a harmnonic function u in D such that

kM (ru)kL2()  C (m)kf kL2(), and @u=@n = f a.e. on , in the sense of non-tangential convergence. Any two such harmonic functions di er by a constant. Let g = T ;1f . If n 4 one such function is given by S (g). If n = 3 one such solution is given by Z u(X ) = g(Q)fjX ; Qj2;n ; jX0 ; Qj2;ngd (Q) @D where X0 is any xed point in Rn n D . Proof. The existence part and the representation formulas follow from the invertibility of T , and the results of Coifman-McIntosh-Meyer 2] mentioned before. It remains to establish uniqueness. We present here an argument which will be very useful for us later on in treating the Lp case 1 < p < 2. LetR ! be harmonic in D, with M (r!) 2 L2(), and @!=@n = 0 a.e. on . Note rst that DR jr!j2dX  CR, where DR = D \ f(x y) : jxj2 + y2 < R2g. Consider now the bi-Lipschitzian mapping  : D ! Rn n D given by

(x y) = (x '(x) ; y ; '(x)]) = (x 2'(x) ; y): Dene ! on Rn n D by the formula ! = !  ;1. A simple calculation shows that in Rn n D , ! veries (in the weak sense) the equation div (A(x y)r!) = 0, where 0t 0 A(x y) = 1=J(X )] (X ) (X ), where X = ;1(x y), 0 is the Jacobian matrix of , and J the Jacobian determinant of . It is easy to see that A 2 L1 (Rn n D ), and hA(x y) i C jj2, where C = C (m). Let now B (x y) =

I for (x y) 2 D A(x y) for (x y) 2 Rn n D

and extend !(x y) to all of Rn by setting it equal to !(x y) in Rn n D . Since M (r!) 2 L2(), and @!=@n = 0 a.e. on , it is easy to see that the extended ! is a weak solution in all of Rn of the divergence form elliptic equation with bounded R measurable coecients div B (x y)r! = 0. The extended ! also satises the estimate jxj 0 such that for all n-dimensional balls B centered in Rn n f(x y) : jxj < 2 jy ; '(x)j < 2g, of radius rR0 = r0(m), we have that jfX 2 B : !(X ) = 0gj d. Furthermore, on such balls B , B jr!j2  C , by the L2 estiamte for the non-tangential maximal of ru. Therefore, by a standard variant of the Poincar inequality, we R function 2 have B !  C . The sub-mean value inequality for L-subsolutions (19]) now establishes (2.8) in the caseR n 4. It remains to show (2.8) when n = 3. In this case, we let, for X 2 D u(X ) =  g(Q)fjX ; Qj2;n ;jX0 ; Qj2;ngd (Q). Schwarz's inequality now shows that ju(X )j  C log(2+dist (X )) whenever dist (X ) 1. Let now X = (x y) and set !(X ) = max(ju(X )j; c 0) for y > '(x) + 1, and 0 otherwise. If c is chosen large enough, then ! is zero in a neighborhood of f(x y) : y = '(x) + 1g, and so ! is a subharmonic function in all of Rn. If ;0 = f(x y) : y + y0 > ;2mjxjg, then ! is 0 on @ ;0 if y0 is large 115

enough, and ! has logarithmic growth at 1. By the Phragmen-Lindelf Theorem (see 17]), ! is identically 0, which yields (2.8) as in the case n 4. Let g be the fundamental solution of L in Rn, with pole at 0. As is well-known, there are constants C1, and C2 which depend only on the ellipticity constants of L (and hence only on m), such that C1jX j2;n  g(X )  C2jX j2;n (see 18]). By (2.8) and the asymptotic expansion of Serrin and Weinberger (Theorem 7 of 22]), there are constants     and R0 0 <  R0, such that u(X ) = g(X ) +  + v(X ) for X 2 Rn n f(x '(x)) : jxj  1g where jv(X )j   jX j2;n; for jX j R0, where  ;1 can be bounded in terms of m, while    and R0 can be bounded in terms of  and kukL1(DnD). We next claim that R = 0. To show this, we recall that if L = div B r, Theorem 7 of 22] shows that  = b hB ru ri, where b is a constant that depends on g, and  is any C 1 function on Rn, which is 0 for jxj  R0 and equals 1 for jxj R1 > R0. Let R be large, and set, for 0 <  , D( ) = f(x y) : '(x) +  < y < '(x) + R jxj  Rg: Clearly, for R large enough,  1 on A( ) = @D( ) ; B ( ) where B ( ) = f(x '(x) +  ) : jxj  Rg. Hence,

Z

Z

hBru ri = Dhru ri = lim !0

Z

hru ri @u = lim @u = lim  (  ; 1) !0 @D( ) @n  !0 @D( ) @n Z Z @u = ( ; 1)ad = 0: = lim (  ; 1) !0 B ( ) @n  R R By our construction of B (see the proof of Theorem 2.5), R nD hB ru ri = D hru r~i, where ~ =   . This quanity is also zero. Let Z a(R) = M (ru)2d  (R) D

Z

DZ( )

n

where

(R) = f(x '(x)) : R  jxj  2Rg

and let

a(1) =

Z f(x '(x)):jxj2g

116

M (ru)2d :

By Theorem 2.5, a(1)  C kak2L2()  C . For Q 2 , let  (Q) = fX 2 D : jX ; Qj < 2 dist (X )g and for Q 2 (R) set 1(Q) = fX 2  (Q) : jX ; Qj < Rg 2(Q) =  (Q) n 1(Q), and Mi(Q) = supfjru(X )j : X 2 i(Q)g i = 1 2. Observe that if X 2 2(Q), then u is harmonic in B = fY : jY ; X j  Rg,;where and supY 2B ju(Y ) ;  j  CR2;n; . Since R = (m) is small enough, 1 ; n 1 ; n ;  jru(X )j  CR B ju(y) ; jdY  CR , it follows that

Z

(2.9)

(R)

(M2(ru))2d  CR1;n;2 :

For  2 I = 1=4 1=2], set ;1  = f(x y ) : '(x) < y < '(x) + R R < jxj <  Rg where  is chosen so that for Q 2 (R) 1(Q)   . From the L2 Neumann theory for bounded Lipschitz domains (13]), it follows that Z Z ; @u  Z 2 2 M1(jruj) d  C d  C jr uj2d  @n (R) @ @  \D with C depending only on m, since @u=@n on @  \ @D. Integrating in  on I yields 

Z

(R)

M1(jr

j

u 2)d

r

Z

 jr uj2dX Z1 4 n1 2  CR;3 C1RjxjC2 R u2dX  CR1;n;2  CR;1

=

=

where the next to the last inequality follows from the inequality of Caciopolli for solutions of Lu = 0 (see 12], for example). Putting this together with (2.9), we see that

Z

(R)

M (ru)2d  CR1;n;2 

which easily yields the lemma. We shall next study the boundary value properties of harmonic functions in D, with M (ru) 2 L1().

Lemma 2.10. Suppose that u is harmonic in D, and M (ru) 2 L1(). Then, ru has

non-tangential limits a.e. on , and if @u=@n = hru(Q) n(Q)i, then @u=@n 2 Hat1 (), with

(2.11)

@u @n H 1 ()  C kM (ru)kL1(): at

117

Proof. We rst remark that by the extension theorem of Varoupoulos (24]), given g continuous and with compact support in , there is a continuous function G in D , which agrees with g on , which has bounded support, and such that jrGjdX is a Carleson measure, with Carleson norm bounded by C (m)kgkBMO(), i.e., for all Q 2 , and all r 0, we have

Z

jr GjdX  C (m)rn;1kgkBMO() fX 2D:jX ;Qj 0 let B

u (X ) = u(X + (0  )). Then,  Z  Z @u @u   g d  = lim g d   !0  @n  @n  Z Z  hrG ru idX   C kgkBMO()  M (ru)d  = lim !0 D by the basic property of Carleson measures (see 11]). Recall now that VMO() is the closure in BMO () of the space of continuous functions with compact support, and that the dual space of VMO () is Hat1 () (see 3]). This concludes the proof of the lemma. We are now in position to solve the Neumann problem on a Lipschitz graph, with data in Hat1 ().

Theorem 2.12. Let f

2 Hat1 ().

Then there exists a harmonic function u in D with M (ru) 2 and @u=@n = f non-tangentially a.e. on . The function u is unique modulo constants. Furthermore, there are constants C1 = C1(m) C2 = C2(m) C3 = C3(m) such that C1kf kHat1 ()  kM (ru)kL1()  C2kf kHat1 () and

L1(),

@u @ T~j H 1 ()  C3kf kH 1 (): at

at

Proof. The exisatence of u follows directly from Lemma 2.7 (a). In order to show uniqueness, let us assume that ! is harmonic in D M (r!) 2 L1(), and @!=@n = 0 nontangentialy a.e. on . We want to conclude that ! is a constant. From the sub-mean value property of jr!j it follows that jr!(X )j  C fdist (X )g1;n :

118

Therefore, adding a suitable constant to !, we have j!(X )j  C fdist (X )g2;n: For  > 0, let ! (X ) = !(X + (0  )). By the estimates above, Sobolev's inequality and the assumption on M (r!), we have that

Z

j ! j(n;1)=(n;2)d  C

with C independent of  . Let now L be the divergence form operator used in the proof of Lemma 2.7, and let G(X Y ) be its fundamental solution in Rn . Fix X 2 D jX j < R, and let  2 C01(Rn), be identically 1 for jX j < R, and 0 for jX j > 2R. We can also assume that Rjrj + R2jj  C , where C is independent of R. Let G(Y ) = G(X Y )+ G(X  Y ), where X  is the re ection of X . Then, Z Z @!  ! (X ) = G   @n d + G! @ @n d

 Z + G  f2hr r! i + ! gdy D = I + II + III: Set F (R) = fX : R < jX j < 2Rg. Then

Z

; 1;n Z j! jd  C R

 jII j  j ! j(n;1)=(n;2)d (n;2)=(n;1) ! 0 \F (R) \F (R) as R ! 1, and Z Z 1 ; n jIII j  C D\F (R) R jr! jdY + C D\F (R) R;n j! j Z ; ;n Z 2 ; n  CR  M (r!)d + C R D\F (R) j! j(n;1)=(n;2)dY (n;2)=(n;1) ! 0 as R ! 1: CR1;n

Hence

! (X ) =

Z 

 G  @! @n d :

Since G 2 L1 (), the dominated convergence theorem shows that !(X ) = lim !0 ! (X ) = 0. The estimate kM (ru)kL1()  C2kf kH 1 () follows by construction, while the estimate C1kf kH 1 ()  kM (ru)kL1() follows by Lemma 2.10. Finally, it is enough to establish the k@u=@T~jkH 1 ()  C3kf kH 1 () when f is an atom. One rst shows that R estimate (@u=@ T~j )d = 0. We see that this follows by considering the functions u   > 0, and then passing to the limit. This fact, togther with (b) of Lemma 2.7 shows that @u=@ T~j is a molecule and the estimate follows by the general theory of 3]. at

at

at

at

We shall now treat the Neumann problem with Lp data. 119

Theorem 2.13. There exists a positive number " = "(n m) such that for all f

2

there is a harmonic function u in D with kM (ru)kL ()  C kf kL () and @u=@n = f non-tangentially a.e. on . Furthermore, u is unique modulo constants.

Lp() 1 < p < 2 + "

p

p

Proof. We rst remark that uniqueness for 1 < p < (n ; 1) follows by the same argument as in the uniqueness part of Theorem 2.11. The case n = 3 p = 2 of uniqueness is proved in Theorem 2.5. The case n = 3 2 < p < 2 + " of uniqueness will be treated later on. Next we note that existence, in the range 1 < p < 2 follows by interpolation. We shall now treat existence in the case p > 2. We remark that this in fact follows from an abstract argument of A. P. Caldern (1.B]). We present here an alternative proof, which also yields uniqueness. Let f 2 L1 () have compact support and let u be the L2 Neumann solution with data f , given by Theorem 2.5. Let H = f(x y) 2 Rn y > 0g and  : H ! D be given by (x y) = (x y + '(x)) where  is a bi-Lipschitzian mapping between H and D. Put v = ru  , and for x 2 Rn;1, set m(x) = sup(x) jvj, where  (x) = f(x0 y) 2 H : y > jx0 ; xjg. Similarly, set  (x) = f(x0 y) 2 H : y > jx0 ; xjg, and m(x) = sup (x) jvj, where  2 (0 1) is a number to be chosen later. Our aim is to show that there exists " = "(n m) > 0 such that, if 0 <  ", then

Z

C

Z

f 2+ d :  R This clearly yields the desired existence results, in the range 2 < p  2 + ". In order to establish (2.14), we needR to introduce a bit more notation. Let g(x) = f (x 0) x 2 Rn;1, and h(x) = supI (1=jI j I g2dx)1=2, where the sup is taken over all cubes I in Rn;1, that contain x. Finally, for  > 0 let E = fx 2 Rn;1 : m(x) > g. We will show that, for  suciently small, Z Z 2 (2.15)  C jE j + C m2dx: (2.14)

;1

n

m2+ dx

fm >  h g

fm > g

Let us assume (2.15) for the time being, and use it to establish (2.14). From (2.15) it immediately follows that, if  is chosen suceintly small,

Z

E

m2dx

 jE j + C C2

Let now and N be positive numbers. Then,

Z

minfm N g ;1

; ;1

fh> g\E

m2dx:

Z

 m2dx d fm> g 0Z Z N ;Z N  +1  C  jE jd + C ;1

]2+dx =

Rn

ZN

Z

0

0

By a classical argument (see 11]), jE j  Cjfm > gj, and so

Z

Rn;1

minfm N g

]2+dx

 C =Z (2 + )

+C

Rn;1

120

Z

Rn;1

fh> g

 m2dx d:

min(m N )]2+dx

min(h N ) m2dx:

Thus, if we choose small enough, we have

Z

R

;1

n

minfm N g

C

]2+dx

Z R

;1

n

h m2dx:

The boundedness of f implies that h is also bounded, and hence the right-hand side is nite. By monotone convergence, we see that

Z

C

Z

h m2dx < +1 R R R R (2.14) now follows from Hlder's inequality and the fact that R ;1 h2+  C  f 2+ d . It remains to establish (2.15). Let fIk g be a Whitney decomposition on E , such that the cubes 3Ik  E , and f3Ik g has bounded overlap (see 25]). Here 3Ik denotes the cube with the same center as Ik , and sides 3 times those of Ik . We are only interested in Whitney cubes Ik such that Ik \ fh  g 6= . For 2 <  < 3, let Ik  = f(x y) : x 2 Ik  0 < y < l(Ik)g, where l(Ik) denotes the side length of Ik . For a set F  H , we let F~ = (F ). Clearly I~k  is a Lipschitz domain, with Lipschitz constants bounded independently of k  . Since Ik is a Whitney cube, there exists a point xk 2 Rn;1 n E with dist (xk  Ik )  Cn l(Ik). Put Ak  = @Ik  \  (xk ) Bk  = (@Ik  \ H ) n Ak  . Since  2 (2 3), the height of Bk  is bounded by Cl(Ik ), i.e., supfy : (x y) 2 Bk  g  Cl(Ik ). Also, jvj   on Ak  . Since Ik \ fh  g 6= , we have Z Z 2 f d  c g2dx  C2jIkj: ;1

n

m2+ dx

;1

n

n

g 3I

3Ik

k

From the L2 Neumann theory for bounded Lipschitz domains, applied to I~k  (13]), we nd that Z Z ; @u  2 2 m1dx  C d

I Z@I @n  C jruj2d + C2jIk j

]

k

k

g

Bk

where m1(x) = supfjv(x0 y)j : jx ; x0j < y < l(Ik)g, and  > 0 is chosen so small that fX 2 H : jX ; (x 0)j < l(Ik )g is contained in Ik  , for all x 2 Ik   2 (2 3). If  is chosen small enough, then, for all x 2 Ik , we have that

 (X ) n f(x0 y) : jx ; x0j < y < l(Ik)g   (xk )

and so, for all  2 (2 3), we have

Z

Ik

m2dx

Z

 c Bg jruj2d + C2jIk j: k

121

Integration in  from 2 to 3 gives

Z

Ik

m2dx

 C

Z 3Ik

m2dx + C2jIkj

which gives (2.14) by additon of k. For the uniqueness, in the case n = 3 2 < p < 2 + ", note that, keeping the notation we used above, if M (ru) 2 Lp 2 < p < 2 + ", and @u=@n = 0 non-tangentially a.e. on , the argument used before shows that

Z

E

But then,

Z

Z1

;

;1

m2dx  C2jE j:

Z

  0 m2dx d E Z1 Z  ; 1  ; 1  C 0  jfm > gjd  C (2 + ) which shows that m is 0 if is small enough. m2+dx

y

m2+dx

3. Regularity properties for the Dirichlet problem on graphs We continue treating domains of the form D = f(x y) 2 Rn : y > '(x) x 2 Rn;1g, where ' : Rn;1 ! R is Lipschitz continuous, i.e., j'(x) ; '(x0)j  mjx ; x0j. We will say that f 2 Lp1() 1 < p < 1, if g(x) = f (x '(x)) has a gradient in Lp(Rn;1). It is easy to check that this is equivalent to the fact that if F is any extension of f to Rn jrT F j (dened as in the remarks following (2.2)) belongs to Lp(). It is also easy to see that this is equivalent to the fact that, for any extension F of f @F=@ T~j  j = 1 : : :  n ; 1, belong to Lp(), where T~1 : : :  T~n;1 are the vector elds introduced in Lemma 2.6. Moreover, @F=@ T~j is independent of the particular choice of the extension F , and depends only on f . We put kf kL1 () = krxgkL (R ;1). Clearly, Lp1() is a Banach space modulo constants. We start out by studying the properties of the single layer potential on the L21() spaces. The following lemma is the graph version of results of G. Verchota (25]). p

p

n

Lemma 3.1. The single layer potential S maps L2() onto L21() boundedly, and has a bounded inverse.

122

Proof. The boundedness follows by the denition of L21 (), and the theorem of Coifman, McIntosh and Meyer (2]). From (2.3) and (2.4), it follows that

@ krT S(f )kL2() C1 @n S (f ) 2 C kf kL2(): L ()

(3.2)

The lemma now follows as in the L2-Neumann case.

Theorem 3.3. For every f 2 L21() there is a harmonic function u in D with M (ru) 2 L2(), and such that @u=@ T~j = @f=@ T~j non-tangentially a.e. on  1 Furthermore, u is unique modulo constants, and

 j  n ; 1.

kM (ru)kL2()  ckf kL21()

where C = C (n m). Proof. The existence follows from Lemma 3.1. To show uniqueness, it is enough to show that if u is harmonic in D M (ru) 2 L2(), and @u=@ T~j = 0 j = 1 : : :  n ; 1 nontantentialy a.e. on , then u is a constant. By our assumption on u jrT uj = 0 a.e. on . By the uniqueness in the Neumann problem (Theorem 2.5),

u(X ) = C +

Z



g(Q)fjX ; Qj2;n ; jX0 ; Qj2;n gd (Q)

where X0 2 Rn n D , and g 2 L2(). By (3.2), g 0, and so u is a constant. A vector-valued function A :  ! RN is a vector-valued atom if A is supported on a ;1=2 surface R ball B = fP 2  : jP ; Qj < 1rg for some Q 2 , and r > 0 kAkL22()  f (B )g and  Ad = 0. We say that f 2 H1 at() if there are functions fj 2 L1() with

@

@ @ f  f  : : :  ~ fj @ T~1 j @ T~2 j @ Tn;1

being vector-valued atoms and 1 X @ @ f  k = 1 : : :  n ; 1 X j j < 1: (3.3) f =  j j ~ j @ T~k j =1 @ Tk P We also set kf kH11 () = inf jj j, where the j 's are as in (3.4). Note that H11 at() is a Banach space modulo constants. at

123

Lemma 3.5. The single layer potential S maps Hat1 () into H11 at() boundedly. Proof. The proof is standard and will be omitted.

Theorem 3.6. Given f 2 H11 at() there is a harmonic function u in D with M (ru) in

L1(), and @u=@ T~j = @f=@ T~j  j = 1 : : :  n ; 1 non-tangentially a.e. on . Moreover u is unique modulo constants, and

kM (ru)kL1()  C kf kH11

at

() :

Proof. For the existence part of the theorem, it is enough to assume '(0) = 0, and to show that if ((@f=@ T~1) : : :  (@f=@ T~n;1)) is a vector-valued atom supported in fQ 2  : jQj < 1g and u is the L21 solution of the Dirichlet problem with data f , given in Theorem 3.3, then

kM (ru)kL1()  C

where C depends only on the Lipschitz constant of . By adding a suitable constant to f , we may assume that f has support in B1 = fQ 2  jQj < R0g where R0 depends only on the Lipschitz constant of . Furthermore by Sobolev's inequality, kf kL2()  C . By the L2 theory for the Dirichlet problem (see 5]) ju(X )j  C = C (m) for X 2 D jX j > 2R0 , and u(X ) takes the boundary value zero continuously on  n B1. Let !(x) = 0 for Rn n D, and !(X ) = ju(X )j for X 2 D, so that ! is subharmonic in Rn n B 1. By the Phragmen-Lindelf theory (see 17]) we have j!(X )j  C jX j2;n;, where C and  only depend on m, and jxj > 2R0. Arguing as in the corresponding Neumann problem, we obtain the existence and the estiamte in Theorem 3.7. We remark that instead of the Phragmen-Lindelf theory we could have used an odd re ection of u to extend u as a solution of Lu = 0 in Rn n B1, and use the Serrin-Weinberger asymptotic expansion just as in the case of the Neumann problem. To show uniqueness, we assume that u is harmonic in D M (ru) 2 L1(), and @u=@ T~j = 0 j = 1 : : :  n ; 1, non-tngentially a.e. on . We must conclude that u is a constant in D. As in the corresponding uniqueness theorem for the Neumann problem, we have jru(X )j  C fdist (X )g1;n , and after Rwe add a suitable constant ju(X )j  C fdist (x )g2;n. Thus, by Sobolev's inequality,  ju j(n;1)=(n;2)d  c, and so u = 0 a.e. on . By the uniqueness in the Neumann problem, it is enough to show that f = 0 a.e., where f = @u=@n. Let b be a Lipschitz function on , with compact support. Let ! be the harmonic extension of b to D. By the Phragmen-Lindelf principle, j!(X )j  C jX j2;n; for X 2 D, X large, where C > 0 > 0. We will now show that for s t > 0 we have Z @! Z @u t !s @n d = ut @ns d :

 124

In fact, let R > 0 be large, and let R = f(x y ) : jxj < R '(x) < y < '(x) + Rg: We then have @ R = R  SR  TR, where R = f(x '(x)) : jxj < Rg SR = f(x y) : jxj = R '(x) < y < '(x) + Rg and TR = f(x y) : jxj < R y = '(x) + Rg: Applying Green's theorem in R, we see that Z Z @u t s !s @n d = ut @! @n d : Since !s 2

@ R

@ R

N (ru) 2 Z Z !s (@ut=@n)d ! !s(@ut=@n)d :

L1 (D ), and

L1(),

R

Also,



Z   !s @ut d   CR2;n; R1;n Rn;1 ;;;! 0 R!1 @n T while  Z Z @u t  !s d   R2;n; M (ru)d ;;;! 0: R!1 @n  n S Similarly, we know that, for s xed jr!s(X )j  C jX j2;n; , and it isR locally bounded in (n;1)=(n;2) () ut (@!s =@n) 2 L1 (), and so D  ut (@!s =@n)d ! R . Thus, since ut 2 L  ut (@!s =@n)d . Also, Z  @! s  ut d   CR2;n  R1;n;  Rn;1 ! 0 @n T while   1 Z 2RZ u @!s d   C Z R R  S t @n  R 2 n Zjutjjr!sjdX 2;n;  C R R  n jutjdX 2

(n;2)=(n;1) Z 2 ; n ;  R ( n ; 1) = ( n ; 2)  C R  jutj dx  Rn=n;1 2 2;n; R ( n  C R  R ;2)=(n;1)Rn=n;1 = CR3;n; ;;;! 0: R!1 Thus, by a choice of an appropriate sequence of Rj 's tending to 1, theR claim follows. R Letting t # 0 we see that  !sfd = 0, and then letting s # 0 we have  bfd = 0 as R

R

R

R

R

R

R

R

R

R

R

desired.

125

Corollary 3.7. The single layer potential S is a bounded operator from Hat1 () onto

H11 at(). It has a bounded inverse, whose norm depends only on the Lipschitz constant of . We now turn to regularity results for the solution of the Dirichlet problem when the data are in Lp1(). This is a new proof of results of Verchota (25]).

Theorem 3.8. There exists a positive number " = "(n m) such that for all f 2 Lp1() 1 < p < 2 + ", there is a harmonic function u in D with M (ru) in Lp(), and @u=@ T~j = @f=@ T~j  j = 1 : : :  n ; 1 non-tangentially a.e. on . Moreover, u is unique modulo

constants and

kM (ru)kL ()  C kf kL1() p

p

were C depends only on p n and m. Proof. The case 2 < p < 2 + " follows in the same way as in the Neumann case. Since S is invertible from Hat1 () onto H11 at() and from L2() onto L21(), it follows by interpolation that S is invertible from Lp () onto Lp1() 1 < p < 2, which gives existence for 1 < p < 2. Uniqueness follows in the same way as in the H11 at() case.

We conclude this section by giving the invertibility properties of layer potentials.

Theorem 3.9. There exists a number " = "(n m) > 0 such that S maps Lp() boundedly p

onto L1(), with a bounded inverse, for 1 < p < 2 + ". Furthermore S is a bounded invertible mapping from Hat1 () onto H11 at(). The operators @S=@n and D are bounded and invertible on Lp () for 1 < p < 2 + " and 2 ; " < p < 1 respectively. Furthermore @S=@n is a bounded invertible mapping on Hat1 (), and D is a bounded invertible mapping on BMO ().

4. Bounded Lipschitz domains In this seciton we will sketch the localization arguments which are necessary to extend the results in the last two sections to the case of general bounded Lipschitz domains in Rn. The L2 theory in the Neumann problem and the L2-regularity in the Dirichlet problem have been treated in 13] and 25]. The Lp regularity in the Dirichlet problem has been treated in 25]. From now on we will assume that D  Rn n 3, is a bounded Lipschitz domain such that D = Rn n D is connected. Atoms are dened as in the graph case, and the atomic 126

Hardy space Hat1 (@D) is also dened as in the graph case. We say that f 2 Lp1(@D) if f 2 Lp(@D d ) and for each coordinate chart (Z '), there are Lp(Z \ @D) functions g1 : : :  gn;1 so that Z Z @ h(x)f (x '(x))dx h(x)gj (x '(x))dx = R ;1 R ;1 @xj for all h 2 C01(Z \ Rn;1). It is easy to see that given f 2 Lp1(@D), it is possible to dene a unique vector rT f 2 Rn, at almost every Q 2 @D so that krT f kL (@D d) is equivalent to the sum over alla the coordinate cylinders in a given covering of @D of the Lp norms of the locally dened functions gj for f , occurring in the denition of Lp1(@D). The resulting vector eld, rT f , will be called the tangential gradient of f . If F is a function dened on Rn rT F is orthogonal to the normal vector n, and rF = rT F + (@F=@n)  n. In local coordinates, rT f may be realized as (g1(x '(x)) g2(x '(x)) : : :  gn;1(x '(x)) 0) ; h(g1(x '(x)) : : :  gn;1(x '(x)) 0) n(x '(x))in(x '(x)) Lp1(@D) may be normed by kf kL1 (@D) = kf kL (@D) + krT f kL (@D). Before we proceed to dene the space H11 at(@D), we will make a few remarks about it in the graph case. We say that f is an H11 at() ; L2 atom if f is in L21(), it is supported in automatically veries R a surface ball B , and A = ((@=@;T~11=)2f : : :  (@=@ T~n;1)f ) ~(which 1 () if f 2 L(n;1)=(n;2) (), 2 . We say that f 2 H Ad

= 0) veries k A k 

( B ) L () 1 at  1 () ; L2 atoms fj and numbers j with P jj j < +1, such that and there exist H 1 at P f = 1j=1 j fj , where the sum is taken in the sense of L(n;1)=(n;2)(). Moreover, if f 2 H11 at(), there exists a constant c such that f ; c 2 H~ 11 at(). Let R : Rn;1 !  be given by (x) = (x '(x)). Then f 2 H~ 11 at(Rn;1) if and only if g(x) = Cn R ;1 (h(y)=jx ; yjn;2)dy, where h 2 H11 at(Rn;1). In fact, such g(x) clearly belong to L(n;1)=(n;2)(Rn;1), and Lemma 3.5 shows that in H~ 11 at(Rn;1). Conversely, if g 2 H~ 11 at(Rn;1), then R they are in fact P g(x) = Cn R ;1 (h(y)=jx ; yjn;2)dy, where h(y) = nj=1 Rj (@=@y )g, where Rj are the classical Riesz transforms. Note that if we dene H~ 11 at(Rn;1 ) by using H11 at(Rn;1) ; Lp atoms, 1 < p  1, we obtain the same characterization of H~ 11 at(Rn;1), which shows that all these spaces coincide, and have comparable norms. The same fact of course remains true for H~ 11 at() This allows one to show in a very simple fashion that if  is a Lipschitz function with compact support in , and f 2 H~ 11 at(), then f also belongs to H~ 11 at(). Our nal remark is that if f 2 H~ 11 at(), and u is the solution to the Dirichlet problem constructed in Theorem 3.7, then uj= f , in the sense of non-tangential convergence, R ( n ; 1) = (n;2) d  C , and ju(X )j  C fdist (X )g2;n . Moreover, the uniqueness  (ju j) then follows without the addendum 'modulo constants'. We are now ready to dene H11 at(@D). We say that f is an H11 at(@D) ; L2 atom if f is supported in a coordinate cylinder (Z '), and if  is the graph of ', f is an H11 at() ; L2 atom. The space H11 at(@D) is then dened as the absolutely convergent sums of H11 at(@D); n

n

p

p

p

p

n

n

j

127

L2 atoms, where the convergence of the sum takes place in the L(n;1)=(n;2)(@D) norms. It is a Banach space, and if we replace L2 atoms by Lp atoms, 1 < p  1, we obtain the same space, with an equivalent norm. Also, if  2 Lip (@D), and f 2 H11 at(@D) f 2 H11 at(@D), and, if f 2 H11 at(@D), then f 2 L(n;1)=(n;2)(@D). Also, Lp1(@D)  H11 at(@D), for any 1 < p  1. The non-tangential regions ;(Q) Q 2 @D, are dened as ;(Q) = fX 2 D : jX ; Qj < (1 + )dist (X @D)g, while the non-tangential maximal function M (!)(Q) = supX 2;1(Q) j!(X )j. Finally, we recall that a bounded Lipschitz domain is called a starlike Lipschitz domain (with respect to the origin) if there exists ' : S n;1 ! R, where ' is strictly positive, and j'();'(0)j  mj ;0j  0 2 S n;1 such that, in polar coordinates (r ) = f(r ) : 0  r < '()g. Note that if D is an arbitrary bounded Lipschitz domain, and (Z ') is a coordinate chart, with kr'k1  m, then, for appropriate > 0 a > 0 b > 0 which depend only on m, the domain D \ U is a starlike Lipschitz domain with respect to X0 = (0 b ), where U = f(x y) : jxj <  jtj < a g.

Lemma 4.1. Let be a starlike Lipschitz domain, and let u be the L2-solution of the

Neumann problem with data an atom a, centred at Q0 2 @ . Then, there exists a constant C , which depends only on the Lipschitz constants of D such that

(a) (b) (c)

kM (ru)kL1(@)  C M (ru)2jQ ; Q0jn;1 d  C @ kukH11 (@)  C

Z

at

if we subtract from u an appropriate constant. Proof. We may assume that the size of the support of a is small. We may also assume that  fy < '(x)g = D, where ' : Rn;1 ! R is Lipschitz with norm depending only on the Lipschitz character of , that @ \ @D  fjX ; Q0j < r0g\ @ , where r0 depends only on the Lipschitz characater of @ , that Q0 is the origin and that supp a  fjX ; Q0j < r0g. Let v be the solution of the Neuman problem in D, with data a, given by Lemma 2.7, and let ! be the L2-solution of the Neumann problem in , with data @!=@n = 0 on @ \ @D, and @!=@n = ;@v=@nj@ on @ n (@ \ @D). We clearly have u = v + !, and from this the lemma follows.

Lemma 4.2. Let be a bounded, starlike Lipschitz domain, and let u be harmonic in , with M (ru) 2 L1 (@ ) and either Then, u is a constant.

rT u = 0 or @u=@n = 0 non-tangentially a.e. 128

on @ .

Proof. Assume rst that @u=@n = 0 non-tangentially a.e. on @ . We can show that u is a constant using a variant of the uniqueness proof in Theorem 2.11, using a radial re ection across our starlike surface. If rT u = 0, is constant on @ , we have, if b 2 Lip (@ ), and ! is its harmonic extension, that (with ur(x) = u(rx))

Z

Z @! @u r !s @n d = ur @ns d : @ @ R If we let r ! 1, the right-hand side tends to 0, while the left-hand side tends to @ b(@u=@n) d = 0, and so @u=@n = 0 a.e. on @ . Therefore u is constant by the previous result. We are now in a position to give the solution of the Neumann problem with Hat1 (@D) data, for a general bounded Lipschitz domain D.

Theorem 4.3. Let D  Rn be a bounded Lipschitz domain. If u is harmonic in D, with

M (ru) 2 L1(@D), then @u=@n 2 Hat1 (@D) and @u 1 (4.4) @n H 1 (@D)  C kM (ru)kL (@D): at

If f 2 Hat1 (@D), then there is a harmonic function u with M (ru) 2 L1 (@D) and @u=@n = f non-tangentially a.e. on @D. Furthermore, u is unique modulo constants, and

(4.5)

u can be chosen so that (4.6)

kM (ru)kL1(@D)  C kf kH 1 (@D) at

kujjH11

at

(@D)

 C kf kH 1 (@D): at

Proof. As in the proof of Lemma 2.10, the estimate (4.4) follows from Green's formula, the extension theorem of Varopoulos (24]) and the fact that the dual of VMO (@D) is Hat1 (@D). (See 8] for the exact form of the Varopoulos extension theorem that is needed here.) In the case when D is a bounded starlike Lipschitz domain, the rest of the theorem follows from Lemma 4.1 and Lemma 4.2.

We now pass to the general case. We rst establish uniqueness in the general case. Thus, M (ru) 2 L1(@D), and @u=@n = 0 a.e. on @D. We can cover a neighborhood of @D in D, 129

with nitely many bounded starlike Lipschitz domains i  D, such that M (ru), the non-tangential maximal function relative to the domain i, is in L1(@ i). Thus, if vi = uj , we have @vi=@n 2 Hat1 (@ i). If also @ i  B (Qi 3r) \ @D, for some r > 0 Qi 2 @D, we can take the atoms in the atomic decomposition of @vi=@n to have supports that are so small that they are all contained in @ i=B (Qi 2r) (since @vi=@n = 0 on B (Qi 3r) \ @D). It then follows from (b) in Lemma 4.1, and the uniqueness for starlike Lipschitz domains, that M (ru) 2 L2(B (Qi r) \ @D). Since iB (Qi r) \ @D can be taken to be @D, it follows that M (ru) 2 L2(@D), and hence u is a constant by the L2-theory (see 13] or 25]). To show (4.5), it is enough to show that if a is an atom with support contained in a ball of radius r, with r  r0 = r0(D), then kM (ru)kL1(@D)  C (D), where u is the solution of the L2-Neumann problem with data a. For  2 ( 41  10) let D() be a domain of the form f(x y) : '(x) < y < '(x) + 1 jxj < 2g, where ' is a Lispchitz function. We can choose numbers 1 2 and coordinate systems so that the domains D() are starlike Lipschitz domains contained in D, for 14 <  < 10. The number r0 is chosen in such a way that there are nitely many D () 1    N = N (D) such that @D (1=4) \ @D = @D, and such that, for any  we have that either the support of a is contained in @D \ @D (4) or supp a \ @D (3) = . We rst claim that for each compact set K  D, we have i

i

sup jruj  C = C (K D):

(4.7)

K

R

R

To see this, pick  2 L1(D) supp  R K and (X )dX = 0. Letting !(X ) = Cn jX ; Y j2;n (Y )dY , we have that ! =  @D (@!=@n)d = 0, and k@!=@nkL1(@D)  C (K D)kkL1(K). R Let h solve the Neumann problem in D with data @!=@n and D h(x)dx = 0. Then,

Z

D

udX =

Z

D

u(! ; h) =

Z

@D

@u : (! ; h) @n

If we now note that the normal derivative of ! ; h is 0 on @D, and we use locally the graph re ection argument that we used in the proof of Lemma 2.7, it follows that k! ; hkL1 (@D)  C (K D)kkL1(K) which yields (4.7). Let M   be the non-tangential maximal operator associated to the domain D (). We can choose a suitable compact set K  D so that, for all  2 (1=4 10) we have (4.8)

Z

@D

M (ru)d 

XZ v

j M   (ru)jd + C sup jruj: K @D () 

In order to apply (4.8), we shall rst study the case when (supp a) \ D (3) = . From the 130

L2-Neumann theory, it follows that for 1=4 <  < 3 we have Z

2 Z  1 = 4 M (ru)d  c M   (ru)2d

@D (1=4) Z@D () @u 2  C @D ()n@D @n d  Z C jru(X )jd : 





@D ()n@D

Integrating in  from 1/2 to 1 now gives

Z

@D (1=4)\@D

M  1=4(

2

Z

ru)d  C D (2) jru(X )j2dX Z

2  C D (3) jru(X )jdX : 



The last inequality follows from the graph re ection and the reversed Hlder inequality for the gradient of the solution of a uniformly elliptic equation in divergence form (see 12]) together with the fact that one can lower the exponent on the right-hand side of such a reversed Hlder inequality. This last fact was proved by the present authors see 10]. to use the R It is  possible R graph re ection because supp @u=@n \ D (3) = . Hence, 1 = 4 (ru)d  C D (3) jru(X )jdX , and therefore, given " > 0 there is a compact D (1=4) M K"  D such that 

(4.9)



Z

D (1=4)

Z

ru)d  C"

M  1=4(

@D

M (ru)d + C (")

Z

K"

jruj:

If supp a  D (4), we let v solve the Neumann problem in D (4), with data a on @D (4) \ @D, and 0 elsewhere on @D (4). Let ! = u ; v. Since @!=@n = 0 on @D (4) \ @D we have, from the argument leading to (4.9), that

Z

@D (1=4)\@D

and therefore (4.10)

Z @D (1=4)\@D

r!)d  C

M  1=4(

ru)d  C"

M  1=4(

Z

Z

+ C (")

@D Z

D (3)

jr!jdX

M (ru)d

K"

jru(X )jdX + C:

Using now (4.8), (4.9) and (4.10), and the weak estimate (4.7), we see that

Z

@D

M (ru)d  C"

Z

@D

M (ru)d + C (") + C

131

and so, if we choose " small enough

Z

M (ru)d  C = C (D)

@D

which yields (4.5). Finally, note that because of (4.7), we can subtract a constant C from u so that supK ju ; C j  CK for all K compact in D. Let v = u ; C . We claim that kvkRH11 (@D (1))  C . In fact, we know that, because of the Poincar inequality on @D (1) @D (1) jvj(n;1)=(n;2)d  C . But, byR Lemma 4.1, there exists a constant C so that kv ; C kH11 (@D (1))  C . But then, @D (1) jv ; C j(n;1)=(n;2)d  C , and thus jC j  C . Therefore kvkH11 (@D (1))  C , and (4.6) follows for the case of atoms. The general case follows from this. We shall next study the regularity in the Dirichlet problem with H11 at(@D) data. 

at



at







at

Lemma 4.11. Let f be an H11 at(@D) ; L2 atom. If u solves the Dirichlet problem with

boundary values f , then

Z

(a) (b)

Z

(

@D

M (r

; u)2d )

@D

Z

@D

M (ru)d  C:

M (ru)2jQ ; Q0j("+1)(n;1)d

1="

 C

where Q0 is the center of the support of f .

(c)

Z

@D

M (ru)2jQ ; Q0jn;1d  C:

Here C and " > 0 are independent of the H11 at (@D) ; L2 atom f . Proof. If we perform a change of scale so that the support of f is of size 1, we see that the arguments in the graph case (Theorem 3.7), yield the proof of Lemma 4.11.

Theorem 4.12. Let D  Rn be a bounded Lipschitz domain. If u is harmonic in D, with M (ru) 2 L1(@D), then u 2 H11 at(@D), and

(4.13)

kukH11

at

(@D)

 C kM (ru)kL1(@D):

If f 2 H11 at (@D), then there is a harmonic function u with M (ru) 2 L1(@D) and u = f non-tangentially a.e. on @D. Furthermore, u is unique, kM (ru)kL1(@D)  C kf kH11at(@D):

132

Proof. Uniqueness follows from the uniqueness in Theorem 4.3. Next note that existence in the range 1 < p < 2 follows by interpolation between Theorem 4.3 and the L2 results. Existence in the case p > 2 follows by a minor modication of the corresponding part of the proof of Theorem 2.12. In fact, the main dierence is that in the bounded case there are two kinds of Whitney cubes Ik , the small ones and the big ones. The small ones are treated just as in 2.12, while the big ones are of diameter comparable to that of D, and hence m1 is comparable to m on them. The rest of the proof is identical, and is therefore omitted.

Our next theorem deals with regularity in the Dirichlet problem with Lp1(@D) data. It was rst proved in 25].

 Rn be a bounded Lipschitz domain. There exists a positive number " = "(D) such that for all f 2 Lp1 (@D) 1 < p < 2+ ", there is a harmonic function u in D, with M (ru) in Lp(@D), and u = f non-tangentially a.e. on @D. Moreover, u is Theorem 4.14. Let D

unique and

kM (ru)kL (@D)  C kf kL1(@D) p

p

where C depends only on p and D. Proof. Uniqueness follows from Theorem 4.12. Existence follows just as in Theorem 4.13 in the range 2 < p < 2 + ", while the case 1 < p < 2 follows by interpolation.

We will now study the Neumann problem and regularity in the Dirichlet problem for the domain D = Rn n D . The L2 theory for D can be found in 25]. We will let M  be the non-tangential maximal operator associated to D , where the non-tangential regions are truncated. We let H~ at1 (@D) be dened as Hat1 (@D), but add the constant 1 to the atoms.

2 H~at1 (@D), there exists a harmonic function u in D with @u=@n = f non-tangentially a.e. on @D u(X ) = o(1) at 1, kM (ru)kL1(@D)  C kf kH~ 1 (@D) and kukH11 (@D)  C kf kH~ 1 (@D): Moreover, u is unique. There exists " = "(D) > 0 such that if f 2 Lp (@D) 1 < p < 2 + ", then kM  (ru)kL (@D)  C kf kL (@D) where C = C (p D).

Theorem 4.15. Given f

at

at

p

at

p

Proof. The uniqueness reduces to the L2-uniqueness just as in Theorem 4.3. For existence in the H~ at1 (@D) case, the atom 1 is taken care of by the L2-theory. The existence and the

133

estimate kM (ru)kL1(@D)  C for the other atoms are the same as in the proof of Theorem 4.3, the only dierence being that the estiamte

Z   u  C (K D )kkL1(K) D

is valid for all  2 L1 (D) supp   K , since u(X ) = o(1) at 1. This fact also shows, by a small variation of the argument used in Theorem 4.3, that kukH11 (@D)  C . The case 1 < p < 2 + " of the theorem follows in the same way as in Theorem 4.13. Note also that if M  (ru) 2 L1(@D), and u(X ) = o(1) at 1, then @u=@n 2 H~ at1 (@D). This is proved in a similar way to (4.4) in Theorem 4.3. at

Theorem 4.16. Given f 2 H11 at(@D), there exists a harmonic function u in D with u = f on @D non-tangentially a.e., u(X ) = o(1) at 1, @u  kM (ru)kL1(@D)  C kf kH11 (@D) and @n H~ 1 (@D)  C kf kH11 at

at

(@D):

at

Moreover, u is unique. There exists " = "(D) > 0 such that if f 2 Lp1 (@D) 1 < p < 2 + ", then kM (ru)kLp(@D)  C kf kLp(@D), where C = C (p D). Proof. Uniqueness follows as in the proof of uniqueness in Theorem 4.12. Existence for atoms follows in the same way as in Lemma 4.11. The estimate k@u=@nkH~ 1 (@D)  C kf kH11 (@D) follows because of the remark before the statement of Theorem 4.16. The case 1 < p < 2 + " follows in the same way as in Theorem 4.14. at

at

We are now ready to prove the sharp invertibility properties of the layer potentials. For P Q 2 @D P 6= Q, let K (P Q) = !1 hQ ; P n(Q)i n where !n is the surface area of the unit sphere in Rn, and put Tf (P ) = p:v: K (P Q)f (Q)d (Q). Also, let

R

@D

Z 1 Sf (P ) = ! (n ; 2) jP ; Qj2;nf (Q)d (Q): n @D

The boundedness properties of these operators are the same as for the corresponding operators in the graph case. 134

Theorem 4.17. There isp a number q0 p= q0(D) q0 2 (2 1), such thatp 12 I ; T  is an

invertible Rmapping from L0 (@D) onto L0(@D) for 1 < p < q0, where L0(@D) = ff 2 Lp(@D) : @D fd = 0g. Also, 21 I ; T  is invertible from Hat1 (@D) onto Hat1 (@D). There is a number p0 = p0 (D) p0 2 (1 2) such that 21 I + T is an invertible mapping of Lp (@D) onto Lp(@D) for p0 < p < 1. Also, 21 I + T is invertible from BMO (@D) onto BMO (@D). There is a number r0 = r0(D) r0 2 (2 1) such that S is an invertible mapping of Lp (@D) onto Lp1(@D). Also, S is an invertible mapping from Hat1 (@D) onto H11 at(@D). Proof. The proof of this theorem is the same as the corresponding L2 case presented in 25], using the results of this ection. Finally, we give representation formulas for the solutions of the Dirichlet and Neumann problem, using layer potential.

 Rn be a bounded Lipschitz domain, whose complement is p connected. Let q0  p0 r0 be the numbers given in Theorem 4.17. Let f 2 L (@D) p0 < p < 1, and let Ru(X ) be the unique solution of the Dirichlet problem given in 5]. Then u(X ) = (1=!n ) @D(hX ; Q n(Q)i=jX ; Qjn)( 12 I + T );1(f )(Q)d (Q). The same holds when f 2 BMO (@D), and uRis the unique solution of the Dirichlet problem given in 9]. Let f 2 Lp(@D) 1 < p < q0 @D fd = 0, and let u(X ) be the unique (modulo constants) Theorem 4.18. Let D

solution of the Neumann problem given in Theorem 4.13. Then, Z ; 1 u(X ) = ! (n ; 2) jX ; Qj2;n 21 I ; T );1 (f )(Q)d (Q): n @D

The same holds when f 2 Hat1 (@D), and u is as in Theorem 4.3. Let f 2 Lp1 (@D) 1 < p < r0 and let u(X ) be the unique solution of the Dirichlet problem given in Theorem 4.14. Then, Z 1 u(X ) = ! (n ; 2) jX ; Qj2;nS ;1(f )(Q)d (Q): n @D The same holds when f 2 H11 at (@D), and u is as in Theorem 4.12.

Proof. The proof follows from well-known properties of layer potentials (see 25], for example), the uniqueness in all the theorems mentioned and Theorem 4.17.

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