mathematica. Chicago ON THE THEORY OF HARMONIC FUNCTIONS OF SEVERAL VARIABLES II. Behavior near the boundary

ON THE THEORY OF HARMONIC FUNCTIONS OF SEVERAL VARIABLES

II. Behavior near the boundary

BY E L I A S M. S T E I N

Chicago

Contents

Page

I n t r o d u c t i o n . . . 137

C h a p t e r I 1. P r e l i m i n a r i e s . . . 141

2. R e g u l a r i z a t i o n of t h e r e g i o n R . . . 143

3. B a s i c l e m m a s . . . 145

4. T h e g e n e r a l i z e d a r e a t h e o r e m . . . 147

C h a p t e r I I 5. A n a u x i l i a r y t h e o r e m . . . 156

6. G e n e r a l t h e o r e m s a b o u t n o n - t a n g e n t i a l l i m i t s of c o n j u g a t e f u n c t i o n s . . . 161

C h a p t e r I I I 7. V a r i o u s e x a m p l e s . . . 163

8. R e l a t i o n w i t h g e n e r a l i z e d H i l b e r t t r a n s f o r m s . . . 168

I n t r o d u c t i o n (1)

T h i s p a p e r is a c o n t i n u a t i o n o f t h e s e r i e s b e g u n i n [9]. H e r e , a s i n t h e p r e v i o u s p a p e r , w e a r e c o n c e r n e d w i t h t h e f o l l o w i n g p r o b l e m : T o e x t e n d , a s f a r a s p o s s i b l e t o t h e g e n e r a l c a s e of s e v e r a l v a r i a b l e s , p r o p e r t i e s of h a r m o n i c f u n c t i o n s i n t w o v a r i a b l e s w h i c h r e s u l t f r o m t h e i r close c o n n e c t i o n t o a n a l y t i c f u n c t i o n s i n o n e v a r i a b l e .

(1) The main results of this paper were a n n o u n c e d in a b s t r a c t s no. 566-35 a n d 566-36, Notices of the A.M.S., 1960.

1 0 - 61173060. Acta mathematica. 106. Imprim6 le 20 ddeembre 1961.

138 ~LZ_AS M. ST~l~

W e shall be concerned with the local behavior of harmonic functions near the b o u n d a r y . To explain t h e m a i n ideas of this paper we begin b y recalling some results f r o m t h e classical case.

There we deal with a function u (x, y) h a r m o n i c in t h e upper-half plane y > 0 . W e are concerned with t h e behavior of u (x, y) near the x-axis, or more precisely, near a general measurable set E located on t h e x-axis. T h e s t u d y of this behavior is i n t i m a t e l y related with t h a t of the conjugate f u n c t i o n v (x, y), a n d t h u s t h e a n a l y t i c function F (z) = u + i v, z = x + i y. A basic concept in this connection is t h a t of a " n o n - t a n g e n t i a l " limit at a p o i n t (x, 0) located on t h e x-axis. T h e results of t h e "local t h e o r y " in t h e classical setup which concern us are then: (1)

(A) u (x, y) has a non-tangential limit /or a.e. x E E i/ u (x, y) is non-tangentially bounded /or a.e. x E E.

(B) I / u (x, y) has a non-tangential limit /or a.e. x E E then the same is true /or v (x, y), and conversely. (~)

T h e p r o p e r t y of h a v i n g a n o n - t a n g e n t i a l limit (or more generally of being non- t a n g e n t i a l l y bounded) is of a n elusive n a t u r e a n d t h u s difficult to pin d o w n a n a l y t - ically. I t is therefore desirable to reexpress this p r o p e r t y in a more t r a c t a b l e b u t logically equivalent form. This r e s t a t e m e n t m a y be accomplished f r o m results of Marcinkiewicz a n d Z y g m u n d a n d Spencer. We shall use the following definition.

F (x0) will denote a s t a n d a r d triangular neighborhood which lies in the u p p e r half plane a n d whose vertex is a t t h e p o i n t (x0, 0). More precisely,

r ( x 0 ) ={(x, y):

Ix-x0l<~y,

0 < y < h }

for t w o fixed c o n s t a n t s ~ a n d h. We t h e n define the so-called area integral

~ u 2 I \ a y ]

I ~ ( x o )

with represents t h e area (points c o u n t e d according t o their multiplicity) of t h e image of F (x0) u n d e r F (z) = u § i v. T h e t h e o r e m of Marcinkiewicz, Z y g m u n d a n d Spencer can be stated, in this context, as follows:

(C) u (x, y) has a non-tangential limit /or a.e. x E E i/ and only i/ the area integral A (x) is /inite /or a.e. x E E.

(1) W e use the a b b r e v i a t i o n a.e. t h r o u g h o u t to m e a n " a l m o s t e v e r y " or " a l m o s t e v e r y w h e r e "

w i t h r e s p e c t to L e b e s g u e m e a s u r e .

(2) These a n d o t h e r r e s u l t s of the classical t h e o r y m a y be f o u n d in [12, Chap. 14], w h e r e refer- ences to the o t h e r original w o r k s m a y also be f o u n d .

O N T H E T H E O R Y O F I{ARMO:bTIC F U N C T I O N S O F S E V E R A L V A R I A B L E S 139 I t should be noted that in this form, the proposition (C) implies (B), because

8 y] \8 x/ \~ y] \8 x/

by the Cauchy-Riemann equations. We add here that the concept of "non-tangential"

limits and the corresponding notion of non-tangential boundedness are basic for the conclusions (A), (B), and (C). For example, the approach to the boundary b y the normal direction only would not do as a substitute notion. (1)

We now turn to the situation in a n y number of variables. The generalization of (A) to harmonic functions of several variables has been known for some time, see [1].

I t is the purpose of this paper to obtain the extension of theorems (B) and (C) to several variables.

We begin b y considering the extension of (C). If

u ( X , y), X =

(xl, x 5 .. . . ,

Xn)

is harmonic in the upper half space y > 0 , as a function of the n § variables

(X,y),

then we set

A(Xo)= ffy ~ ' V u , ~ d X d y , (*)

F(x0)

n

where ] V u ]e = ]~ u/~ y ]5 § k~l] 8 u / 8 x k ]5 and F (Z0) is the truncated "cone"

( ( X , y ) :

I X - X o l < o c y ,

0 < y < h } ,

for fixed a and h. I n the folloving theorem E denotes an arbitrary measurable subset of E~, where En is considered as the boundary hyper-plane of our half-space.

THWO~EM 1.

I n order that u ( X , y) have a non-tangential limit /or a.e. X EE, it is necessary and su//icient that the generalized area integral, A (X), be/inite/or a.e. X E E.

The proof of the theorem, which is contained in section 4, is based on the ele- mentary lemmas of Section 3. The necessity of the finiteness of the integral in (*) was previously known, see [2]. The method we use leads to a simplification of the proof of that part of the theorem. The sufficiency, which is our principal object, makes use of some similar ideas, but is more difficult. We add two remarks: (a) A different approach leading to the proof of Theorem 1 was found independently by Calderon (b). The generalized area integral was considered in a different context by us in [8].

B y the use of Theorem 1 we can obtain a generalization of proposition (B) to (1) See in particular the example in [I1], Chap. 14, p. 204.

140 ELIAS M~ STEI:N

a n y n u m b e r of variables. F o r this purpose let us recall the s y s t e m of h a r m o n i c func- tions considered in C h a p t e r I. The so-called Riesz s y s t e m is m a d e up of n § 1 func- tions, u, Vl, v2, ..., vn, satisfying

~u_ ~ ~v~ ~ u ~vk ~v k ~vj

~Y ~- k=l ~ = 0 , ~x k ~y , ~x~ - - = ~xk .

I t can be characterized alternatively as arising as the gradient of a h a r m o n i c f u n c t i o n H ( x ; y); t h a t is

( u , v 1 , v 2, . . . ~ v n ) = ~ H - , ~ H ~ , . . , . ' ~x 1 ~x2

Our generalization of Chapter I I can t h e n be s t a t e d as follows (see Section 7).

THEOREM 4. I / U has a non-tangential limit /or a.e. X C E , then so do the conjugates

~1, v2, ..., Vn, and conversely.

I t m u s t be r e m a r k e d t h a t this t h e o r e m does not follow directly from T h e o r e m 1, a s in the case n = l . This is due to the fact t h a t if n > l t h e n there is no simple a p p r o p r i a t e relation between ~ IV vk [2 a n d IV u [3.

k - 1

T h u s an extra step is needed to deduce T h e o r e m 4. This step is given in Sec- t i o n 5, a n d it allows us to obtain a wide generalization of T h e o r e m 4. The n a t u r e .of this generalization m a y be u n d e r s t o o d as follows. T h e s y s t e m of harmonic functions .satisfying the M. Riesz equations above represents one possible extension of t h e C a u c h y - R i e m a n n equations to several variables. There are other generalizations--al- t h o u g h less direct---which are of importance. Some of these systems are discussed in Sec- tions 7 a n d 8. A systematic discussion of these extensions c a n n o t be given here, b u t will b e t h e subject of a future paper in this series. W i t h o u t discussing t h e general problem 9 exhaustively, we can give a definition of c o n j u g a c y - - w h i c h a l t h o u g h t e n t a t i v e in n a t u r e - - i s significant technically in view of its inclusiveness a n d its applicability.

W e shall s a y t h e h a r m o n i c function u ( X , y) is conjugate to v (X, y) if there exists positive integer r a n d a differential polynomial P (D) homogeneous of degree r in

~ / ~ y , ~ / ~ x 1 . . . . , ~/~x~ (with c o n s t a n t coefficients) so t h a t u a n d v are related b y

~ru

- - = P ( D ) v .

~ yr

This definition can be extended to the case when u a n d v are respectively vec- t o r s of harmonic functions with /c a n d m components, a n d P ( D ) i s t h e n a k • m a t r i x whose entries are differential operators of the t y p e described. Our generaliza- t i o n of T h e o r e m 4 is t h e n (see Section 6).

ON T H E T H E O R Y OF H A R M O N I C F U N C T I O : N S OF S E V E R A L V A R I A B L E S 141 THEOREM 3. I / v ( X , y ) has a non-tangential limit /or a.e. X E E then so doe~

u ( i , y).

Examples illustrating this notion of conjugacy and Theorem 3, are given in Sec- tion 7. I n Section 8 the meaning of this conjugacy is further examined in terms o f harmonic functions which are Poisson integrals. I t then turns out t h a t this notion is~

equivalent with that arising from singular integrals (i.e., generalization of the Hilbert transform) whose "symbols", when restricted to the unit sphere, are (harmonic) poly- nomials. This fact is summarized in Theorem 7 of Section 8 below.

We wish now to discuss briefly the possibility of further extensions of the above.

The first generalization is immediate: we need not assume that our functions are de- fined and harmonic in the entire upper-half space, but only in an appropriate region about our set E. For example, we could restrict our consideration to the "cylinder'"

{X, y):

X

E El,

0 < y < he} where E 1 is the set of all points at distance not greater than hi from E, and h v h e are two fixed positive constants. I n all our proofs below we actually do not go outside such a cylinder, and we shall therefore assume once and for all that all our theorems are considered with this slight unstated generalization in mind.

Our sets E lie on the boundary, which is a hyper-plane ( y = 0 ) . I t would b e desirable to extend these results b y considering non-tangential behavior for sets lying on more general hyper-surfaces. Presumably this could be done without too much difficulty if the bounding hyper-surface were smooth enough. I t would be of definite interest, however, to allow the most general bounding hyper-surface for which non- tangential behavior is meaningful. Hence, extension of these results to the case when the bounding surfaces are, for example, of class C 1 would have genuine merit. W h e t h e r this can be done is an open problem.

Chapter I

The main purpose of this chapter is the proof of Theorem 1 in Section 4. Section I contains various definitions and statements of known facts. Section 2 deals with a technical device useful for the proof of Theorem 1. Section 3 contains several lemmas needed in the proof of the theorem.

1. Preliminaries

We shall follows as far as possible the notation of the previous paper in this series, which we now summarize.

142 E L I A S M . S T E I N

E~ will d e n o t e t h e E u c l i d e a n space of n dimensions. P o i n t s in t h i s space will be d e n o t e d b y c a p i t a l l e t t e r s X , X0, Y, Z, a n d in c o o r d i n a t e n o t a t i o n we will set X = (x 1, x2, ..., xn) ... etc. E~+I+ will d e n o t e t h e E u c l i d e a n n + l d i m e n s i o n a l u p p e r h a l f space: I t s p o i n t s will be d e n o t e d b y t h e p a i r ( X , y ) , w h e r e X E E n a n d 0 < y < co.

F o r X E En, we h a v e (X, 0 ) E E n + I , t h u s we consider E n as e m b e d d e d in E~+I a s t h e b o u n d a r y h y p e r - p l a n e of _En+l. +

W e shall also use t h e following c o n v e n t i o n . I n t e g r a l s o v e r a n (n + 1) d i m e n s i o n a l s u b s p a c e of E~++x will be d e n o t e d b y d o u b l e integrals, such as j'~ ( . ) d x d y . I f we inte- g r a t e o v e r a n n - d i m e n s i o n a l subset, such as o v e r E~, we shall i n d i c a t e t h i s b y a single i n t e g r a l like ~ ( . ) d X .

m (E) will d e n o t e t h e n - d i m e n s i o n a l m e a s u r e of a set in E~ (all sets o c c u r r i n g will be a s s u m e d t o be m e a s u r a b l e ) , a a n d a ' will d e n o t e p o i n t s i n E~+I, a n d ~ will d e n o t e a s p h e r e whose c e n t e r is ~.

L e t X o d e n o t e a p o i n t in E~. W e d e n o t e b y F (X0) t h e i n t e r i o r of a t r u n c a t e d cone in E~++I w i t h v e r t e x a t X 0. T h u s

r(Xo) ={(x, y): ]x--Xo]<~y, O<y<h},

for some f i x e d ~ a n d h. W h e n we wish t o i n d i c a t e t h e p a r a m e t e r s zr a n d h we shall w r i t e

P (X 0) = P (X0; ~, h).

I n w h a t follows we shall refer to t h e i n t e r i o r of t r u n c a t e d cones s i m p l y as cones.

F o r a n y set E c E ~ , a n d ~ a n d h f i x e d we shall a s s o c i a t e a r e g i o n R in En++l.

T h e r e g i o n R is t h e u n i o n of all cones F (X0; ~j h) w h e r e X 0 r a n g e s o v e r t h e p o i n t s of E . T h u s

R = LI F ( X 0 ; ~, h).

XveE

T h e following t w o l e m m a s a r e k n o w n a n d we t a k e t h e m for g r a n t e d . T h e first is of a n e l e m e n t a r y c h a r a c t e r ; t h e second, however, is deep. (1)

E~ + I. we are bounded

LEMMA 1. Let u ( X , y ) be continuous in + Suppose given a

set E o c E~ with the /ollowing property. Whenever X o E Eo, u (X, y) is bounded as ( X , y) ranges in some cone F (Xo). (The shape o/ the cone and bound m a y depend on Xo. ) For a n y ~ > O, then there exists a closed subset E, E c E o. so that

(1) The first lemma is contained, although not stated explicitly, in [1]. The second lemma, in a more general form, is the main result of that paper.

ON THE THEORY OF HARMOIqIC FUNCTIONS OF SEVERAL VARIABLES 1 4 3

(1) m ( E o - E ) < r

(2) I / o: and h are /ixed, u (X, y) is uni/ormly bounded in R = [J F (Xo; o~, h).

Xoe E

For the statement of the next lemma we shall need the following definition. If u ( X , y ) is defined in E~+~, we shall say t h a t it has a + non-tangential limit l at the point Xo, (X 06En), if for every fixed ~, u ( X , y ) - + l , as y - + 0 , w i t h l X - X ol < ~ y . LEMMA 2. Suppose that u ( X , y) is harmonic (as a /unction o/ the n § 1 variables) in E +~+1, and that /or every point X o belonging to a set E ~ E ~ , u (x~ y) is bounded in a cone F (Xo) whose vertex is at X o. Then u ( X , y) has a non-tangential limit /or a.e.

point X o E E.

2. Regularization of the region R

Given a closed bounded subset E of En and fixed positive quantities ~ and h we associate with it, as before, the open region R = U F (X0; ~, h). I t is to be noted

XoE E

t h a t the region R is not necessarily connected. (1)

We add a marginal comment. This type of region has been considered for some time in the study of non-tangential behavior of harmonic functions, especially when n = 1. I n t h a t case the boundary of R is a rectifiable curve and thus the study of harmonic functions in R is greatly facilitated b y the use of conformal transformations of R. (2) Needless to say, these considerations are rLot applicable ill the general case.

The boundary B of R consists o~ two pieces, B =B~U B 2. To describe them we introduce ~ the distance function d ( X , E ) = distance of X from E. Then B 1 is the

"surface" y = a - l . d ( X , E ) , lying over those points X so t h a t d ( X , E ) < o : h . B 2 is t h a t portion of the hyperplane, y = h , lying over those X for which d ( X , E)<~o:h.

A basic step in the argument t h a t follows is the application of Green's Theorem to certain integrals extended over the region R. This requires t h a t we approximate our given regior~ by a family of smooth regions for which Green's Theorem is appli- cable. This is accomplished in the lemma below.

LEMMA 3. There exists a /amily o/ regions R~, e > 0 , with the/ollowing properties.

(1) R ~ c R

(2) R ~ c R ~ , i/ e~<el

(3) Re --> R as e --> 0 (i.e., U R~ = R)

1 E v e n t h o u g h i t m a y b e m a d e c o n n e c t e d b y a n i n e s s e n t i a l m o d i f i c a t i o n . See f o o t n o t e (1) on p. 138.

144 : E L I A S M . S T E I N

(4) the boundary B, o/ R, is at a positive distance /tom E~ (y = 0), and consists o/

two pieces B~ and B~ so that

~ (X)

(5)

B~ is a portion o/ the sur/ace y = o~-l.(~(X), where ~ <~ 1, e > O, k = l . . . n, and (~,(X) 6 C ~.

(6)

B~ is a portion o/ the hyperplane y =h.

Proo/.

L e t

(5 (X) = d (X, E)

when d (X, E) ~< h, otherwise let (~ (X) = h. Then (~ (X) is defined on all of

En,

and as is easily seen satisfies the Lipschitz condition

I ~ ( x ) - ~ ( y > l < l x - yl.

L e t q, (X) be a C ~ " a p p r o x i m a t i o n to the identity". I t m a y be constructed as follows.

Take

q~(X) 6 C :r q~(X)>~O, q~(X)-O

if [X[>~I, and

f ~(XldX=l.

En

Set

q ~ ( X ) = u - ~ q ~ ( X / u ) .

L e t / v ( X ) =

f ( ~ ( X - Y ) q~,(Y)dY.

Then b y the usual argu-

. 2 E n

ments,

/~ (X) 6 C ~,

and ]~ (X) --> ~ (X) uniformly as ~1 -+ 0. L e t U = U (e) be so small so t h a t

I ], (X) - ~ (X)] < s, and set ~ (X) = f~ (X) + 2 s.

Taking a subset of the collection {~(X)} (with possible reindexing of the sub- script e) we obtain

(a) ~ (X) > ,~ (X)

(b) (~ (X) ~> (~ (X), if % > sl

(c) ~ (x) -+ ~ (x).

Define now the regions R, to be

R ~ = { ( X , y ) :

5 ~ ( X ) < ~ y , 0 < y < h } .

I n view of the fact t h a t 6 ( X ) = m i n {d(X, E),

h},

(a), (b) and (c) imply conclusions (1), (2) and (3) of the lemma.

The boundary B~ of Re is the union of two sets, B~ and B~:

and

B ~ = ( ( X , y ) :

1 ~ y = ~ ( X ) , 0 < y < h }

B , = { ( X , y ) : y = h , (5,(X)<~h}.

2

Ol~I T H E T H E O R Y O F H A R M O N I C : F U N C T I O N S OF S E V E R A L V A R I A B L E S 1 4 5

Clearly B~] is a portion of the smooth surface = y = 8 ~ ( X ) , while By is a portion of the hyper-plane y = h . In fact, the region R~ consists exactly in the set of points lying above B~ and below B~.

In order to conclude the proof of the lemma it remains to be shown that

~ ( X )

~ - x k ~<1"

In view of the definition of ~ (X) it is sufficient to prove a similar inequality for / 7 ( X ) = f d ( x - Y ) ~ % ( Y ) d Y .

En

I7 (x1) - l~ (x2) = f [8 ( x 1 - y ) - ~ ( x 2 - Y)] ~ (Y)

Now, d r .

(Xl) - 17 (X2) [ ~< [ X~ - X 2 [ f ~ (Y) d Y = [ X~ - X 21 owing to the fact t h a t ~ (X) Hence

117

satisfies the above discussed Lipsehitz condition. Therefore

~ l T ( x ) < 1, x k

~ (x)

and hence ~ ~<1 q.e.d.

3. Basic l e m m a s In all that follows IV u l will denote

We let fl, ~, k, and h be given positive quantities with /~ > ~, and k > h.

L]~MMA 4:. Let z e ( X , y ) be harmonic in the cone F(X0; fi, k) and suppose that l u (X, y) ] 4 1 there. Then

y l V u I < ~ A in the cone F ( X 0 ; ~ , h ) ,

where A = A (fl, ~, k, h) depends only on the indicated parameters but not on X o or u. (1) (i) The constants A, a, c , . . . need to be the same in different contexts.

146 WLL~S M. STEIN

Proo/.

We shall need the following fact: If u is harmonic in a sphere ~ of radius one in En+l, and its absolute value is bounded b y one there, then the value of I V u I at the center ~ of :E is bounded b y fixed constant A, which does not depend on u.

This m a y be read off from the familiar Poisson integral representation of harmonic functions in a sphere in terms of their boundary values. Alternatively, we m a y use the following indirect argument. Assuming the contrary, there would then exist a sequence un of functions harmonic in ~ and bounded by one in absolute value so t h a t ]Vun(a) l-+ co. B y a well-known property of harmonic functions, we can select a subsequence of the u= which converge together with all derivatives uniformly on every closed set interior to ~. This is a contradiction and proves the existence of the re- quired A.

If now :E is the sphere of radius @ and l u] is still bounded by one there, then I Vu(~)I~<A/@. This follows from our previous observation b y making a change of scale which expands each coordinate by a factor of @.

We now consider

u ( X , y)

which is harmonic in the cone F(fi, k ) = { ( X , y ) :

I i - X o l < f l y ,

0 < y < h } .

Let

(X, y)

be a n y point in the smaller cone P (~, h). Notice t h a t since ~ < fi, and h < k, there exists a fixed constant c > 0 , so t h a t the sphere of radius

c y

whose center is (X, y) lies entirely in P (fl, It).

We now apply the previous fact to the case where ~ is the sphere of radius

c y

whose center a is (X, y), and obtain

IV u (X, y) [ < A / c y, (X, y) e F (~, h)

t h a t is,

y l V u ( X , Y ) ] < ~ A / c ,

for (X,y) e F ( ~ , h ) , q.e.d.

LEM~A 5.

Suppose that u ( X , y) is harmonic in the cone

P(X0; fi, k)

and f f Y I - ~ ' V u ' 2dXdy<~l"

F (X0; fl, k)

Then y l v u ( X , Y ) I<-A

in F ( X 0 ; ~ , h ) ,

< fi, h < k. The constant A depends only on ~, fl, h and k and not on u or X o.

Proo/.

Let ~ denote a sphere located in

E~+I,+

and let a denote its center. Then b y the mean-value theorem

ax-~ = [ ~ d X d y ,

/c=O, 1 . . . . , n,

O N T H E T H E O R Y O F H A R M O N I C : F U N C T I O N S O F SEV:ERA~L V A R I A B L E S 147 where y = x0, a n d

I EI

d e n o t e s t h e n + 1 d i m e n s i o n a l v o l u m e of ~ . H e n c e , b y S c h w a r z ' s i n e q u a l i t y

~ u ( ~ ) 1 2 < ~ f f l ~ x ~u ~ d X d y .

E

A d d i n g , we o b t a i n

^{[v u} l ff lVnl dXdy.

2;

A r g u i n g as in t h e p r o o f of t h e p r e v i o u s l e m m a , we t a k e a = ( X , y ) t o be a n y p o i n t in F ( X 0 ; a, h); t h e n if Z is t h e s p h e r e of r a d i u s

cy

whose c e n t e r is

a,

~ F (X0; fi, k). N o t i c e t h a t [ ~ [ =

cy ~+1.

W e t h e r e f o r e h a v e

,.if/

I V u ( ~ ) l 2 = l V u ( X , y ) 1 2 < ~

cl I V u l 2 d x d y

2;

<-.c y ffy-n+llVul"dXdy c y- ffy- +'lVuI dXdy,

2; F

w h e r e F = F (X0;

fi, Ic).

This p r o v e s t h e l e m m a .

4. The generalized area theorem

T h e t h e o r e m w h i c h we shall p r o v e can be f o r m u l a t e d as follows.

TH]~OR]~M 1.

Let u ( X , y) be harmonic in E ~

n + l .

(a)

Suppose that /or every point X o belonging to a set E, u (X, y) is bounded in a cone F (Xo) whose vertex is X o. Then the generalized area integral (1)

f f y l - n [ V u l 2 d X d y

(4.1)

17 (X0)

is /inite for a.e. X o C E

(b)

Conversely, suppose that /or every X o EE, the integral

(4.1)

is finite, then u ( X , y) has a non-tangential limit /or a.e. X o E E.

Proof.

W e consider first p a r t (a).

W e m a y assume, w i t h o u t loss of g e n e r a l i t y , t h a t t h e set E h a s finite m e a s u r e , a n d b y t h e use of L e m m a 1, n e g l e c t i n g a set of a r b i t r a r i l y s m a l l m e a s u r e , we m a y (1) We use the terminology of "generalized area integral" although (4.1) when n > 1, no longer can be interpreted as an actual area or volume.

148 also t h e region.

: E L I A S M . S T E Y N

assume t h a t E is closed a n d b o u n d e d on t h a t u (X, y) is u n i f o r m l y b o u n d e d in

/~ = U r (Xo; fi, k) (4.2)

Xoe E

w h a t e v e r fixed fi a n d k we choose.

We shall show t h a t

A(Xo)= ff yl-~lVulXdXd

Y ( X ~

y is finite for a.e. X o E E ,

f A (Xo) dXo< ~.

E

L e t ~ (Xo; X, y) be t h e characteristic function of F(Xo; cr h). T h a t is, K~(Xo; X , y) = 1 if

otherwise tF (X0; X, y) = 0.

We must, therefore, show t h a t

is finite H o w e v e r f / F ( X o ; X, y) E

T h u s it suffices to show t h a t

jJ ,

R

where R is as in (4.3).

IX-Xol<Zcy

^{a n d}

O<y<h,

f f {f W(Xo; X,y) dXo}yl-nIvu(X,Y)]XdXdy

R~ E

dXo <~ f dXo=cyL

I x o - x l < ~ y

V u(X, y)]2dXdy< oo,

(4.4}

a n d t h u s R c / ~ , a n d hence u is u n i f o r m l y b o u n d e d in R also. I n order to show t h a t A ( X o ) < ~ , for a.e. X 0 E E, it suffices to show t h a t

where F (X0)= F (Xo; a, h) a n d h are fixed quantities chosen once a n d for all, a n d t a k e n so t h a t f l > a ,

k>h.

L e t R = U F ( X 0 ~, h) (4.3)

Xoe E

ON T H E T H E O R Y O F HARMO:NIC ~ U l q C T I O N S OF S E V E R A L V A R I A B L E S 149 We shall transform the integral (4.4) b y Green's theorem. I n order to do this we shall use the approximating smooth regions R~ discussed in Section 2. B y the properties listed in L e m m a 3 it m a y be seen t h a t (4.4) is equivalent with

f ^{f yIVu(x,} y) 12 d X dy<~c < ~ ,

R .

(4.5)

where the constant c is independent of e.

Since the region Re has a sufficiently smooth Green's theorem in the form

b o u n d a r y B~ we apply to it

B e B e

Here

8/8~1e

indicates the directional derivative along the outward normal to BE.

dye is the element of " a r e a " of Be.

I n the above formula we take

F = u ~,

and

G=y.

A simple calculation shows t h a t A ( u e)

= 2 1 V u r ,

since u is harmonic, white i~ is clear t h a t A ( y ) = 0 . Therefore, we obtain

B e R e

I t is therefore sufficient to prove t h a t

f [ ~u ~ u 2 aY'~

Be

(4.6)

Notice t h a t Be c R c / ~ . Hence, u and therefore u 2 is bounded uniformly there.

Moreover I ~ Y/~ ne I ~< I; notice also t h a t 8 u2/8 n = 2 u 8 u / 8 n. Thus

8u <21 I'Y" "Y" Ivul;

therefore b y L e m m a

3 y 8u2/~ne

is uniformly bounded in Be c R = [3 F (X0; r162

h),

be-

X~e E

cause u is bounded in R = [3 F (Xo; •, k).

Xoc: E

Hence the integral in (4.6) is uniformly bounded b y a constant multiple of

fdve.

B~

1 5 0 E L I A S M . S T E I N

However,

1 2

Be B e B~

N o w B~ is a p o r t i o n of t h e surface y = g - 1 . ~ e ( X ) . Therefore

t h e r e However, I ~ , ( X ) h x ~ l < l . Thus d ~ < ( l § Also B~ is a portion of t h e hyper-plane y = h . Since b o t h B~ a n d B~ are included in a fixed sphere, it follows t h a t [ d ~ is u n i f o r m l y hounded. This proves (4.5), a n d hence p a r t (a) of t h e t h e o r e m .

B~

We n o w pass t o t h e proof of p a r t (b). We t e m p o r a r i l y relable the set on which t h e integral (4.1) is finite b y calling it E 0. B y simple a r g u m e n t s we m a y reduce the hypotheses to:

(1) I f y~-~lVupdXdy

is u n i f o r m l y h o u n d e d as X0 ranges over E0, where

p (X,: 3, k)

a n d k are some fixed positive quantities, (2) t h e set E 0 is bounded.

Given n o w a n y ~/> 0, we m a y pick a

closed

set E, E c E 0 which satisfies t h e following two additional properties

(3) m ( E 0 - E ) <

(4) there exists a fixed ~,, so t h a t

m ( { y :

IX-Yl<<o}nEo)>~89 I / - r l < e } , if XeE,

0 < ~ < ~ ) , .

This can be done as follows. A l m o s t every X 6 E 0 is a p o i n t of density of Eo;

t h e n for such X we have

lim ~ ({ r: I x - Y I < ~o} n E0) = 1.

o+0 ~ ( { r : l x - r l < e )

Hence a simple a r g u m e n t shows t h a t for a n y ~/ we can find an a p p r o p r i a t e subset E of E 0 to satisfy (3) a n d (4).

We n o w fix the set E f o u n d in this way. I t will suffice to show t h a t u ( X , y) has a non-tangential limit for a.e. X 6 E. (Thus at t h e conclusion of t h e proof we let m ( E o - E) --~ 0.)

O N T H E T H E O R Y O F H A R M O N I C F U N C T I O N S O F S E V E R A L V A R I A B L E S 151

First step

We consider the region

R = U F (X0; ~, h),

XoeE

where ~</3,

h < k ,

a n d B is the boundary. The first step in the proof of p a r t (b) will be to show, in effect,

f

l ~ , l ~ d ~ < ~ . (4.7)

B

Of course, (4.7) as it stands is not meaningful, because u is not defined for all of B and neither is the element of " a r e a " d~.

To bypass these technical difficulties we consider again the approximating regions Re with their boundaries B~ discussed in section 2, and we show t h a t

f lul2 d ~,<<_c < ~,

(4.7")

Be

where the constant c is independent of e.

The proof of (4.7*) is in some ways a reversal of the argument used to prove p a r t (a). We begin b y showing t h a t

f f ylVul2 dX dy< ~.

(4.8)

R

This is done as follows. B y (1) we have

f f Y ~ - ~ I V u I ~ d X d y < ~ A < ~ , X o E E o.

I ~ (xo; fl, k)

Integrating over

E o

we obtain

fffz .(Xo) ~ ( X o ; X , y ) y l - n , V u ( X , y ) [ 2 d X d y d X o < ~ .

^(4.9) Here Zz. is the characteristic function of Eo, and ~F is the characteristic function of the cone F (X0;/3, k). We shall show t h a t

f

^{~I z (X0;}

X , y) d X o >~ c y~,

Eo

(4.]0) where (X,

y) E R, c > O.

1 5 2 F~LIAS M. S T E I N

Recall t h a t R = [J F ( Z ; : c , h ) . Thus (X,y) ER means t h a t there exists a Z E E zE~

so t h a t

]X-Z]<o~y,

0 < y < h . Since ~F is the characteristic function of the set where I X - X 0]</~y, 0 < y < k , we see t h a t

f

~F(X0; X,

y)dXo>~ f dX o.

E0 E0 n (IXo- Zl <(fl-~)y}

Since Z E E, an application of (4) shows t h a t the second integral exceeds c y n, (if 0 < y < h ) , for some appropriate constant c, c > 0 . This proves (4.10). Applying (4.10) in (4.9) proves (4.8). We now replace (4.8) b y an equivalent s t a t e m e n t

f

y l V u l ~ d X d y < ~ c < ~ .

f

R e

(4.11)

R~ are the approximating regions of R, and c is independent of r

We now transform (4.11) b y Green's t h e o r e m - - a s in the proof (a) of the theorem.

We obtain

f [ ~u~ 2 ~Y~

0<~ [Y~n~ - u ~n-~) dT:~<~c< ~.

(4.12)

B e

Now the b o u n d a r y B~ is the union of two parts B~ and B~. However, B~ is a portion of the hyperplane y = h (and thus at a positive distance from the b o u n d a r y hyper- plane y = 0).

Moreover, B2~ is contained in a fixed sphere. Thus the total contribution of the integral (4.12) over B~ is uniformly bounded. We therefore have

(4.13)

We claim t h a t

~y/~n,<~ -o~ (r162 +n)-89

I n fact, ~/~n~ is the ( o u t w a r d ) n o r m a l deriva- tive to the surface whose equation is F~ (X: y ) = c c y - ~ ( X ) = 0 . A set of (unnormal- ized) direction numbers for this direction is

~ y ' ~x 1' ~x 2 . . . ~x~/ - ~ ' a x 1 ' ~x 2 . . . ~x~ / "

However, I ~ ~ (X)/~ x k ] ~< 1. This shows t h a t ~ y / a ne ~< - ~ ( ~ § n ) - 89

ON T H E T H E O R Y OF HARMO:NIC FU1WCTIONS OF S E V E R A L V A R I A B L E S 153

t ^{au~l= 2 au}

B u t y ~ u . y . ~ ~< 2 1 u l" Y" I V u I. Moreover b y L e m m a 4 y IV u I is b o u n d e d in U F (Xo; a, h) a n d hence in BI~. Combining these facts in (4,13) we o b t a i n

Xoc E

We have seen in the proof of p a r t (a) t h a t ~ d r ~ is u n i f o r m l y bounded. L e t t i n g

. 1

J~ ⁼ u 12 d re,

we o b t a i n J~ ~< c 3 J~ + c 2.

Since c a a n d c 2 are independent of e, we t h e n have t h a t J~ is u n i f o r m l y b o u n d e d hence

f l~l~dv~<c< ~ .

(4.14)

.1o

Second step

W e n e x t seek to majorize t h e function

u(X, y)

b y another,

v(X,

y), whose non- tangential behavior is k n o w n to us. We proceed as follows. T h e " s u r f a c e " B~ is a p o r t i o n of t h e surface

y = ~-1. ~ (X).

L e t

/~(X)

be t h e function defined on y = 0, whose values are the projection on y = 0 of t h e value of u (X, y) on B~, a n d otherwise zero. T h a t is,

l~ ( x ) = u ( x , ~-1 ~ (x)),

for those (X, 0) lying below B~, otherwise

/ ( X ) =

0. W e claim

f l/~(x) l~ dX <c <

^{~ .}

E n

(4.15)

I n fact, since d T~ >~ d X, we have

f ll~(X)l~dXV fl.l"d~-<c

E n B~ 1

b y (4.14).

1 1 - 61173060. A c t a mathematica. 106. I m p r i m 6 lo 20 d 4 c e m b r o 1961.

< c o

1 5 4 E L I A S lVI. S T E I N

We now let v, (X, y) be the harmonic function which is the Poisson integral of the function ]]~(X)]. Thus

v,(X,

y ) =

[ P ( X - Z, y)Jh(Z)JdZ,

E n

where P (X, y) is the Poisson kernel P (X, y) = c; 1

(d + Ix

(For the needed properties of Poisson integrals, see the previous paper in this series, Section 3.)

We shall show t h a t there exists two constants cx and c 2 so t h a t

]U (X,

Y)] <

Cl

ve (X, y) + % (X, y) e R,. (4.16) B y the m a x i m u m principle for harmonic functions it is sufficient to show t h a t the inequality (4.16) holds for (X, y) belonging to the boundary B,. Now

B~=B~ O B~,

where B~ is a subset of the hyperplane y = h, lying in a fixed sphere. Since

v,(X,y)>~O,

we can satisfy (4.16) on B~ b y choosing % large enough (and independ- ent of e).

I t remains to consider (X, y ) E B1,. L e t us call a = (X, y). Since B1, c R = [J F ( X 0 ; ~ , h ) ,

X o e E

we can find a constant c > 0 , with the following property: The sphere ~ whose center is ~ = (X, y), and whose radius to

cy

lies entirely in [J F (X0;/~*, k*), where

X~r E

< fi* < fi, and h < It* </c. Recall t h a t the cones F (Xo; fi, ~) have ~the p r o p e r t y t h a t

f Y~ nlVu]2dXdy<~A<oo , XoEE.

:P (Xo; ~, k)

Making use of L e m m a 5, it follows t h a t

yIVulKAl<c~

for

(X,y) E U P(Xo;fl*,k*).

(4.17)

X o e E

L e t now a ' be another point in the sphere ~ , and let 1 be the line segment joining

~' with o. Then

ON T H E T H E O R Y OF I t A R M O N I C : F U N C T I O N S O:F S E V E R A L V A R I A B L E S ] 5 5

Since, however, l a ' - ~ l ~< radius of

~ = c y ,

it follows from (4.17) t h a t

lu(cr')-u(c;)l<~A,

if a ' E ~ . (4.18)

L e t S = t h a t portion of the surface B~ which lies in the sphere ~ . L e t IS[ denote its area

Nn B~

Since

d ~ >~ d X,

and ~ is a sphere of radius

c y,

we obtain after a simple geometric argument,

Isl> ay =,

where a is an appropriate constant, a > 0 . Using (4.18) we obtain

lu(o)l 1

s

I n view of our definition of

/~(X),

our estimate for I S I, and the fact t h a t

dv~<~(l+n~ 2)89

(see the proof of p a r t (a)), we then get

I ~ ( ~ ) l < b y -n

f I/~(Z)IdZ+A.

I X - Zl<cy

The Poisson kernel has the p r o p e r t y t h a t

P(Z,y)>~(b/cx)y -~

for

I X l < c y ,

where c 1 is an appropriate constant. We therefore obtain

lu(cr)l=lu(X, yll<cl f P(x-Z, yll/~(z)ldZ+A

^for

(X,y) e B 1.

This proves our desired estimate (4.16) on B 1. We have already remarked that~

on B~ it is semi-trivial. Hence, we have the estimate on B~ and therefore on R~.

Thus (4.16) is completely proved.

:Because of the uniform estimate (4.15) on the norms of

/~(X),

we can select a subsequenee ( x ) I ) of the functions {[/~ (X)I} which converge to

I/(x) l

e L 2 (En), weakly in L 2.

1 5 6 ELIAS M. STEIN

L e t v ( X , y ) = J R ( X - Z , y ) f i / ( Z ) l a Z be the Poisson integral of If

i.

each (X, y), y > 0 ,

v~k(X, y)-->v (X, y).

Since

R~--~R,

we then have

lu(X, y ) i < c l v ( X ,

y ) + c 2, (X,

y) ER.

This is the decisive majorization of u (X, y).

Then for

(4.19)

Final step

Because of the known behavior of Poisson integrals near the boundary, we can assert t h a t v (X, y) is bounded non-tangentially for almost every X 0 in En. More precisely, for a.e.

X o E E~, (X, y)

is bounded in the cone P (X0)= F (X0; a, h). (For these facts see I, Section 3.)

Because R = (J F(X0; ~, h), and (4.19), it follows t h a t for a.e.

XoEE, u ( X , y)

XoE E

is bounded in F(X0; ~, h). I n view of L e m m a 2, this shows t h a t u has a non- tangential limit for a.e. X 0 E E. This concludes the proof of the theorem.

Chapter II

The main purpose of this chapter will be the proof of Theorem 3 in Section 6.

Actually this will be an easy result of Theorem 1 proved in the previous chapter and a n auxiliary result, Theorem 2, which is contained in Section 5.

5. An Auxiliary theorem

Theorem 1 we have just proved is useful because--disregarding sets of measure z e r o - - i t shows t h a t the existence of non-tangential limits for harmonic functions is equivalent with the finiteness of certain integrals. I n m a n y cases these integrals are easier to deal with. We shall see t h a t this is the case in the following theorem which is of particular interest in terms of its applications considered in the following paragraphs.

I n w h a t follows u ( X , y) will denote a vector of k components (u 1 (X, y), u 2 (X, y) . . . . uk (X, y)),

where each component is harmonic. Similarly v (X, y) will denote a vector of m components (/c4m, in general), each component being harmonic. We shall set

O N T H E T H E O R Y O F H A R M O N I C F U N C T I O N S O F S E V E R A L V A R I A B L E S 1 5 7

similarly for v. W h e n we say t h a t

u (X, y)

has a non-tangential limit at a given point X0, we shall mean t h a t each component has, etc.

T~EOREM 2.

Let u (X, y), v (X, y) be harmonic in the cone

F (2(o;

fl, k). Let P

(D}

be a k x m matrix, each o/ whose entries is a homogeneous di//erential operator in ~/~y~

~/~xl, ..., ~/~xn o/ degree r, with constant coe//icients. Assume that u and v are re- lated by

~r u X

-~y~ ( , y) = P (D) v.

(5.1)

Assume also that f f yl-~ lvi2 d X d y <

oo.

F(X,; fl, k)

Then i/ ~ < fl, h < k, we can conclude that

f f y l - n l u l ~ d X d y < oo.

r(Xo; ~, h)

The equation (5.1) m a y be viewed as a relation between a harmonic function and its conjugates, in its most general form. Examples and interpretations of (5.1}

will be discussed in Sections 7 a n d 8.

Before proving the theorem we derive from it a particular consequence of in- terest.

COROLLARY.

Let the cones F

(Xo;/7, k)

and

F (Xo;

0% h) be as in the above theorem.

Suppose that H

(X,

y) is harmonic in the cone

F (Xo;/~, It).

( a ) . ff

^{y l - n}

OH dXdy<oo. _{a y}

I~(Xo; fl. k)

then Y ~xs d X d y < 0% ] = 1 , 2 . . . n.

f(Xo; ~, h)

then,

(b)

I / each o/ the integrals

~ f yl-n O H ~ d X d y < oo, ]=1, 2 . . .

^~,,

F(XD; fl. k)

f f y 1-n O H i ~ d X ~-~y[ d y < ~r

F(X.: r162 h)

158 E L I A S M . S T E I N

P a r t (a) of t h e c o r o l l a r y is p r o v e d b y t a k i n g

u = O H / O x s , v = O H / O y ;

t h e n (5.1)

fly1 ^~yy d y < ~ ,

for a.e.

X o E E ,

I ' ( X . )

o r

f f y l - n ( ~ ~U 2 1 d X d y < ~ '

for a.e.

X o E E . j=l ~Xj J

F ( X . )

T h i s is a d e f i n i t e s t r e n g t h e n i n g of T h e o r e m ], p a r t (b).

F o r t h e case n = 1 t h i s f a c t has a l r e a d y f o u n d a p p l i c a t i o n in c e r t a i n p r o b l e m s i n one r e a l v a r i a b l e , see S t e i n a n d Z y g m u n d [10]. I n t h a t case ( n = 1) t h e c o r o l l a r y follows from a t h e o r e m of F r i e d r i c h s , see [4]. A s far as t h e case of g e n e r a l n is concerned, F r i e d r i c h s has p r o v e d in [5] a g e n e r a l i z a t i o n of his p r e v i o u s r e s u l t . B u t t h i s does n o t o v e r l a p w i t h our r e s u l t for g e n e r a l n.

T h e p r o o f of T h e o r e m 2 will r e q u i r e t w o p r e l i m i n a r y l e m m a s of a n e l e m e n t a r y c h a r a c t e r .

LEMMA 6.

Let 9 (s) = "|el (t) dt. Then

. . ! t

8

0 0

T h i s is a w e l l - k n o w n i n e q u a l i t y of H a r d y , see [6].

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

b

F(8)= f /(t)dt.

as

b b

Yhe, f lF f ll(t) 3 dt (5.2)

0 0

LEMMA 7.

Let O ~ a o ~ a < ~ , and

b e c o m e s

~ u / ~ y = a v / a x j .

To p r o v e (b) we t a k e

u = ~ H / ~ y, a n d

v = (v~, v 2 .. . . , Vn) = (~ H / ~ xj, ..., ~ U / ~ x~).

T h e n (5.1) b e c o m e s

~ u / ~ y = - ( ~ v ~ / ~ x 1 + ~v~/~x~ ... + ~v,,/~x,~).

T h e m e a n i n g of this c o r o l l a r y in c o n n e c t i o n w i t h T h e o r e m 1 is clear. I n o r d e r t o p r o v e t h a t u (X, y) has a n o n - t a n g e n t i a l l i m i t a.e. in a set E E E~, i t suffices t o

O ~ T H E T I - I E O R Y O F I-IAP~lYIO:bTIC F U I ~ C T I O ~ S O F S E V E R A L V A R I A B L E S 159

Proo/.

First, if qP (s)= J ~ -

a s 0 0

from L e m m a 6 b y the change of variable

s-+as.

N e x t

b b b

t

a s a s a s

Applying the previous inequality proves the lemma.

We now come to the proof of Theorem 2. We shall assume for simplicity t h a t the vertex is X 0 = 0; this involves no loss of generality. We then relable the cones P (X0; fi, k) and F (X0; ~, h) as P (/~, k) and P (~, h) respectively.

N o w let @ denote t h a t segment of a r a y passing through the origin and lying in the cone P (~, h). L e t s be the parametrization of the segment @ according to its length, with s = 0 corresponding to the origin. With u (X, y) given, we shall define

% (s) by, % (s) = restriction of u ( X , y) to the r a y segment @.

We shall show t h a t

h P

J sl%(s) 12ds<A<~,

(5.3)

0

where the bound is independent of the r a y @ lying in F (~, h). If we prove this in- equality, then an integration of it over all Q of the t y p e specified will then prove our theorem. We therefore t u r n to the proof of (5.3).

B y (5.1) we obtain

h

- 1 1)! f T ) r - 1 (X, T ) ] d T + R . (5.4)

u(X, Y)~(r (Y- [P(D)v

Y

Since R involves only the values of u (X, y),

au(X, y)/ay

. . . ar 1 (u(X,

y))/ayr-1

at

y=h,

this t e r m is uniformly bounded.

We now examine the t e r m

P(D)v(X, ~)

in (5.4). We use again a fact used several times before: We can find a constant c, c > 0, so t h a t if Z (X, ~) is the sphere whose center is (X, ~), with

(X,

T) E P (~, h) a n d whose radius is c ~, then Z c F (/3, k).

We fix this constant c in the rest of this proof. We also need the following fact.

L e t P (D) be a fixed m a t r i x of differential polynomials, homogeneous of degree r, and let ~ be the sphere whose center is a and radius in & Then (if v (X, y) is harmonic)

IP(D) (5.5)

Z

160 : E L I A S M . S T ] ~

This m a y be seen as follows. First consider the case when Z is of radius one.

Notice t h a t the class of harmonic functions which satisfy

I'['[vl2dXdy<~l,

are uni-

Z

formly bounded on interior compact subsets, b y the mean value property. I t then follows b y an indirect a r g u m e n t (of the t y p e used in the proof of L e m m a 4) t h a t there exist a constant A so t h a t

[P(D)v((~)[~A,

for this class of functions. The general inequality (5.5) then follows b y a homogeneity a r g u m e n t which involves stretching each component b y the factor 5. Alternatively, (5.5) can be proved directly from the Poisson integral representation for spheres.

We then take (5.5) with ~ being the sphere • (X, ~) whose center is (X, T) a n d whose radius is c v, and substitute this estimate in (5.4). This gives

lu(X, y)l<B f z_~(n+8){

h

f f lvl2dXdy)~dT+A. ^(5.6)

y Z(X,~)

We now call S~ the " l a y e r " in the cone P(fl, k) contained between

"~-cv

and T + c ~ ; i.e.,

Then clearly, since ~ (X, T ) c 1 ~ (/~, It) we have ~ (X, T ) c ST, and therefore

ff,

Next, call 0 the angle t h a t the r a y ~ makes with the y-axis. Since the r a y is contained in the cone

IXl<~y,

0 < y , it follows t h a t 1/> cos

0>~ao=(1+~2)-89

Notice t h a t y = s cos 0, where s is the p a r a m e t e r of arc-length along ~. Recalling the definition of

ue (s),

(5.6) gives

h

lu~(s)l<~B J T-89

(5.7)

s c o s 0

We now invoke (5.2) of L e m m a 7. We therefore obtain

h h

0 0

However,

O N T H E T H E O R Y O F H 3 - R M O N I C F U N C T I O N S O F S E V E R A L V A R I A B L E S 161

h h

o o s~ r(~, k)

d X d Y ,

where Z (T,

X, y)

is the characteristic function of the layer S~ in F (fi, It).

y / ] - c

But fT-nZ(T, X, y)dT= f T - n d T ~ f T-nd-T=cly -n+l.

T--cT<y<T+CT; 0 < y < k Y l I + r

This shows t h a t

h

0 F(fl, k)

which proves (5.3).

I n t e g r a t i n g (5.3) over all r a y ' s • lying in F (~, h) shows the finiteness of the integral

f f yl . ]ul2 d X dy.

Here 15 is t h a t portion of the cone I X I < ~y, in the upper half-space, which is truncated b y the sphere I X ]2 + y2 = h 2. This " c o n e " differs from our original cone F (a, h) b y a set which lies a t a positive distance from the exterior of F (fl, It). Since u was assumed harmonic in F (fl, k) it is certainly bounded in F (~, h ) - 1 ~. Thus the integral over 1 ~ (~, h) is also finite. This concludes the proof of theorem.

6. General theorems about non-tangential limits o f conjugate functions We now come to the principal result of this paper.

u and v will denote, as in the previous section, vectors of harmonic functions of k a n d m components respectively.

THEOREM 3.

Let u (X, y) and v (X, y) be harmonic in E+n+~. Suppose that they satis/y the relation

Dr u

- - = P (D) v, (6.1)

~y~

where P (D) is a k• matrix whose entries are di]]erential polynomials (with constant

coe]/icients) homogeneous o] degree r, r >1 1. Suppose that/or a given set E, E c E,, v (X, y)

1 6 2 E L I A S M . S T E I N

has a non-tangential limit /or every X EE. Then u (X, y ) h a s a non-tangential limit /or a.e. X 6E.

Proo/.

I n order to show t h a t u ( X , y) has a non-tangential limit a.e. in E it suffices, b y p a r t (b) of Theorem 1, to show t h a t

f f y l - ' ~ l V u l 2 d X d y < o o

for ^{a . e .}

Xo6E.

F(X0; ~. h)

However, b y p a r t (a) of t h a t theorem is follows t h a t

f y~-'~lVv[2dXdy< ~ f

for a.e.

XoEE.

F(Xo: 3, k)

L e t now U denote the vector whose components are (0 u / 0 y, D u / 0 x 1 .. . . .

0 u/aXn).

Actually since u itself is a vector of k components, then U is a vector of / c ( n + l ) components, suitable arranged. Similarly let V = (~

v/O y, ~ v/~ xl, ..., 0 v/O xn)

be the indicated vector, of m ( n + l ) components. I f we differentiate the relation (6.1) suc- cessively with respect to

O/Oy, O/Ox 1 .. . . . O/axn

then we obtain the relation

0 r U _ / 5 (D) V. (6.2)

~yr

Here /5(D) is a

I c ( n + l ) x m ( n + l )

matrix which consists of n + l copies of P ( D ) arranged down the diagonal. Or p u t in another way, the m a t r i x ]3 (D) is the tensor product of P ( D ) b y In+l, where In+l is the (n + 1)x (n + 1) identity matrix.

Notice t h a t b y our definitions I V I = I V v ] and I U I = ] V u ] " We are thus in a position to apply Theorem 2, with (6.2) in place of (5.1). F r o m the finiteness of

F ( x , ; 3. k) F(X0; 3. k)

follows the finiteness of

f/ Yl-nlVul2dXdy=ffYl nlUI2dXdy;

l~(X0; ar h)

and thus by w h a t has been said above, we obtain the proof of the theorem.

O N T H E T H E O R Y O F H A R M O N I C F U N C T I O N S O F S E V E R A L V A R I A B L E S 163

Chapter lII

7. Various examples

W e consider first t h e g e n e r a l i z a t i o n of t h e C a u c h y - R i e m a n n e q u a t i o n s s t u d i e d in p a p e r I, a n d also d i s c u s s e d in t h e i n t r o d u c t i o n of t h e p r e s e n t p a p e r . L e t us c h a n g e t h e n o t a t i o n s l i g h t l y ( m a k i n g i t s y m m e t r i c in all v a r i a b l e s ) b y calling y = x o. T h u s t h e u n d e r l y i n g v a r i a b l e s arc x 0, x 1 . . . . , x~. S i m i l a r l y l e t us call u 0 = u, a n d u 1 = vl, uz =v2, . . . , u~ =v~. T h e n t h e e q u a t i o n s b e c o m e

W e t h e n h a v e

•

~ u k = 0

k = o ~ X k '

~ u k ~ u j

~ x j Oxk' o<~j, k ~ n .

(7.1)

T H E O R E M 4. L e t u0, ul, . . . , u~ the s y s t e m o/ /,unctions s a t i s / y i n g (7.1).

(a) S u p p o s e that u o has a non-tangential l i m i t /or each p o i n t (xl, x2, . . . , x~) be- longing to a set E c E~ ( = the h y p e r p l a n e x o = 0). T h e n / o r a.e. (xl, x 2 . . . x~) C E , the s a m e is true /or each uk, k = 1, 2 . . . . , n.

(b) Conversely, s u p p o s e that ul, u 2 . . . . , u~ each have non-tangential l i m i t s i n a se E c E ~ . T h e n /or a.e. p o i n t i n E the s a m e is true /or % .

P r o o / . To p r o v e (a) we use a u ~ / a y = a u o / a x k. F o r (b) use u o / a y = - (a U l / a xi + a u 2 / a x2 + . . . +

x=).

T h u s a n a p p l i c a t i o n of T h e o r e m 3 p r o v e s t h e t h e o r e m i m m e d i a t e l y .

I t is e v i d e n t t h a t t h i s t h e o r e m g e n e r a l i z e s t h e c o r r e s p o n d i n g classical r e s u l t for a n a l y t i c f u n c t i o n s of P r i v a l o v a n d Plessner.

I t m a y be seen t h a t t h i s t h e o r e m i s t b e s t possible in t h e following sense.

(a) I f we w a n t we e x i s t e n c e of n o n - t a n g e n t i a l l i m i t s for t h e n + 1 c o m p o n e n t s

% , u 1 . . . u~ (a.e. on a set E ) b y a s s u m i n g i t for o n l y one of t h e m , t h e n t h i s one m u s t be %.

(b) H o w e v e r , if we do n o t m a k e a n y a s s u m p t i o n s on u0, we m u s t a s s u m e t h a t t h e r e m a i n i n g n c o m p o n e n t s , ul, u2, . . . , u~, h a v e n o n - t a n g e n t i a l a.e. in E in o r d e r t o o b t a i n t h e conclusion for all t h e n + l c o m p o n e n t s . To show t h i s consider a n F (z) = u 0 (x 1 + i y) + i u 1 @1 + i y) which is a n a l y t i c for y > 0, b u t does n o t h a v e b o u n d a r y

1 6 4 E L I A S M. STEIN

values for y = 0 . Then the set (u0, ux, 0, 0 . . . 0) satisfies (7.1) b u t does not have non-tangential limits.

I t is to be recalled t h a t the system (7.1) is locally equivalent with one arising out of a single harmonic function H (X, y) via

~ H

u j = - - , X o = Y , O ~ ] < ~ n . (7.1") The system (7.1) (or alternatively (7.1")) m a y be thought of as the most direct generalization of the Cauchy-Riemann equations. However, there are notions of "con- j u g a c y " which have no direct analogue to the classical case b u t which are never- theless of interest in higher dimensions. A systematic approach to the possible no- tions of conjugacy (i.e., appropriate generalizations of the Cauchy-Riemann equations) involves the study of how these systems transforms under r o t a t i o n s - - a n d thus is in- t i m a t e l y connected with the theory of representations of the group of rotations in n + 1 variables. This problem will be treated in a future paper of Guido Weiss and the author. Here we shall consider only briefly some of the possible systems which arise.

F o r every integer r we shall consider the "gradient of order r " - - t h a t is, the system of harmonic functions obtained from a single harmonic function H (xl, x 2 .. . . . xn, y) as follows:

{ ~:H (Xl, _x~, :::, y) )

0 y~" ~ x~' ... ~ x~" ' (i~ + il -~ "'" -~ in : r) . (7.2) This system m a y also be characterized b y a set of equations like (7.1. We now state the following theorem which generalizes Theorem 4.

T H E O ~ E ~ 5. S u p p o s e that /or each p o i n t (xl, x 2 . . . . , xn) belonging to a set E c En, the ]unction ~ r H / a y r has a non-tangential limit. T h e n the same is true a.e. i n E /or each other derivative o/ order r (i.e., other component o] (7.2)), a n d conversely.

This theorem, like Theorem 4, is an immediate consequence of the general Theo- r e m 3.

RemarIc. While this theorem is clearly a generalization of Theorem 4, it has only a secondary interest relative to Theorem 4. This is because the assumptions of the converse are to a large measure redundant. This m a y already be seen in the case r = 2 , n = 2 . The existence of the non-tangential limits of each of the following three sets of components implies t h a t a.e. all the other second order components have non- tangential limits:

O~N T I t E T H E O R Y O F I t A R M O : N I C I~UYCCTIO:NS O F S E V E R A L V A R I A B L E S 165

~ H {

a s H ~ H~

(1) ~ y ~ \ ~x~ ~ x ~ ~ ]

~ H ~ H

(2) - - and - - Oy ~xl ~ y ~ x 2

~2 H ~2 H ~2 H

- - and

(3)

~xl~x2 ~x~ ~x~"

T h a t (1) does is of course contained in Theorem 5. F r o m Theorem 4 it m a y be shown t h a t the existence of non-tangential limits of (2) also implies the other second- order components. (3) m a y be proved b y similar arguments.

This leads us to the following general question. Suppose

P1 (D), P2 (D) ... Pk (D)

are k given differential polynomials in a/~ y, ~/~ x~ . . . a/~ x~, homogeneous of degree r.

Question:

W h a t conditions m u s t be imposed on P1, Pz . . . Pk, so t h a t they are

determining

in the following sense: I f H is harmonic in @1, x2, x3, " " , x~, y ) a n d /)1 ( D ) H , P2 ( D ) H , . . . , Pk ( D ) H have non-tangential limits in E, then a.e. in E so does a n y derivative of order r of H.

We shall now a t t e m p t to answer this question.

Suppose t h a t

P(D)

is a homogeneous polynomial of degree r in

~/~y, ~/~x 1, .... ~/~x=,

we shall consider with in the associated polynomial,

p(X),

which is a polynomial of the n variables xl, x 2 .. . . . x~ of degree ~<r.

First, there exists a homogeneous polynomial of degree r in

~/~y, ~/~x 1 .. . . .

~/~x,~, P* (D),

so t h a t

(a) P* (D) H = P (D) H, whenever H is harmonic, (b)

P*(D)

is of degree ~<1 in

~/ay.

P* (D) is obtained from P (D) b y replacing aJ/a yJ b y ( - )89 (a2/a x~ + a~/a x~... + a2/a x~)}J if j is even; and replacing

~J/~x j

b y

~/~y(-) 89 1)(~2/~x~+ ...

+~2/~x~)89 ' if j is odd. Thus

( ~ + ~ t

where s and t are respectively homogeneous polynomials of degrees r and r - 1 , and

~/~ z ⁼ (a/~ xl, ~/~ x~ . . . ~/~ x.).

We then define p (X) b y

p ( x ) = s ( x ) + t ( x ) . Examples of this definition are as follows: