SOME RESULTS O N HP(D 2)

By JOSEPH A. CIMA AND ALAN S. COVER in Chapel Hill, N. C,, U.S.A. in Clemson, S. C., U.S.A.

1.

Introduction. This paper studies functions in Hardy spaces of several complex variables.

We will list and outline proofs for some of the standard representation theorems which hold in the several variable case. Our final and principal theorem is one concerning removable singularities of H p functions. In some cases we feel the theorems of section 3 to be known but since we have no specific references for their proofs and for coherence of the later section we include them.

2.

N o t a t i o n and d e f i n i t i o n s of the B a n a e h space

HP(D2).

For notational purposes we consider D 2 = D • D rather than D". Let F 2 be the distinguished boundary of D 2 and let dm be the measure on [0, 2n) x [0, 2n) given by dxidx2 (2~) 2 . I f f is holomorphic in D 2 and r j < 1 (j = 1,2) we set for p > l

M(f: p,

= f

[f(r e

rze'~)

IPdm.

A holomorphic function f is said to be in HP(D 2) if sup M(f: p, r j) is finite for rj < 1. For n = 1, M ( f ; p, r) is an increasing function of r and this fact can be used to prove that M ( f , p ; r 1, r2) is increasing with r I and r 2 . Setting

llsll.

=

lim ( M ( f ; p ; r l , r 2 ) ) lip = rl-~l

sup ( M ( f ; p ; r l , r 2 ) ) lip rt,r2 < 1

r2-~l

333

334

JOSEPH A. CIMA AND ALAN S. COVER

it is clear that HP(D 2) is a normed linear space. We quote two theorems of A. Zygmund (6, page 210) which are useful in our proofs, T h e o r e m A. I f f is in HP(D 2) and if A(xl) and A(x2) are triangular domains at e ix~ and ei~2 respectively (lying in D) then for almost all xl

and x2 the limit of f ( z , w ) exists for (z,w) in A ( x l ) x A(x2) and z ~ e ix~, w ~ e i~2. (The limit will be written as f * ( x l , x2) and is a complex number.) T h e o r e m B.

I f f is in H~'(D z) then

lim rl--~l r2-~l

exists and is zero. Finally for z = I zl

f If(rj% r2 e~') - f * ( x ~ , x2)['din

e'~ the

Poisson kernel will be written

(1-1zl

P:(t) = 1 - 2 1 z l c o s ( 0 3.

as

Pz(t)

2) t) + Izl 2"

HP(D 2) as a B a n a c h s p a c e and a c h a r a c t e r i z a t i o n .

We comment on an example before proving our results. Assume f ( z , Wo) is in HP(D) for each wo ~ D and that f(zo, w) is in HP(D) for each z o ~ D . We know that

such an f

is holomorphic in D 2 but the example

f ( z , w) = (1 - zw)-1 shows that f need not be in HP(D2). L e m m a 1.

I f f is in HP(D 2) then for each fixed z o in D , f ( z o , w) is in

HP(D). Proof.

The function ]f(z,w)l p is subharmonic in z for w fixed in D.

Thus 211l

I f(z~

)1 P

,f

<= 2re -

o

IS(:o +

pe",r:e":ll"dO

SOME RESULTS ON HP(D 2)

335

for p small. Results of R. M. Gabriel (4; p. 130) show that if 1 > r~ > ([ Zo I + P) we have 2~

2n

f

0

f ]f(rle ~', r2e'~2)['dx 0 .

0

Integrating with respect to

dx2

yields

2=kf If(zo,r2e,2)l,ax=< 2(iis11.r 0

Let f be holomorphic in D 2. Then if f is in

Theorem 1.

HP(D 2)

we

have f(z, w) = f P~(xl)Pw(xz)f*(xl, x2)dm. Proof.

Lemma 1 and the representation theorem for functions in

H~(D)

show that the theorem is true for f analytic on the closure of D z . Consider (z, w) in a compact set of D 2 say (1 z [ < rl) x ([ w [ < r2) and let p be sufficiently close to 1. Then

li(=, w) - f e=(xl)ew(x2)f*(xa, x2)dm [ < [ f (e(z/')(xl)e(w/P)(x2)-e'(xl)e'(x2))'f(a~ +

f

Pz(xl)Pw(x2)(f(peiX',pe'X2)

-- f*(xl,

x2))dm .

The first absolute value on the right tends to zero as p tends to one and Theorem B shows that the second absolute value tends to zero. Theorem 2.

HP(D 2)

is a Banach space.

336

JOSEPH A. CIMA A N D A L A N S. COVER

Proof.

Assume

{fk}

is a Cauchy sequence. The representation of Theo-

rem 1 shows that the family

(fk}

is uniformly bounded on compact subsets

of D 2 . A theorem of A. Julia (4; p. 58) shows that it is normal. Proceeding as in the one variable case one can show that a subsequence (f,k} converges to a holomorphic function F uniformly on compact subsets. Also F will be

IIf , - F II. + 0

in HP(D 2) and

u(z,w)

A function

Definition.

defined in D 2 is biharmonic if it is har-

monic in each of its variables separately (for the remaining variable fixed).

Let f be holomorphie in D z. Then f is in HP(Dz) if and only if there is a biharmonie funetion u satisfying If(z,w) l, ____.(.,w) o n D'. T h e o r e m 3.

Proof.

If such a u exists then

M(p;rl, r2) < Now if

feHP(D 2)

(u(0,0)) p.

and P~ is the Poisson kernel for the disk I zl < r then

we know that the functions

U,,,,~(z,W)

f P'~' (x~)P~2 (x2)]f(rae 'x', r2e'~) [Pdm

a r e b i h a r m o n i c i n ( i z I < rl) x

2
Iwol
= U(z,w)

exists.

To show U is biharmonic it suffices to show that the convergence is uniform on compact polydisks. But if we have

rl, r'l, rz, r'l

all greater than r o and

(z, w) e D,Zo.,o then

[ u,,.,~(z, ~) - u,,,.,,,(z, w) I =<

,(0 ~sup I P" (x,)P:: x j =<2~z + C"

P:', (x,)P;':

I1,

( f if(rleiXi, r2eiX2)iP-if(r'ieiXl, r'zei~')iP}dm I

SOME R E S U L T S O N

HV(D 2)

337

where C is a constant. Again Theorem B shows that the last integral is small and the sup on the right tends to zero as rj and r) tend to 1. 4.

Removable

singularities

for H p functions.

If C is a domain in ~2 and g is holomorphic in C we let z(g) be the zero set of g. It is known L. Bers (2; pp. 51-52) that i f f is holomorphic and in H ~~ of C - z(g) then f has a holomorphic extension to all of C. Using the equivalence of Theorem 3 we say that a f u n c t i o n f holomorphic on a domain C is in Hv(C) if there exists a biharmgnic function

U in C such that

If(z,w)[V< U(z,w) in C. With this agreement we have the following extension of the above result.

Let g be holomorphic in D 2 ( g ~ 0 )

T h e o r e m 4.

and assume f is in

HP(D z - z(g)), then f can be extended to holomorphic function on D z . Proof.

Before beginning our p r o o f we note that our result does not tell

us that f is in HV(D2). We have D E , to

0 2 --

z(g) a domain. It is sufficient to construct a function F on

show that F is holomorphic in each variable and that F(z, w) = f ( z , w)

on D E - z(g). Let us choose then a fixed z o and define F on (z0, w) for I w ] < 1 and then show that F is holomorphic as a function of w. Further we assume z o = 0 and by a transformation that

g(z, w) = (z" + A,_ l(w)z"- 1 + ... + Ao(w)" z + for

I zl _< ~

I wl z

and

~ (where f~ is a unit). Now set

A = {z~Olg(z,w)--O B

=

Ao(W))f~(z,w)

{w~olg(z,w)-O

all

Iwl
all

[z]
We claim A and B are countable with no accumulation points in D. For if not there would exist a sequence {zn} in A with z, distinct points and Zn--* Z*, I Z* 1 < 1. Choosing w, distinct and w, ~ w*, I w*l < 1

we

have

that g(z,,Wk) = 0 all n and all k. This would imply that g(z, w) = 0 on D 2 and so A (and B) are discrete. For zo = 0 not im A the set {w I g(O, w) = 0} is at most countable. We may assume then that our ~ and ~/are so small that g has precisely say k roots

338

JOSEPH A. CIMA AND ALAN S. COVER

I wl---n

in

for each

I zl =< ~.

Let Izll < ct and designate the roots by

wl, "", Wk with wj = w/z~) and g(z~, w) # 0 if w is not equal to some wi.

I zl _-< ~ and

Also we may assume that no z in A lies in

no w in B lies on

[ w [ = t/. We will define F on the points (0, wj) and we consider only say 0 = wl, the definition for F on the other points (0, wj) being similar, f(0, w) is holomorphic on

Iwl _-
except at

wt,'",wk. By assumption [f(0,w)] p

< U(O,w) for w # Wk implies by M. Parreau [5; p. 14] t h a t f i s holomorphic (can be extended) to the points wt, ..., Wk. Setting "f(0,e)

F(O, w) =

if w # wj

f f(O, w)dw ~--~_~-~j)

if w = wj; where

d Cj

Cj is a sufficiently "small" circle about wj, F is the holomorphic extension of f . We show that F is uniquely defined at ( 0 , w l ) = (0,0). Consider Wo fixed on Ca say ]Wo[ =

p where again no w in B lies on lw I = p. For lzl _-<

f(z, Wo) holomorphic except perhaps at a finite number of points zl,...,zs with IzJI = ~ (g is essentially a polynomial in Izl z ~ with

we have say

w = Wo). That is, again by M. Parreau, the function

f f(z, Wo)

if z ~ z,, I~1 __s

/

hwo(Z) = ] ~ f(z_,W__o)dz l j z-zj

if z = zj; where

t. D j

Di is a "small" circle about z j, is holomorphic in [z [ < a and

hwo(Z)=f(z,wo)

almost everywhere on ]z I = a. Thus since wo is arbitrary,

F(O, O) =

f f(O, w)dw Cj

hw(z)dz cl

w = j

[zl =~

[z I = a

z

zy,,z

w)dw w

c~

= f f(z,O)dZz Iz;

=~

and F is well defined at (0, wl) = (0, 0). We have used the fact that if 0 # zj we have hwo(O) = f(O, Wo) and if 0 = z l , we have

SOME RESULTS ON HP(D2) hwo(0) =

f f(z, WO)zd z = Dj

f

339

f(z, wo)dZ.z

Izl =~

By the definition then F(0, w) is holomorphic for

Iwl

~ in particular at

wl = 0 so that F is our extension. This result implies that for each z not in A we can define the function F(z, w) as a holomorphic extension o f f . A repetition of this proof also shows that for w not in B we can define F as a holomorphic extension of f . Now assume again Zo = 0 but 0 is in A. There is an a > 0 such that each z in 0 < [z [ < a is not in A. Thus F(z, w) is holomorphic in w (1 w [ < 1) if z is in this annulus. Now if w is not in B we know that F(z, w) is holomorphic in

I zl ___~.

The points (0,w*) for w* in

B are the only points of

([ z I < ~) x (l w [ < 1) where F is not holomorphic. These points are isolated and so F is holomorphic there. That is F is defined at z = 0 for [ w [ < 1 and is holomorphic. This defines F holomorphically on the polydisk. Now the boundary values f * of those f in HP(D 2) form a closed subspace of LP(F2). A result in the one variable case states that HP(F)is the LP(F) closure of those functions having negative Fourier coefficients equal to zero. Let us define then for f * in L (F 2) the Fourier coefficients

C,,k(f*) = Cm,k = f f * ( x l , xz)e-*t=xa+~a)dm where k and m are integers. We have then the following theorem.

Theorem 5.

Let f* be in LP(F2) (p > 1) and assume the Fourier coef-

ficients off* are zero if m or k is less than zero. Then f(z, w) = f Pz(xl)Pw(x2)f*(xl, xz)dm is in HP(D2). The converse is valid. Proof.

Let f,,,,2 be the function defined by

f~,,,e(xl, x2) = f(rle'*', r2e'*~).

340

JOSEPH A. CIMA A N D ALAN S. COVER

f,,,,2 is continuous for ri, r 2 < 1. Using the facts that L p and

Lq(1/p 4- 1/q = 1)

are dual spaces and that translation is continuous at (0, 0) one can prove as in the one variable case (by applying the Hahn-Banach Theorem) that 11/,,,,2-/*11,-+ 0 as r j ~ 1. We already know that f(z,

w) is harmonic in the polydisk and that if

Cz(0 = ~(( - z)-1 is the Cauchy kernel for the unit disk the function

f(z,w) = f Cz(Ci)'Cw(C2)'f*(xl, x2)dm is holomorphic in the polydisc. If z = +oo

f(z, w) =

+oo

]~

~,, m=-Oo

k=-oo

f =

rle ~~ and w = rze io~ we see that

~']kl"[m] "I / 2

e z(k*, + m d ' 2 ) f * ( ~ l ,

9

~2)KFF/((~ 1 , t~2)

oo 2~o k m f ei(k*'+m*2)f*(~l,qbz)dm = f(z,w) ~.~ F1F2 k=0 m=0

or that f is actually holomorphic. The converse follows immediately from the work of Zygmund quoted earlier. He showed that for a.e. x2 that f ( z , e*~2) is in

HP(D'). T h u s f * ( x l , x ~ is in H p

of the circle and its negative Fourier coefficients are zero. So that if say n > 0 we have 2~

if

2--~

f*(xl'

x2)ei"X'dxl = 0

a.e. x2.

0

Thus 2~t

2~

f f f*(xi'x2)einxlelkx2dxldx2 o o if n or k is a positive integer.

=0

SOME RESULTS ON HP(D 2)

341

REFERENCES 1. S. Bergman and J. Marcinkiewicz, Sur les fonctions analytiques de deux variables complexes, Fundamenta Mathematicae, 33 (1939-1946), 75-94. 2. L. Bers, Introduction to several complex variables Notes Courant Institute of Mathematical Sciences, 1964 N.Y. University. 3. R. M. Gabriel, An inequality concerning the integrals of positive subharmonic functions along certain curves, Journal London Math. Soc., 5 (1930), 129-131. 4. G. Julia, Sur les familles de fonctions analytiques de plusieurs variables, Acta Mathematica, 47, (1926) 53-115. 5. M. Parreau, Sur les mo3,ennes des fonctions harmoniques et analytiques et la classification des surfaces de Riemann, Annales de l'Institut Fourier, 3 (1951) 103-197. 9. A. Zygmund, On the boundary values of functions of several complex variables, I. Fundamenta Mathematicae, 36 (1949) 207-235. THE UNIVERSITYOF NORTH COROLINA

CHAPELHILL, N.C., U.S.A. AND CLEMSON UNIVERSITY CLEMSON, S.C., U.S.A.

(Received July 28, 1968)