Journal of 191, 224–240 (2002) doi:10.1006/jfan.2001.3851

Norm Approximation by in Some Weighted Bergman Spaces1

Ali Abkar

Department of Mathematics, Imam Khomeini International University, P.O. Box 288, Qazvin 34194, Iran; and Department of Mathematics, Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-1795, Tehran, Iran E-mail: [email protected]

Communicated by D. Sarason

Received November 2, 2000; revised August 1, 2001; accepted August 1, 2001

The polynomials are shown to be dense in weighted Bergman spaces in the unit View metadata,disk whose citation weights and aresimilar superbiharmonic papers at core.ac.uk and vanish in an average sense at the brought to you by CORE

boundary. This leads to an alternative proof of the Aleman–Richter–Sundberg provided by Elsevier - Publisher Connector Beurling-type theorem for zero-based invariant subspaces in the classical Bergman . Additional consequences are deduced. © 2002 Elsevier Science (USA)

1. INTRODUCTION

We denote by D the unit disk, and by T its boundary in the complex plane. A weight function, or simply a weight,inD is any continuous posi- . p tive function w: D Q ]0,+ [. The weighted Bergman space La (D, w), 0

1/p 1 p 2 ||f|| p = F |f(z)| w(z) dA(z) <+., La(D, w) D

where dA(z)=p −1 dx dy is the normalized area measure on D.If . p 1 [ p<+ , it follows that La (D, w) is a Banach space with the above 2 , and for 0

2 Of, gP 2 =F f(z) g(z) w(z) dA(z), f, g ¥ L (D, w). La(D, w) a D

1 This research was supported in part by a grant from Institute for Theoretical Physics and Mathematics (IPM), Tehran, Iran. 224 0022-1236/02 $35.00 © 2002 Elsevier Science (USA) ⁄ All rights reserved. APPROXIMATION IN BERGMAN SPACES 225

In case that the weight w is identically 1, the corresponding space is called p the Bergman space, and we write La (D) for it, dropping the indication of the weight. A function u defined on D is said to be superbiharmonic provided that D2u \ 0, where D stands for the Laplace operator in the complex plane:

1 “2 “2 D=D = 1 + 2, (z=x+iy). z 4 “x2 “y2

In this paper, we shall consider a class of (non-radial) weights w satisfying the following conditions: (i) D2w \ 0, the weight function w is superbiharmonic, and > (ii) limr Q 1 T w(rz) ds(z)=0, where ds is the normalized arc-length measure on T. The main result of this paper (Theorem 2.6) states that the polynomials p . are dense in the weighted Bergman space La (D, w),0

p Background. A closed subspace M of the Bergman space La (D) is said to be invariant provided that zM … M. Assume that M is an invariant 2 subspace of La (D) with index one; this means that the quotient space M/zM has dimension one. In [6], Hedenmalm considered the following extremal problem,

(j) sup{Re f (0) : f ¥ M, ||f|| 2 [ 1}, La(D) where j is the multiplicity of the common zero at the origin of all the func- tions in M. The unique solution to this problem is called the Hedenmalm extremal function for M and is denoted by jM . The extremal functions in the Bergman spaces play the same role as the inner functions play in the Hardy spaces. Hedenmalm further showed that there is a function F 5 . jM ¥ C(D) C (D) such that F F 2 1. jM =0 on T and D jM =(|jM | −1) inside D, F 2 2. 0 [ jM (z) [ 1 − |z| , for z ¥ D. A particularly simple class of invariant subspaces is the one characterized by a zero sequence; a subset L … D is said to be a Bergman space zero 226 ALI ABKAR sequence provided that there is a function in the Bergman space which 2 vanishes precisely on L. Let M be the invariant subspace of La (D) consist- ing of all functions which vanish on L, and let jM be the corresponding extremal function for M. Now, the following question is natural: Does jM generate the invariant subspace M? Hedenmalm [6, Corollary 2.4] obtained the following reduction of the problem. Denoting by [jM ] the invariant subspace generated by jM,he found that

2 [jM ]=jM · A0 (jM), 2 where A0 (jM ) is the closure of the polynomials in the Hilbert space

2 3 2 F F 2 .4 A (jM )= f ¥ La (D): jM (z) D |f(z)| dA(z) < + , D with the following norm

2 2 2 2 2 F F D ||f|| A (jM) =||f|| L (D) + jM (z) |f(z)| dA(z). a D

2 By [6, Theorem 4.2], we know that M … jM · A (jM ). Now, the question if jM generates M becomes the question of whether the polynomials are 2 2 dense in the space A (jM ). We will show that the norm in A (jM ) is comparable to

2 2 F 2 ||f|| g =|f(0)| + wjM (z) D |f(z)| dA(z), D 2 2 F where wjM (z)=(1− |z| ) + jM (z). Hence the polynomials are dense in 2 A (jM ) if and only if they are dense in the weighted Bergman space 2 La (D, wjM ). It follows that jM generates M if the polynomials are dense in 2 the weighted Bergman space La (D, wjM ), where the weight function wjM has the following properties: 2 1. D wjM >0, and 2 2. 0

As an application of Theorem 2.6, we prove that jM generates the invariant subspace M (see Theorem 3.1), a result which was proved in a similar fashion by Aleman, Richter, and Sundberg (see [2, Propositions 2 4.4, 4.5]). Indeed, they proved that the polynomials are dense in A (jM ), 2 2 2 so that A0 (jM )=A (jM ) and hence [jM ]=jM · A (jM ). This answers the question of polynomial approximation for the weights wjM considered above. They showed that

F F 2 F F 2 jM (z) D |f(rz)| dA(z) Q jM (z) D |f(z)| dA(z), as r Q 1, D D APPROXIMATION IN BERGMAN SPACES 227 the proof of which uses the representation formula

F F 2 jM (z)= C(z, z) D |jM(z)| dA(z), z ¥ D, D whereby C(z, z) is the biharmonic Green function for the operator D2 in the unit disk:

z−z 2 C(z, z)=|z − z|2 log : : +(1 − |z|2)(1−|z|2), (z, z) ¥ D × D. 1−zz¯

Weights w satisfying the conditions (i) and (ii) need not have as simple a F representation as the function jM above. Nevertheless, according to a result due to the author and Hedenmalm [1], the weight function w can be represented by

(1.1) w(z)=F C(z, z) D2w(z) dA(z)+F H(z, z)dm(z), z ¥ D, D T where dm is a positive Borel measure on T and

(1−|z|2)2 H(z, z)= , (z, z) ¥ T × D, |1 − z¯z|2 is called the harmonic compensator (indeed, here we are using a special case of a representation formula established in [1]). To prove the main approximation theorem (Theorem 2.6), we consider fr —the dilation of f by r—defined by fr (z)=f(rz) for 0 [ r<1 and z ¥ D. We then prove that fr Q f in norm, as r Q 1, that is,

F p lim |fr (z) − f(z)| w(z) dA(z)=0, r Q 1 D from which the result follows.

2. AN APPROXIMATION THEOREM

In [2], the authors studied properties of a function which is closely related to the biharmonic Green function C(z, z), namely,

(1 − |z|2)2 (1−|z|2)2 C2(z, z)= , (z, z) ¥ D × D. |1 − z¯z|2 228 ALI ABKAR

Among other things, they proved that C2(z, z) has some kind of monotoni- city property. More precisely, for fixed z ¥ D, the function r W rC2(z, z/r) is increasing on the interval (|z|, 1) (see [2, p. 285]). In the following lemma, we shall see that for fixed z ¥ T, the harmonic compensator

(1−|z|2)2 H(z, z)= , (z, z) ¥ T × D, |1 − z¯z|2 enjoys the same monotonicity property; the function rH(z, z/r) is an increasing function of r for |z|

Lemma 2.1. Let z ¥ T be fixed and consider

(1 − |z|2)2 H(z, z)= , z ¥ D. |1 − z¯z|2

z Then for 0

r−a 2 b−a 2 |b−a|2 2 Re(b − a) : : =:1+ : =1+ + r−b r−b (r−b)2 r−b is a decreasing function of r. To prove the lemma, we note that

r+|z| 2 |z| 2 1 2 1`r − 2 +4 |z| z (r2 −|z|2)2 `r `r rH 1z, 2= = = . r r|r−z¯z|2 : r−z¯z :2 : r−z¯z :2 r−|z| r−|z| APPROXIMATION IN BERGMAN SPACES 229

According to the above observation, the denominator is a decreasing func- tion of r (with a=z¯z and b=|z|

Lemma 2.2. Let C(z, z) denote the biharmonic Green function for the unit disk. Then: (a) C(z, z) has the power series representation

. [(1 − |z|2)(1 − |z|2)]n+2 C(z, z)= C , (z, z) ¥ D × D. 2(n+1) n=0 (n+1)(n+2) |1 − z¯z|

(b) C(z, z)>0, for every (z, z) ¥ D × D. (c) For every (z, z) ¥ D × D we have

1 (1−|z|2)2 (1−|z|2)2 (1−|z|2)2 (1−|z|2)2 [ C(z, z) [ . 2 |1 − z¯z|2 |1 − z¯z|2

(d) For every (z, z) ¥ D × D we have

1 2 2 2 2 2 2 2 (1 − |z|) (1−|z| ) [ C(z, z) [ (1+|z|) (1−|z| ) . Proof. First of all we recall the elementary identity

z−z 2 (1 − |z|2)(1 − |z|2) (2.1) 1−: : = (z, z) ¥ D × D. 1−z¯z |1 − z¯z|2

Proof of (a). Putting

(1−|z|2)(1 − |z|2) (2.2) x= , (z, z) ¥ D × D, |1 − z¯z|2 we get |z − z|2=|1 − z¯z|2 (1−x). We now use the defining formula for the biharmonic Green function to obtain

(2.3) C(z, z)=|1 − z¯z|2 (1−x)log(1 − x)+(1 − |z|2)(1 − |z|2) =|1 − z¯z|2 [(1 − x) log(1 − x)+x]. 230 ALI ABKAR

Now, a manipulation with the power series expansion of log(1−x) on the right-hand side of (2.3) yields

. xn+1 . xn . xn+2 (2.4) (1−x)log(1 − x)+x= C − C = C · n=1 n n=2 n n=0 (n+1)(n+2)

It then follows from (2.2), (2.3), and (2.4) that

. (1 − |z|2)n+2 (1−|z|2)n+2 C(z, z)=|1 − z¯z|2 C 2(n+2) n=0 (n+1)(n+2) |1 − z¯z| . (1 − |z|2)n+2 (1 − |z|2)n+2 = C · 2(n+1) n=0 (n+1)(n+2) |1 − z¯z|

Hence (a) is proved. Part (b) follows immediately from (a). To verify (c), we note that the representing series of C(z, z) in part (a) above consists of nonnegative terms, so that the first term for n=0 can be served as a lower bound for the biharmonic Green function. Hence the first inequality in (c) follows. To obtain an upper bound for C(z, z), we use part (a) to write

. [(1 − |z|2)(1 − |z|2)]n+2 (2.5) C(z, z)= C 2(n+1) n=0 (n+1)(n+2) |1 − z¯z| (1 − |z|2)2 (1−|z|2)2 . xn = C , 2 |1 − z¯z| n=0 (n+1)(n+2) where x is given by (2.2). Since 0 [ x [ 1, it follows that the sum of the last series in (2.5) is no bigger than 1, so that the second inequality in (c) follows. As in [2], part (d) follows from part (c) and the inequalities

1 1 1 [ [ . (1+|z|)2 |1 − z¯z|2 (1−|z|)2

The proof is complete. L

We now proceed to prove that the biharmonic Green function C(z, z) has the same monotonicity property as the harmonic compensator H(z, z), refuting a statement made in [2, p. 307].

Lemma z 2.3. For 0

Proof. We use the following formula, due to Hadamard, to represent the biharmonic Green function C(z, z) in terms of the harmonic compen- sator H(z, z) (see [5, p. 74]):

1 1 p z z C(z, z)= F F H 1eih, 2 H 1eih, 2 sdh ds, (z, z) ¥ D × D. p max {|z|, |z|} −p s s

Hence for |z|

z 1 1 p z z r C 1z, 2= F F H 1eih, 2 rs H 1eih, 2 dh ds. |z| r p max {|z|, r } −p s rs

Note that the interval

|z| 5max 3|z|, 4,16 r

|z| gets bigger, as r increases. On the other hand, since r

z rs H 1eih, 2 rs is an increasing function of r, according to Lemma 2.1. The proof is complete. L

Proposition 2.4. Let w be a superbiharmonic weight function satisfying the condition (ii). Then the function rw(z/r), for 0

Lemma 2.5. Suppose that m is a finite positive measure on a measure . space X,0

F p F p . lim sup |fn | dm [ |f| dm <+ , n Q +. X X and fn Q f almost everywhere with respect to the measure m. Then

F p lim |f−fn | dm=0. n Q +. X 232 ALI ABKAR

We can now state the main result of this paper.

Theorem 2.6 (0

||f −f|| p Q 0, as r Q 1. r La(D, w) In particular, the polynomials are dense in the weighted Bergman space p La (D, w). p Proof. Let f be a function in the weighted Bergman space La (D, w), . 0

||f || p Q ||f|| p , as r Q 1. r La(D, w) La(D, w) We start by writing

p p p F ||fr || L (D, w) = |fr (z)| w(z) dA(z), a D and then replace z by z/r on the right side to obtain

p 1 p 1z 2 ||f || p = F |f(z)| rw dA(z). r La(D, w) 3 r rD r

By Proposition 2.4, the integrand is an increasing function of r, and therefore the monotone convergence theorem applies:

p p ||f || p Q ||f|| p , as r Q 1. r La(D, w) La(D, w) The proof is complete. L An alternative proof for Theorem 2.6. It is possible to prove Theorem 2.6 without any resort to Lemma 2.3. To do this, we use Fubini’s theorem and formula (1.1) to write

p p 2 p F F (2.6) ||fr || L (D,w) = |fr (z)| C(z, z) dA(z) D w(z) dA(z) a D D

F F p + |fr (z)| H(z, z) dA(z) dm(z). T D APPROXIMATION IN BERGMAN SPACES 233

Replacing z by z/r, we see that

1 z F F p F F p 1 2 |fr (z)| H(z, z) dA(z) dm(z)= 3 |f(z)| rH z, dA(z) dm(z). T D r T rD r

Together with Lemma 2.1 and the monotone convergence theorem, this implies that

(2.7)

F F p F F p lim |fr (z)| H(z, z) dA(z) dm(z)= |f(z)| H(z, z) dA(z) dm(z). r Q 1 T D T D

As in the proof of Theorem 2.6, we want to show that

||f || p Q ||f|| p , as r Q 1. r La(D, w) La(D, w)

To this end, by (2.6) and (2.7), it suffices to prove that

F F p 2 (2.8) lim |fr (z)| C(z, z) dA(z) D w(z) dA(z) r Q 1 D D

=F F |f(z)|p C(z, z) dA(z) D2w(z). D D

Recall that by Lemma 2.2(c)

1 2 2 (2.9) 2 C(z, z) [ C(z, z) [ C(z, z), (z, z) ¥ D × D, where C2 is given by

(1 − |z|2)2 (1−|z|2)2 C2(z, z)= , (z, z) ¥ D × D. |1 − z¯z|2

2 z Moreover, according to [2, p. 285], for |z| < r, the function rC(r , z) is an increasing function of r. It follows from (2.9) and a change of variables that

(2.10) 1 z F p 2 F p 2 1 2 2 |fr (z)| C(z, z) dA(z) D w(z) [ 3 |f(z)| rC , z dA(z) D w(z). D r rD r 234 ALI ABKAR

1 We may assume that 2 [ r<1. It follows from (2.9), (2.10), the monotone convergence theorem, and Lemma 2.2 that

F p 2 F p 2 |fr (z)| C(z, z) dA(z) D w(z) [ 16 |f(z)| C(z, z) dA(z) D w(z). D D

This shows that the dominated convergence theorem can be applied to (2.8), so that the equality (2.8) follows if we prove that

F p F p (2.11) lim |fr (z)| C(z, z) dA(z)= |f(z)| C(z, z) dA(z). r Q 1 D D

As for (2.11), we observe that by a change of variables,

1 F p 2 2 F p 2 2 2 |fr (z)| (1 − |z| ) dA(z)= 6 |f(z)| (r −|z| ) dA(z), D r rD so that the monotone convergence theorem applies:

F p 2 2 F p 2 2 . lim |fr (z)| (1−|z| ) dA(z)= |f(z)| (1−|z| ) dA(z) < + . r Q 1 D D

Since

p 2 2 p 2 2 Fr (z)=|fr (z)| (1−|z| ) Q F(z)=|f(z)| (1−|z| ) , as r Q 1,

1 it follows from Lemma 2.5 (with p=1) that Fr Q F in L (D, dA),asr Q 1; or

F p p 2 2 lim ||fr (z)| − |f(z)| |(1−|z| ) dA(z)=0. r Q 1 D

This, together with Lemma 2.2(d), implies that

F p p 0 [ ||fr (z)| − |f(z)| | C(z, z) dA(z) D

2 F p p 2 2 [ (1+|z|) ||fr (z)| − |f(z)| |(1−|z| ) dA(z) Q 0, as r Q 1. D

It now follows that

:F p p : (|fr (z)| − |f(z)| ) C(z, z) dA(z) D

F p p [ ||fr (z)| − |f(z)| | C(z, z) dA(z) Q 0, as r Q 1. D APPROXIMATION IN BERGMAN SPACES 235

Hence (2.11) is proved. Combining (2.11), (2.8), (2.7) and (2.6) we conclude that

||f || p Q ||f|| p , as r Q 1, r La(D, w) La(D, w) which completes the alternative proof of Theorem 2.6. Remark 2.7. In case that the weight function w is radial, that is, w depends only on |z|, the statement that the polynomials are dense in such weighted Bergman spaces is well-known; for a proof see [8, p. 343; 9, p. 131].

3. SOME APPLICATIONS

As a consequence of Theorem 2.6, we have:

Theorem 3.1 (Aleman–Richter–Sundberg). Let L be a zero sequence in 2 2 the Bergman space La (D), and let M be the invariant subspace of La (D) consisting of all functions vanishing on L. Then M is generated by its extremal function jM . 2 Proof. Let [jM ] denote the invariant subspace of La (D) generated by 2 jM. According to [6, Corollary 2.4] we know that [jM]=jM · A0 (jM ), 2 where A0 (jM ) is the closure of the polynomials in the space

2 3 2 F F 2 .4 A (jM )= f ¥ La (D): jM (z) D |f(z)| dA(z) < + , D with the norm

2 2 2 2 2 F F D ||f|| A (jM) =||f|| L (D) + jM (z) |f(z)| dA(z). a D

F Recall that jM was defined in Section 1. It is clear that [jM ] … M,so what we need to know is the inclusion

2 M … [jM ]=jM · A0 (jM ).

2 According to [6, Theorem 4.2], we have M … jM · A (jM), therefore it 2 2 suffices to show that A0 (jM )=A (jM ), that is the polynomials are dense 2 2 in the space A (jM ). To this end, we shall verify that the norm in A (jM ) is comparable to

2 2 F 2 (3.1) ||f|| g =|f(0)| + wjM (z) D |f(z)| dA(z), D 236 ALI ABKAR where

2 2 F wjM (z)=(1− |z| ) + jM (z).

It then follows from Theorem 2.6 that the polynomials are dense in the 2 weighted Bergman space La (D, wjM ), because the weight wjM is super- 2 biharmonic and 0 [ wjM [ 2(1 − |z| ). We start by verifying that for 2 f ¥ La (D) with f(0)=0 the following estimates hold:

2 2 2 2 2 2 2 F 2 (3.2) 3 ||f|| L (D) [ (1−|z| ) D |f(z)| dA(z) [ 2||f||L (D) . a D a

Indeed, an application of Green’s formula yields

(1−|z|2)2 F D |f(z)|2 dA(z) D 2

=F (2 |z|2 − 1) |f(z)|2 dA(z) D

2 2 2 2 2 =2 ||zf|| 2 −||f|| 2 [ 2||f|| 2 −||f|| 2 =||f|| 2 . La(D) La(D) La(D) La(D) La(D)

Hence the right-hand side inequality in (3.2) follows. As for the left-hand ;. n side inequality in (3.2), we note that for f(z)= n=1 an z (here we use the fact that f(0)=0) we have

|a |2 3 |a |2 n [ n , for n \ 1. n+1 2 n+2

This implies that

4 2 2 ||f|| 2 [ 2 ||zf|| 2 , 3 La(D) La(D) and finally,

2 2 (1 − |z| ) 2 2 2 F 2 2 D |f(z)| dA(z)=2 ||zf||L (D) −||f||L (D) D 2 a a

4 2 2 1 2 \ ||f|| 2 −||f|| 2 = ||f|| 2 . 3 La(D) La(D) 3 La(D) APPROXIMATION IN BERGMAN SPACES 237

Hence (3.2) is proved. We now consider the function

U(z)=−log |z|2+|z|2 −1, z¥ D.

Note that U(z) > 0 for every z ¥ D. It follows from Green’s formula that 2 for f ¥ La (D) we have

2 2 2 2 F (3.3) ||f|| L (D) =|f(0)| + U(z) D |f(z)| dA(z). a D

2 It now follows from (3.2) and (3.3) that for every f ¥ La (D) with f(0)=0 the following holds:

2 F 2 F 2 2 2 (3.4) 3 U(z) D |f(z)| dA(z) [ (1−|z| ) D |f(z)| dA(z) D D

[ 2 F U(z) D |f(z)|2 dA(z). D

Replacing f by f − f(0) in (3.4), we see that (3.4) is valid for every 2 f ¥ La (D). We now use (3.3) and (3.4) to obtain

2 2 2 2 2 F F D ||f|| A (jM) =||f||L (D) + jM (z) |f(z)| dA(z) a D

2 F 2 F F 2 =|f(0)| + U(z) D |f(z)| dA(z)+ jM (z) D |f(z)| dA(z) D D

3 3 2 F 2 2 F 2 4 [ 2 |f(0)| + ((1 − |z| ) + jM (z)) D |f(z)| dA(z) D

3 2 =2 ||f|| g . Similarly, we prove that

2 1 2 2 ||f|| A (jM) \ 2 ||f|| g ,

2 proving that the norm in A (jM ) is equivalent to ||·||g given by (3.1). This completes the proof. L Lemma 2.3 has further applications. It is shown in [2, Proposition 2.6] that for a given positive function v in the unit disk, and 0 [ r [ 1, the 3 inequality C[r vr ](z) [ 2 C[v](z) holds throughout D, where vr denotes the dilation of v by r, and C[v] is defined as

C[v](z)=F C(z, z)v(z) dA(z), z ¥ D. D 238 ALI ABKAR

We sharpen this estimate by dropping the constant 2 in the following proposition:

Proposition 3.2. Let v be a nonnegative function in D. Then for 0 [ r [ 1 and z ¥ D, we have

3 C[r vr ](z) [ C[v](z).

Proof. Together with Lemma 2.3, a change of variables gives

3 F 3 F z C[r vr ](z)= C(z, z)rvr (z) dA(z)= rC(z, r )v(z) dA(z) D rD

[ F C(z, z)v(z) dA(z)=C[v](z), D which is the desired result. L

The following definition of a Bergman space inner function is now standard (see [2, 6]).

Definition 3.3. An analytic function j in the unit disk is said to be an p La (D)-inner function provided that

F |j(z)|p h(z) dA(z)=h(0), D for every bounded harmonic function h in D.

In a Banach space of analytic functions X, the dilation operator

Rs :XQ X, 0 [ s [ 1, is defined by Rs f=fs , where fs is the dilation of f by s. p . Let j be an La (D)-inner function, 0

Theorem 3.4 . . p (1 [ p<+ ) Let j be an La (D)-inner function. Then 2 p p the operator Rs (f)=sfs is a contraction in P (|j| ), the closure of the polynomials in the weighted space Lp(D,|j|p). APPROXIMATION IN BERGMAN SPACES 239

Proof. Suppose f ¥ Pp(|j|p). It follows from the definition that there . exists a sequence of polynomials {pn }n=1 such that

F p p F p p . |pn (z)| |j(z)| dA(z) Q |f(z)| |j(z)| dA(z), as n Q + . D D

p This means that fj ¥ [j], the invariant subspace of La (D) generated by the inner function j. It is known [2] that

(3.5) p p p p p p F F ||fj|| L (D) =||f||L (D) + C(z, z) Dz |j(z)| Dz |f(z)| dA(z) dA(z). a a D D

Replacing f by fs in (3.5), we obtain

p p p p p p F F ||fs j|| L (D) =||fs || L (D) + C(z, z) Dz |j(z)| Dz |fs (z)| dA(z) dA(z). a a D D

z We now replace z by s in the last integral, and then multiply both sides by s to get

(3.6) p p 1 z 2 p p s||fj|| p =s ||f || p +F F sC z, D |j(z)| D |f(z)| dA(z) dA(z). s La(D) s La(D) z z sD D s

It follows from Lemma 2.3 and the monotone convergence theorem that the second term on the right side of (3.6) is no bigger than the second term on the right side of (3.5). This together with the fact that for 0

p p ||sf j|| p [ ||fj|| p , for 0

This completes the proof. L

ACKNOWLEDGMENTS

First of all, I express my sincere thanks to professor Håkan Hedenmalm (Lund). I am also grateful to the editor, Professor Donald Sarason (Berekely), whose invaluable suggestions improved the presentation of this paper. During the preparation of this paper I have had some oral/written communications with the following people: Alexander Borichev (Bordeaux), Stefan Richter (Tennessee), Stefan Jakobsson (KTH, Stockholm), and Alexandru Aleman (Lund). Here, I record my thanks to all of them. 240 ALI ABKAR

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