Essential norm of products of multiplication composition and differentiation operators on weighted Bergman spaces

Applied Mathematics and Computation 218 (2011) 2386–2397

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Essential norm of products of multiplication composition and differentiation operators on weighted Bergman spaces Stevo Stevic´ a,⇑, Ajay K. Sharma b, Ambika Bhat b a b

Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia School of Mathematics, Shri Mata Vaishno Devi University, Kakryal, Katra 182320, J&K, India

a r t i c l e

i n f o

a b s t r a c t Let w be a holomorphic function on the open unit disk D and u a holomorphic self-map of D. Let C u ; Mw and D denote the composition, multiplication and differentiation operator, respectively. We find an asymptotic expression for the essential norm of products of these operators on weighted Bergman spaces on the unit disk. This paper is a continuation of our recent paper concerning the boundedness of these operators on weighted Bergman spaces. Ó 2011 Elsevier Inc. All rights reserved.

Keywords: Product operator Multiplication operator Bergman space Essential norm Unit disk

1. Introduction Let D be the open unit disk in the complex plane C, @D its boundary, HðDÞ the space of all functions holomorphic on D and H1 ðDÞ ¼ H1 the space of all bounded holomorphic functions with the norm kf k1 ¼ supz2D jf ðzÞj. Let dmðzÞ ¼ p1 dxdy be the normalized area measure on D (i.e. mðDÞ ¼ 1). For each a 2 ð1; 1Þ, we set

dma ðzÞ ¼ ða þ 1Þð1  jzj2 Þa dmðzÞ;

z 2 D:

Let Lpa ðDÞ ¼ Lpa , p > 0, a > 1, be the weighted Lebesgue space containing all measurable functions f on D such that jf ðzÞjp dma ðzÞ < 1. By Apa ðDÞ ¼ Apa we denote the space Lpa ðDÞ \ HðDÞ, which is called the weighted Bergman space. For D 1 6 p < 1 the space is Banach with the norm R

kf kApa ¼

Z

1=p jf ðzÞjp dma ðzÞ :

D

For the case a ¼ 0, space Apa will be denoted simply by Ap . It is well known that f 2 Apa if and only if f 0 ðzÞð1  jzj2 Þ 2 Lpa . Moreover the following asymptotic relation holds

kf kpAp  jf ð0Þjp þ a

Z

jf 0 ðzÞjp ð1  jzj2 Þp dma ðzÞ:

ð1Þ

D

Let u be a holomorphic self-map on D. The composition operator C u induced by u is defined by ðC u f ÞðzÞ ¼ ðf  uÞðzÞ; f 2 HðDÞ. For w 2 HðDÞ the multiplication operator M w is defined on HðDÞ by M w f ðzÞ ¼ wðzÞf ðzÞ; f 2 HðDÞ. The differentiation operator denoted by D is defined by Df ¼ f 0 ; f 2 HðDÞ. Some results on products of concrete linear operators can be found, e.g. in [4–11,16–18,21–49] (see also the references therein). ⇑ Corresponding author. E-mail address: [email protected] (S. Stevic´). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.06.055

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The products of composition, multiplication and differentiation operators can be defined in the following six ways

ðM w C u Df ÞðzÞ ¼ wðzÞf 0 ðuðzÞÞ; ðM w DC u f ÞðzÞ ¼ wðzÞu0 ðzÞf 0 ðuðzÞÞ; ðC u M w Df ÞðzÞ ¼ wðuðzÞÞf 0 ðuðzÞÞ;

ð2Þ

ðDMw C u f ÞðzÞ ¼ w0 ðzÞf ðuðzÞÞ þ wðzÞu0 ðzÞf 0 ðuðzÞÞ; ðC u DMw f ÞðzÞ ¼ w0 ðuðzÞÞf ðuðzÞÞ þ wðuðzÞÞf 0 ðuðzÞÞ; ðDC u Mw f ÞðzÞ ¼ w0 ðuðzÞÞu0 ðzÞf ðuðzÞÞ þ wðuðzÞÞu0 ðzÞf 0 ðuðzÞÞ

for z 2 D and f 2 HðDÞ. From operator M w C u D for wðzÞ ¼ 1, we get operator C u D, while for wðzÞ ¼ u0 ðzÞ, we get operator DC u , which have been studied, for example, in [4,6,9,11,14,17,28,30]. To treat operators in (2) in a unified manner, we introduced in [38] the following operator

T w1 ;w2 ;u f ðzÞ ¼ w1 ðzÞf ðuðzÞÞ þ w2 ðzÞf 0 ðuðzÞÞ;

f 2 HðDÞ;

ð3Þ

where w1 ; w2 2 HðDÞ and u is a holomorphic self-map of D. It is clear that all products of composition, multiplication and differentiation operators in (2) can be obtained from the operator T w1 ;w2 ;u by fixing w1 and w2 . Indeed, we have M w C u D ¼ T 0;w;u ; M w DC u ¼ T 0;wu0 ;u ; C u M w D ¼ T 0;wu;u ; DM w C u ¼ T w0 ;wu0 ;u ; C u DM w ¼ T w0 u;wu;u ; DC u M w ¼ T ðw0 uÞu0 ; ðwuÞu0 ;u . Let 0 < b < 1. Recall that a positive Borel measure l on D is called a b-Carleson measure if

klkb :¼ sup I@D

lðSðIÞÞ jIjb

< 1;

where SðIÞ ¼ fz : 1  jIj 6 jzj < 1; z=jzj 2 Ig is a Carleson box based on the arc I  @D of length jIj > 0. Measure l is a vanishing b-Carleson measure if

lim

jIj!0

lðSðIÞÞ jIjb

¼ 0:

This paper is continuation of its part one (see [38]), where we characterized the boundedness of operator (3) on weighted Bergman spaces by proving the following result. Theorem 1. Let 1 6 p < 1; a 2 ð1; 1Þ; w1 ; w2 2 HðDÞ and u be a holomorphic self-map of D. (1) If w1 2 H1 , then the following statements are equivalent: (i) T w1 ;w2 ;u is bounded on Apa . (ii) The pull-back measure lw2 ;u;a;p ¼ mw2 ;a;p  u1 of mw2 ;a;p induced by u is an ða þ 2 þ pÞ-Carleson measure. R 2 aþ2 Þ jw2 ðwÞjp dma ðwÞ < 1. (iii) supz2D D j1ð1jzj zuðwÞj2ðaþ2Þþp (2) If w2 satisfies the condition

M :¼ sup z2D

jw2 ðzÞj 1  juðzÞj2

< 1;

ð4Þ

then the following statements are equivalent: (i) T w1 ;w2 ;u is bounded on Apa . (ii) The pull-back measure lw1 ;u;a;p ¼ mw1 ;a;p  u1 of R 2 aþ2 Þ (iii) supz2D D j1ð1jzj jw1 ðwÞjp dma ðwÞ < 1. zuðwÞj2ðaþ2Þ

mw1 ;a;p induced by u is an ða þ 2Þ-Carleson measure.

Here we estimate the essential norm of the operator, and apply these results to concrete operators listed in (2). Recall that the essential norm kTke of a bounded linear operator T on a Banach space X is given by

kTke ¼ inffkT þ KkX : K

is compact on Xg;

that is, its distance in the operator norm from the space of compact operators on X. The essential norm provides a measure of non-compactness of T. Clearly, T is compact if and only if kTke ¼ 0. For some results in the topic see, e.g. [1–3,15,24,25,31,42], and the related references therein. Throughout this paper, constants are denoted by C, they are positive and not necessarily the same at each occurrence. The notation A  B means that there is a positive constant C such that B=C 6 A 6 CB.

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2. Auxiliary results Here we quote some results which are used in the proof of the main result. Let aþ2

ð1  jzj2 Þ

kz ðwÞ ¼

p

2ðaþ2Þ p

ð5Þ

:

ð1  zwÞ

The next two theorems can be found in [1]. Theorem 2. Let 1 6 p < 1, a 2 ð1; 1Þ and equivalent:

l be a finite positive Borel measure on D. Then the following statements are

(a) The inclusion map i : Apa ! Lp ðlÞ is bounded. (b) The measure l is an ða þ 2Þ-Carleson measure. R (c) supz2D D jkz ðwÞjp dlðwÞ < 1. Theorem 3. Let 1 6 p < 1, a 2 ð1; 1Þ and equivalent:

l be a finite positive Borel measure on D. Then the following statements are

(a) The inclusion map i : Apa ! Lp ðlÞ is compact. (b) The measure l is a vanishing ða þ 2Þ-Carleson measure. R (c) limjzj!1 D jkz ðwÞjp dlðwÞ ¼ 0. Since every holomorphic self-map u of D induces a bounded composition operator on Apa for all 0 < p < 1 and a > 1, by Theorem 2 we get the following result. Lemma 1. For every holomorphic self-map u of D, the pull-back measure

lu;a ¼ ma  u1 is an ða þ 2Þ-Carleson measure.

We also need the next results related to Theorems 2 and 3 ([12]). Before we formulate it we define a Dirichlet-type space with a finite positive Borel measure l

Dl ðDÞ ¼

  Z f 2 HðDÞ : kf k2Dl :¼ jf 0 ðzÞj2 dlðzÞ < 1 : D

Theorem 4. Let 1 6 p < 1, a 2 ð1; 1Þ and equivalent: (a)

R D

l be a finite positive Borel measure on D. Then the following statements are

jf 0 ðwÞjp dlðwÞ 6 Ckf kpAp for all f 2 Apa . a

(b) Measure l is an ða þ 2 þ pÞ-Carleson measure. R 0 (c) supz2D D jkz ðwÞjp dlðwÞ < 1. Moreover, the next asymptotic relation klkaþ2þp  C holds. Theorem 5. Let 1 6 p < 1, a 2 ð1; 1Þ and equivalent:

l be a finite positive Borel measure on D. Then the following statements are

(a) The inclusion i : Apa ! Dl is compact. (b) Measure l is a vanishing ða þ 2 þ pÞ-Carleson measure. R 0 (c) limjzj!1 D jkz ðwÞjp dlðwÞ ¼ 0. The generalized Nevanlinna counting function N u;k , is defined for k > 0 by

Nu;k ðwÞ ¼

k X  1 log ; jzj z2u1 ðwÞ

w 2 D n fuð0Þg;

where z 2 u1 ðwÞ is repeated according to the multiplicity of the zeros of the function u1 ðzÞ :¼ uðzÞ  w. We need the next lemma regarding function N u;k .

s be a holomorphic self-map of D, and let k P 1. If sð0Þ–0 and 0 < r < jsð0Þj, then

Lemma 2 [15]. Let

Ns;k ð0Þ 6

1 r2

Z rD

Ns;k ðzÞdmðzÞ:

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Lemma 3 ([20]). If g is a positive measurable function on D and u is a holomorphic self-map of D, then

 k Z 1 ðg  uÞðzÞju0 ðzÞj2 log dmðzÞ ¼ gðzÞNu;k ðzÞdmðzÞ: jzj D D

Z

The next formula follows from Lemma 3 ([19]).

kf kpAp  jf ð0Þjp þ a

Z

jf ðwÞjp2 jf 0 ðwÞj2 ð1  jwjÞ2 dma ðwÞ:

ð6Þ

D

Definition. We say that u has a finite angular derivative at a point f 2 @D if there is a point x 2 @D such that the difference quotient ðuðzÞ  xÞ=ðz  fÞ has a finite limit, as z tends non-tangentially to f. The next criteria for compactness of C u , were proved in [13]. Theorem 6. Let 1 6 p < 1, a 2 ð1; 1Þ and u be a holomorphic self-map of D. Then the following conditions are equivalent: (a) C u is compact on Apa . (b) u has angular derivative at no point on @D. (c) The pull back measure ma  u1 is a vanishing ða þ 2Þ-Carleson measure on D. Recall that

bðz; wÞ ¼

 þ jz  wj 1 j1  zwj log   jz  wj 2 j1  zwj

is the Bergman metric on D. Throughout this paper we fix some positive radius 0 < r < 1 and consider disks Dðz; rÞ in the Bergman metric. The set Dðz; rÞ ¼ fw 2 D : bðz; wÞ < rg; z 2 D, is called the hyperbolic disk or Bergman disk of radius r with hyperbolic center z. It is well known that Dðz; rÞ is a Euclidean disk whose Euclidean center and Euclidean radius are

ð1  s2 Þz 1  s2 jzj

2

and

ð1  jzj2 Þs 1  s2 jzj2

;

where s ¼ tanh r 2 ð0; 1Þ, respectively. The next lemmas list some additional properties of the Bergman disks (see, for example, [45]). Lemma 4. Let r, s and R be positive numbers. Then  when bðz; wÞ < r; (a) 1  jzj2  1  jwj2  j1  zwj, (b) mðDðw; sÞÞ  mðDðz; rÞÞ, when bðz; wÞ < R.

Lemma 5. Let r 2 ð0; 1 be fixed. Then there is a positive integer N and a sequence ðan Þn2N in D such that : (a) Disk D is covered by fDðan ; rÞgn2N . (b) Every point in D belongs to at most N sets in fDðan ; 2rÞgn2N . (c) If n–m, then bðan ; am Þ P 2r . It is well-known that a positive Borel measure

Ml :¼ sup z2D

lðDðz; rÞÞ ð1  jzj2 Þb

l on D is a b-Carleson measure if

<1

and that M l  klkb , while it is a vanishing b-Carleson measure if

lim

jzj!1

lðDðz; rÞÞ ð1  jzj2 Þb

¼ 0:

ð7Þ

Recall that if T w1 ;w2 ;u is bounded on Apa , then by taking the test functions f0 ðzÞ ¼ 1 and f1 ðzÞ ¼ z it follows that w1 ; w2 2 Apa [38]. Let

dmu;a;p ðzÞ ¼ ð1  jzj2 Þa juðzÞjp dmðzÞ: Note that if w1 and w2 are in Apa , then

mw1 ;a;p and mw2 ;a;p are finite measures.

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3. Essential norm In this section, we prove our main result in this paper, that is, estimate the essential norm of the operator T w1 ;w2 ;u on Apa and give its numerous applications. In order to give some upper and lower bounds for the essential norm of operator T w1 ;w2 ;u on weighted Bergman spaces, we need two more lemmas. The next lemma is motivated by Lemmas 1 and 2 in [2]. Lemma 6. Let 0 < r < 1, p P 1, a > 1,

Z

  M 1 r ¼ sup jzjPr

ð1  jzj2 Þaþ2 dl1 ðwÞ j1  zwj2ðaþ2Þ

D

and

Z

  M 2 r ¼ sup jzjPr

ð1  jzj2 Þaþ2 dl2 ðwÞ: j1  zwj2ðaþ2Þþp

D

f1 Þr and ð l f2 Þr denote the restrictions of l1 and l2 to D n Dð0; rÞ, respectively. Then, if l1 is an ða þ 2Þ-Carleson measure and Let ð l f1 Þr and ð l f2 Þr , respectively. Moreover, l2 is an ða þ 2 þ pÞ-Carleson measure for Apa , so are ð l

  f1 Þr kaþ2 6 N 1 M 1 r kð l

  f2 Þr kaþ2þp 6 N2 M 2 r ; and kð l

where N 1 and N 2 are some constants. Proof. Let i 2 f1; 2g and

li ðSðIÞÞ

ðM i Þr ¼ sup

jIjbi

0
;

where b1 ¼ a þ 2 and b2 ¼ a þ 2 þ p. Let I  @D be such that jIj > 0. Then jIj ¼ cð1  rÞ for some c 2 ð0; 1=ð1  rÞ. If 0 < c 6 1, then SðIÞ  D n Dð0; rÞ and so

ðf li Þr ðSðIÞÞ ¼ li ðSðIÞÞ 6 ðMi Þr jIjbi : If c > 1, then 1 < ð½c þ 1Þ=c 6 2. Let m ¼ ½c þ 1. Then I can be covered by m arcs I1 ; . . . ; Im such that jIk j ¼ 1  r, k ¼ 1; . . . ; m. Since bi > 1, i 2 f1; 2g, we have m X

ðf li Þr ðSðIÞÞ 6

li ðSðIk ÞÞ 6 ðMi Þr

k¼1

m X

jIk jbi ¼ ðMi Þr

k¼1

jIjbi

cbi

ð½c þ 1Þ 6 2ðMi Þr jIjbi :

This implies kðf li Þr kbi 6 2ðMi Þr .   To complete the proof, we prove ðM i Þr 6 N M i r for some N > 0. Let I be an arc such that jIj 6 1  r. Let a ¼ ð1  jIjÞeih , ih where e is the center of arc I. Then jaj ¼ 1  jIj P r. By a standard procedure we get a C > 0 such that

ð1  jzj2 Þaþ2 C P b ; bi þaþ2  jIj i j1  zwj

z; w 2 SðIÞ:

Hence

li ðSðIÞÞ jIjbi

6

1 C

Z SðIÞ

ð1  jzj2 Þaþ2 1 dli ðwÞ 6 C j1  zwjbi þaþ2

Z D

  Mi r ð1  jzj2 Þaþ2 d l ðwÞ 6 : i C j1  zwjbi þaþ2

  From this and by taking the supremum over I, we get ðM i Þr 6 M i r =C. h Let f ðzÞ ¼

P1

Q m f ðzÞ ¼

k¼0 ak z

m1 X

k

2 HðDÞ,

ak zk

and Rm f ðzÞ ¼ ðI  Q m Þf ðzÞ;

k¼0

where I is the identity map. Then clearly

Rm f ðzÞ ¼

1 X

ak zk :

k¼m

To estimate the essential norm of T w1 ;w2 ;u , we need the next lemma which is proved similar to Proposition 5.1 in [15]. Hence the proof is omitted.

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Lemma 7. Let T be a bounded operator on Apa . Then there is a C > 0 such that

C lim sup kTRm k 6 kTke 6 lim inf kTRm k: m!1

m!1

Theorem 7. Let p 2 ð1; 1Þ, a 2 ð1; 1Þ, w1 , w2 2 HðDÞ and u be a holomorphic self-map of D. Suppose that T w1 ;w2 ;u is bounded on Apa . (a) If w1 2 H1 and u has angular derivative at no point on @D when w1 X0, then there is an absolute constant C > 0 such that

C lim sup

Z

jzj!1

Z

ð1  jzj2 Þaþ2 1 jw2 ðwÞjp dma ðwÞ 6 kT w1 ;w2 ;u kpe 6 lim sup C jzj!1 j1  zuðwÞj2ðaþ2Þþp

D

D

ð1  jzj2 Þaþ2 jw2 ðwÞjp dma ðwÞ: j1  zuðwÞj2ðaþ2Þþp

(b) If w2 satisfies condition (4) and u has angular derivative at no point on @D when w2 X0, then there is an absolute constant C > 0 such that

C lim sup

Z

jzj!1

ð1  jzj2 Þaþ2 1 jw1 ðwÞjp dma ðwÞ 6 kT w1 ;w2 ;u kpe 6 lim sup C jzj!1 j1  zuðwÞj2ðaþ2Þ

D

Z D

ð1  jzj2 Þaþ2 jw1 ðwÞjp dma ðwÞ: j1  zuðwÞj2ðaþ2Þ

Proof. (a) Upper bound. We have

Z

kðT w1 ;w2 ;u Rm Þf kpAp ¼ a

jw1 ðwÞðRm f ÞðuðwÞÞ þ w2 ðwÞðRm f Þ0 ðuðwÞÞjp dma ðwÞ Z  Z jw1 ðwÞjp jðRm f ÞðuðwÞÞjp dma ðwÞ þ jw2 ðwÞjp jðRm f Þ0 ðuðwÞÞjp dma ðwÞ 6C D D   Z Z p p jðRm f ÞðwÞj dðma  u1 ÞðwÞ þ jðRm f Þ0 ðwÞjp dlw2 ;u;a;p ðwÞ ¼ CðI1 ðmÞ þ I2 ðmÞÞ: 6 C kw1 k1 D

D

ð8Þ

D

We have

jRm f ðwÞj ¼ jhRm f ; K w ij ¼ jhf ; Rm K w ij 6 kf kApa kRm K w kAqa ; 1 p

ð9Þ

1 q

where þ ¼ 1. We also have

  jðRm f Þ0 ðwÞj ¼ j Rm f ; K 0w j ¼ j f ; Rm K 0w j 6 kf kApa kRm K 0w kAqa :

ð10Þ

Let 0 < r < 1 and w 2 D be such that jwj 6 r. Consider the Taylor series expansion of K w ðzÞ ¼ we get the estimates

jRm K w ðzÞj 6

1 X ðk þ 1Þr k

and jðRm K w Þ0 ðzÞj 6

k¼m

1 X

kðk þ 1Þr k :

P1

k¼0 ðk

 k . Using this þ 1ÞðwzÞ

ð11Þ

k¼m

From (9) and the first inequality in (11) we get

sup jðRm f ÞðwÞj 6 Ckf kApa

jwj6r

1 X ðk þ 1Þr k ;

ð12Þ

k¼m

while from (10) and the second inequality in (11) we get

sup jðRm f Þ0 ðwÞj 6 Ckf kApa

jwj6r

1 X

kðk þ 1Þr k :

ð13Þ

k¼m

We have

I2 ðmÞ ¼

Z

þ

Dð0;rÞ

Z DnDð0;rÞ

! jðRm f Þ0 ðwÞjp dlw2 ;u;a;p ðwÞ:

ð14Þ

Using (13) it is easy to show that for a fixed r

lim sup

m!1 kf k 61 p A

Z Dð0;rÞ

jðRm f Þ0 ðwÞjp dlw2 ;u;a;p ðwÞ ¼ 0:

ð15Þ

a

Denote by lw2 ;u;a;p r ¼ lw2 ;u;a;p jDnDð0;rÞ . Since T w1 ;w2 ;u is bounded on Apa , by Theorem 1 we have that dlw2 ;u;a;p is an ða þ 2 þ pÞCarleson measure for Apa . Hence by Theorem 4 and Lemma 6, we have

Z

DnDð0;rÞ

  jðRm f Þ0 ðwÞjp dlw2 ;u;a;p ðwÞ 6 Cklw2 ;u;a;p r kaþ2þp kRm f kpAp 6 CN2 M2 r kf kpAp ; a

a

    where C and N 2 are constants and M 2 r ¼ M 2 r ðlw2 ;u;a;p Þ is defined as in Lemma 6.

ð16Þ

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If w1 0, then taking the supermum in (16) over all analytic functions f in the unit ball of Apa , then using such obtained inequality in (14), letting m ! 1 and employing (15), we get

    lim sup sup kðT w1 ;w2 ;u Rm Þf kpAp 6 C lim sup N2 M 2 r ¼ CN2 M 2 r : m!1

a

kf kAp 61

ð17Þ

m!1

a

Letting r ! 1 in (17) we get

lim sup kT w1 ;w2 ;u Rm kpe 6 CN2 lim sup m!1

Z

jzj!1

ð1  jzj2 Þaþ2 jw2 ðwÞjp dma ðwÞ; j1  zuðwÞj2ðaþ2Þþp

D

which by Lemma 7 gives the desired upper bound in this case. Now assume w1 X0, w1 2 H1 and u has angular derivative at no point on @D. To complete the proof we only need to show that supkf k p 61 I1 ðmÞ ! 0 as m ! 1. By (12) we can easily show that for each fixed r 2 ð0; 1Þ A

a

lim sup

m!1 kf k 61 p A

Z

jðRm f ÞðwÞjp dðma  u1 ÞðwÞ ¼ 0:

ð18Þ

Dð0;rÞ

a

Since u has angular derivative at no point on @D, by Theorem 6, we have that ma  u1 is a vanishing ða þ 2Þ-Carleson measure on Apa . Thus by (7) for each fixed q 2 ð0; 1Þ and every e > 0, we can choose an r 0 > 0 such that

ðma  u1 ÞðDðw; qÞÞ < eð1  jwj2 Þaþ2 for all w 2 D such that jwj > r0 . Let ðwn Þn2N be a sequence as in Lemma 5 such that jwn j 6 jwnþ1 j; n 2 N. Then ðma  u1 ÞðDðwn ; qÞÞ < eð1  jwn jÞaþ2 for all wn 2 D such that jwn j > r0 . Thus by Lemma 5, we have that for some r1 2 ðr 0 ; 1Þ and some k 2 N; D n Dð0; r 1 Þ # [n>k Dðwn ; qÞ and

Z

jðRm f ÞðwÞjp dðma  u1 ÞðwÞ 6

DnDð0;r 1 Þ

Z 1 X

6

1 X

jðRm f ÞðwÞjp dðma  u1 ÞðwÞ

Dðwn ;qÞ

n¼kþ1

ðma  u1 ÞðDðwn ; qÞÞ sup jðRm f ÞðzÞjp z2Dðwn ;qÞ

n¼kþ1

Z 1 X ðma  u1 ÞðDðwn ; qÞÞ

6C

ð1  jwn jÞaþ2

n¼kþ1

6 eCN

Z

D

jðRm f ÞðwÞjp dma ðwÞ Dðwn ;2qÞ

jðRm f ÞðwÞjp dma ðwÞ 6 eCNkf kpAp :

ð19Þ

a

Since e is an arbitrary positive number, from (18) with r 1 ¼ r and (19) the claim follows, and we get the desired upper bound. Lower bound. Consider the function kz ðwÞ. Then kkz kApa ¼ 1 and kz ! 0 uniformly on compact subsets of D as jzj ! 1. Fix a compact operator K on Apa . Then kKkz kApa ! 0 as jzj ! 1, since p > 1. Therefore,

 kT w1 ;w2 ;u þ Kk P lim sup kðT w1 ;w2 ;u þ KÞkz kApa P lim sup kT w1 ;w2 ;u kz kApa  kKkz kApa ¼ lim sup kT w1 ;w2 ;u kz kApa :

ð20Þ

kT w1 ;w2 ;u kpe ¼ inf kT w1 ;w2 ;u þ Kkp P lim sup kT w1 ;w2 ;u kz kpAp :

ð21Þ

jzj!1

jzj!1

jzj!1

Hence K

a

jzj!1

From the boundedness of T w1 ;w2 ;u it easily follows that there exists a constant C > 0 such that

Z D

ð1  jzj2 Þaþ2 jw2 ðwÞjp dma ðwÞ 6 C kT w1 ;w2 ;u kz kpAp þ a j1  zuðwÞj2ðaþ2Þþp

Z D

! ð1  jzj2 Þaþ2 p jw1 ðwÞj dma ðwÞ : j1  zuðwÞj2ðaþ2Þ

ð22Þ

If w1 0, then from (22) we get the desired lower bound. Now assume w1 X0; w1 2 H1 and u has angular derivative at no point on @D. Then by Theorem 6, ma  u1 is a vanishing ða þ 2Þ-Carleson measure. Hence we have

Z D

ð1  jzj2 Þaþ2 jw1 ðwÞjp dma ðwÞ 6 kw1 kp1 j1  zuðwÞj2ðaþ2Þ

Z D

ð1  jzj2 Þaþ2 dma ðwÞ ¼ kw1 kp1 j1  zuðwÞj2ðaþ2Þ

Z D

ð1  jzj2 Þaþ2 dðma  u1 ÞðwÞ ! 0; j1  zwj2ðaþ2Þ

as jzj ! 1, where in the last line we use Theorem 3. From this and by (20)–(22) we get the desired lower bound. (b) Upper bound. Similar to (a), we have

kðT w1 ;w2 ;u Rm Þf kpAp 6 C a

Z

jw1 ðwÞjp jðRm f ÞðuðwÞÞjp dma ðwÞ þ

D

¼ CðJ 1 ðmÞ þ J 2 ðmÞÞ:

Z

 jw2 ðwÞjp jðRm f Þ0 ðuðwÞÞjp dma ðwÞ

D

ð23Þ

S. Stevic´ et al. / Applied Mathematics and Computation 218 (2011) 2386–2397

2393

Let

Z

J 1 ðmÞ ¼

þ Dð0;rÞ

!

Z DnDð0;rÞ

jðRm f ÞðwÞjp dlw1 ;u;a;p ðwÞ:

ð24Þ

From (12) easily follows that for a fixed r

Z

lim sup

m!1 kf k p 61 A

Dð0;rÞ

jðRm f ÞðwÞjp dlw1 ;u;a;p ðwÞ ¼ 0:

ð25Þ

a

Let

lw1 ;u;a;p r ¼ lw1 ;u;a;p jDnDð0;rÞ . Thus by Lemma 6, we get Z

DnDð0;rÞ

  jðRm f ÞðwÞjp dlw1 ;u;a;p ðwÞ 6 Cklw1 ;u;a;p r kaþ2 kRm f kpAp 6 CN1 M1 r kf kpAp ; a

ð26Þ

a

    where C and N 1 are absolute constants and M 1 r ¼ M1 r ðlw1 ;u;a;p Þ is defined as in Lemma 6. If w2 0, then taking the supermum in (26) over all analytic functions f in the unit ball of Apa , using such obtained inequality in (24), letting m ! 1 and using (25), from (23) we get

    lim sup sup kðT w1 ;w2 ;u Rm Þf kpAp 6 C lim sup N1 M 1 r ¼ CN1 M 1 r : a

m!1 kf k p 61 A

m!1

a

Letting r ! 1, we get

kT w1 ;w2 ;u Rm kpe 6 CN1 lim sup jzj!1

Z D

ð1  jzj2 Þaþ2 jw1 ðwÞjp dma ðwÞ j1  zuðwÞj2ðaþ2Þ

from which by Lemma 7 we get the upper bound. Now assume w2 X0. We show supkf k p 61 J 2 ðmÞ ! 0 as m ! 1. As in (a), by Theorem 6 we have that ma  u1 is a vanishing A a ða þ 2Þ-Carleson measure on Apa . Thus for each fixed q1 2 ð0; 1Þ and every e > 0, we can choose an r2 > 0 such that

ðma  u1 ÞðDðw; q1 ÞÞ < eð1  jwj2 Þaþ2 for all w 2 D such that jwj > r 2 . Using assumption (4) we get

J 2 ðmÞ 6 M p

Z

jðRm f Þ0 ðuðwÞÞjp ð1  juðwÞj2 Þp dma ðwÞ:

ð27Þ

D

From (13) easily follows that that for each fixed r 2 ð0; 1Þ

lim sup

m!1 kf k p A 61

Z

jðRm f Þ0 ðuðwÞÞjp ð1  juðwÞj2 Þp dma ðwÞ ¼ 0:

ð28Þ

Dð0;rÞ

a

We have

Z

jðRm f Þ0 ðuðwÞÞjp ð1  juðwÞj2 Þp dma ðwÞ ¼

DnDð0;rÞ

Z

jðRm f Þ0 ðwÞjp ð1  jwj2 Þp dðma  u1 ÞðwÞ:

ð29Þ

DnDð0;rÞ

Let q1 2 ð0; 1Þ be fixed and ðwn Þn2N be a sequence as in Lemma 5 such that jwn j 6 jwnþ1 j, n 2 N. Then

ðma  u1 ÞðDðwn ; q1 ÞÞ < eð1  jwn jÞaþ2 for all wn 2 D such that jwn j > r2 . Similar to the first part, we have that for some r3 2 ðr2 ; 1Þ and k 2 N

Z DnDð0;r 3 Þ

Z 1 X ðma  u1 ÞðDðwn ; q1 ÞÞ

jðRm f Þ0 ðwÞjp ð1  jwj2 Þp dðma  u1 ÞðwÞ 6 C

ð1  jwn jÞaþ2

n¼kþ1

6 eC

Z

jðRm f Þ0 ðwÞjp ð1  jwj2 Þp dma ðwÞ

Dðwn ;2q1 Þ

jðRm f Þ0 ðwÞjp ð1  jwj2 Þp dma ðwÞ

D

6 eCkRm f kpAp 6 eCkf kpAp : a

a

ð30Þ

Applying (28) and (29) with r ¼ r3 , and (30) in (27), the claim easily follows, which implies the desired upper bound. Lower bound. Using the facts that w2 satisfies condition (4) and u has angular derivative at no point on @D, and proceeding as in the corresponding part of (a) we can obtain the desired lower bound. We omit the details. h

S. Stevic´ et al. / Applied Mathematics and Computation 218 (2011) 2386–2397

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Corollary 1. Let p 2 ð1; 1Þ; a 2 ð1; 1Þ; w 2 HðDÞ and u be a holomorphic self-map of D. (a) If Mw C u is bounded on Apa , then there is an absolute constant C > 0 such that

C lim sup

Z

jzj!1

D

ð1  jzj2 Þaþ2 1 jwðwÞjp dma ðwÞ 6 kMw C u kpe 6 lim sup C jzj!1 j1  zuðwÞj2ðaþ2Þ

Z D

ð1  jzj2 Þaþ2 jwðwÞjp dma ðwÞ: j1  zuðwÞj2ðaþ2Þ

p

(b) If C u M w is bounded on Aa , then there is an absolute constant C > 0 such that

C lim sup

Z

jzj!1

D

ð1  jzj2 Þaþ2 1 jwðuðwÞÞjp dma ðwÞ 6 kC u Mw kpe 6 lim sup C jzj!1 j1  zuðwÞj2ðaþ2Þ

Z D

ð1  jzj2 Þaþ2 jwðuðwÞÞjp dmðwÞ: j1  zuðwÞj2ðaþ2Þ

Corollary 2. Let p 2 ð1; 1Þ; a 2 ð1; 1Þ; w 2 HðDÞ and u be a holomorphic self-map of D. (a) If Mw C u D is bounded on Apa , then there is an absolute constant C > 0 such that

C lim sup

Z

jzj!1

D

ð1  jzj2 Þaþ2 1 jwðwÞjp dma ðwÞ 6 kM w C u Dkpe 6 lim sup C jzj!1 j1  zuðwÞj2ðaþ2Þþp

Z D

ð1  jzj2 Þaþ2 jwðwÞjp dma ðwÞ: j1  zuðwÞj2ðaþ2Þþp

(b) If M w DC u is bounded on Apa , then there is an absolute constant C > 0 such that

C lim sup

Z

jzj!1

D

ð1  jzj2 Þaþ2 jwðwÞu0 ðwÞjp dma ðwÞ 6 kMw DC u kpe j1  zuðwÞj2ðaþ2Þþp Z 1 ð1  jzj2 Þaþ2 6 lim sup jwðwÞu0 ðwÞjp dma ðwÞ: C jzj!1 zuðwÞj2ðaþ2Þþp D j1  

(c) If C u M w D is bounded on Apa , then there is an absolute constant C > 0 such that

C lim sup jzj!1

Z D

ð1  jzj2 Þaþ2 1 jwðuðwÞÞjp dma ðwÞ 6 kC u M w Dkpe 6 lim sup C jzj!1 j1  zuðwÞj2ðaþ2Þþp

Z D

ð1  jzj2 Þaþ2 jwðuðwÞÞjp dma ðwÞ: j1  zuðwÞj2ðaþ2Þþp

Corollary 3. Let p 2 ð1; 1Þ; a 2 ð1; 1Þ; w 2 HðDÞ and u be a holomorphic self-map of D. (1) If C u D is bounded then the following statements are equivalent: (a) C u D is compact on Apa . (b) The pull-back measure lu ¼ ma  u1 induced by u is a vanishing ða þ 2 þ pÞ-Carleson measure. R 2 aþ2 Þ (c) limjzj!1 D j1ð1jzj dma ðwÞ ¼ 0. zuðwÞj2ðaþ2Þþp Moreover, if 2 6 p < 1, then C u D is compact on Apa if and only if

Nu;aþ2 ðwÞ ¼ o

 aþ2þp ! 1 as jwj ! 1: log jwj

ð31Þ

(2) If DC u is bounded then the following statements are equivalent: (a) DC u is compact on Apa . (b) The pull-back measure lu;u0 ¼ mu0 ;a;p  u1 of mu0 ;a;p induced by u is a compact vanishing ða þ 2 þ pÞ-Carleson measure. R 2 aþ2 Þ (c) limjzj!1 D j1ð1jzj ju0 ðwÞjp dmðwÞ ¼ 0. zuðwÞj2ðaþ2Þþp Moreover, if a P 1, then DC u : A2a ! A2a is compact if and only if

Nu;a ðwÞ ¼ o



log

aþ2 ! 1 as jwj ! 1: jwj

Proof. (1) The equivalence of statements (a)–(c) follows from Theorem 7 (a) and Theorem 5. Hence, we have to prove only that (31) is equivalent to (a)–(c) for the case p P 2. Recall that the family of functions kw ðzÞ is not only bounded but converges uniformly on compacts of D as jwj ! 1. Hence if operator C u D is compact on Apa , then

lim kC u Dkw kApa ¼ 0:

jwj!1

ð32Þ

S. Stevic´ et al. / Applied Mathematics and Computation 218 (2011) 2386–2397

2395 0

On the other hand (next few lines are from [38], for the completeness), using (6) to the function kw  u, we get 0

kC u Dkw kpAp  jkw ðuð0ÞÞjp þ a

0

 jkw ðuð0ÞÞjp þ

Z

0

00

jkw ðuðzÞÞjp2 jkw ðuðzÞÞj2 ju0 ðzÞj2 dmaþ2 ðzÞ

D

Z

0

00

jkw ðuÞjp2 jkw ðuÞj2 Nu;aþ2 ðuÞdmðuÞ P Cða; pÞ

Z

D

D

p2

jwjpþ2 ð1  jwj2 Þaþ2 Nu;aþ2 ðuÞdmðuÞ;  2aþpþ6 j1  wuj

2

where Cða; pÞ ¼ ð2ða þ 2Þ=pÞ ð1 þ 2ða þ 2Þ=pÞ .  and using the fact that jb0w ðfÞj2 ¼ ð1  jwj2 Þ2 =j1  wfj  4 , we get By the change u ¼ bw ðfÞ ¼ ðw  fÞ=ð1  wfÞ

kC u Dkw kpAp P Cða; pÞ a

Z

 2aþpþ2 jwjpþ2 j1  wfj 2 aþpþ2

ð1  jwj Þ

D

Nu;aþ2 ðbw ðfÞÞdmðfÞ P Cða; pÞ

Z D=2

 2aþpþ2 jwjpþ2 j1  wfj ð1  jwj2 Þaþpþ2

Nu;aþ2 ðbw ðfÞÞdmðfÞ:

 P 1=2 for f 2 D=2. Using this fact, N u;c ðbw ðfÞÞ ¼ N bw u;c ðfÞ and the sub-mean value property of the NevanlNote that j1  wfj inna counting function, we obtain pþ2 Cða; pÞ jwj N u;aþ2 ðwÞ

22aþpþ2 ð1  jwj2 Þaþpþ2

6 kC u Dkw kpAp :

ð33Þ

a

Letting jwj ! 1 in (33), using (32) and a known asymptotic formula we get (31). Now assume (31) holds. Then for every e > 0 there is an r0 2 ð0; 1Þ such that

 aþ2þp 1 Nu;aþ2 ðwÞ < e log ; jwj

for r 0 < jwj < 1:

ð34Þ

Let ðfn Þn2N be a bounded sequence in Apa , say by L, converging to zero uniformly on compacts of D. From (6) and Lemma 3 we have that

kC u Dfn kpAp a



jfn0 ð

p

uð0ÞÞj þ

Z

þ

r0 D

Z

! jfn0 ðwÞjp2 jfn00 ðwÞj2 Nu;aþ2 ðwÞdmðwÞ:

Dnr0 D

    By the Weierstrass theorem we obtain that fn0 n2N and fn00 n2N also converge to zero uniformly on compacts of D. Hence

lim

n!1

Z

r0 D

jfn0 ðwÞjp2 jfn00 ðwÞj2 Nu;aþ2 ðwÞdmðwÞ ¼ 0:

ð35Þ

On the other hand, from (34), asymptotic relation (6) applied to the function fn0 , and finally by using (1), we obtain

Z

 aþ2þp 1 jfn0 ðwÞjp2 jfn00 ðwÞj2 log dmðwÞ jwj Dnr 0 D Z Z jfn0 ðwÞjp2 jfn00 ðzÞjdmaþpþ2 ðwÞ 6 eC jfn0 ðwÞjp dmaþp ðwÞ 6 eCkfn kpAp 6 eC

jfn0 ðwÞjp2 jfn00 ðwÞj2 Nu;aþ2 ðwÞdmðwÞ 6 e

Dnr 0 D

Z

D p

D

6 eCL :

ð36Þ

From (35) and (36) we get

lim kC u Dfn kApa ¼ 0

n!1

from which the compactness of the operator C u D on Apa follows. The proof of (2) is similar so is omitted. h If u is univalent and uðzÞ ¼ w, then

Nu;a ðwÞ ¼



log

1 jzj

a

 ð1  jzjÞa :

From this and Corollary 3 we obtain the next result. Corollary 4. Let 2 6 p < 1 and u be a univalent holomorphic self-map of D. Then the following statements hold. 2

aþ2

Þ ¼ 0. (a) If a 2 ð1; 1Þ. Then C u D is compact on Apa () limjzj!1 ð1jð1jzj uðzÞj2 Þaþ2þp 2

a

Þ ¼ 0. (b) If a 2 ½1; 1Þ. Then DC u is compact on A2a () limjzj!1 ð1jð1jzj uðzÞj2 Þaþ2

Corollary 5. Let p 2 ð1; 1Þ; a 2 ð1; 1Þ; w 2 HðDÞ and u be a holomorphic self-map of D. (1) The following statements are equivalent. (a) M w C u is compact on Apa .

a

S. Stevic´ et al. / Applied Mathematics and Computation 218 (2011) 2386–2397

2396

(b) The pull-back measure lw;u;a;p ¼ mw;a;p  u1 induced by u is a vanishing ða þ 2Þ-Carleson measure. R 2 aþ2 Þ (c) limjzj!1 D j1ð1jzj jwðwÞjp dma ðwÞ ¼ 0. zuðwÞj2ðaþ2Þ (2) The following statements are equivalent. (a) C u M w is compact on Apa . (b) The pull-back measure lwu;u;a;p ¼ mwu;a;p  u1 induced by u is a vanishing ða þ 2Þ-Carleson measure. R 2 aþ2 Þ jwðuðwÞÞjp dma ðwÞ ¼ 0. (c) limjzj!1 D j1ð1jzj zuðwÞj2ðaþ2Þ

Proof. Since M w C u ¼ T w;0;u and C u M w ¼ T wu;0;u , the proof follows from Theorems 3 and 7.

h

Corollary 6. Let p 2 ð1; 1Þ; a 2 ð1; 1Þ; w 2 HðDÞ and u be a holomorphic self-map of D. (1) The following statements are equivalent. (a) M w C u D is compact on Apa . (b) The pull-back measure lw;u;a;p ¼ mw;a;p  u1 induced by u is a vanishing ða þ 2 þ pÞ-Carleson measure. R 2 aþ2 Þ (c) limjzj!1 D j1ð1jzj jwðwÞjp dma ðwÞ ¼ 0. zuðwÞj2ðaþ2Þþp (2) The following statements are equivalent. (a) M w DC u is compact on Apa . (b) The pull-back measure lwu0 ;u;a;p ¼ mwu0 ;a;p  u1 induced by u is a vanishing ða þ 2 þ pÞ-Carleson measure. R 2 aþ2 Þ (c) limjzj!1 D j1ð1jzj jwðwÞu0 ðwÞjp dma ðwÞ ¼ 0. zuðwÞj2ðaþ2Þþp (3) The following statements are equivalent. (a) C u M w D is compact on Apa . (b) The pull-back measure lwu;u;a;p ¼ mwu;a;p  u1 induced by u is an ða þ 2 þ pÞ-Carleson measure. R 2 aþ2 Þ (c) limjzj!1 D j1ð1jzj jwðuðwÞÞjp dma ðwÞ ¼ 0. zuðwÞj2ðaþ2Þþp References [1] C.C. Cowen, B.D. 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