Abstract. Let ϕ be an analytic self-map of the open unit disk D. The operator given by (C

COMPOSITION OPERATORS FROM WEIGHTED BERGMAN SPACES TO S ^p SPACES

Waleed Al-Rawashdeh

(Received March 1, 2017)

Abstract. Let ϕ be an analytic self-map of the open unit disk D. The operator given by (C

f )(z) = f (ϕ(z)), for z ∈ D and f analytic on D is called compo- sition operator. For each p ≥ 1, let S

be the space of analytic functions on D whose derivatives belong to the Hardy space H

. In this paper, we study composition operators between weighted Bergman space A

and the space S

for 1 ≤ p, q < ∞. We characterize the boundedness and compactness of these composition operators. We also give a lower bound for the essential norm of these operators.

1. Introduction

Let D be the open unit disk in the complex plane C. For 0 < p < ∞ and

−1 < α < ∞, the weighted Bergman space A ^p _α (D) consists of all analytic functions f on D that satisfy

kf k ^p _A

=

Z

D

|f (z)| ^p dA α (z) < ∞, where dA α (z) = (1 + α) 1 − |z| ^2
α

dA(z) is the weighted Lebesgue area measure on D. For 0 < p < ∞, the Hardy space H ^p (D) consists of all analytic functions f on D that satisfy

kf k ^p _H

= sup

0<r<1

Z

∂D

|f (rζ)| ^p dσ(ζ) < ∞,

where σ is the normalized Lebesgue measure on the boundary of the unit disk. Let ϕ be a nonconstant analytic self-map of D. The composition operator C ^ϕ is defined on the space H(D) of all analytic functions on D by (C ϕ f )(z) = f (ϕ(z)), for all f ∈ H(D) and z ∈ D. Let ψ be an analytic function on D, the weighted composi- tion operator W _ψ,ϕ is defined on the space H(D) by (W ψ,ϕ f )(z) = ψ(z)C _ϕ f (z) = ψ(z)f (ϕ(z)), for all f ∈ H(D) and z ∈ D. The boundedness and compactness of the (weighted) composition operators have been studied by many authors, see for example the monographs [3], [6], [9], [18], [21], [22] and the references therein for the overview of the field as of the early 1990s.

For f belongs to H ^p (D), it is well known from Fatou’s theorem that the radial limit f ^∗ (ζ) = lim _r→1

f (rζ) exists for almost all ζ on ∂D. Moreover,

kf k ^p _H

= Z

∂D

|f ^∗ (ζ)| ^p dσ(ζ),

2010 Mathematics Subject Classification Primary 47B33, 47B38, 30H10, 30H20; Secondary 47B37, 30H05, 32C15.

Key words and phrases: Composition operators, Bergman spaces, Hardy spaces, S

spaces, com-

pact operators, essential norm, and vanishing Carleson measure .

for all finite values of p. Moreover, for f ∈ H ^p and ϕ self-analytic map of D, we know that (f ◦ ϕ) ^∗ = f ^∗ ◦ ϕ ^∗ a.e. on ∂D, see for example [17]. We often use the standard abuse of notation of writing f instead of f ^∗ . For 1 ≤ p < ∞, we denote by S ^p the space of all analytic functions f on the unit disk D whose derivative f ⁰ lies in H ^p , endowed with the norm

kf k _S

= |f (0)| + kf ⁰ k _H

.

One can easily show that S ^p is a Banach space with respect to this norm.

The boundedness and compactness of the (weighted) composition operators on the space S ^p have been studied by many authors, see for example [1], [4], [12], [14], [15], [16] and the references therein. In this paper, we characterize boundedness and compactness of composition operators that act from weighted Bergman spaces A ^p _α to S ^q spaces for 1 ≤ p, q < ∞. Moreover, we find a lower bound for the essential norm of composition operator from A ^p _α into S ^q spaces, 1 < p ≤ q.

The results we obtain about composition operators will be given in terms of a certain measure, which we define next. For 1 ≤ p < ∞, we define for any ϕ ∈ S ^p a finite positive Borel measure µ ϕ,p on D as:

µ ϕ,p (E) = Z

ϕ

(E)∩∂D

|ϕ ⁰ | ^p dσ,

where E is a Borel subset of the closed unit disk D, and where σ is the normal- ized Lebesgue measure on the boundary of the unit disk ∂D. Thus by using ([8], Theorem III.10.4) we get the following change of variable formula,

Z

D

gdµ ϕ,p = Z

∂D

|ϕ ⁰ | ^p g(ϕ)dσ, where g is an arbitrary measurable positive function on D.

For r ∈ (0, 1) and w ∈ D, the pseudo-hyperbolic metric ρ on D is defined by ρ(z, w) = |ϕ w (z)| where ϕ w (z) = _1−wz ^w−z , for z ∈ D. Moreover the pseudo-hyperbolic disk, whose pseudo-hyperbolic center is w and pseudo-hyperbolic radius is r, is de- fined as D(w, r) = {z ∈ D : ρ(z, w) < r}. For further details on pseudo-hyperbolic metric see [7], for example. Throughout this paper the notation A ≈ B means that there is a positive constant C independent of A and B such that C ⁻¹ B ≤ A ≤ CB.

Lemma 1.1. Suppose 0 < r < 1, p > 0, and α > −1. Then there exists a positive constant C = C(p, r, α) such that

|f (z)| ^p ≤ C (1 − |z| ² ) ^2+α

Z

D(z,r)

|f (w)| ^p dA α (w), for all f ∈ H(D) and all z ∈ D.

The proof of Lemma 1.1 can be easily seen by using Lemma 2.12 and Lemma 2.14 in [9], so the proof is not given here. In what follows, for a ∈ D, D(a) denotes the disk whose pseudo-hyperbolic center is a and pseudo-hyperbolic radius is r.

The following lemma is a standard estimate in Bergman space, which is a result of

Luecking ([11], Lemma 2.1).

Lemma 1.2. Let a ∈ D and let r be fixed, 0 < r < 1. There exists a constant C such that if f is analytic in D and p ≥ 1 then

|f ⁽ⁿ⁾ (a)| ^p ≤



C2 ⁿ (n + 2)n!

(1 − r) ⁿ⁺²

 ^p R

D(a) |f | ^p dA (r(1 − |a| ² )) ^2+nq .

Luecking characterized positive measures µ with the property kf ⁽ⁿ⁾ k L

(µ) ≤ kf k _A

. The following result is a special case of Luecking’s result ([11], Theorem 2.2) for n = 1.

Theorem 1.3. Let 1 ≤ p ≤ q and let α > −1. Let µ ≥ 0 be a finite measure on D.

The following are equivalent.

(1) There exists a constant C such that kf ⁰ k L

(µ) ≤ Ckf k _A

for all f ∈ A ^p _α . (2) µ(D(a)) = O

 

1 − |a| ²  ^q(α+2+p)/p 

as |a| → 1.

Luecking ([10], Theorem 1), for the case 1 ≤ q < p, used Khinchine’s inequality to obtain a version of Theorem 1.3 for f ⁽ⁿ⁾ , where f in the unweighted Bergamn space A ^p ₀ . Here we are interested in the case n = 1 and f ∈ A ^p _α . Therefore, we present next Theorem 1.4 which is a slight modification of Luecking’s result, so the proof’s details are omitted.

Theorem 1.4. Let 1 ≤ q < p and let α > −1. Let µ ≥ 0 be a finite measure on D.

Let L(z) = 1 − |z| ²
−(α+2+q)

µ(D(z)). The following are equivalent.

(1) There exists a constant C such that kf ⁰ k L

(µ) ≤ Ckf k _A

for all f ∈ A ^p _α . (2) L ∈ L ^p/(p−q) (A ^p _α ).

Finally before stating our results, we show the following lemma whose proof can be obtained by adapting the proof of ([3], Proposition 3.11).

Lemma 1.5. For 1 ≤ p, q < ∞ and α > −1. Let ϕ be an analytic self-map of D such that C _ϕ is bounded from A ^p _α (D) into S ^q . Then C _ϕ is compact if and only if whenever {f _n } is bounded sequence in A ^p _α (D) and f n → 0 uniformly on compact subsets of D, then kC ϕ (f _n )k _S

→ 0.

2. Boundedness and Compactness of Composition Operators

In the following results we show the boundedness and compactness of the com- position operators C ϕ : A ^p _α → S ^q can be characterized in terms of the finite positive Borel measure µ ϕ,p defined in the previous section.

Theorem 2.1. Let 1 ≤ p ≤ q, and let α > −1. Let ϕ be an analytic self-map of D and ϕ ∈ S ^q . Then C ϕ is bounded from A ^p _α into S ^q if and only if µ ϕ,q (D(a)) = O

 

1 − |a| ²  ^(α+2+p)q/p 

as |a| → 1.

Proof. Suppose f ∈ A ^p _α and ϕ ∈ S ^q . Then, by using change of variables formula

we get

kC ϕ f k ^q _S

≥ k(f ◦ ϕ) ⁰ k ^q _H

= Z

∂D

|f ⁰ (ϕ(z))| ^q |ϕ ⁰ (z)| ^q dσ(z)

= Z

D

|f ⁰ (w)| ^q dµ _ϕ,q (w)

= kf ⁰ k ^q _L

(µ

) .

If C ϕ : A ^p _α → S ^q is bounded, then there exists a positive constant C such that kf ⁰ k _L

(µ

) ≤ kC ϕ f k S

≤ Ckf k _A

.

Therefore we get the desired result by using Theorem 1.3.

Conversely, by the closed graph theorem, it suffices to show that C ϕ (f ) ∈ S ^q whenever f ∈ A ^p _α . From the hypothesis on µ ϕ,q , Theorem 1.3 implies

kf ⁰ k L

(µ

) ≤ C 1 kf k _A

, (1) for some constant C ₁ . On the other hand, we can find a constant C ₂ such that

kC ϕ (f )k _S

= |f (ϕ(0))| + k(f ◦ ϕ) ⁰ k H

≤ k(f ◦ ϕ) ⁰ k H

+ C 2 kf k _A

. Therefore, we get

kC _ϕ (f )k _S

≤ kf ⁰ k _L

(µ

) + C ₂ kf k _A

. (2) From (1) and (2), we get kC _ϕ (f )k _S

≤ (C 1 + C ₂ )kf k _A

. Which completes the

proof. 

Recall that an operator is said to be compact if it takes bounded sets to sets with compact closure. In the next theorem we are following operator-theoretic wisdom: If a “big-oh” condition determines when an operator is bounded, then the corresponding “little-oh” condition determines when it is compact.

Theorem 2.2. Let 1 ≤ p ≤ q and α > −1. Let ϕ be an analytic self-map of D and ϕ ∈ S ^q . Then C _ϕ is compact from A ^p _α into S ^q if and only if

µ _ϕ,q (D (a)) = o 1 − |a| ^{2
(α+2+p)q/p}

as |a| → 1. (3)

Proof. Suppose that C ϕ : A ^p _α → S ^q is compact. For a ∈ D, define K _a (z) = 1 − |a| ^2
(α+2)/p

(1 − az) ^2(α+2)/p .

It is clear that kK _a k _A

≈ 1, and K a converges to zero uniformly on compact subsets of D as |a| → 1. The compactness of C ϕ yields that for  > 0, there exists r ∈ (0, 1) such that kC _ϕ (K _a )k ^q _S

<  for |a| > r. Then for |a| > r

Z

D(a)

|K _a ⁰ (z)| ^q dµ ϕ,q (z) ≤ Z

D

|K _a ⁰ (z)| ^q dµ ϕ,q (z)

= Z

∂D

|K _a ⁰ (ϕ(w))| ^q |ϕ ⁰ (w)| ^q dσ(w)

≤ kC _ϕ (K _a )k ^q _S

< .

On the other hand, it is easy to see for z ∈ D(a)

|K _a ⁰ (z)| ≈ 1

(1 − |a| ² ) ^(α+2+p)/p . Thus, above inequalities yield for all a with |a| > r

µ _ϕ,q (D(a)) <  1 − |a| ^{2
(α+2+p)q/p} , which gives the desired result as |a| → 1.

Conversely, suppose that condition (3) holds. Let {f n } be a bounded sequence in A ^p _α with f n → 0 uniformly compact subsets of D. To show C ϕ is compact, it suffices to show

kC ϕ (f n )k ^q _S

≤ Ckf _n ⁰ k ^q _L

(µ

) → 0 as n → ∞.

Now, by using Lemma 1.2 we have

|f _n ⁰ (z)| ^q ≤ C ₁ (1 − |z| ² ) ^2+q

Z

D(z)

|f n (w)| ^q dA(w).

Integrate the last inequality with respect to µ _ϕ,q and then use Fubini’s Theorem to get

kf _n ⁰ k ^q _L

(µ

) ≤ C 1

Z

D

µ ϕ,q (D(w))

(1 − |w| ² ) ^2+q |f n (w)| ^q dA(w).

From Lemma 1.1, we have for f ∈ A ^p _α and w ∈ D

|f _n (w)| ≤ C 2 kf n k _A

(1 − |w| ² ) ^(2+α)/p . Therefore, we get

kf ⁰ k ^q _L

(µ

) ≤ C 1 C 2 kf n k ^q−p _A

Z

D

|f n (w)| ^p µ ϕ,q (D(w))

(1 − |w| ² ) (αq+2q+pq−αp)/p dA(w).

Condition (3) implies that for a given  > 0, there exists r ∈ (0, 1) such that Z

|w|>r

|f n (w)| ^p µ _ϕ,q (D(w))

(1 − |w| ² ) ^(α+2+p)q/p (1 − |w| ² ) ^α dA(w) ≤ kf _n k ^p _A

α

. (4)

On the other hand, since f n → 0 uniformly on compact subsets of D, namely

|w| ≤ r, we can find constants C 3 and C ₄ such that for large n Z

D

|f n (w)| ^p µ ϕ,q (D(w))

(1 − |w| ² ) (αq+2q+pq−αp)/p dA(w)

≤ C 3

Z

|w|≤r

µ ϕ,q (D(w))dA(w)

≤ C 3

Z

D

µ ϕ,q (D)dA(w)

= C 3 C 4 . (5)

Hence, from (4) and (5) we have

kf ⁰ k ^q _L

(µ

) ≤ C 1 C 2 M ^q−p (M ^p + C 3 C 4 ) ,

which completes the proof. 

The next theorem characterizes boundedness and compactness of C ϕ that maps A ^p _α into S ^q when 1 ≤ q < p.

Theorem 2.3. Let 1 ≤ q < p and α > −1. Let ϕ be an analytic self-map of D and ϕ ∈ S ^q . Let L(z) = 1 − |z| ²
−(α+q+2)

µ ϕ,q (D(z)). Then the following are equivalent.

(1) C ϕ is bounded from A ^p _α into S ^q . (2) C _ϕ is compact from A ^p _α into S ^q . (3) L ∈ L ^p/(p−q) (D, dA ^α ).

Proof. It is clear that (2) implies (1). Using change of variable formula, then use Theorem 1.4 we get the equivalence of (1) and (3). It remains to verify that (3) implies (2). Let {f n } be a bounded sequence in A ^p _α with f n → 0 uniformly on compact subsets of D. As in Theorem 2.2 it suffices to show that

n→∞ lim kf _n ⁰ k ^q _L

(µ

) = 0.

Following similar argument as that in Theorem 2.2 we have kf _n ⁰ k ^q _L

µϕ,q

≤ C Z

D

1 (1 − |z| ² ) ^2+q

Z

D(z)

|f n (w)| ^q dA(w)d µ

(z)

≤ C Z

D

µ _ϕ,q (D(z))

(1 − |z| ² ) ^(α+2+q) |f n (z)| ^q 1 − |z| ^2
α dA(z)

= C Z

D

|f n (z)| ^q L(z)dA α (z).

Suppose L(z) ∈ L ^p/(p−q) (D, dA ^α ). Then for  > 0, there exists r ∈ (0, 1) such that Z

D

L ^p/(p−q) (z)dA α (z) <  ^p/(p−q) . Using H¨ older’s inequality we get,

Z

|z|>r

|f n (z)| ^q L(z)dA α (z)

≤ Z

|z|>r

|f n (z)| ^p dA α (z)

! q/p

Z

|z|>r

L ^p/(p−q) (z)dA α (z)

! (p−q)/p

≤ kf _n k ^q _A

α

. (6)

Since f _n → 0 uniformly on compact subsets of D, |f n (z)| <  for all z such that

|z| ≤ r and for large n. Thus, Z

|z|≤r

|f _n (z)| ^q L(z)dA _α (z) ≤  Z

|z|≤r

L(z)dA _α (z)

=  Z

|z|≤r

µ ϕ,q (D(z))

(1 − |z| ² ) ^(α+2+q) dA α (z)

≤ C Z

|z|≤r

µ _ϕ,q (D(z))dA _α (z)

≤ C Z

D

µ ϕ,q (D)dA ^α (z)

< C, (7) where the finiteness of µ ϕ,q (D) follows from ϕ ∈ S ^q . Hence, from (6) and (7) we

get the desired result. 

3. Essential Norm of a Composition Operator

The essential norm of an operator, denoted by k.k _e , is its distance in the operator norm from the ideal of compact operators. So for any composition operator C _ϕ : A ^p _α → S ^q , we define

kC _ϕ k _e = inf

K∈K kC _ϕ − Kk,

where K = K(A ^p α , S ^q ) is the set of compact operators acting from A ^p _α into S ^q . It is well known that a bounded operator T is compact if and only if kT k _e = 0, so that estimates of essential norm lead for a compactness of a composition operator. The essential norm has been studied by many authors in spaces of analytic functions, see for example [2], [5], [13], [19] and the related references therein.

For any ζ in the unit circle and 0 < r < 1, the Carleson squares are defined as:

S(ζ, r) = {z ∈ D : |1 − hz, ζi| < r}. Let µ be a positive Borel measure on D, and let X be a Banach space of analytic functions on D. Then for q > 0, µ is called vanishing (X, q)−Carleson measure if

lim

k→∞

Z

D

|f k | ^q dµ = 0,

whenever {f k } is a bounded sequence in X that converges to 0 uniformly on compact subsets of D. As a special case of ([20], p.71) we have the next Lemma 3.1.

Lemma 3.1. Suppose 0 < p ≤ q < ∞, α > −1 and µ is a positive Borel measure on D. Then the following are equivalent.

(1) µ is vanishing (A ^p _α , q)−Carleson measure.

(2) The limit

lim

r→0

µ(S(ζ, r)) r ^(2+α)q/p = 0, holds uniformly for ζ ∈ S.

The following theorem gives a lower bound for the essential norm of a composition operator.

Theorem 3.2. Let 1 < p ≤ q and α > −1. Let ϕ be an analytic self-map of D and ϕ ∈ S ^q . If C _ϕ is bounded from A ^p _α into S ^q , then there exists a constant C such that

kC ϕ k ^q _e ≥ C lim sup

|a|→1

Z

∂D

|ϕ ⁰ (w)| ^q (1 − |a| ² ) ^(2+α)q/p

|1 − aϕ(w)| (2α+4+p)q/p dσ(w).

Proof. For any a ∈ D, consider the function

f a (z) = (1 − |a| ² ) ^(α+2)/p (1 − az) ^2(α+2)/p .

It is clear that f a ∈ A ^p _α and kf a k _A

≈ 1. Moreover, f a → 0 uniformly on compact

subsets of D. For the moment fix a compact operator K acting from A ^p α into S ^q .

Then kKf a k S

→ 0 as |a| → 1. Thus, kC ϕ − Kk ≥ lim sup

|a|→1

k(C ϕ − K)f a k S

≥ lim sup

|a|→1

(kC ϕ f a k S

− kKf a k S

)

= lim sup

|a|→1

kC _ϕ f _a k _S

.

Upon taking the infimum of both side of this inequality over all compact operators K, we obtain

kC ϕ − Kk ^q _e ≥ lim sup

|a|→1

kC ϕ f a k S

= lim sup

|a|→1



|f a (ϕ(0))| ^q + Z

∂D

|f _a ⁰ (ϕ(z))| ^q |ϕ ⁰ (z)| ^q dσ(z)



= lim sup

|a|→1

Z

D

|f _a ⁰ (w)| ^q dµ ϕ,q (w), (8)

where the last line follows from the change of variable formula and the fact that f _a (ϕ(0)) → 0 as |a| → 1. Using (8) and the derivative of f _a give a desirable

result. 

We use the essential norm and vanishing Carleson measure to prove our main result in this section, which characterizes the compactness of a composition opera- tor.

Theorem 3.3. Let 1 < p ≤ q and α > −1. Let ϕ be an analytic self-map of D and ϕ ∈ S ^q . Suppose C ϕ is bounded from A ^p _α into S ^q . Then C ϕ is compact if and only if

lim sup

|a|→1

Z

∂D

|ϕ ⁰ (w)| ^q (1 − |a| ² ) ^(2+α)q/p

|1 − aϕ(w)| (2α+4+p)q/p dσ(w) = 0.

Proof. The necessary condition follows from Theorem 3.2. We consider the suf- ficient condition, first we prove that µ ϕ,q is vanishing (A ^p _α , q)−Carleson measure.

Suppose the hypothesis holds. If ζ ∈ ∂D and r ∈ (0, 1), then by using change of variable formula we get for all a ∈ D

lim sup

|a|→1

Z

S(ζ,r)

(1 − |a| ² ) ^(2+α)q/p

|1 − aw| (2α+4+p)q/p dµ ϕ,q (w) = 0.

Hence, for all  > 0, there exists r ∈ (0, 1) such that for |a| > r 

    Z

S(ζ,r)

(1 − |a| ² ) ^(2+α)q/p

|1 − aw| (2α+4+p)q/p dµ ϕ,q (w)     

< . (9)

By taking a = (1 − r)ζ, it is easy to see (1 − |a| ² ) ^(2+α)q/p

|1 − aw| (2α+4+p)q/p ≥ 2 −(2α+4+p)q/p

r ^(α+2)q/p . (10)

So, from (9) and (10) we get for all ζ ∈ ∂D

 >

Z

S(ζ,r)

2 −(2α+4+p)q/p

r ^(α+2)q/p dµ _ϕ,q

≥ C µ ϕ,q (S(ζ, r)) r ^(α+2)q/p , which gives lim

r→0

µ ϕ,q (S(ζ, r))

r ^(α+2)q/p = 0. Therefore, by using Lemma 3.1, µ ϕ,q is a van- ishing (A ^p _α , q)−Carleson measure. Hence using the definition of vanishing Carleson measure, for any bounded sequence {f _n } in A ^p _α with f _n → 0 on compact subsets of D, consider the function g n ∈ S ^q such that g ⁰ _n = f _n and g _n (0) = 0. Then

0 = lim

n→∞

Z

D

|f n (z)| ^q dµ ϕ,q (z)

= lim

n→∞

Z

∂D

|g _n ⁰ (ϕ(z))| ^q |ϕ ⁰ (z)| ^q dσ(z)

= lim

n→∞ kC ϕ (g n )k ^q _S

.

Therefore, by using Lemma 1.5, we get C ϕ is compact as desired.  Acknowledgement. The author is very grateful to the referee for his/her valuable comments and suggestions which improved the presentation of this paper.

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Waleed Al-Rawashdeh,

Department of Mathematical Sciences Montana Tech of The University of Montana Montana 59701, USA.

[email protected]