Weighted composition operators from $F(p,q,s)$ into logarithmic Bloch space

J. Korean Math. Soc. 45 (2008), No. 4, pp. 977–991

WEIGHTED COMPOSITION OPERATORS FROM F (p, q, s) INTO LOGARITHMIC BLOCH SPACE

Shanli Ye

Reprinted from the

Journal of the Korean Mathematical Society Vol. 45, No. 4, July 2008

c

°2008 The Korean Mathematical Society

J. Korean Math. Soc. 45 (2008), No. 4, pp. 977–991

WEIGHTED COMPOSITION OPERATORS FROM F (p, q, s) INTO LOGARITHMIC BLOCH SPACE

Shanli Ye

Abstract. We characterize the boundedness and compactness of the weighted composition operator uC

from the general function space F (p, q, s) into the logarithmic Bloch space β

on the unit disk. Some neces- sary and sufficient conditions are given for which uC

is a bounded or a compact operator from F (p, q, s), F

(p, q, s) into β

, β

respectively.

1. Introduction

Let D = {z : |z| < 1} be the open unit disk in the complex plane C, and H(D) denote the set of all analytic functions on D. For f ∈ H(D), let

kf k _β

= sup{(1 − |z| ² ) ln( 2

1 − |z| )|f ⁰ (z)| : z ∈ D}.

As in [6], the logarithmic Bloch space β L consists of all f ∈ H(D) satisfying kf k _β

< +∞ and the little logarithmic Bloch space β _L ⁰ consists of all f ∈ H(D) satisfying lim _|z|→1

(1 − |z| ² ) ln( _1−|z| ² )|f ⁰ (z)| = 0. The β L is a Banach space under the norm

kf k L = |f (0)| + kf k β

.

In [6], the author proved that β _L ⁰ is a closed subspace and coincides with the closure of polynomials under the norm. Yoneda [8] studied the composition operator in the β L space and the β _L ⁰ space.

We write β α (α > 0) for the space of analytic function f for which kf k β

=

|f (0)| + sup _z∈D (1 − |z| ² ) ^α |f ⁰ (z)| < ∞ and β _α ⁰ (α > 0) for f ∈ β α such that lim |z|→1

(1 − |z| ² ) ^α |f ⁰ (z)| = 0. The functions in β α and β _α ⁰ will be referred to as α-Bloch functions, little α-Bloch functions respectively. It is easily proved that for 0 < α < 1, β α $ β L $ β 1 . For more information about the β α (see [12]).

Received November 12, 2006; Revised February 28, 2007.

2000 Mathematics Subject Classification. Primary 47B38; Secondary 30H05.

Key words and phrases. weighted composition operator, F (p, q, s) space, logarithmic Bloch space.

The research was supported by NNSF of China (Grant No.10771130) and NSF of Fujian Province (Grant No. 2006J0201).

c

°2008 The Korean Mathematical Society

977

Let dm denote the Lebesgue area measure in D. Let 0 < p, s < ∞ and

−2 < q < ∞, then a function f ∈ H(D) is said to belong to the general function space F (p, q, s) (see [11]) if

kf k _{F (p,q,s)} = |f (0)| + © sup

a∈D

Z

D

|f ⁰ (z)| ^p (1 − |z| ² ) ^q (1 − |ϕ a (z)| ² ) ^s dm(z) ª

p

< ∞

and the little general function space F 0 (p, q, s) if f ∈ F (p, q, s) and

|a|→1 lim

Z

D

|f ⁰ (z)| ^p (1 − |z| ² ) ^q (1 − |ϕ a (z)| ² ) ^s dm(z) = 0.

Here ϕ a is a conformal automorphism defined by ϕ a (z) = _1−¯ ^a−z _az for a ∈ D. We can get many function spaces if we take some specific parameters of p, q, s.

For example (see [11]), F (p, q, s) = β

p

and F 0 (p, q, s) = β ⁰

p

for s > 1;

F (p, q, s) ⊂ β

p

and F 0 (p, q, s) ⊂ β ⁰

for 0 < s ≤ 1; F (2, 0, s) = Q s and F 0 (2, 0, s) = Q s,0 ; F (2, 0, 1) = BM OA and F 0 (2, 0, 1) = V M OA. For more information about the BM OA and V M OA (see [1]). If q + s ≤ −1, then F (p, q, s) is the space of constant functions.

An analytic self-map ϕ : D → D induces the composition operator C ϕ on H(D), defined by C ϕ (f ) = f (ϕ(z)) for f analytic on D. It is a well known con- sequence of Littlewood’s subordination principle that the composition operator C ϕ is bounded on the classical Hardy and Bergman spaces (see, for example, [2, 13]).

Recall that a linear operator is said to be bounded if the image of a bounded set is a bounded set, while a linear operator is compact if it takes bounded sets to sets with compact closure. It is interesting to provide a function theoretic characterization of when ϕ induces a bounded or compact composition operator on various spaces. The book [2] contains plenty of information on this topic.

Let u be a fixed analytic function on the open unit disk. Define a linear operator uC ϕ on the space of analytic functions on D, called a weighted com- position operator, by uC ϕ f = u · (f ◦ ϕ), where f is an analytic function on D.

We can regard this operator as a generalization of a multiplication operator and a composition operator.

In [4, 5], Ohno, Stroethoff and Zhao have characterized the boundedness and compactness of weighted composition operators between α-Bloch spaces, between H ^∞ and the Bloch space β 1 , and from the little F 0 (p, q, s) into the little Bloch space β ₁ ⁰ . The boundedness and compactness of weighted composition operators from F (p, q, s) into the Bloch space β α were investigated in [9].

In this paper we study the weighted composition operators from the general

function spaces F (p, q, s) into the logarithmic Bloch space β L and the little

F 0 (p, q, s) into the little logarithmic Bloch space β _L ⁰ . Throughout this paper,

constants are denoted by C, they are positive and may differ from one occur-

rence to the other.

2. Boundedness and compactness of uC ϕ : F (p, q, s) → β L

In this section we characterize the boundedness and compactness of the weighted composition operator uC ϕ : F (p, q, s) → β L . For this purpose we need some lemmas. The following lemma can be found in [11].

Lemma 2.1. Suppose that 0 < p, s < ∞, −2 < q < ∞, q + s > −1 and f ∈ F (p, q, s). Then there is a positive constant C such that kf k β

≤ Ckf k _{F (p,q,s)} , moreover, if f ∈ F 0 (p, q, s), then f ∈ β _α ⁰ , where α = ^q+2 _p .

Lemma 2.2 ([7]). Suppose t > 0 and f ∈ H(D). Then sup _z∈D (1−|z|) ^t |f (z)| <

+∞ if and only if sup _z∈D (1 − |z|) ^t+1 |f ⁰ (z)| < +∞.

Lemma 2.3. Let 0 < p, s < ∞, −2 < q < ∞ and q + s > −1. The operator uC ϕ : F (p, q, s) → β L is compact if and only if for any bounded sequence {f n } in F (p, q, s) which converges to zero uniformly on compact subsets of D, we have kuC ϕ f n k L → 0 as n → ∞.

Proof. Using [6, Lemma 2.1] and Montel’s Theorem, one may prove the lemma.

The details are omitted here. ¤

Lemma 2.4 ([7]). Let α > 0 and f ∈ β α . Then (1) |f (z)| ≤ Ckf k β

, where α < 1;

(2) |f (z)| ≤ C log( _1−|z| ²

)kf k β

, where α = 1;

(3) |f (z)| ≤ (α−1)(1−|z|) ^C

kf k β

, where α > 1.

Theorem 2.1. Let 0 < p, s < ∞, −2 < q < ∞, q + s > −1, u ∈ H(D) and ϕ be an analytic self-map of D.

(i) If 2 + q > p, then uC ϕ is a bounded operator from F (p, q, s) to β L if and only if

(1) sup

z∈D

(1 − |z| ² ) log _1−|z| ²

(1 − |ϕ(z)| ² ) ^α |u(z)ϕ ⁰ (z)| < +∞

and

(2) sup

z∈D

(1 − |z| ² ) log _1−|z| ²

(1 − |ϕ(z)| ² ) ^α−1 |u ⁰ (z)| < +∞, where α = ^q+2 _p .

(ii) If 2 + q < p, then uC ϕ is a bounded operator from F (p, q, s) to β L if and only if u ∈ β L and (1) holds.

(iii) If 2 + q = p, s > 1, then uC ϕ is a bounded operator from F (p, q, s) to β L if and only if (1) holds and

(3) sup

z∈D |u ⁰ (z)|(1 − |z| ² ) log ¡ 2

1 − |z| ) log ¡ 2

1 − |ϕ(z)| ² ) < ∞.

(iv) Let 2+q = p, 0 < s ≤ 1. Then uC ϕ is a bounded operator from F (p, q, s)

to β L provided that (1) and (3) hold.

Conversely, if uC ϕ is a bounded operator from F (p, q, s) to β L , then u ∈ β L

and (1) holds.

Proof. Suppose that uC ϕ is bounded from F (p, q, s) to β L . Then we can easily obtain u ∈ β L by taking f (z) = 1. Let α = ^q+2 _p . Fixed w ∈ D, we consider the function

(4) f w (z) = (z − ϕ(w))(1 − |ϕ(w)| ² ) (1 − ϕ(w)z) ^α+1 .

Then kf w k F (p,q,s) ≤ C by [9], where C is not depended on w. Since that f w (ϕ(w)) = 0 and f _w ⁰ (ϕ(w)) = _(1−|ϕ(w)| ¹

)

, it follows that

(1 − |w| ² ) log _1−|w| ²

(1 − |ϕ(w)| ² ) ^α |u(w)ϕ ⁰ (w)|

= (1 − |w| ² ) log( 2

1 − |w| )|u(w)f _w ⁰ (ϕ(w))ϕ ⁰ (w)|

≤ kuC ϕ f w k L ≤ kuC ϕ kkf w k F (p,q,s) ≤ CkuC ϕ k < +∞, which showing that (1) is necessary for all case.

(i) Let α = ^q+2 _p > 1. Fix w ∈ D, we take again the test function

(5) g w (z) = (α + 1)(1 − |ϕ(w)| ² )

(1 − ϕ(w)z) ^α − α(1 − |ϕ(w)| ² ) ² (1 − ϕ(w)z) ^α+1 .

Then we can easily prove that kg w k _{F (p,q,s)} ≤ C by using the same methods in [9], where C is not depended on w. Since g ⁰ _w (ϕ(w)) = 0 and g w (ϕ(w)) =

1

(1−|ϕ(w)|

)

, it follows that (1 − |w| ² ) log( 2

1 − |w| )|u ⁰ (w)g w (ϕ(w))|

= (1 − |w| ² ) log 2

1 − |w| |(uC ϕ g w ) ⁰ (w)|

≤ kuC ϕ g w k L ≤ kuC ϕ kkg w k _{F (p,q,s)} ≤ CkuC ϕ k < +∞.

So,

sup

w∈D

(1 − |w| ² ) log _1−|w| ²

(1 − |ϕ(w)| ² ) ^α−1 |u ⁰ (w)| < +∞.

This proves that (2) is also necessary.

Conversely, suppose that u and ϕ satisfy the condition in (i). For arbitrary f ∈ F (p, q, s), by Lemma 2.1 and 2.2 we have

sup

z∈D (1 − |z| ² ) ^α |f ⁰ (z)| ≤ Ckf k F (p,q,s)

and sup _z∈D (1 − |z| ² ) ^α−1 |f (z)| ≤ Ckf k _{F (p,q,s)} . It follows that kuC ϕ f k β

≤ sup

z∈D (1 − |z| ² ) log( 2

1 − |z| )|u ⁰ (z)f (ϕ(z))|

+ sup

z∈D

(1 − |z| ² ) log( 2

1 − |z| )|u(z)||f ⁰ (ϕ(z))ϕ ⁰ (z)|

= sup

z∈D

(1 − |ϕ(z)| ² ) ^α−1 |f (ϕ(z))| (1 − |z| ² ) log _1−|z| ² (1 − |ϕ(z)| ² ) ^α−1 |u ⁰ (z)|

+ sup

z∈D

(1 − |ϕ(z)| ² ) ^α |f ⁰ (ϕ(z))| (1 − |z| ² ) log _1−|z| ²

(1 − |ϕ(z)| ² ) ^α |ϕ ⁰ (z)u(z)|

≤ Ckf k _{F (p,q,s)} and by Lemma 2.4,

|u(0)f (ϕ(0))| ≤ C|u(0)|

(α − 1)(1 − |ϕ(0)|) ^α−1 kf k β

≤ Ckf k _{F (p,q,s)} . Hence uC ϕ is bounded from F (p, q, s) to β L .

Next we will prove (ii). Let α = ^q+2 _p < 1. Suppose u ∈ β L and that (1) holds. Assume that f ∈ F (p, q, s). Then we have sup _z∈D |f (z)| ≤ Ckf k F (p,q,s)

by Lemma 2.1 and 2.4. It follows that kuC ϕ f k β

≤ sup

z∈D (1 − |z| ² ) log( 2

1 − |z| )|u ⁰ (z)f (ϕ(z))|

+ sup

z∈D

(1 − |z| ² ) log( 2

1 − |z| )|u(z)||f ⁰ (ϕ(z))ϕ ⁰ (z)|

≤ Ckuk β

kf k _{F (p,q,s)} + Ckf k _{F (p,q,s)} sup

z∈D

(1 − |z| ² ) log _1−|z| ²

(1 − |ϕ(z)| ² ) ^α |ϕ ⁰ (z)u(z)|

≤ Ckf k _{F (p,q,s)}

and |u(0)f (ϕ(0))| ≤ C|u(0)|kf k F (p,q,s) . Hence uC ϕ is bounded from F (p, q, s) to β L .

Conversely, we have proved that (1) holds and u ∈ β L above.

Finally let α = ^q+2 _p = 1 and s > 1, we set (6) h w (z) = 2 log 2

1 − ϕ(w)z − 1 log _1−|ϕ(w)| ²

(log 2 1 − ϕ(w)z ) ² for z, w ∈ D. Then

h ⁰ _w (z) = 2ϕ(w) 1 − ϕ(w)z − 2 ¡

log 2

1 − ϕ(w)z ) ϕ(w) (1 − ϕ(w)z)

1 log _1−|ϕ(w)| ²

.

By a direct calculation we get sup _w∈D kh w k β

≤ C < +∞. Noting s > 1, thus we have h w ∈ F (p, q, s) and sup _w∈D kh w k _{F (p,q,s)} ≤ C < +∞ by introduction.

Since h ⁰ _w (ϕ(w)) = 0 and h w (ϕ(w)) = log _1−|ϕ(w)| ²

, it follows that sup

w∈D |u ⁰ (w)|(1 − |w| ² ) log( 2

1 − |w| ) log( 2 1 − |ϕ(w)| ² )

= sup

w∈D (1 − |w| ² ) log( 2

1 − |w| )|u ⁰ (w)h w (ϕ(w))|

≤ kuC ϕ h w k L ≤ kuC ϕ kkh w k _{F (p,q,s)} ≤ CkuC ϕ k < +∞, which showing that (3) is necessary.

Conversely, suppose (1) and (3) hold. Assume that f ∈ F (p, q, s). Then f ∈ β 1 by Lemma 2.1. From Lemma 2.4, it follows that

kuC ϕ f k β

≤ sup

z∈D

(1 − |z| ² ) log( 2

1 − |z| )|u ⁰ (z)f (ϕ(z))|

+ sup

z∈D (1 − |z| ² ) log( 2

1 − |z| )|u(z)||f ⁰ (ϕ(z))ϕ ⁰ (z)|

≤ Ckf k _{F (p,q,s)} sup

z∈D

|u ⁰ (z)|(1 − |z| ² ) log( 2

1 − |z| ) log( 2 1 − |ϕ(z)| ² ) + kf k _{F (p,q,s)} sup

z∈D

(1 − |z| ² ) log _1−|z| ²

(1 − |ϕ(z)| ² ) |ϕ ⁰ (z)u(z)| ≤ Ckf k _{F (p,q,s)} and

|u(0)f (ϕ(0))| ≤ C|u(0)| log( 2

1 − |ϕ(0)| ² )kf k β

≤ Ckf k _{F (p,q,s)} . Hence uC ϕ is bounded from F (p, q, s) to β L .

(iv) The results have been proved in case (ii) and (iii). ¤ Corollary 2.1. Let 0 < p, s < ∞, −2 < q < ∞, q + s > −1 and ϕ be an analytic self-map of D. Then C ϕ is a bounded operator from F (p, q, s) into β L

if and only if

sup

z∈D

(1 − |z| ² ) log _1−|z| ²

(1 − |ϕ(z)| ² )

|ϕ ⁰ (z)| < +∞.

In the formulation of the corollary, we will use the notation M (X, Y ) to denote the set of all multipliers of X into Y : M (X, Y ) = {u : f u ∈ Y for all f ∈ X}. By M u we denote the operator of multiplication by u: M u f = uf, f ∈ X.

Corollary 2.2. Let 0 < p, s < ∞, −2 < q < ∞, q + s > −1.

(1) If 2 + q ≥ p, then M (F (p, q, s), β L ) = {0}.

(2) If 2 + q < p, then M (F (p, q, s), β L ) = β L .

Proof. If 2 + q < p, then M u is bounded from F (p, q, s) to β L if and only if u ∈ β L and

(7) sup

z∈D (1 − |z| ² ) ¹⁻

log 2

1 − |z| |u(z)| < ∞ by Theorem 2.1. However, if u ∈ β L , by [6, Lemma 2.1], we get

|u(z)| ≤ (2 + log(log 2

1 − |z| ))kuk L .

Hence (7) holds, since ^q+2 _p < 1. ¤

Theorem 2.2. Let 0 < p, s < ∞, −2 < q < ∞, q + s > −1, u ∈ H(D) and ϕ be an analytic self-map of D.

(i) If 2 + q > p, then uC ϕ is a compact operator from F (p, q, s) to β L if and only if u, uϕ ∈ β _L ,

(8) lim

|ϕ(z)|→1

(1 − |z| ² ) log _1−|z| ²

(1 − |ϕ(z)| ² ) ^α |u(z)ϕ ⁰ (z)| = 0 and

(9) lim

|ϕ(z)|→1

(1 − |z| ² ) log _1−|z| ²

(1 − |ϕ(z)| ² ) ^α−1 |u ⁰ (z)| = 0, where α = ^q+2 _p .

(ii) If 2 + q < p, then uC ϕ is a compact operator from F (p, q, s) to β L if and only if u, uϕ ∈ β L and (8) holds.

(iii) If 2 + q = p, s > 1, then uC ϕ is a compact operator from F (p, q, s) to β _L if and only if u, uϕ ∈ β _L , (8) holds and

(10) lim

|ϕ(z)|→1

|u ⁰ (z)|(1 − |z| ² ) log( 2

1 − |z| ) log( 2

1 − |ϕ(z)| ² ) = 0.

Proof. Suppose that uC ϕ is compact from F (p, q, s) to β L . It is easily obtained that u, uϕ ∈ β _L by taking f (z) = 1 and f (z) = z respectively. Let α = ^q+2 _p and {z n } be a sequence in D such that |ϕ(z n )| → 1 ⁻ as n → ∞. We take the test functions

f n (z) = (z − ϕ(z n ))(1 − |ϕ(z n )| ² ) (1 − ϕ(z n )z) ^α+1 .

Then f n ∈ F (p, q, s) and sup _n kf n k F (p,q,s) ≤ C < ∞ by using the same methods

in [9]. Then {f n } is a bounded sequence in F (p, q, s) which converges to 0

uniformly on compact subsets of D. Note that f n (ϕ(z n )) = 0 and f _n ⁰ (ϕ(z n )) =

1

(1−|ϕ(z

)|

)

, it follows that kuC ϕ f n k L ≥ kuC ϕ f n k β

≥ (1 − |z n | ² ) log( 2

1 − |z n | )|u ⁰ (z n )f n (ϕ(z n ))+u(z n )f _n ⁰ (ϕ(z n ))ϕ ⁰ (z n )|

= (1 − |z n | ² ) log _1−|z ²

n

|

(1 − |ϕ(z _n )| ² ) ^α |u(z n )ϕ ⁰ (z n )|.

Then, by Lemma 2.3, (8) holds for all case.

(i) Case α = ^q+2 _p > 1. We consider the functions g n defined by (11) g n (z) = (α + 1)(1 − |ϕ(z n )| ² )

(1 − ϕ(z n )z) ^α − α(1 − |ϕ(z n )| ² ) ² (1 − ϕ(z n )z) ^α+1 .

Then g n ∈ F (p, q, s) and sup _n kg n k F (p,q,s) ≤ C < ∞, so {g n } is a bounded sequence in F (p, q, s) which converges to 0 uniformly on compact subsets of D.

Since g ⁰ _n (ϕ(z n )) = 0 and g n (ϕ(z n )) = _(1−|ϕ(z ¹

n

)|

)

, it follows that kuC ϕ g n k L ≥ kuC ϕ g n k β

≥ (1 − |z n | ² ) log( 2

1 − |z n | )|u ⁰ (z n )g n (ϕ(z n ))+u(z n )g _n ⁰ (ϕ(z n ))ϕ ⁰ (z n )|

= (1 − |z _n | ² ) log( 2

1 − |z n | )|u ⁰ (z _n )| 1

(1 − |ϕ(z n )| ² ) ^α−1 . Then (9) holds by Lemma 2.3.

(ii) Case α = ^q+2 _p < 1. The necessity in condition (ii) has been proved above.

(iii) Case α = ^q+2 _p = 1 and s > 1. We take the other test functions h n (z) = 3

a n (log 2

1 − ϕ(z n )z ) ² − 2

a ² _n (log 2 1 − ϕ(z n )z ) ³ , where a n = log _1−|ϕ(z ²

n

)|

. Clearly h n (z) → 0 uniformly on compact subsets of D. By a direct calculation we get sup _n kh n k β

≤ C < +∞. Noting s > 1, thus we have h n ∈ F (p, q, s) and sup _n kh n k F (p,q,s) < ∞ by introduction. Then {h n } is a bounded sequence in F (p, q, s) which converges to 0 uniformly on compact subsets of D. Note that h ⁰ _n (ϕ(z n )) ≡ 0 and h n (ϕ(z n )) = a n , it follows that

kuC ϕ h n k L ≥ kuC ϕ h n k β

≥ (1− |z n | ² ) log( 2

1 − |z n | )|u ⁰ (z n )h n (ϕ(z n ))+u(z n )h ⁰ _n (ϕ(z n ))ϕ ⁰ (z n )|

= (1− |z n | ² ) log( 2

1 − |z n | ) log 2

1 − |ϕ(z n )| ² |u ⁰ (z n )|.

Then (10) holds by Lemma 2.3.

Now we prove the sufficient conditions.

Case α = ^2+q _p > 1. Assume that (8) and (9) hold and u, uϕ ∈ β L . Then it is not difficult to obtain that (1), (2) and

(12) sup

z∈D (1 − |z| ² ) log( 2

1 − |z| )|u(z)ϕ ⁰ (z)| < ∞

hold. Let {f n } be a bounded sequence in F (p, q, s) which converges to 0 uni- formly on compact subsets of D. Let M = sup _n kf n k F (p,q,s) < +∞. We only prove lim n→∞ kuC ϕ (f n )k L = 0 by Lemma 2.3. This amounts to showing that both

sup

w∈D (1 − |w| ² ) log( 2

1 − |w| )|u(w)f _n ⁰ (ϕ(w))ϕ ⁰ (w)| −→ 0 and

sup

w∈D (1 − |w| ² ) log( 2

1 − |w| )|u ⁰ (w)f n (ϕ(w))| −→ 0 as n → ∞.

If |ϕ(w)| ≤ r < 1, by (12), then (1 − |w| ² ) log( 2

1 − |w| )|u(w)f _n ⁰ (ϕ(w))ϕ ⁰ (w)| ≤ C max

|z|≤r |f _n ⁰ (z)|.

If |ϕ(w)| > r, then by Lemma 2.1 we get (1 − |w| ² ) log( 2

1 − |w| )|u(w)f _n ⁰ (ϕ(w))ϕ ⁰ (w)|

≤ CM (1 − |w| ² ) log( _1−|w| ² )

(1 − |ϕ(w)| ² ) ^α |ϕ ⁰ (w)u(w)|.

Thus

sup

w∈D (1 − |w| ² ) log( 2

1 − |w| )|u(w)f _n ⁰ (ϕ(w))ϕ ⁰ (w)|

≤ C max

|w|≤r |f _n ⁰ (w)| + CM sup

|ϕ(w)|>r

(1 − |w| ² ) log( _1−|w| ² )

(1 − |ϕ(w)| ² ) ^α |ϕ ⁰ (w)u(w)|.

First letting n tend to infinity and subsequently r increase to 1, one obtains that

sup

w∈D (1 − |w| ² ) log( 2

1 − |w| )|u(w)f _n ⁰ (ϕ(w))ϕ ⁰ (w)| −→ 0 as n → ∞. The other statement is proved similarly.

If |ϕ(w)| ≤ r < 1, by u ∈ β L , then (1 − |w| ² ) log( 2

1 − |w| )|u ⁰ (w)f n (ϕ(w))| ≤ kuk L max

|z|≤r |f n (z)|.

If |ϕ(w)| > r, by Lemma 2.1 and 2.2, then (1 − |w| ² ) log( 2

1 − |w| )|u ⁰ (w)f n (ϕ(w))| ≤ CM (1 − |w| ² ) log( _1−|w| ² )

(1 − ϕ(w)| ² ) ^α−1 |u ⁰ (w)|.

Thus

sup

w∈D

(1 − |w| ² ) log( 2

1 − |w| )|u ⁰ (w)f _n (ϕ(w))|

≤ kuk L max

|w|≤r |f n (w)| + CM sup

|ϕ(w)|>r

(1 − |w| ² ) log( _1−|w| ² ) (1 − |ϕ(w)| ² ) ^α−1 |u ⁰ (w)|, which also implies that

sup

w∈D (1 − |w| ² ) log( 2

1 − |w| )|u ⁰ (w)f n (ϕ(w))| −→ 0 as n → ∞.

Case α = ^2+q _p < 1. Similarly, let {f n } be a bounded sequence in F (p, q, s) which converges to 0 uniformly on compact subsets of D. By Lemma 2.1 and [10, Lemma 3.2] we have sup _z∈D |f n (z)| −→ 0 as n → ∞. Then

sup

w∈D (1 − |w| ² ) log( 2

1 − |w| )|u ⁰ (w)f n (ϕ(w))| ≤ kuk L sup

z∈D |f n (z)| −→ 0 as n → ∞. As the same proof as in case α = ^2+q _p > 1, we get

sup

w∈D

(1 − |w| ² ) log( 2

1 − |w| )|u(w)f _n ⁰ (ϕ(w))ϕ ⁰ (w)| −→ 0.

Hence uC ϕ is a compact operator from F (p, q, s) to β L .

Case α = ^2+q _p = 1 and s > 1. Let {f n } be a bounded sequence in F (p, q, s) which converges to 0 uniformly on compact subsets of D. Let M = sup _n kf n k F (p,q,s) < +∞. By Lemma 2.1 and 2.4, we get

(13) sup

n |f n (z)| ≤ CM log 2 1 − |z| ² . If |ϕ(w)| ≤ r < 1, by u ∈ β L , then

(1 − |w| ² ) log( 2

1 − |w| )|u ⁰ (w)f _n (ϕ(w))| ≤ kuk _L max

|z|≤r |f _n (z)|.

If |ϕ(w)| > r, by (13), we have (1 − |w| ² ) log( 2

1 − |w| )|u ⁰ (w)f n (ϕ(w))|

≤ CM (1 − |w| ² ) log( 2

1 − |w| ) log( 2

1 − |ϕ(w)| ² )|u ⁰ (w)|.

Thus sup

w∈D (1 − |w| ² ) log( 2

1 − |w| )|u ⁰ (w)f n (ϕ(w))|

≤ kuk L max

|w|≤r |f n (w)| + CM (1 − |w| ² ) log( 2

1 − |w| ) log( 2

1 − |ϕ(w)| ² )|u ⁰ (w)|,

which also implies that sup

w∈D (1 − |w| ² ) log( 2

1 − |w| )|u ⁰ (w)f n (ϕ(w))| −→ 0 as n → ∞. We also similarly prove that

sup

w∈D

(1 − |w| ² ) log( 2

1 − |w| )|u(w)f _n ⁰ (ϕ(w))ϕ ⁰ (w)| −→ 0

as n → ∞. Hence lim n→∞ kuC ϕ (f n )k L = 0. This completes the proof of

Theorem 2.2. ¤

3. Boundedness and compactness of uC ϕ : F 0 (p, q, s) → β _L ⁰ In this section we characterize the boundedness and compactness of the weighted composition operator uC ϕ : F 0 (p, q, s) → β _L ⁰ . For this purpose we need another lemmas.

Lemma 3.1. Let 0 < s < ∞, 1 ≤ p < ∞, −2 < q < ∞ and q + s > −1. For 0 < t < 1, z ∈ D, let f t (z) = f (tz). If f ∈ F (p, q, s), then f t ∈ F (p, q, s) and kf t k F (p,q,s) ≤ kf k F (p,q,s) .

Proof. For f ∈ F (p, q, s) and 0 < t < 1, by the Poisson formula, we have f t (z) =

Z _2π

0

f (ze ^iθ ) 1 − t ²

|e ^iθ − t| ² dθ

2π , z ∈ D.

Then by H¨older’s inequality we get, Z

D

|f _t ⁰ (z)| ^p (1 − |z| ² ) ^q (1 − |ϕ a (z)| ² ) ^s dm(z)

≤ Z

D

Z _2π

0

|f ⁰ (e ^iθ z)| ^p 1 − t ²

|e ^iθ − t| ² dθ

2π (1 − |z| ² ) ^q (1 − |ϕ a (z)| ² ) ^s dm(z)

= Z _2π

0

Z

D

|f ⁰ (e ^iθ z)| ^p (1 − |z| ² ) ^q (1 − |ϕ a (z)| ² ) ^s dm(z) 1 − t ²

|e ^iθ − t| ² dθ 2π

≤ kf k ^p _p,q,s Z _2π

0

1 − t ²

|e ^iθ − t| ² dθ

2π = kf k ^p _p,q,s , where kf k ^p _p,q,s = sup _a∈D R

D |f ⁰ (z)| ^p (1 − |z| ² ) ^q (1 − |ϕ a (z)| ² ) ^s dm(z). That is, kf t k F (p,q,s) ≤ kf k F (p,q,s) .

¤

Lemma 3.2. Let u be an analytic function on the unit disk D and ϕ an analytic

self-map of D. Let 0 < s < ∞, 1 ≤ p < ∞, −2 < q < ∞ and q + s > −1. If

uC ϕ is a bounded operator from F 0 (p, q, s) into β _L ⁰ , then uC ϕ is bounded from

F (p, q, s) to β L .

Proof. Suppose uC ϕ is bounded from F 0 (p, q, s) into β ⁰ _L and for f ∈ F (p, q, s), we have f t ∈ F 0 (p, q, s) for all 0 < t < 1. Then, according to Lemma 3.1,

kuC ϕ (f t )k _{F (p,q,s)} ≤ kuC ϕ kkf t k _{F (p,q,s)} ≤ kuC ϕ kkf k _{F (p,q,s)} < +∞.

Hence kuC ϕ (f )k _{F (p,q,s)} ≤ kuC ϕ kkf k _{F (p,q,s)} < +∞, which shows uC ϕ is bound-

ed from F (p, q, s) to β L . ¤

Theorem 3.1. Let u be an analytic function on the unit disc D and ϕ an analytic self-map of D. Let 0 < p, s < ∞, −2 < q < ∞ and q + s > −1.

Then uC ϕ : F 0 (p, q, s) −→ β ⁰ _L is a bounded operator provided that u ∈ β _L ⁰ , uC ϕ : F (p, q, s) −→ β L is bounded and

(14) lim

|z|→1

(1 − |z| ² ) log 2

1 − |z| |u(z)ϕ ⁰ (z)| = 0.

Conversely, if uC ϕ : F 0 (p, q, s) −→ β ⁰ _L is a bounded operator, then u ∈ β _L ⁰ and (14) holds; moreover, if 1 ≤ p < ∞, then uC ϕ : F (p, q, s) −→ β L is bounded.

Proof. Assume that u ∈ β ⁰ _L and lim |z|→1

(1 − |z| ² ) log _1−|z| ² |u(z)ϕ ⁰ (z)| = 0.

Then, for each polynomial p(z), we have that (1−|z| ² ) log 2

1 − |z| |(uC ϕ p) ⁰ (z)|

≤ (1−|z| ² ) log 2

1 − |z| |u ⁰ (z)||p(ϕ(z))|+(1 − |z| ² ) log 2

1 − |z| |u(z)ϕ ⁰ (z)p ⁰ (ϕ(z))|, from which it follows that uC ϕ p ∈ β _L ⁰ . Since the set of all polynomials is dense in F 0 (p, q, s), we have that for every f ∈ F 0 (p, q, s) there is a sequence of polynomials {p n } such that kf − p n k _{F (p,q,s)} → 0 as n → ∞. Note that uC _ϕ : F (p, q, s) −→ β _L is bounded, then

kuC ϕ f − uC ϕ p n k L ≤ kuC ϕ kkf − p n k F (p,q,s) −→ 0 as n → ∞. Since β _L ⁰ is closed in subset of β L , we obtain

uC ϕ (F 0 (p, q, q)) ⊂ β ⁰ _L . So uC ϕ : F 0 (p, q, s) −→ β _L ⁰ is bounded.

Conversely, assume that uC ϕ : F 0 (p, q, s) −→ β _L ⁰ is bounded. Taking the functions f (z) = 1 and f (z) = z respectively, we obtain that u ∈ β _L ⁰ and lim _|z|→1

(1 − |z| ² ) log _1−|z| ² |u(z)ϕ ⁰ (z)| = 0. If p ≥ 1, then uC _ϕ : F (p, q, s) −→

β L is bounded by Lemma 3.2. ¤

Lemma 3.3. Let U ⊂ β _L ⁰ . Then U is compact if and only if it is closed, bounded and satisfies

|z|→1 lim

sup

f ∈U

(1 − |z| ² ) log( 2

1 − |z| )|f ⁰ (z)| = 0.

The proof is similar to [3, Lemma 1], so we omitted it.

Theorem 3.2. Let 0 < p, s < ∞, −2 < q < ∞, q + s > −1, u ∈ H(D) and ϕ be an analytic self-map of D.

(i) If 2 + q > p, then uC ϕ is a compact operator from F 0 (p, q, s) to β _L ⁰ if and only if

(15) lim

|z|→1

(1 − |z| ² ) log _1−|z| ²

(1 − |ϕ(z)| ² ) ^α |u(z)ϕ ⁰ (z)| = 0 and

|z|→1 lim

(1 − |z| ² ) log _1−|z| ²

(1 − |ϕ(z)| ² ) ^α−1 |u ⁰ (z)| = 0, where α = ^q+2 _p .

(ii) If 2 + q < p, then uC ϕ is a compact operator from F 0 (p, q, s) to β _L ⁰ if and only if u ∈ β _L ⁰ and (15) holds.

(iii) If 2 + q = p, s > 1, then uC ϕ is a compact operator from F 0 (p, q, s) to β _L ⁰ if and only if (15) holds and

|z|→1 lim

|u ⁰ (z)|(1 − |z| ² ) log( 2

1 − |z| ) log( 2

1 − |ϕ(z)| ² ) = 0.

Proof. First we consider the case that α = ^q+2 _p > 1. Suppose that u and ϕ satisfy the condition in (i). By Theorem 2.1 and Theorem 3.1, we know that uC ϕ is bounded from F 0 (p, q, s) to β _L ⁰ . Suppose that f ∈ F 0 (p, q, s) is such that kf k _{F (p,q,s)} ≤ 1. By Lemma 2.1, we know that f ∈ β _α ⁰ and kf k β

≤ Ckf k _{F (p,q,s)} . Then from Lemma 2.2 it follows that

(1 − |z| ² ) log 2

1 − |z| |(uC ϕ f ) ⁰ (z)|

≤ C (1 − |z| ² ) log _1−|z| ²

(1 − |ϕ(z)| ² ) ^α−1 |u ⁰ (z)| + C (1 − |z| ² ) log _1−|z| ²

(1 − |ϕ(z)| ² ) ^α |ϕ ⁰ (z)u(z)|, thus

sup{(1 − |z| ² ) log 2

1 − |z| |(uC ϕ f ) ⁰ (z)| : f ∈ F 0 (p, q, s), kf k F (p,q,s) ≤ 1}

≤ C (1 − |z| ² ) log _1−|z| ²

(1 − |ϕ(z)| ² ) ^α−1 |u ⁰ (z)| + C (1 − |z| ² ) log _1−|z| ²

(1 − |ϕ(z)| ² ) ^α |ϕ ⁰ (z)u(z)|, and it follows that

|z|→1 lim

sup{(1−|z| ² ) log 2

1 − |z| |(uC ϕ f ) ⁰ (z)| : f ∈ F 0 (p, q, s), kf k F (p,q,s) ≤ 1} = 0, so that uC ϕ is compact from β ⁰ _α to β _L ⁰ by Lemma 3.3.

Conversely, suppose that uC ϕ is compact from F 0 (p, q, s) to β _L ⁰ . By Lem- ma 3.3, we have

|z|→1 lim

sup{(1−|z| ² ) log 2

1 − |z| |(uC ϕ f ) ⁰ (z)| : f ∈ F 0 (p, q, s), kf k _{F (p,q,s)} ≤ M } = 0

for some M > 0. Note that the proof of Theorem 2.1 and the fact that the functions given in (4) are in F 0 (p, q, s) and have norms bounded uniformly in w, we get

|w|→1 lim

(1 − |w| ² ) log _1−|w| ²

(1 − |ϕ(w)| ² ) ^α |u(w)ϕ ⁰ (w)| = 0.

Similarly, note that the functions given in (5) are in F 0 (p, q, s) and have norms bounded uniformly in w, we get

|w|→1 lim

(1 − |w| ² ) log _1−|w| ²

(1 − |ϕ(w)| ² ) ^α−1 |u ⁰ (w)| = 0.

This completes the proof of (i).

Next, we consider the case α = ^q+2 _p < 1. Assume that u ∈ β ⁰ _L and (14) holds. Suppose that f ∈ F 0 (p, q, s) is such that kf k F (p,q,s) ≤ 1. By Lemma 2.1 and 2.4, we get

(1 − |z| ² ) log _1−|z| ² |(uC ϕ f ) ⁰ (z)|

≤ C(1 − |z| ² ) log _1−|z| ² |u ⁰ (z)| + C ^(1−|z| _(1−|ϕ(z)|

^{) log}

)

|ϕ ⁰ (z)u(z)|, thus

|z|→1 lim

sup{(1−|z| ² ) log 2

1 − |z| |(uC ϕ f ) ⁰ (z)| : f ∈ F 0 (p, q, s), kf k _{F (p,q,s)} ≤ 1} = 0.

So that uC ϕ is compact from F 0 (p, q, s) to β ⁰ _L by Lemma 3.3. The fact that the conditions in (ii) are necessary for compactness of operator uC ϕ as an operator from F 0 (p, q, s) to β _L ⁰ is proved in Theorem 3.1 and the case that α > 1.

Finally, we will prove (iii). Similarly, suppose that u and ϕ satisfy the condi- tion in (iii). By Theorem 2.2, we know that uC ϕ is bounded from F 0 (p, q, s) to β _L ⁰ . Suppose that f ∈ F 0 (p, q, s) is such that kf k F

(p,q,s) ≤ 1. By Lemma 2.1 and 2.4, we get

(1 − |z| ² ) log 2

1 − |z| |(uC ϕ f ) ⁰ (z)|

≤ C(1 − |z| ² ) log 2

1 − |z| log 2

1 − |ϕ(z)| ² |u ⁰ (z)|

+ C (1 − |z| ² ) log _1−|z| ²

1 − |ϕ(z)| ² |ϕ ⁰ (z)u(z)|, thus

|z|→1 lim

sup{(1 − |z| ² ) log 2

1 − |z| |(uC _ϕ f ) ⁰ (z)| : f ∈ F ₀ (p, q, s), kf k _F

_(p,q,s) ≤ 1} = 0,

so that uC ϕ is compact from F 0 (p, q, s) to β ⁰ _L by Lemma 3.3.

Conversely, noting that the functions given in (6) are in F 0 (p, q, s) and have norms bounded uniformly in w, we get

|w|→1 lim

|u ⁰ (w)|(1 − |w| ² ) log( 2

1 − |w| ) log( 2

1 − |ϕ(w)| ² ) = 0.

The another necessity in condition (iii) is similarly proved in (i). This completes

the proof of Theorem 3.2 ¤

Acknowledgment. The author is greatly indebted to the referee for his(her) valuable advice.

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Department of Mathematics Shantou University

Shantou 515063, China and

School of Mathematics and Computer Science Fujian Normal University

Fuzhou 350007, China

E-mail address: [email protected]; ye [email protected]