The generalized Libera transform is bounded on the Besov mixed-norm, BMOA and VMOA spaces on the unit disc

doi:10.1016/j.amc.2009.03.022

Applied Mathematics and Computation

Volume 213, Issue 2, 15 July 2009, Pages 304-311

https://doi.org/10.1016/j.amc.2009.03.022 Get rights and content

Abstract

The main results of this note prove that the generalized Libera operator is bounded on the Besov mixed-norm space $B_{α}^{p, q} (D)$ as well as on the spaces BMOA and VMOA on the unit disk. The compactness of the operator on $B_{α}^{p, q} (D)$ is also studied.

Section snippets

Introduction and preliminaries

Let $D$ denote the unit disc in the complex plane $C, \partial D$ its boundary, $H (D)$ the set of all analytic functions on $D$ and $dm (\cdot) = \frac{1}{π} r dr d θ$ the normalized Lebesgue area measure on $D$ . For each complex $γ$ with $R γ > - 1$ and for each nonnegative integer k, let $A_{k}^{γ}$ be defined as the $k th$ coefficient in the expression $(1 - x)^{- (γ + 1)} = \sum_{k = 0}^{\infty} A_{k}^{γ} x^{k},$ so that $A_{k}^{γ} = \frac{Γ (γ + k + 1)}{Γ (γ + 1) Γ (k + 1)} .$

Let $z_{0} \in D$ be fixed, then the following operator $Λ_{z_{0}} (f) (z) = \frac{1}{z - z_{0}} \int_{z_{0}}^{z} f (t) dt, z \in D,$ where $f \in H (D),$ is one of the most natural averaging operators on $H (D)$ ,

Auxiliary results

The following lemma, regarding the boundedness of the composition operator on the mixed-norm space, was proved in [41]. We sketch its proof here for the completeness and for benefit of the reader.

Lemma 1

Let $p, q \in (0, \infty), α > - 1, φ : D \to D$ be a nonconstant analytic function. Then the composition operator $C_{φ} (f) = f \circ φ$ on $A_{α}^{p, q} (D)$ satisfies the following inequality: $‖ C_{φ} (f) ‖_{A_{α}^{p, q}}^{q} ⩽ 3^{\frac{q}{p}} {(\frac{‖ φ ‖_{\infty} + | φ (0) |}{‖ φ ‖_{\infty} - | φ (0) |})}^{\frac{q}{p} + α + 1} ‖ f ‖_{A_{α}^{p, q}}^{q} .$

Proof

Let $a = | φ (0) |$ and $b = ‖ φ ‖_{\infty} = \sup_{z \in D} | φ (z) |$ . By a well-known consequence of the Schwarz’s Lemma (see, for

Boundedness of the generalized Libera transform on $B_{α}^{p, q}, BMOA$ and VMOA

In this section we prove the main results of this paper. Let $d μ_{γ} (t) = (γ + 1) (1 - t)^{γ} dt and d μ_{k, α, q} (r) = (1 - r)^{q (k - α) - 1} dr .$

Theorem 1

(i)
For $z_{0} \in \bar{D}$ fixed, the generalized Libera transform (3) is bounded on the Besov mixed-norm space $B_{α}^{p, q}$ if $p, q \in [1, \infty), α > 0$ .
(ii)
For $z_{0} \in D$ fixed, the generalized Libera transform (3) is bounded on the Besov mixed-norm space $B_{α}^{p, q}$ if $p, q \in (0, \infty), α > 0$ .

Proof

(i)
We may assume that $γ$ is a real number. Applying Minkowski’s inequality twice, Lemma 1, with $φ = ϕ_{t}$ and the fact that $‖ ϕ_{t} ‖_{\infty} = (1 - t) | z_{0} | + t$ , we obtain $‖ Λ_{z_{0}}^{γ} (f) ‖_{B_{α}^{p}}$

Compactness of the generalized Libera transform on $B_{α}^{p, q}$

In this section we find some sufficient conditions for the generalized Libera transform (3) to be compact on the Besov mixed-norm space $B_{α}^{p, q}$ . Compactness of the operator (3) on $A_{α}^{p, q}$ is studied in [41].

Theorem 3

For $z_{0} \in D$ , the generalized Libera transform (3) is compact on the Besov mixed-norm space $B_{α}^{p, q}$ if $1 ⩽ p < \infty, 0 < q < \infty, α > 0$ .

Proof

Similarly to Lemmas 4 and 5 of [41] we can show that the operator $Λ_{z_{0}}^{γ} : B_{α}^{p, q} \to B_{α}^{p, q}$ is compact if and only if for every bounded sequence $(f_{m})_{m \in N}$ in $B_{α}^{p, q}$ which converges to zero

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The generalized Libera transform is bounded on the Besov mixed-norm, BMOA and VMOA spaces on the unit disc

Abstract

Section snippets

Introduction and preliminaries

Auxiliary results

Boundedness of the generalized Libera transform on Bαp,q,BMOA and VMOA

Compactness of the generalized Libera transform on Bαp,q

J. Math. Anal. Appl.

Appl. Math. Comput.

Acta Math. Sci. Ser. B Engl. Ed.

J. Math. Anal. Appl.

J. Math. Anal. Appl.

J. Math. Anal. Appl.

Appl. Math. Comput.

J. Math. Anal. Appl.

Appl. Math. Comput.

J. Math. Anal. Appl.

An integral operator on Hp and Hardy’s inequality

J. Anal. Math.

Holomorphic Sobolev spaces on the ball

Diss. Math.

On some integral operators on the unit polydisk and the unit ball

Taiwanese J. Math.

Estimates of an integral operator on function spaces

Taiwanese J. Math.

The generalized Cesàro operator on the unit polydisk

Taiwanese J. Math.

Addendum to the paper “A note on weighted Bergman spaces and the Cesàro operator”

Nagoya Math. J.

Metrical and topological properties of a generalized Libera transform

Arch. Math.

Bounded Analytic Functions

Theory of Bergman Spaces

Banach Spaces of Analytic Functions

A class of nonlinear integral operators preserving double subordinations

Abstr. Appl. Anal.

Boundedness of the generalized Libera transform on $B_{α}^{p, q}, BMOA$ and VMOA

Compactness of the generalized Libera transform on $B_{α}^{p, q}$

An integral operator on $H^{p}$ and Hardy’s inequality