The generalized Libera transform is bounded on the Besov mixed-norm, BMOA and VMOA spaces on the unit disc
Section snippets
Introduction and preliminaries
Let denote the unit disc in the complex plane its boundary, the set of all analytic functions on and the normalized Lebesgue area measure on . For each complex with and for each nonnegative integer k, let be defined as the coefficient in the expressionso that
Let be fixed, then the following operatorwhere is one of the most natural averaging operators on ,
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 be a nonconstant analytic function. Then the composition operator on satisfies the following inequality: Proof Let and . By a well-known consequence of the Schwarz’s Lemma (see, for
Boundedness of the generalized Libera transform on and VMOA
In this section we prove the main results of this paper. Let Theorem 1 For fixed, the generalized Libera transform (3) is bounded on the Besov mixed-norm space if . For fixed, the generalized Libera transform (3) is bounded on the Besov mixed-norm space if .
Proof
- (i)
We may assume that is a real number. Applying Minkowski’s inequality twice, Lemma 1, with and the fact that , we obtain
Compactness of the generalized Libera transform on
In this section we find some sufficient conditions for the generalized Libera transform (3) to be compact on the Besov mixed-norm space . Compactness of the operator (3) on is studied in [41]. Theorem 3 For , the generalized Libera transform (3) is compact on the Besov mixed-norm space if . Proof Similarly to Lemmas 4 and 5 of [41] we can show that the operator is compact if and only if for every bounded sequence in which converges to zero
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