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p-Summing Bloch mappings on the complex unit disc

M. G. Cabrera-Padilla, A. Jiménez-Vargas, D. Ruiz-Casternado

Abstract

The notion of $p$-summing Bloch mapping from the complex unit open disc $\mathbb{D}$ into a complex Banach space $X$ is introduced for any $1\leq p\leq\infty$. It is shown that the linear space of such mappings, equipped with a natural seminorm $π^{\mathbb{B}}_p$, is Möbius-invariant. Moreover, its subspace consisting of all those mappings which preserve the zero is an injective Banach ideal of normalized Bloch mappings. Bloch versions of the Pietsch's domination/factorization Theorem and the Maurey's extrapolation Theorem are presented. We also introduce the spaces of $X$-valued Bloch molecules on $\mathbb{D}$ and identify the spaces of normalized $p$-summing Bloch mappings from $\mathbb{D}$ into $X^*$ under the norm $π^{\mathbb{B}}_p$ with the duals of such spaces of molecules under the Bloch version of the $p$-Chevet--Saphar tensor norms $d_p$.

p-Summing Bloch mappings on the complex unit disc

Abstract

The notion of -summing Bloch mapping from the complex unit open disc into a complex Banach space is introduced for any . It is shown that the linear space of such mappings, equipped with a natural seminorm , is Möbius-invariant. Moreover, its subspace consisting of all those mappings which preserve the zero is an injective Banach ideal of normalized Bloch mappings. Bloch versions of the Pietsch's domination/factorization Theorem and the Maurey's extrapolation Theorem are presented. We also introduce the spaces of -valued Bloch molecules on and identify the spaces of normalized -summing Bloch mappings from into under the norm with the duals of such spaces of molecules under the Bloch version of the -Chevet--Saphar tensor norms .
Paper Structure (12 sections, 15 theorems, 111 equations)

This paper contains 12 sections, 15 theorems, 111 equations.

Table of Contents

  1. $p$-Summing Bloch mappings on the unit disc
  2. Inclusions
  3. Injective Banach ideal property
  4. Möbius invariance
  5. Pietsch domination
  6. Pietsch factorization
  7. Maurey extrapolation
  8. Banach-valued Bloch molecules on the unit disc
  9. Pairing
  10. Projective norm
  11. $p$-Chevet--Saphar Bloch norms
  12. Duality

Key Result

Proposition 1.1

Let $1\leq p<q\leq\infty$. Then $\Pi^{\mathcal{B}}_p(\mathbb{D},X)\subseteq\Pi^{\mathcal{B}}_q(\mathbb{D},X)$ with $\pi^{\mathcal{B}}_q(f)\leq\pi^{\mathcal{B}}_p(f)$ for all $f\in\Pi^{\mathcal{B}}_p(\mathbb{D},X)$. Moreover, $\Pi^{\mathcal{B}}_\infty(\mathbb{D},X)=\mathcal{B}(\mathbb{D},X)$ with $\p

Theorems & Definitions (29)

  • Proposition 1.1
  • proof
  • Proposition 1.2
  • proof
  • Proposition 1.3
  • Theorem 1.4
  • proof
  • Lemma 1.5
  • proof
  • Theorem 1.6
  • ...and 19 more