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A new criterion for the normalized Haar measure to be a Pietsch measure

Alexandre Bispo, Renato Macedo, Joedson Santos

Abstract

In this paper we present a new criterion to determine when the normalized Haar measure on a compact topological group is a Pietsch measure for nonlinear summing mappings. As a consequence, we provide a partial answer to a problem raised by Botelho et al. in \cite{haar botelho} motivated by a question posed to the authors by J. Diestel. It is explicitly shown that this criterion encompasses recent and new results as particular cases.

A new criterion for the normalized Haar measure to be a Pietsch measure

Abstract

In this paper we present a new criterion to determine when the normalized Haar measure on a compact topological group is a Pietsch measure for nonlinear summing mappings. As a consequence, we provide a partial answer to a problem raised by Botelho et al. in \cite{haar botelho} motivated by a question posed to the authors by J. Diestel. It is explicitly shown that this criterion encompasses recent and new results as particular cases.

Paper Structure

This paper contains 7 sections, 8 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Notations and background
  3. Main Results
  4. A partial answer to Open Problem
  5. Other applications
  6. Lipschitz $p$-semi-integral multilinear operators
  7. $\Phi$-summing multilinear operators, where $\Phi$ is an Orlicz function

Key Result

Theorem 3.2

Let $G$ be a compact Hausdorff topological group and $D: G \to [0,\infty)$ be a continuous function. If there is a regular Borel probability measure $\nu$ on $G$ such that $\nu$ is $D$-dominating, then $\sigma_{G}$ is $D$-dominating.

Theorems & Definitions (19)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • ...and 9 more