When is the Haar measure a Pietsch measure for nonlinear mappings?

TL;DR

This paper demonstrates that the Haar measure serves as a Pietsch measure for certain nonlinear summing mappings on translation invariant subspaces of continuous functions, extending known linear results to nonlinear contexts.

Contribution

It establishes the Haar measure as a Pietsch measure for nonlinear summing mappings, answering a question by J. Diestel and applying to various classes of nonlinear mappings.

Findings

Haar measure is a Pietsch measure for nonlinear summing mappings on certain subspaces.

The result generalizes the linear case to nonlinear settings.

Application to well-studied classes of nonlinear summing mappings.

Abstract

We show that, as in the linear case, the normalized Haar measure on a compact topological group $G$ is a Pietsch measure for nonlinear summing mappings on closed translation invariant subspaces of $C(G)$. This answers a question posed to the authors by J. Diestel. We also show that our result applies to several well-studied classes of nonlinear summing mappings. In the final section some problems are proposed.