A Marcinkiewicz maximal-multiplier theorem

TL;DR

This paper establishes a boundedness result for a maximal operator involving multipliers with bounded variation norm, extending the understanding of multiplier theorems in harmonic analysis.

Contribution

It introduces a Marcinkiewicz maximal-multiplier theorem for r < 2, demonstrating boundedness of a maximal operator over multipliers with controlled variation.

Findings

Proves boundedness of the maximal operator for r < 2.

Extends classical multiplier theorems to a maximal setting.

Provides new tools for analyzing multipliers with bounded variation.

Abstract

For r < 2, we prove the boundedness of a maximal operator formed by applying all multipliers m with $\|m\|_{V^r} \leq 1$ to a given function.