Endpoint sparse domination for classes of multiplier transformations

TL;DR

This paper establishes endpoint sparse domination results for translation invariant multiscale operators using Fourier multiplier classes, leading to optimal bounds for oscillatory and radial bump multipliers.

Contribution

It introduces new endpoint sparse domination results for classes of Fourier multipliers, extending the theory to multiscale operators with optimal bounds.

Findings

Optimal sparse bounds for oscillatory multipliers

Sparse domination for multi-scale radial bump multipliers

Framework based on localized M^{p→q} norms

Abstract

We prove endpoint results for sparse domination of translation invariant multiscale operators. The results are formulated in terms of dilation invariant classes of Fourier multipliers based on natural localized $M^{p\to q}$ norms which express appropriate endpoint regularity hypotheses. The applications include new and optimal sparse bounds for classical oscillatory multipliers and multi-scale versions of radial bump multipliers.