New $L^p$ bounds for Bochner-Riesz multipliers associated with convex planar domains with rough boundary

TL;DR

This paper introduces new $L^p$ bounds for Bochner-Riesz multipliers linked to convex planar domains with rough boundaries, based on properties like additive energy and directional maximal operators, improving previous results.

Contribution

It identifies key geometric properties of convex domains that lead to sharper $L^p$ bounds for Bochner-Riesz multipliers, extending prior work by Seeger and Ziesler.

Findings

Domains with low additive energy have improved $L^p$ bounds.

Asymptotically good $L^q$ bounds for Nikodym-type maximal operators imply better Bochner-Riesz bounds.

New geometric criteria enhance understanding of multiplier bounds.

Abstract

We consider generalized Bochner-Riesz multipliers of the form $(1-\rho(\xi))_+^{\lambda}$ where $\rho(\xi)$ is the Minkowski functional of a convex domain in $\mathbb{R}^2$, with emphasis on domains for which the usual Carleson-Sj\"{o}lin $L^p$ bounds can be improved. We produce convex domains for which previous results due to Seeger and Ziesler are not sharp. We identify two key properties of convex domains that lead to improved $L^p$ bounds for the associated Bochner-Riesz operators. First, we introduce the notion of the "additive energy" of the boundary of a convex domain. Second, we associate a set of directions to a convex domain and define a sequence of Nikodym-type maximal operators corresponding to this set of directions. We show that domains that have low higher order additive energy, as well as those which have asymptotically good $L^q$ bounds for the corresponding sequence of Nikodym-type maximal operators where $q=(p^{\prime}/2)^{\prime}$, have improved $L^p$ bounds for the associated Bochner-Riesz operators over those proved by Seeger and Ziesler.