Bochner-Riesz means at the critical index: Weighted and sparse bounds

TL;DR

This paper investigates Bochner-Riesz means at the critical index on weighted spaces, establishing new endpoint sparse bounds and extending weak type inequalities, with optimal results in certain dimensions.

Contribution

It introduces new endpoint sparse domination results for Bochner-Riesz means at the critical index, extending weak type inequalities to weighted spaces and achieving optimal bounds in specific cases.

Findings

Extended Vargas' weak type (1,1) inequality to weighted spaces for certain p.

Established new endpoint sparse domination results, nearly optimal in dimension 2.

Proved fully optimal sparse bounds for means of index (d-1)/(2d+2).

Abstract

We consider Bochner-Riesz means on weighted $L^p$ spaces, at the critical index $\lambda(p)=d(\frac 1p-\frac 12)-\frac 12$. For every $A_1$-weight we obtain an extension of Vargas' weak type $(1,1)$ inequality in some range of $p>1$. To prove this result we establish new endpoint results for sparse domination. These are almost optimal in dimension $d=2$; partial results as well as conditional results are proved in higher dimensions. For the means of index $\lambda_*=\frac{d-1}{2d+2}$ we prove fully optimal sparse bounds.