Sparse Endpoint Estimates for Bochner-Riesz Multipliers on the Plane

Abstract

For $0<\lambda <\frac{1}{2}$, let $B_{\lambda }$ be the Bochner-Riesz multiplier of index $\lambda$ on the plane. Associated to this multiplier is the critical index $1<p_{\lambda }=\frac{4}{3+2\lambda }<\frac{4}{3}$. We prove a sparse bound for $B_{\lambda }$ with indices $(p_{\lambda },q)$, where $p_{\lambda }^{\prime }<q<4$. This is a further quantification of the endpoint weak $L^{p_{\lambda }}$ boundedness of $B_{\lambda }$, due to Seeger. Indeed, the sparse bound immediately implies new endpoint weighted weak type estimates for weights in $A_1\cap R{H}_{\rho }$, where $\rho >\frac{4}{4-3p_{\lambda }}$.

10 pages

References in corpus (2)

• Sparse bilinear forms for Bochner Riesz multipliers and applications
• Sparse Bounds for Bochner-Riesz Multipliers

Cited by in corpus (6)

• Dyadic harmonic analysis and weighted inequalities: the sparse revolution
• Sparse Bounds for Bochner-Riesz Multipliers
• Multi-scale sparse domination
• Vector-valued extensions of operators through multilinear limited range extrapolation
• Endpoint sparse domination for classes of multiplier transformations
• Bochner-Riesz means at the critical index: Weighted and sparse bounds