Boundedness of projection operators and Ces\`aro means in weighted $L^p$ space on the unit sphere

TL;DR

This paper establishes sharp local bounds for projection operators and demonstrates the convergence of Cesàro means in weighted $L^p$ spaces on the sphere, ball, and simplex, with implications for harmonic analysis in weighted settings.

Contribution

It provides new sharp local estimates for projection operators and proves convergence of Cesàro means in weighted spaces, extending classical results to more general weights.

Findings

Sharp local estimates for projection operators

Convergence of Cesàro means below critical index

Results applicable to sphere, ball, and simplex

Abstract

For the weight function $\prod_{i=1}^{d+1}|x_i|^{2\k_i}$ on the unit sphere, sharp local estimates of the orthogonal projection operators are obtained and used to prove the convergence of the Ces\`aro $(C,\delta)$ means in the weighted $L^p$ space for $\delta$ below the critical index. Similar results are also proved for corresponding weight functions on the unit ball and on the simplex.