Sharp $L^p$-$L^q$ estimates for the spherical harmonic projection

TL;DR

This paper establishes sharp $L^p$-$L^q$ bounds for spherical harmonic projections and applies these results to prove off-diagonal Carleman estimates for the Laplacian, extending previous work by Jerison, Kenig, and Stein.

Contribution

The paper provides the first sharp $L^p$-$L^q$ estimates for spherical harmonic projections and uses them to extend Carleman estimates for the Laplacian.

Findings

Sharp $L^p$-$L^q$ bounds for spherical harmonic projections

Extension of Carleman estimates for the Laplacian

Improved understanding of harmonic analysis on spheres

Abstract

We consider $L^p$-$L^q$ estimates for the spherical harmonic projection operators and obtain sharp bounds on a certain range of $p$, $q$. As an application, we provide a proof of off-diagonal Carleman estimates for the Laplacian, which extends the earlier results due to Jerison and Kenig \cite{JK}, and Stein \cite{St-append}.