$(L^{r}, L^{s})$ Resolvent Estimate for the Sphere off the Line $\frac{1}{r}-\frac{1}{s}=\frac{2}{n}$

TL;DR

This paper extends resolvent estimates on the sphere to exponents off the critical line, using interpolation techniques and spherical harmonic projections, highlighting the limitations of uniform bounds outside the line.

Contribution

It introduces a new $(L^{r}, L^{s})$ estimate for spherical harmonic projections off the critical line, expanding the understanding of resolvent bounds on the sphere.

Findings

Extended resolvent estimates off the critical line

Applied interpolation techniques to spherical harmonic operators

Identified limitations of uniform bounds outside the line

Abstract

We extend the resolvent estimate on the sphere to exponents off the line $\frac{1}{r}-\frac{1}{s}=\frac{2}{n}$. Since the condition $\frac{1}{r}-\frac{1}{s}=\frac{2}{n}$ on the exponents is necessary for a uniform bound, one cannot expect estimates off this line to be uniform still. The essential ingredient in our proof is an $(L^{r}, L^{s})$ norm estimate on the operator $H_{k}$ that projects onto the space of spherical harmonics of degree $k$. In showing this estimate, we apply an interpolation technique first introduced by Bourgain [2]. The rest of our proof parallels that in Huang-Sogge [8].