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

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].