(Lr,Ls) Resolvent estimate for the sphere off the line 1 r −1 s = 2 n

Tianyi Ren

doi:10.1142/s1664360719500036

Abstract

We extend the resolvent estimate on the sphere to exponents off the line [Formula: see text]. Since the condition [Formula: see text] 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 [Formula: see text] norm estimate on the operator [Formula: see text] that projects onto the space of spherical harmonics of degree [Formula: see text]. In showing this estimate, we apply an interpolation technique first introduced by Bourgain [J. Bourgain, Estimations de certaines fonctions maximales, C. R. Acad. Sci. Paris Sér. I Math. 301(10) (1985) 499–502.]. The rest of our proof parallels that in Huang–Sogge [S. Huang and C. D. Sogge, Concerning [Formula: see text] resolvent estimates for simply connected manifolds of constant curvature, J. Funct. Anal. 267(12) (2014) 4635–4666].

Highlights

We extend the resolvent estimate on the sphere to exponents off the line
Since Kenig–Ruiz–Sogge’s classical result [9] on resolvent estimate in the Euclidean space, there has been a lot of interest and endeavor in extending this work to manifolds
In 2014, Huang and Sogge [8] proved the resolvent estimate on spaces of constant positive and negative curvature for exponents lying on the full line segment (1)

Summary

Introduction

Since Kenig–Ruiz–Sogge’s classical result [9] on resolvent estimate in the Euclidean space, there has been a lot of interest and endeavor in extending this work to manifolds. Shao and Yao [11] proved the resolvent estimate on compact manifolds for exponents that do not necessarily lie on the line of duality. In their theorem, the exponents r and s need only satisfy. In 2014, Huang and Sogge [8] proved the resolvent estimate on spaces of constant positive and negative curvature for exponents lying on the full line segment (1). Real interpolation yields the desired estimate on the segment between the endpoints We already followed this path in [10], to prove an endpoint version of Stein–Tomas Fourier restriction theorem

Proof of Proposition 1

Proof of Theorem 1