(L ^r , l ^s ) resolvent estimate for the sphere off the line ¹ r ⁻¹ s ^{= 2} n

We extend the resolvent estimate on the sphere to exponents off the line ¹ r ⁻¹ s ^{= 2} n ^. Since the condition ¹ r ⁻¹ s ^{= 2} on the exponents is necessary for a uniform bound, one n 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 [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 l ^p resolvent estimates for simply connected manifolds of constant curvature, J. Funct. Anal. 267(12) (2014) 4635–4666].