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In this joint work with Jean Bourgain,Peng Shao and Xiaohua Yao,we study $L^p$ estimates for the resolvent and address a question raised in recent work of Dos Santos Ferreira,Kenig and Salvo.We obtain sharp estimates for the round sphere and Zoll manifolds,and establish necessary conditions for favorable estimates in terms of the density of eigenvalues.We also are able to obtain necessary and sufficient conditions for favorable resolvent estimates in terms of improved $L^p$ estimates for shrinking spectral projection operators,which are related to the question as to how close in the discrete case one can come to the Stein-Tomas restricition theorem involving continuous spectrum.Using this result we improve earlier estimates of Z.Shen for the torus,using recent techniques of Bourgain based in part on multilinear estimates of Bennett,Carbery and Tao.We also are able to obtain improvements for manifolds with negative curvature using a variation of a classical argument of Berard.The fact more favorable estimates hold in these two cases is not surprising given that,unlike the sphere,their eigenvalues do not cluster.