Resolvent Estimates for the Laplacian in the Euclidean Space and on the Sphere

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Johns Hopkins University
The thesis consists of two parts. In the first part, we prove an endpoint version of the uniform Sobolev inequalities in Kenig-Ruiz-Sogge [13]. Although strong inequality no longer holds for the pairs of exponents that are endpoints in the classical theorem of Kenig-Ruiz-Sogge [13], they enjoy restricted weak type inequality. The key ingredient in our proof is a method of interpolation first introduced by Bourgain [2]. Along with the proof of the endpoint uniform Sobolev inequalities, we give a complete description of the boundary case of Sogge's version of the Stein-Tomas restriction theorem in [18]. In the second part of the thesis, we turn to the sphere. More specifically, We extend the resolvent estimate on the sphere to exponent pairs off the line 1/r-1/s=2/n. Since the condition 1/r-1/s=2/n on the exponent pairs is necessary for a uniform bound in the Euclidean case, one should not expect estimates off this line to be uniform for manifolds with constant curvature. The crucial step in our proof is an oscillatory integral theorem which easily leads to an (L^{r}, L^{s}) norm estimate on the operator H_{k} that projects onto the space of spherical harmonics of degree k. The rest of our proof then parallels that in Huang-Sogge [12].
resolvent estimates for the Laplacian, Stein-Tomas restriction theorem, sphere