Semiclassical L^p Estimates of Quasimodes on Submanifolds

TL;DR

This paper establishes sharp semiclassical L^p estimates for quasimodes restricted to submanifolds on compact manifolds, extending Strichartz estimates to this geometric setting.

Contribution

It provides new sharp L^p restriction estimates for quasimodes on submanifolds, advancing understanding of semiclassical analysis and PDEs on manifolds.

Findings

Sharp L^p estimates for quasimodes on submanifolds

Extensions of Strichartz estimates in a semiclassical context

Results applicable to a broad class of pseudodifferential operators

Abstract

Let M be a compact manifold and P = P(h) a semiclassical pseudodifferential operator on M . Suppose that u(h) is a L^2 normalised family of functions such that P(h)u(h) is O(h) in L^2, as h goes to 0. Then, for any compact submanifold Y contained in M, we obtain estimates on the L^p norm of u(h) restricted to Y, with exponents that are sharp for h goes to 0. As part of the technical development we prove some extensions of the abstract Strichartz estimates of Keel and Tao.