Strichartz estimates for equations with structured Lipschitz coefficients

TL;DR

This paper establishes sharp Strichartz estimates for Schrödinger and wave equations with Lipschitz and structured coefficients, using Phillips functional calculus, and applies these results to nonlinear equations and spectral multipliers.

Contribution

It introduces a novel approach using Phillips functional calculus to derive dispersive estimates for equations with structured Lipschitz coefficients, extending classical results.

Findings

Sharp Strichartz estimates for equations with Lipschitz coefficients

Extension of estimates to structured coefficients of bounded variation

Application to nonlinear Schrödinger equations and spectral multipliers

Abstract

Sharp Strichartz estimates are proved for Schr\"odinger and wave equations with Lipschitz coefficients satisfying additional structural assumptions. We use Phillips functional calculus as a substitute for Fourier inversion, which shows how dispersive properties are inherited from the constant coefficient case. Global Strichartz estimates follow provided that the derivatives of the coefficients are integrable. The estimates extend to structured coefficients of bounded variations. As applications we derive Strichartz estimates with additional derivative loss for wave equations with H\"older-continuous coefficients and solve nonlinear Schr\"odinger equations. Finally, we record spectral multiplier estimates, which follow from the Strichartz estimates by well-known means.