On extremizers for Strichartz estimates for higher order Schrödinger equations

Abstract

For an appropriate class of convex functions $\varphi$, we study the Fourier extension operator on the surface $\{(y,|y{|}^2+\varphi (y)):y\in {\mathbb{R}}^2\}$ equipped with projection measure. For the corresponding extension inequality, we compute optimal constants and prove that extremizers do not exist. The main tool is a new comparison principle for convolutions of certain singular measures that holds in all dimensions. Using tools of concentration-compactness flavor, we further investigate the behavior of general extremizing sequences. Our work is directly related to the study of extremizers and optimal constants for Strichartz estimates of certain higher order Schrödinger equations. In particular, we resolve a dichotomy from the recent literature concerning the existence of extremizers for a family of fourth order Schrödinger equations, and compute the corresponding operator norms exactly where only lower bounds were previously known.

39 pages, 1 figure; v2: typos corrected

References in corpus (5)

• Some sharp restriction inequalities on the sphere
• Existence of Extremals for a Fourier Restriction Inequality
• Optimal constants and extremisers for some smoothing estimates
• On the extremizers of an adjoint Fourier restriction inequality
• The linear profile decomposition for the fourth order Schrödinger equation

Cited by in corpus (4)

• A comparison principle for convolution measures with applications
• Extremals for $\alpha$-Strichartz inequalities
• Sharp Fourier extension on fractional surfaces
• On profile decomposition for Airy type equation