Extremals for $\alpha$-Strichartz inequalities

TL;DR

This paper characterizes when extremal sequences for one-dimensional $\alpha$-Strichartz inequalities are precompact, providing conditions for the existence of extremals and extending results to asymmetric cases using profile decompositions and the van der Corput Lemma.

Contribution

It establishes a necessary and sufficient condition for precompactness of extremal sequences and proves the existence of extremals for non-endpoint $\alpha$-Strichartz inequalities, including asymmetric cases.

Findings

Characterization of precompactness of extremal sequences

Existence of extremals for non-endpoint inequalities

Extension to asymmetric cases

Abstract

A necessary and sufficient condition on the precompactness of extremal sequences for one dimensional $\alpha$-Strichartz inequalities, equivalently $\alpha$-Fourier extension estimates, is established based on the profile decomposition arguments. One of our main tools is an operator-convergence dislocation property consequence which comes from the van der Corput Lemma. Our result is valid in asymmetric cases as well. In addition, we obtain the existence of extremals for non-endpoint $\alpha$-Strichartz inequalities.