Asymptotic expansions and extremals for the critical Sobolev and Gagliardo-Nirenberg inequalities on a torus

TL;DR

This paper thoroughly investigates interpolation inequalities for periodic zero-mean functions, focusing on the critical logarithmic Sobolev inequality in two dimensions, including extremals, best constants, and asymptotic behaviors.

Contribution

It provides new results on the existence, asymptotic expansions, and best constants for these inequalities, especially in the critical two-dimensional case.

Findings

Existence of extremals for the inequalities.

Asymptotic expansions for extremals and constants.

Results on the algebraic case and higher dimensions.

Abstract

We give a comprehensive study of interpolation inequalities for periodic functions with zero mean, including the existence of and the asymptotic expansions for the extremals, best constants, various remainder terms, etc. Most attention is paid to the critical (logarithmic) Sobolev inequality in the two-dimensional case, although a number of results concerning the best constants in the algebraic case and different space dimensions are also obtained.