Sharp weighted Sobolev and Gagliardo-Nirenberg inequalities on half space via mass transport and consequences

TL;DR

This paper develops sharp weighted Sobolev and Gagliardo-Nirenberg inequalities on half spaces using mass transport methods, characterizes extremal functions, and extends results to more general weighted domains with applications to logarithmic Sobolev inequalities.

Contribution

It introduces a novel application of mass transport to derive sharp weighted inequalities on half spaces and extends these results to complex weighted domains.

Findings

Characterization of extremal functions in weighted Sobolev inequalities

Reproduction of known sharp Gagliardo-Nirenberg inequalities via dimension reduction

Derivation of a weighted $L^p$-logarithmic Sobolev inequality

Abstract

By adapting the mass transportation technique of Cordero-Erausquin, Nazaret and Villani, we obtain a family of sharp Sobolev and Gagliardo-Nirenberg (GN) inequalities on the half space $\mathbf{R}^{n-1}\times\mathbf{R}_+$, $n\geq 1$ equipped with the weight $\omega(x) = x_n^a$, $a\geq 0$. It amounts to work with the fractional dimension $n_a = n+a$. The extremal functions in the weighted Sobolev inequalities are fully characterized. Using a dimension reduction argument and the weighted Sobolev inequalities, we can reproduce a subfamily of the sharp GN inequalities on the Euclidean space due to Del Pino and Dolbeault, and obtain some new sharp GN inequalities as well. Our weighted inequalities are also extended to the domain $\mathbf{R}^{n-m}\times \mathbf{R}^m_+$ and the weights are $\omega(x,t) = t_1^{a_1}\dots t_m^{a_m}$, where $n\geq m$, $m\geq 0$ and $a_1,\cdots,a_m\geq 0$. A weighted $L^p$-logarithmic Sobolev inequality is derived from these inequalities.