Gagliardo-Nirenberg-Sobolev inequalities for convex domains in $\mathbb{R}^d$

TL;DR

This paper establishes Gagliardo-Nirenberg-Sobolev inequalities for convex domains in , providing explicit constants and improved bounds for cubes, which are essential for various mathematical proofs and applications.

Contribution

It proves GNS inequalities for convex domains with explicit geometric-dependent constants and improves these constants specifically for cubes using Rumin's discrete method.

Findings

GNS inequalities are established for convex domains with explicit constants.

Improved GNS constants are obtained for cubes using Rumin's method.

The results have implications for proofs of Lieb-Thirring inequalities.

Abstract

A special type of Gagliardo-Nirenberg-Sobolev (GNS) inequalities in $\mathbb{R}^d$ has played a key role in several proofs of Lieb-Thirring inequalities. Recently, a need for GNS inequalities in convex domains of $\mathbb{R}^d$, in particular for cubes, has arised. The purpose of this manuscript is two-fold. First we prove a GNS inequality for convex domains, with explicit constants which depend on the geometry of the domain. Later, using the discrete version of Rumin's method, we prove GNS inequalities on cubes with improved constants.