Gagliardo–Nirenberg–Sobolev inequalities for convex domains in Rd

Abstract

A special type of Gagliardo–Nirenberg–Sobolev (GNS) inequalities in Rd has played a key role in several proofs of Lieb–Thirring inequalities. Recently, a need for GNS inequalities in convex domains of Rd, in particular for cubes, has arisen. 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 ofthe domain. Later, using the discrete version of Rumin’s method, we prove GNS inequalities on cubes with improved constants.