Interpolation inequalities in function spaces of Sobolev-Lorentz type

TL;DR

This paper investigates interpolation inequalities in Sobolev-Lorentz type spaces, establishing general conditions and deriving Gagliardo-Nirenberg inequalities for fractional derivatives, including limiting cases.

Contribution

It introduces new interpolation inequalities in Triebel-Lizorkin-Lorentz and Besov-Lorentz spaces under broad assumptions, and proves Gagliardo-Nirenberg inequalities for fractional derivatives in Lorentz spaces.

Findings

Established general interpolation inequalities for Sobolev-Lorentz spaces.

Derived Gagliardo-Nirenberg inequalities for fractional derivatives.

Included results for limiting parameter cases.

Abstract

Interpolation inequalities in Triebel-Lizorkin-Lorentz spaces and Besov-Lorentz spaces are studied for both inhomogeneous and homogeneous cases. First we establish interpolation inequalities under quite general assumptions on the parameters of the function spaces. Several results on necessary conditions are also provided. Next, utilizing the interpolation inequalities together with some embedding results, we prove Gagliardo-Nirenberg inequalities for fractional derivatives in Lorentz spaces, which do hold even for the limiting case when one of the parameters is equal to 1 or $\infty$.