Weighted Gagliardo-Nirenberg Interpolation Inequalities

TL;DR

This paper establishes weighted Gagliardo-Nirenberg interpolation inequalities involving fractional derivatives using harmonic analysis techniques, generalizing classical inequalities and accommodating various weight functions.

Contribution

It introduces a flexible harmonic analysis approach to prove weighted inequalities with fractional derivatives for a broad class of weights, including non-homogeneous and power-law types.

Findings

Proved weighted Gagliardo-Nirenberg inequalities with fractional derivatives.

Generalized Stein-Weiss and Caffarelli-Kohn-Nirenberg inequalities.

Extended results to non-homogeneous weights and Japanese bracket weights.

Abstract

In this paper, we prove weighted versions of the Gagliardo-Nirenberg interpolation inequality with Riesz as well as Bessel type fractional derivatives. We use a harmonic analysis approach employing several methods, including the method of domination by sparse operators, to obtain such inequalities for a general class of weights satisfying Muckenhoupttype conditions. We also obtain improved results for some particular families of weights, including power-law weights $|x|^\alpha$. In particular, we prove an inequality which generalizes both the Stein-Weiss inequality and the Caffarelli-Kohn-Nirenberg inequality. However, our approach is sufficiently flexible to allow as well for non-homogeneous weights and we also prove versions of the inequalities with Japanese bracket weights $\langle x \rangle^\alpha=(1+|x|^2)^{\alpha/2}$.