On the weighted inequality between the Gagliardo and Sobolev seminorms

TL;DR

This paper establishes new weighted inequalities connecting Gagliardo, Sobolev seminorms, and Marcinkiewicz quasi-norms, improving previous results for $A_1$ weights.

Contribution

It introduces improved weighted inequalities between key seminorms and quasi-norms, extending earlier work with sharper bounds for $A_1$ weights.

Findings

Weighted inequalities between Gagliardo and Sobolev seminorms established.

Enhanced bounds for Marcinkiewicz quasi-norm and Sobolev seminorm with $A_1$ weights.

Generalization of inequalities to broader classes of weights.

Abstract

We prove weighted inequalities between the Gagliardo and Sobolev seminorms and also between the Marcinkiewicz quasi-norm and the Sobolev seminorm. With $A_1$ weights we improve earlier results of Bourgain, Brezis, and Mironescu.