Sobolev Anisotropic inequalities with monomial weights

TL;DR

This paper establishes new anisotropic Sobolev inequalities in multi-dimensional space with monomial weights, expanding the understanding of weighted inequalities in rearrangement invariant spaces.

Contribution

It introduces a novel approach using integral oscillation inequalities in multiplicative form to derive these weighted anisotropic Sobolev inequalities.

Findings

Derived anisotropic Sobolev inequalities with monomial weights.

Extended inequalities to general rearrangement invariant spaces.

Provided a new integral oscillation inequality framework.

Abstract

We derive some anisotropic Sobolev inequalities in $\mathbb{R}^{n}$ with a monomial weight in the general setting of rearrangement invariant spaces. Our starting point is to obtain an integral oscillation inequality in multiplicative form.