Conductor inequalities and criteria for Sobolev-Lorentz two-weight inequalities

TL;DR

This paper introduces integral conductor inequalities that relate Lorentz norms of gradients to capacitance integrals, leading to new criteria for Sobolev-Lorentz inequalities with two measures.

Contribution

It generalizes previous Sobolev norm inequalities to Lorentz norms and establishes necessary and sufficient conditions for two-measure Sobolev-Lorentz inequalities.

Findings

Established integral conductor inequalities for Lorentz norms

Derived criteria for Sobolev-Lorentz inequalities with two measures

Extended previous Sobolev norm inequalities to Lorentz spaces

Abstract

In this paper we present integral conductor inequalities connecting the Lorentz p,q-(quasi)norm of a gradient of a function to a one-dimensional integral of the p,q-capacitance of the conductor between two level surfaces of the same function. These inequalities generalize an inequality obtained by the second author in the case of the Sobolev norm. Such conductor inequalities lead to necessary and sufficient conditions for Sobolev-Lorentz type inequalities involving two arbitrary measures.