On the necessity of the assumptions used to prove Hardy-Littlewood and Riesz Rearrangement Inequalities 1

TL;DR

This paper demonstrates that supermodularity and kernel monotonicity are essential assumptions for the validity of generalized Hardy-Littlewood and Riesz rearrangement inequalities, clarifying the foundational conditions of these mathematical results.

Contribution

It establishes the necessity of supermodularity and kernel monotonicity assumptions for the inequalities, advancing the theoretical understanding of their underlying conditions.

Findings

Supermodularity is necessary for the inequalities.

Monotonicity of kernels is essential in Riesz-type integrals.

Clarifies foundational assumptions of rearrangement inequalities.

Abstract

We prove that supermodularity is a necessary condition for the generalized Hardy- Littlewood and Riesz rearrangement inequalities. We also show the necessity of the monotonicity of the kernels involved in the Riesz{type integral.