Rearrangements and Leibniz-type rules of mean oscillations

TL;DR

This paper establishes a rearrangement inequality in probability measure spaces to derive sharp Leibniz-type rules for mean oscillations in various function spaces, enhancing understanding of oscillation behavior.

Contribution

It introduces a new rearrangement inequality that leads to precise Leibniz-type rules for mean oscillations in Lp and rearrangement invariant Banach spaces.

Findings

Proved a rearrangement inequality in probability measure spaces.

Derived sharp Leibniz-type rules for mean oscillations.

Applicable to Lp-spaces and rearrangement invariant Banach spaces.

Abstract

We shall prove a rearrangement inequality in probability measure spaces in order to obtain sharp Leibniz-type rules of mean oscillations in Lp-spaces and rearrangement invariant Banach function spaces.