Littlewood--Paley--Rubio de Francia inequality in Morrey--Campanato spaces

TL;DR

This paper extends Rubio de Francia's Littlewood--Paley inequality to Morrey--Campanato spaces, including H"older and BMO spaces, using adapted methods beyond the classical $L^p$ setting.

Contribution

It develops and applies Rubio de Francia's techniques to establish Littlewood--Paley inequalities in Morrey--Campanato spaces, broadening their applicability.

Findings

Proved Littlewood--Paley inequality in H"older spaces.

Extended inequality to BMO space.

Demonstrated methods beyond classical $L^p$ spaces.

Abstract

Rubio de Francia proved the one-sided Littlewood--Paley inequality for arbitrary intervals in $L^p$, $2 \le p < \infty$. In this article, his methods are developed and employed to prove an analogue of such an inequality "beyond the index $p=\infty$", i.e., for spaces of H\"older functions and BMO.