A generalization of the Littlewood-Paley inequality for the fractional Laplacian $(-\Delta)^{\alpha/2}$

TL;DR

This paper establishes a parabolic version of the Littlewood-Paley inequality tailored for the fractional Laplacian operator, extending classical harmonic analysis tools to fractional order derivatives.

Contribution

It introduces a generalized parabolic Littlewood-Paley inequality specifically for the fractional Laplacian with  in (0,2), broadening the scope of harmonic analysis techniques.

Findings

Proves a new inequality for fractional Laplacian operators.

Extends classical Littlewood-Paley theory to fractional order derivatives.

Provides tools for analyzing PDEs involving fractional Laplacians.

Abstract

We prove a parabolic version of the Littlewood-Paley inequality for the fractional Laplacian $(-\Delta)^{\alpha/2}$, where $\alpha\in (0,2)$.