Littlewood-Paley equivalence and homogeneous Fourier multipliers

TL;DR

This paper explores Littlewood-Paley operators and their role in characterizing function spaces, extending results to weighted spaces and applying findings to Sobolev space theory.

Contribution

It generalizes H"ormander's theorem to weighted Lebesgue spaces and connects Littlewood-Paley theory with Fourier multipliers and Sobolev spaces.

Findings

Characterization of function spaces via Littlewood-Paley operators

Extension of H"ormander's theorem to weighted spaces

Applications to Sobolev space theory

Abstract

We consider certain Littlewood-Paley operators and prove characterization of some function spaces in terms of those operators. When treating weighted Lebesgue spaces, a generalization to weighted spaces will be made for H\"ormander's theorem on the invertibility of homogeneous Fourier multipliers. Also, applications to the theory of Sobolev spaces will be given.