A Generalization of Littlewood-Paley Type Inequality for Evolution Systems Associated with Pseudo Differential Operators

TL;DR

This paper extends Littlewood-Paley inequalities to evolution systems linked with pseudo-differential operators, establishing boundedness of associated g-functions and their sharp functions using harmonic analysis techniques.

Contribution

It introduces a generalized Littlewood-Paley inequality for evolution systems related to pseudo-differential operators, including boundedness results and maximal function estimates.

Findings

Boundedness of the Littlewood-Paley g-function in L^q spaces.

Sharp function of g-function is controlled by a maximal function.

Established Littlewood-Paley inequality for evolution systems with pseudo-differential operators.

Abstract

In this paper, we first prove that the Littlewood-Paley $g$-function, related to the convolution corresponding to the composition of pseudo-differential operator and evolution system associated with pseudo-differential operators, is a bounded operator from $L^{q}((a,b)\times \mathbb{R}^{d};V)$ with a Hilbert space $V$ into $L^{q}((a,b)\times \mathbb{R}^{d})$. Secondly, we prove that the sharp function of the Littlewood-Paley $g$-function is bounded by some maximal function. Finally, by applying Fefferman-Stein theorem and Hardy-Littlewood maximal theorem, we prove the Littlewood-Paley type inequality for evolution systems associated with pseudo-differential operators.