Weakly canceling operators and singular integrals

TL;DR

This paper introduces an elementary harmonic analysis approach to canceling and weakly canceling differential operators, extending these concepts to anisotropic Fourier multipliers with mild regularity, and establishes key inequalities in this broader setting.

Contribution

It generalizes canceling and weakly canceling operators to anisotropic Fourier multipliers, providing new inequalities and a simplified analytical framework.

Findings

Established $L_{ ext{infinity}}$ bound for weakly canceling operators of order $d$

Proved $L_2$ bound for canceling operators of order $d/2$

Extended notions to anisotropic Fourier multipliers with mild regularity

Abstract

We suggest an elementary Harmonic Analysis approach to canceling and weakly canceling differential operators, which allows to extend these notions to anisotropic setting and also replace differential operators with Fourier multiplies with mild smoothness regularity. In this more general setting of anisotropic Fourier multipliers, we prove the inequality $\|f\|_{L_{\infty}} \lesssim \|Af\|_{L_1}$ if $A$ is a weakly canceling operator of order $d$ and the inequality $\|f\|_{L_2} \lesssim \|Af\|_{L_1}$ if $A$ is a canceling operator of order $\frac{d}{2}$, provided $f$ is a function in $d$ variables.