Weighted jump and variational inequalities for rough operators

TL;DR

This paper establishes weighted jump and variational inequalities for rough singular integral and averaging operators, advancing understanding of their boundedness properties in harmonic analysis.

Contribution

It provides the first systematic study of weighted jump and variational inequalities for rough operators, including new bounds for truncated singular integrals and averaging operators.

Findings

Weighted jump inequalities established for rough operators.

Variational inequalities proved for families of truncated singular integrals.

Results extend known bounds to rough kernels in harmonic analysis.

Abstract

In this paper, we systematically study weighted jump and variational inequalities for rough operators. More precisely, we show some weighted jump and variational inequalities for the families $\mathcal T:=\{T_\varepsilon\}_{\varepsilon>0}$ of truncated singular integrals and $\mathcal M_{\Omega}:=\{M_{\Omega,t}\}_{t>0}$ of averaging operators with rough kernels, which are defined respectively by $$ T_\varepsilon f(x)=\int_{|y|>\varepsilon}\frac{\Omega(y')}{|y|^n}f(x-y)dy$$ and $$M_{\Omega,t} f(x)=\frac1{t^n}\int_{|y|<t}\Omega(y')f(x-y)dy, $$ where the kernel $\Omega$ belongs to $L^q(\mathbf S^{n-1})$ for $q>1$.