Jump and variational inequalities for rough operators

TL;DR

This paper investigates jump and variational inequalities for rough operators, specifically truncated singular integrals and averaging operators with rough kernels, establishing results under optimal conditions for boundedness.

Contribution

It provides new jump and variational inequalities for families of rough operators with kernels in specific function spaces, extending and sharpening previous results.

Findings

Established inequalities for kernels in $L ext{log}^+L$, $H^1$, and $ ext{G}_eta$ spaces.

Results are sharp under the best known conditions for maximal operator boundedness.

Extended the theory of jump and variational inequalities to rough operators with less regular kernels.

Abstract

In this paper, we systematically study jump and variational inequalities for rough operators, whose research have been initiated by Jones {\it et al}. More precisely, we show some jump and variational inequalities for the families $\mathcal T:=\{T_\varepsilon\}_{\varepsilon>0}$ of truncated singular integrals and $\mathcal M:=\{M_t\}_{t>0}$ 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 $$M_t f(x)=\frac1{t^n}\int_{|y|<t}\Omega(y')f(x-y)dy, $$ where the kernel $\Omega$ belongs to $L\log^+\!\!L(\mathbf S^{n-1})$ or $H^1(\mathbf S^{n-1})$ or $\mathcal{G}_\alpha(\mathbf S^{n-1})$ (the condition introduced by Grafakos and Stefanov). Some of our results are sharp in the sense that the underlying assumptions are the best known conditions for the boundedness of corresponding maximal operators.