Weighted jump and variational inequalities for rough operators

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 $q>1$.