  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 $q>1$.

Classical Analysis and ODEs