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 $q>1$.
Categories: math.CA