Jump and variational inequalities for rough operators

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.