Abstract:

In this paper, we will prove that the variation of singular integral operators with rough variable kernels are bounded on $L^{2}({\mathbb S} ^{n})$ if $\Omega\in L^{\infty}({\mathbb R} ^{n})\times L^{q}({\mathbb S}^{n-1})$ for $q>2(n-1)/n$, and $n\geq2$. Moreover, we can also obtain the weighted variational inequalities for singular integral operators with smooth variable kernels. Finally, we extend the result to the Morrey spaces.

Key words: Variational inequalities, Variable kernel, Singular integral, A_p-weight, Morrey space