Certain Operators with Rough Singular Kernels

We study the singular integral operator T(Omega),(alpha) f(x) = p.v. ... b(|y|) Omega (yʹ)|y|^(-n-(alpha)) f(x-y) dy1, defined on all test functions f, where b is a bounded function, alpha >= 0, Omega(yʹ) is an integrable function on the unit sphere S^(n-1) satisfying certain cancellation conditions. We prove that, for 1 < p < infinity, T((Omega),(alpha)) extends bounded operator from the Sobolev space Lp(alpha) to the Lebesgue space Lp with (Omega) being a distribution in the Hardy space Hq(S^(n-1)) with q= (n-1)/(n-1+(alpha)). The result extends some known results on the singular integral operators. As applications, we obtain the boundedness for T((Omega),(alpha)) on the Hardy spaces, as well as the boundedness for the truncated maximal operator T*(Omega),mʹ.