Certain operators with rough singular kernels

We study the singular integral operator TOmega, alpha f(x) = P.v. integral(Rn) b(\y\)Omega(y')\y\(-n-alpha)f(x - y)dy, defined on all test functions f,where b is a bounded function, a greater than or equal to 0, Omega(y') is an integrable function on the unit sphere S(n-1) satisfying certain cancellation conditions. We prove that, for 1< p < infinity, TOmega, extends to a bounded operator from the Sobolev space L(alpha)(p) to the Lebesgue space L(p) with Omega being a distribution in the Hardy space H(q)(S(n-1)) where q = n-1/n-1+alpha. The result extends some known results on the singular integral operators. As applications, we obtain the boundedness for TiOmega, alpha on the Hardy spaces, as well as the boundedness for the truncated maximal operator T*(Omega, m).