ROUGH SINGULAR INTEGRALS AND MAXIMAL OPERATOR WITH RADIAL-ANGULAR INTEGRABILITY

In this paper, we study the rough singular integral operator T(Omega)f(x) - p.v. integral(Rn) f(x - y)Omega(y')/vertical bar y vertical bar(n)dy, and the corresponding maximal singular integral operator T-Omega*f(x) = sup(epsilon>0)vertical bar integral(vertical bar y vertical bar >=epsilon) f(x - y)Omega(y')vertical bar y vertical bar(n)dy vertical bar, where the kernel Omega is an element of H-1(Sn-1) has zero mean value and n >= 2. We prove that T-Omega and T-Omega* are bounded on the mixed radial-angular spaces L-vertical bar x vertical bar(p) L-theta((p) over tilde)(R-n) for some suitable indexes 1 < p, <(p)over tilde> < infinity. The corresponding vector-valued versions are also established.