On the existence of double singular integrals for kernels without smoothness

Abstract

Calderón and Zygmund have proved the pointwise convergence of singular integrals in R n {R^n} for locally integrable homogeneous kernels whose even part is locally in L L log L L by change to polar coordinates and use of the boundedness in L p {L^p} of the maximal operator of the one-dimensional Hilbert transformation. The present note shows how analogous results for double singular integrals can be derived from boundedness of the maximal operator of the double Hilbert transform.