On the existence of double singular integrals for kernels without smoothness

Abstract:

Calderón and Zygmund have proved the pointwise convergence of singular integrals in ${R^n}$ for locally integrable homogeneous kernels whose even part is locally in $L$ log $L$ by change to polar coordinates and use of the boundedness in ${L^p}$ 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.