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. Calder6n and Zygmund have proved the pointwise convergence of singular integrals in Rn 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 LP 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.

On the existence of double singular integrals for kernels without smoothness