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On the existence of double singular integrals for kernels without smoothness

Author: T. Walsh
Journal: Proc. Amer. Math. Soc. 28 (1971), 439-445
MSC: Primary 47.70; Secondary 42.00
MathSciNet review: 0276838
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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.

References [Enhancements On Off] (What's this?)

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Keywords: Double singular integrals, kernels without smoothness, maximal function
Article copyright: © Copyright 1971 American Mathematical Society