Abstract
Calderón-Zygmund singular integral operators have been extensively studied for almost half a century. This paper provides a context for and proof of the following result: If a Calderón-Zygmund convolution singular integral operator is bounded on the Hardy space H1 (Rn), then the homogeneous of degree zero kernel is in the Hardy space H1(Sn−1) on the sphere.
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References
Calderón, A.P. and Zygmund, A. (1956). On singular integrals,Am. J. Math.,78, 289–309.
Coifman, R. and Weiss, G. (1977). Extensions of Hardy spaces and their use in analysis,Bull. Amer. Math. Soc,83, 569–645.
Colzani, L. (1982). Hardy and Lipschitz spaces on spheres, Thesis, Washington University, St. Louis, MO.
Connett, W. (1979). Singular integrals near L1,Proc. Symp. Pure Math. AMS I,35, 163–166.
Daly, J. (1983). On the necessity of the Hormander condition for multipliers for Hp(R n),Proc. AMS,88, 321–325.
Daly, J. and Phillips, K. (1989). On the classification of homogeneous multipliers bounded on H1 (R2),Proc. AMS,106, 685–696.
Daly, J. and Phillips, K. (1983). On singular integrals, multipliers, H1 and Fourier series—a local field phenomenon,Math. Ann.,265, 181–219.
Erdelyi, A., Magnus, W., Oberhettinger, F., and Ticomi, F.G. (1953).Bateman Manuscript Project, Table of Integrals II, McGraw-Hill, New York.
Grafakos, L. and Stefanov, A. Convolution Calderon-Zygmund singular integral operators with rough kernels, preprint.
Hörmander, L. (1983).The Analysis of Partial Differential Operators I, Springer-Verlag, Berlin.
Ricci, F. and Weiss, G. (1979). A characterization of H1 (Σn−1),Proc. Symp. Pure Math. AMS I,35, 289–294.
Seeger, A. (1996). Singular integral operators with rough convolution kernels,J. AMS,103, 95–105.
Stefanov, A. Characterization of H1 and applications to singular integral operators, preprint.
Stein, E. (1970).Singular Integrals and Differentiability Property of Functions, Princeton University Press, Princeton, NJ.
Watson, G.N. (1966).Theory of Bessel Functions, University of Cambridge Press, Cambridge, England.
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Communicated by Fulvio Ricci
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Daly, J.E. A necessary condition for Calderón-Zygmund singular integral operators. The Journal of Fourier Analysis and Applications 5, 303–308 (1999). https://doi.org/10.1007/BF01259372
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DOI: https://doi.org/10.1007/BF01259372