Abstract
In this paper simple proofs are given for several propositions about continuity of singular integral operators with Cauchy kernel. Some of these propositions turn out to be consequences of more general tests for continuity of operators of the form
under the condition that
is a continuous operator (in a given pair of spaces). As the functions a and h one considers, as a rule, functions of the form l/(eit —eis) and
respectively.
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Literature cited
M. Sh. Birman and M. Z. Solomyak, “Dual Stieltjes operator integrals. III,” in: Problems of Mathematical Physics [in Russian], Vol. 6, Leningrad State Univ. (1973), pp. 27–53.
V. P. Khavin, “Boundary properties of Cauchy-type integrals and harmonically conjugate functions in domains with rectifiable boundaries,” Mat. Sb.,68, No. 4, 499–517 (1965).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 73, pp. 24–34, 1977.
The author thanks V. P. Khavin for helpful discussions of this paper.
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Vinogradov, S.A. Continuity of perturbations of integral operators, Cauchy-type integrals, maximal operators. J Math Sci 34, 2033–2039 (1986). https://doi.org/10.1007/BF01741577
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DOI: https://doi.org/10.1007/BF01741577