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Singular integrals and fractional powers of operators


Author: Michael J. Fisher
Journal: Trans. Amer. Math. Soc. 161 (1971), 307-326
MSC: Primary 47.70; Secondary 46.00
DOI: https://doi.org/10.1090/S0002-9947-1971-0285935-1
MathSciNet review: 0285935
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Abstract: Recently R. Wheeden studied a class of singular integral operators, the hypersingular integrals, as operators from $ L_p^\alpha (H)$ to $ {L_p}(H);L_p^\alpha (H)$ is the range of the $ \alpha$th order Bessel potential operator acting on $ {L_p}(H)$ with the inherited norm. The purposes of the present paper are to extend the known results on hypersingular integrals to complex indices, to extend these results to operators defined over a real separable Hilbert space, and to use Komatsu's theory of fractional powers of operators to show that the hypersingular integral operator $ {G^\alpha }$ is $ {\smallint _H}{( - {A_y})^\alpha }f\,d\mu (y)$ when $ {\mathop{\rm Im}\nolimits} (\alpha ) \ne 0$ or when $ {\mathop{\Re}\nolimits} (\alpha )$ is not a positive integer where $ {A_y}g$ is the derivative of g in the direction y. The case where $ {\mathop{\rm Im}\nolimits} (\alpha ) = 0$ and $ {\mathop{\Re}\nolimits} (\alpha )$ is a positive integer is treated in a sequel to the present paper.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0285935-1
Keywords: Hypersingular integral, fractional powers of operators, Bessel potential, singular integral operator, Calderon-Zygmund operator
Article copyright: © Copyright 1971 American Mathematical Society

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