A Necessary Condition on a Singular Kernel for the Continuity of an Integral Operator in Hölder Spaces

We prove that a condition of boundedness of the maximal function of a singular integral operator, that is known to be sufficient for the continuity of a corresponding integral operator in Hölder spaces, is actually also necessary in case the action of the integral operator does not decrease the regu...

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Bibliographic Details
Published in:Mediterranean journal of mathematics 2024-03, Vol.21 (3), Article 47
Main Author: Lanza de Cristoforis, Massimo
Format: Article
Language:English
Subjects:
Continuity
Mathematics
Mathematics and Statistics
Operators (mathematics)
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Summary:We prove that a condition of boundedness of the maximal function of a singular integral operator, that is known to be sufficient for the continuity of a corresponding integral operator in Hölder spaces, is actually also necessary in case the action of the integral operator does not decrease the regularity of a function. We do so in the frame of metric measured spaces with a measure satisfying certain growth conditions that include nondoubling measures. Then we present an application to the case of an integral operator defined on a compact differentiable manifold.
ISSN:1660-5446
1660-5454
DOI:10.1007/s00009-023-02562-4