A necessary condition on a singular kernel for the continuity of an integral operator in Hölder spaces

2023/05/05 by de Cristoforis, Massimo Lanza
#47G10 #47G40 #FOS: Mathematics #Functional Analysis (math.FA) #Primary 42B20 #Secondary 31B10

paper · doi:10.48550/arxiv.2305.03656

Abstract

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 the 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.

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