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We prove that a condition of boundedness of the maximal function of asingular integral operator, that is known to be sufficient for the continuityof the corresponding integral operator in H\"{o}lder spaces, is actually alsonecessary in case the action of the integral operator does not decrease theregularity of a function. We do so in the frame of metric measured spaces witha measure satisfying certain growth conditions that include nondoublingmeasures. Then we present an application to the case of an integral operatordefined on a compact differentiable manifold.

A necessary condition on a singular kernel for the continuity of an integral operator in H\"{o}lder spaces