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 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.