Frostman lemma revisited

Abstract

We study sharpness of various generalizations of Frostman's lemma. These generalizations provide better estimates for the lower Hausdorff dimension of measures. As a corollary, we prove that if a generalized anisotropic gradient \((\partial_1^{m_1} f, \partial_2^{m_2} f,\ldots, \partial_d^{m_d} f)\) of a function \(f\) in \(d\) variables is a measure of bounded variation, then this measure is absolutely continuous with respect to the Hausdorff \(d-1\) dimensional measure.