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 $({\partial }_1^{m_1}f,{\partial }_2^{m_2}f,\dots ,{\partial }_d^{m_d}f)$ 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.