Frostman lemma revisited

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