Frostman lemma revisited

TL;DR

This paper examines the sharpness of generalized Frostman's lemma, providing improved estimates for measure dimensions and establishing conditions for absolute continuity of measures derived from anisotropic gradients.

Contribution

It refines the understanding of Frostman's lemma generalizations and links anisotropic gradient measures to absolute continuity with respect to Hausdorff measures.

Findings

Improved bounds for lower Hausdorff dimension estimates

Conditions for measures from anisotropic gradients to be absolutely continuous

Enhanced understanding of Frostman's lemma generalizations

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.