Lacunary measures and self-similar probability measures in function spaces
Let \(\mu\) be a Radon measure with bounded support of Lebesgue measure zero. In multifractal geometry, one consideres the quantity \[ \mu^\lambda_{p,q}= \biggl(\sum_j 2^{j\lambda q} \biggl(\sum_m \mu(Q_{j,m})^p\biggr)^{q/p}\biggr)^{1/q}, \] \(\lambda\in {\mathbb R}\) and \(Q_{j,m}\) denotes a standard dyadic cube. Let \(t=\frac 1 p\) and \(\lambda_\mu(\frac 1 p)=\sup\{\lambda:\, \mu^\lambda_{p,q}<\infty\}\). One can consider an analytical characterization of the measure \(\mu\) of the form \(s_\mu(\frac 1 p)=\sup\{s: \mu\in B^s_{p,q}({\mathbb R}^n)\}\). The Radon measure \(\mu\) is called tame if \(s_f(\frac 1 p)= \lambda^f(\frac 1 p)+ n(\frac 1 p - 1)\). The aim of the paper under review is to find sufficient conditions for a Radon measure \(\mu\) to be tame. It is proven that some lacunary measures and special self-similar probability measures are tame.
scientific article; zbMATH DE number 2130966