Commun. Korean Math. Soc. 2019; 34(1): 213-230
Online first article December 14, 2018 Printed January 31, 2019
https://doi.org/10.4134/CKMS.c180030
Copyright © The Korean Mathematical Society.
Najmeddine Attia, Bilel Selmi
Faculty of Sciences of Monastir; Faculty of Sciences of Monastir
We construct new metric outer measures (multifractal analogues of the Hewitt-Stromberg measure) ${ H}^{q,t}_{\mu}$ and ${ P }^{q,t}_{\mu}$ lying between the multifractal Hausdorff measure ${\mathcal H}^{q,t}_{\mu}$ and the multifractal packing measure ${\mathcal P}^{q,t}_{\mu}$. We set up a necessary and sufficient condition for which multifractal Hausdorff and packing measures are equivalent to the new ones. Also, we focus our study on some regularities for these given measures. In particular, we try to formulate a new version of Olsen's density theorem when $\mu$ satisfies the doubling condition. As an application, we extend the density theorem given in \cite{HHBaek}.
Keywords: multifractal Hausdorff measure, multifractal packing measure, Hewitt-Stromberg measure, regularities, densities, doubling measures
MSC numbers: Primary 28A78, 28A80
2022; 37(4): 1073-1097
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