Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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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.

Regularities of multifractal Hewitt-Stromberg measures

Najmeddine Attia, Bilel Selmi

Faculty of Sciences of Monastir; Faculty of Sciences of Monastir

Abstract

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

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