Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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REGULARITIES OF MULTIFRACTAL HEWITT-STROMBERG MEASURES

  • Attia, Najmeddine (Department of Mathematics Faculty of Sciences of Monastir) ;
  • Selmi, Bilel (Department of Mathematics Faculty of Sciences of Monastir)
  • Received : 2018.01.24
  • Accepted : 2018.07.24
  • Published : 2019.01.31
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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 [3].

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References

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