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The Maximal Operator in Variable Spaces 𝐿 𝑝(Β·)(Ξ©, ρ) with Oscillating Weights

  • Vakhtang Kokilashvili , Natasha Samko and Stefan Samko
Published/Copyright: March 10, 2010
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Georgian Mathematical Journal
From the journal Volume 13 Issue 1

Abstract

We study the boundedness of the maximal operator in the spaces 𝐿 𝑝(Β·)(Ξ©, ρ) over a bounded open set Ξ© in 𝑅𝑛 with the weight , where π‘€π‘˜ has the property that belongs to a certain Zygmund-type class. Weight functions π‘€π‘˜ may oscillate between two power functions with different exponents. It is assumed that the exponent 𝑝(π‘₯) satisfies the Dini–Lipschitz condition. The final statement on the boundedness is given in terms of index numbers of functions π‘€π‘˜ (similar in a certain sense to the Boyd indices for the Young functions defining Orlicz spaces).

Received: 2005-07-08
Published Online: 2010-03-10
Published in Print: 2006-March

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