Georgian Mathematical Journal 13 (2006), No. 1, 109--125
Copyright Heldermann Verlag 2006
The Maximal Operator in Variable Spaces Lp(.)(Ω, ρ) with Oscillating Weights
Vakhtang Kokilashvili
International Black Sea University, 2 Agmashenebeli Alley 13th km, Tbilisi 0131, Georgia
kokil@rmi.acnet.ge
Natasha Samko
University of Algarve, Portugal
nsamko@ualg.pt
Stefan Samko
University of Algarve, Portugal
ssamko@ualg.pt
[Abstract-pdf]
We study the boundedness of the maximal operator in the spaces $L^{p(\cdot)}(\Omega, \rho)$ over a bounded open set $\Omega$ in $R^n$ with the weight $$ \rho(x)=\prod\limits_{k=1}^mw_k(|x-x_k|),\quad x_k\in \overline{\Omega}, $$ where $w_k$ has the property that $r^{\frac{n}{p(x_k)}}w_k(r)$ belongs to a certain Zygmund-type class. The weight functions $w_k$ may oscillate between two power functions with different exponents. It is assumed that the exponent $p(x)$ satisfies the Dini-Lipschitz condition. The final statement on the boundedness is given in terms of index numbers of functions $w_k$ (similar in a certain sense to the Boyd indices for the Young functions defining Orlicz spaces).
Keywords: Maximal functions, weighted Lebesgue spaces, variable exponent, potential operators.
MSC: 42B25, 47B38
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