Vakhtang Kokilashvili, Natasha Samko, Stefan Samko

The maximal Operator in Variable Spaces $L^{p(\cdot)}(\Omega,\rho)$
with Oscillating Weights

We study the boundedness of the maximal operator in the spaces $L^{p(\cdot)}(\Om,\rho)$ over a bounded open set $\Omega$ in $R^n$ with the weight $\rho(x)=\prod\limits_{k=1}^mw_k(|x-x_k|)$, $x_k\in \ol{\Omega}$, where $w_k$ has the property that $r^{\frac{n}{p(x_k)}}w_k(r)$ belongs to a certain Zygmund-type class. 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).

Maximal functions, weighted Lebesgue spaces, variable exponent, potential operators.

MSC 2000: 42B25, 47B38