**Vakhtang Kokilashvili, Natasha Samko, Stefan Samko**

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

with Oscillating Weights

**Abstract:**

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).

**Keywords:**

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

**MSC 2000:** 42B25, 47B38