Maximal and Fractional Operators in Weighted $L^{p(x)}$ Spaces

Abstract

We study the boundedness of the maximal operator, potential type operators and operators with fixed singularity (of Hardy and Hankel type) in the spaces $L^{p(\cdot )}(\rho ,\Omega )$ over a bounded open set in ${\mathbb{R}}^n$ with a power weight $\rho (x)=|x-x_0{|}^{\gamma }$, $x_0\in \overline{\Omega }$, and an exponent $p(x)$ satisfying the Dini-Lipschitz condition.