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.