Local Hardy-Littlewood maximal opeator in variable Lebesgue spaces

We investigate the class $\mathcal{B}^{loc}(\mathbb{R}^{n})$ of exponents $p(\cdot)$ for with local Hardy-Littlewood maximal operator is bounded in $L^{p(\cdot)}(\mathbb{R}^{n})$ space. Littlewood-Paley square-function characterization of $L^{p(\cdot)}(\mathbb{R}^{n})$ spaces with the above class of exponent are also obtained.