  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.
