Integrability of maximal functions and Riesz potentials in Orlicz spaces of variable exponent (Q961038)

Let \(L^{p(\cdot )}(\mathbb{R}^{n})\) denote the space of all measurable functions \(f\) on \(\mathbb{R}^{n}\) satisfying \(\int |f(x)/\lambda|^{p(x)}dx<\infty\) for some \(\lambda >0\), where \(p(\cdot)\) denotes a positive continuous function on \(\mathbb{R}^{n}\). This space is endowed with the norm \(\| \cdot \|_{p(\cdot)}\), where \(\| f\|_{p(\cdot)}\) is defined to be the infimum of all \(\lambda \) for which the above integral does not exceed \(1\). The authors establish a result concerning boundedness of the Hardy-Littlewood maximal operator on \(L^{p(\cdot)}(\mathbb{R}^{n})\), and give applications to Sobolev-type, Hardy-type and Trudinger inequalities for Riesz potentials of functions in Orlicz spaces of variable exponent.