On Sobolev Theorem for Riesz-Type Potentials in Lebesgue Spaces with Variable Exponent

Abstract

The Riesz potential operator of variable order $\alpha(x)$ is shown to be bounded from the Lebesgue space $L^{p(\cdot)}({\Bbb R}^n)$ with variable exponent $p(x)$ into the weighted space $L^{q(\cdot)}_\rho({\Bbb R}^n)$, where $\rho(x) = (1 + 'x')^{-\gamma}$ with some $\gamma > 0$ and ${1 \over q(x)} = {1 \over p(x)} - {\alpha(x) \over n}$ when $p$ is not necessarily constant at infinity. It is assumed that the exponent $p(x)$ satisfies the logarithmic continuity condition both locally and at infinity and $1 < p(\infty) \le p(x) \le P < \infty$ $(x \in {\Bbb R}^n)$.