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 $L^{p(\cdot )}({\mathbb{R}}^n)$ with variable exponent $p(x)$ into the weighted space $L_{\rho }^{q(\cdot )}({\mathbb{R}}^n)$, where $\rho (x)=(1{+}^{\prime }{x}^{\prime }{)}^{-\gamma }$ with some $\gamma >0$ and $\frac{1}{q(x)}=\frac{1}{p(x)}-\frac{\alpha (x)}{n}$ 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 $1<p(\infty )\le p(x)\le P<\infty$ $(x\in {\mathbb{R}}^n)$.