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On Sobolev theorem for Riesz-type potentials in Lebesgue spaces with variable exponent. (English) Zbl 1040.42013
Summary: The Riesz potential operator of variable order $\alpha(x)$ is shown to be bounded from the Lebesgue space $L^{p(\cdot)}(\bbfR^n)$ with variable exponent $p(x)$ into the weighted space $L^{q(\cdot)}_\rho(\bbfR^n)$, where $\rho(x)=(1+\vert x\vert)^{-\gamma}$ 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 $1< p(\infty)\le p(x)\le P<\infty$ $(x\in \bbfR^n)$.

42B20Singular and oscillatory integrals, several variables
47B38Operators on function spaces (general)
42B25Maximal functions, Littlewood-Paley theory