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Zeitschrift für Analysis und ihre Anwendungen


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Volume 22, Issue 4, 2003, pp. 899–910
DOI: 10.4171/ZAA/1178

Published online: 2003-12-31

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

Vakhtang Kokilashvili[1] and Stefan Samko[2]

(1) Georgian Acadademy of Sciences, Tbilisi, Georgia
(2) University of Algarve, Faro, Portugal

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 $1 < p(\infty) \le p(x) \le P < \infty$ $(x \in {\Bbb R}^n)$.

Keywords: Variable exponent, Lebesgue spaces, Riesz potential, weighted estimates, maximal function

Kokilashvili Vakhtang, Samko Stefan: On Sobolev Theorem for Riesz-Type Potentials in Lebesgue Spaces with Variable Exponent. Z. Anal. Anwend. 22 (2003), 899-910. doi: 10.4171/ZAA/1178