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

Abstract

The Riesz potential operator of variable order α(x) is shown to be bounded from the Lebesgue space Lp(̇)(ℝn) with variable exponent p(x) into the weighted space L̇q(ℝ)(Rn), where p(x) = (1 + |x|)-γ with some γ > 0 and 1/q(x) = 1/p(x) - α(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(∞) ≤ p(x) ≤ P < ∞ (x εℝn).