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

Abstract

The Riesz potential operator of variable order fi(x) is shown to be bounded from the Lebesgue space L p(¢) (R n ) with variable exponent p(x) into the weighted space L q(¢) ‰ (R n ), where ‰ = (1 + jxj) i∞ with some ∞ > 0 and 1 q(x) = 1 p(x) i fi(x) n when p(x) is not necessarily constant at inflnity. It is assumed that the exponent p(x) satisfles the logarithmic continuity condition both locally and at inflnity and 1 < p(1) • p(x) • P < 1; x 2R n .