Remarks on compactness results for variable exponent spaces L-p(.)

Given the Lebesgue space with variable exponent L-s(.)(Omega) whose norm is denoted by parallel to.parallel to(s(.)), we show the following equivalence: lim(vertical bar E vertical bar -> 0) parallel to chi(E)parallel to(s(.)) = 0 if and only if lim(p ->+infinity) 1/p [f(Omega)s(x)(p)dx](1/p) = 0, where chi(E) is the characteristic function of the measurable set E and vertical bar E vertical bar its Lebesgue measure. We apply such results to characterize characterize compactness of some inclusions. (C) 2021 Elsevier Masson SAS. All rights reserved.