Some s-numbers of an integral operator of Hardy type on Lp(·) spaces

Let I = [a, b] ⊂ R, let p : I →(1,∞) be either a step-function or strong log-Hölder continuous on I, let Lp(·)(I ) be the usual space of Lebesgue type with variable exponent p, and let T : Lp(·)(I )→Lp(·)(I ) be the operator of Hardy type defined by Tf (x) = x a f (t)dt. For any n ∈ N, let sn denote the nth approximation, Gelfand, Kolmogorov or Bernstein number of T . We show that lim n→∞ nsn = 1 2π I p (t)p(t)p(t)−1 1/p(t) sin π/p(t) dt where p (t) = p(t)/(p(t)−1). The proof hinges on estimates of the norm of the embedding id of Lq(·)(I ) in Lr(·)(I ), where q, r : I →(1,∞) are measurable, bounded away from 1 and∞, and such that, for some ε ∈ (0, 1), r(x) q(x) r(x)+ε for all x ∈ I. It is shown that min 1, |I |ε id ε|I| + ε−ε, a result that has independent interest.