Some s-numbers of an integral operator of Hardy type on L-p(center dot) spaces

摘要

Let I = vertical bar a, b vertical bar subset of R, let p : I -> (1, infinity) be either a step-function or strong log- Holder continuous on I, let L-p(center dot)(I) be the usual space of Lebesgue type with. variable exponent p, and let T : L-p(center dot)(I) -> L-p(center dot)(I) be the operator of Hardy type defined by T f(x) = integral(x)(a) f (t)dt. For any n is an element of N, let s(n) denote the nth approximation,Gelfand, Kolmogorov or Bernstein number of T. We show that lim(n ->infinity) ns(n) = 1/2 pi integral(I) {p'(t)p(t)(p(t)-1)}(1/p(t)) sin(pi/p(t))dt. where p'(t) = p(t)/(p(t) - 1). The proof hinges on estimates of the norm of the embedding id of L-q(center dot)(I) in L-r(center dot) (I), where q, r : l -> (1, infinity) are measurable, bounded away from 1 and infinity, and such that, for some epsilon is an element of (0, 1), r(x) <= q(x) <= r(x) + epsilon for all x is an element of I. It is shown that min(I, vertical bar I vertical bar(epsilon)) <= parallel to id parallel to <= epsilon vertical bar I vertical bar + epsilon(-epsilon), a result that has independent interest.