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

Let I=[a,b]⊂RI=[a,b]⊂R, let p:I→(1,∞)p:I→(1,∞) be either a step-function or strong log-Hölder continuous on I  , let Lp(⋅)(I)Lp(⋅)(I) be the usual space of Lebesgue type with variable exponent p  , and let T:Lp(⋅)(I)→Lp(⋅)(I)T:Lp(⋅)(I)→Lp(⋅)(I) be the operator of Hardy type defined by Tf(x)=∫axf(t)dt. For any n∈Nn∈N, let snsn denote the nth approximation, Gelfand, Kolmogorov or Bernstein number of T. We show thatlimn→∞nsn=12π∫I{p′(t)p(t)p(t)−1}1/p(t)sin(π/p(t))dt where p′(t)=p(t)/(p(t)−1)p′(t)=p(t)/(p(t)−1). The proof hinges on estimates of the norm of the embedding id   of Lq(⋅)(I)Lq(⋅)(I) in Lr(⋅)(I)Lr(⋅)(I), where q,r:I→(1,∞)q,r:I→(1,∞) are measurable, bounded away from 1 and ∞, and such that, for some ε∈(0,1)ε∈(0,1), r(x)⩽q(x)⩽r(x)+εr(x)⩽q(x)⩽r(x)+ε for all x∈Ix∈I. It is shown thatmin(1,ε|I|)⩽‖id‖⩽ε|I|+ε−ε,min(1,|I|ε)⩽‖id‖⩽ε|I|+ε−ε, a result that has independent interest.