Estimates of s-numbers of a Sobolev embedding involving spaces of variable exponent

Abstract

Let Ω be a bounded open subset of Rd, suppose that p(⋅):Ω→(1,∞) is a bounded, log-Hölder continuous function, and let Lp(⋅)(Ω), W∘(Ω)p(⋅)1 be the usual variable exponent Lebesgue space and the corresponding Sobolev space. The natural embedding id:W∘(Ω)p(⋅)1→Lp(⋅)(Ω) is compact; when Ω is a bounded domain it is shown that there are positive constants K1,K2 such that for all n∈N,K1≤n1/dsn(id)≤K2, where sn(id) is the nth approximation, Bernstein, Gelfand or Kolmogorov number of id. When p is constant this result is familiar; for variable p and d>1 it appears to be the first result available for s-numbers of Sobolev embeddings. The paper also contains a sharp estimate of the norm of embeddings between Lp(⋅)(Ω) spaces which is interesting in its own right.