Estimates for entropy numbers of sets of smooth functions on the torus Td

In this paper, we investigate entropy numbers of multiplier operators Λ={λk}k∈Zd and Λ∗={λk∗}k∈Zd, Λ,Λ∗:Lp(Td)→Lq(Td) on the d-dimensional torus Td, where λk=λ(|k|) and λk∗=λ(|k|∗) for a function λ defined on the interval [0,∞), with |k|=k12+⋯+kd21∕2 and |k|∗=max1≤j≤d|kj|. In the first part, upper and lower bounds are established for entropy numbers of general multiplier operators. In the second part, we apply these results to the specific multiplier operators Λ(1)=|k|−γ(ln|k|)−ξk∈Zd, Λ∗(1)=|k|∗−γ(ln|k|∗)−ξk∈Zd, Λ(2)=e−γ|k|rk∈Zd and Λ∗(2)=e−γ|k|∗rk∈Zd for γ>0, 0