Estimates of diameters of Sobolev classes of small smoothness (Q788960)

For Sobolev spaces \(\tilde W^ r_ p\) where \(r>1\), the diameter \(d_ n(\tilde W_ 1,L^ q)\) is bounded above and below by \(n^{-r+1/2}\), but it was known that there is a change if the smoothness is lowered. \textit{B. S. Kashin} showed [Izv. Akad. Nauk Arm. SSR, Mat. 15, 379-394, (1980; Zbl 0452.15023)] that for \(2<q<\infty\), \(1-1/q<r<1\), and \(n\to \infty d_ n(\tilde W^ r_ 1,L^ q)\asymp n^{q/2(-r+1-1/q)}\) which allows one to get estimates for \(d_ n(\tilde W^ r_ p,L^ q)\), \(1<p\leq 2\), \(r<1/p\). The author considers \(2\leq p<q<\infty \quad 1/p- 1/q<r\leq 1/p\) and shows \(Cn^{\gamma}\leq d_ n(\tilde W^ r_ p,L^ q)\leq Cn^{\gamma}\ln^{\beta}n\) where \(\gamma =\max(-r,q/2(-r+1/p- 1/q))\) and \(\beta =3/2\) for r not equal to \((1/p-1/q)/2(1/2-1/q)\) and \(\beta\) equal to \(5/2+r\) if \(r=(1/p-1/q)/2(1/2-1/q).\) He remarks at the end that the result on which he based his paper has been improved and consequently he can show that the diameter is bounded both above and below by \(n^{\gamma}\) in the above cases if \(r\neq \beta\), while for \(1/p-1/q<r<1/p, 1\leq p<2<q<\infty\) the diameter is bounded above and below by \(n^{q/2(-r+1/p-1/q)}\).