Some \(s\)-numbers of an integral operator of Hardy type on \(L^{p(\cdot)}\) spaces
The Hardy-type operator \(Tf(x)=\int_a^x f(t)\,dt\), \(x \in I=[a,b]\), is studied in the space \(L^{p (\cdot)} (I)\), where \(p (\cdot)\) is a variable exponent satisfying certain conditions. The following estimate is established for the relevant \(n\)th approximation number \(s_n\): \[ \lim\limits_{n\to\infty} ns_n=\frac{1}{2\pi}\int_I [p'(t) p(t) ^{p(t)-1}]^{1/p(t)} \sin (\pi/p(t))\,dt, \] where \(p'(t)=p(t)/(p(t)-1)\).
scientific article; zbMATH DE number 5572192
scientific article