Approximation numbers of Hardy-type operators on trees (Q2707672)

This paper studies a Hardy-type operator NEWLINE\[NEWLINE T_a f(x)=u(x)\int _a^x f(t)v(t)dt ,NEWLINE\]NEWLINE where \(u\) and \(v\) are weights defined on a tree. NEWLINENEWLINENEWLINEIn [\textit{W. D. Evans, D. J. Harris} and \textit{L. Pick}, J. Lond. Math. Soc., II. Ser. 52, No. 1, 121-136 (1995; Zbl 0835.26010)] a characterization of boundedness of \( T_a\) was obtained between Lebesgue spaces. In this paper the authors give criteria for compactness. Under a suitable condition on weights \(u\) and \(v\) upper and lower bounds and asymptotic estimates for approximation numbers \( a_n(T_a) \) of \(T_a\) are established. NEWLINENEWLINENEWLINEThey study the relationship between the approximation numbers \(\sigma _k, \sigma _{k,i}\) and \(a_k\). They also provide upper and lower estimates for the \(\ell ^q\) and \(\ell ^q _w\) norms of \((a_n(T_a))\).NEWLINENEWLINEFor the entire collection see [Zbl 0933.00035].