Strict s-numbers of weighted Hardy type operators on trees

Abstract

In this paper we calculate the strict $s$-numbers of Hardy type operators $T_o:L^p(\Upsilon_o)\rightarrow L^p(\Upsilon_o)$ for $1< p <\infty $, defined by
$$ T_of(x)\coloneqq v(x)\int_{o}^{x}f(t)u(t)dt,~~~~ for~~ o\in \Upsilon_o,$$
where $u$ and $v$ are measurable functions on $ \Upsilon_o$ satisfying the conditions $u\in L^{p'}(\kappa), v\in L^p(\Upsilon_o)$, $f\in L^p(\Upsilon_o)$ and $ x\in\Upsilon_o$, for every subtree $\kappa$ of a tree $\Upsilon_o$ such that the closure of $ \kappa$ is compact subset of $\Upsilon_o$. We obtain the equality among strict $s$-numbers.