Some $s$-numbers of an integral operator of Hardy type in Banach function spaces

TL;DR

This paper studies the asymptotic behavior of various s-numbers of a Hardy-type integral operator acting on Banach function spaces, linking their limits to integrals involving the kernel functions and space properties.

Contribution

It provides bounds for the asymptotic s-numbers of the operator in terms of integrals of kernel functions, revealing geometric properties of the underlying Banach space.

Findings

Bounds for the limit superior of s-numbers in terms of integral of u(x)v(x)

Constants depend only on the Banach space E

Conditions on u and v ensure the bounds hold

Abstract

Let $s_{n}(T)$ denote the $n$th approximation, isomorphism, Gelfand, Kolmogorov or Bernstein number of the Hardy-type integral operator $T$ given by $$ Tf(x)=v(x)\int_{a}^{x}u(t)f(t)dt,\,\,\,x\in(a,b)\,\,(-\infty<a<b<+\infty) $$ and mapping a Banach function space $E$ to itself. We investigate some geometrical properties of $E$ for which $$ C_{1}\int_{a}^{b}u(x)v(x)dx \leq\limsup\limits_{n\rightarrow\infty}ns_{n}(T) \leq \limsup\limits_{n\rightarrow\infty}ns_{n}(T)\leq C_{2}\int_{a}^{b}u(x)v(x)dx $$ under appropriate conditions on $u$ and $v.$ The constants $C_{1},C_{2}>0$ depend only on the space $E.$