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

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 $$C_1{\int }_a^{b}u(x)v(x)dx\le \underset{n\to \infty }{\mathrm{limsup}}n{s}_n(T)\le \underset{n\to \infty }{\mathrm{limsup}}n{s}_n(T)\le C_2{\int }_a^{b}u(x)v(x)dx$$ under appropriate conditions on $u$ and $v.$ The constants $C_1,C_2>0$ depend only on the space $E.$