This is an announcement for the paper "Some $s$-numbers of an integral operator of Hardy type in Banach function spaces" by David Edmunds, Amiran Gogatishvili, Tengiz Kopaliani and Nino Samashvili.
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 $E.$
Archive classification: math.FA math.AP math.CA
Mathematics Subject Classification: 35P30, 46E30, 46E35, 47A75 47B06, 47B10, 47B40, 47G10
Submitted from: gogatish@math.cas.cz
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1507.08854
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