Lomakina, Elena; Stepanov, Vladimir On the compactness and approximation numbers of Hardy-type integral operators in Lorentz spaces. (English) Zbl 0853.42013 J. Lond. Math. Soc., II. Ser. 53, No. 2, 369-382 (1996). We characterize the mapping properties such as the boundedness, compactness and measure of noncompactness for those real weight functions \(\varphi\), \(\psi\), \(u\geq 0\), \(v\geq 0\), for which the Hardy-type integral operator of the form \[ Kf(x)= \varphi(x) \int^x_0 k(x, y) \psi(y) f(y) dy,\quad x> 0, \] acts from \(L^{rs}_v\) to \(L^{pq}_u\), when the parameters are restricted to the range \(1< \max(r, s)\leq \min(p, q)< \infty\) and the kernel \(k(x, y)\geq 0\) satisfies the Oinarov condition of the form \[ D^{- 1} (k(x, y)+ k(y, z))\leq k(x, z)\leq D(k(x, y)+ k(y, z)),\quad x> y> z\geq 0. \] For the case \(k(x, y)= 1\) we obtain lower and upper estimates of the approximation numbers. Reviewer: E.Lomakina and V.Stepanov (Khabarovsk) Cited in 10 Documents MSC: 42B25 Maximal functions, Littlewood-Paley theory 47B38 Linear operators on function spaces (general) 46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) Keywords:Lorentz space; boundedness; compactness; measure of noncompactness; Hardy-type integral operator; approximation numbers PDFBibTeX XMLCite \textit{E. Lomakina} and \textit{V. Stepanov}, J. Lond. Math. Soc., II. Ser. 53, No. 2, 369--382 (1996; Zbl 0853.42013) Full Text: DOI