On the compactness and approximation numbers of Hardy-type integral operators in Lorentz spaces. (English) Zbl 0853.42013

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.


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.)
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