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.

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.)
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