Weighted Lorentz spaces and the Hardy operator. (English) Zbl 0784.46022

Carro, María J.; Soria, Javier

J. Funct. Anal. 112, No. 2, 480-494 (1993).

The authors find a new expression for the norm of a function in the weighted Lorentz space, with respect to the distribution function, and obtain as simple consequence a generalization of the classical embeddings \(L^{p,1}\subset\cdots\subset L^ p\subset\cdots\subset L^{p,t}\) and a new definition of the weak space \(\Lambda^{p,t}_ u(w)\). They also give some applications to the boundedness of the Hardy operator \(Sf=\int^ x_ 0 f\) from \(\Lambda^{p_ 0}_{u_ 0}(w_ 0)\) into \(\Lambda^{p_ 1}_{u_ 1}(w_ 1)\) with \(0< p_ 0\leq p_ 1\).

Reviewer: J.Appell (Würzburg)

MSC:

46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47B38	Linear operators on function spaces (general)

Keywords:

weighted Lorentz space; embeddings; weak space; boundedness of the Hardy operator