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

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 \(0< p_ 0\leq p_ 1\).


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)
Full Text: DOI