## 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 $$0< p_ 0\leq p_ 1$$.

### 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)
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