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Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator. (English) Zbl 0538.47020

Let f(x) denote a measurable function defined on \((0,\infty)\). The Hardy operator T is defined by \(Tf(x)=\int^{x}_{0}f(s)ds.\) Suppose w(x) and v(x) are nonnegative weight functions.
What conditions for the pair of w,v are necessary and sufficient in order that the operator T will be bounded from the Lorentz space \(L^{r,s}((o,\infty),vdx)\) to \(L^{p,q}((o,\infty),wdx) (o<p,q,r,s\leq \infty)?\)
This is the problem discussed in this paper. The modified Hardy operators \(T_{\eta}f(x)=x^{-\eta}Tf(x)\) for \(\eta\) real are also treated.
Reviewer: K.Wang

MSC:

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.)
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
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