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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References:

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