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Zbl 0538.47020
Sawyer, Eric
Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator.
(English)
[J] Trans. Am. Math. Soc. 281, 329-337 (1984). ISSN 0002-9947; ISSN 1088-6850/e

Let f(x) denote a measurable function defined on $(0,\infty)$. The Hardy operator T is defined by $Tf(x)=\int\sp{x}\sb{0}f(s)ds.$ Suppose w(x) and v(x) are nonnegative weight functions. \par 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 $L\sp{r,s}((o,\infty),vdx)$ to $L\sp{p,q}((o,\infty),wdx) (o<p,q,r,s\le \infty)?$ \par This is the problem discussed in this paper. The modified Hardy operators $T\sb{\eta}f(x)=x\sp{-\eta}Tf(x)$ for $\eta$ real are also treated.
[K.Wang]
MSC 2000:
*47B38 Operators on function spaces
46E30 Spaces of measurable functions
47A30 Operator norms and inequalities

Keywords: weight; Lorentz space; norm inequalities; Hardy operator

Cited in: Zbl 0813.47033

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