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The weighted Hardy’s inequality for nonincreasing functions. (English) Zbl 0786.26015
Work started by D. W. Boyd and S. G. Krein and E. M. Semenov and developed later by M. Ariño and B. Muckenhoupt led to a proof that \[ \left(\int^ \infty_ 0\left({1\over x} \int_ 0^ x g(t) dt\right)^ p v(x) dx\right)^{1/p}\leq C\left(\int_ 0^ \infty g^ p(x) v(x) dx\right)^{1/p}, \] holds for all \(g\geq 0\), nonincreasing iff \(v\in B_ p\), i.e., \[ \int^ \infty_ 0 v(x)/x^ p dx\leq D/t^ p\int_ 0^ t v(x) dx,\;\forall t>0. \] E. T. Sawyer [Stud. Math. 96, No. 2, 145-158 (1990; Zbl 0705.42014)] gave a necessry and sufficient condition for the two weight case of the above inequality \[ \left(\int^ \infty_ 0 \left({1\over x} \int_ 0^ x g(t) dt\right)^ q w(x) dx\right)^{1/q}\leq C\left(\int^ \infty_ 0 g^ p(x)v(x) dx\right)^{1/p}, \] again when \(g\geq 0\), nonincreasing. Sawyer’s proof was valid for the case \(1<p\leq q<\infty\), and \(1<q<p<\infty\), and was based on a reverse Hölder inequality estimating \[ \left(\int^ \infty_ 0 gv\right)\left/\left(\int^ \infty_ 0 g^ p v\right)^{1/p}\right. \] for nonincreasing function \(g\).
The author gives an alternative direct proof of Sawyer’s result that allows him to extend the results to the cases \(0< q< 1< p<\infty\) and \(0< p\leq q<\infty\), \(0< p<1\).

MSC:
26D15 Inequalities for sums, series and integrals
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
42B25 Maximal functions, Littlewood-Paley theory
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