The weighted Hardy’s inequality for nonincreasing functions. (English) Zbl 0786.26015

Stepanov, Vladimir D.

Trans. Am. Math. Soc. 338, No. 1, 173-186 (1993).

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 \(0< p\leq q<\infty\), \(0< p<1\).

Reviewer: R.Johnson (College Park)

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

Keywords:

weighted Hardy inequality; nonincreasing functions; two weight; reverse Hölder inequality

Citations:

Zbl 0705.42014