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Weighted Hardy inequalities for increasing functions. (English) Zbl 0796.26008

Complete results are known for the two weight inequality for the Hardy operator \[ Pf(x) = {1\over x} \int_ 0^ x f(t) dt, \]
\[ \left( \int_ 0^ \infty \bigl | Pf(x) \bigr |^ q w(x) dx \right)^{1/q} \leq C \left( \int_ 0^ \infty \bigl | f(x) \bigr |^ p v(x) dx \right)^{1/p}, \] where \(f\) is restricted to the class of nonincreasing nonnegative functions.
By a proof similar to one used by the second author [Trans. Am. Math. Soc. 338, No. 1, 173-186 (1993; Zbl 0786.26015)] to prove results for the Hardy operator for nonincreasing functions, the authors show that the Hardy inequality holds for \(1 < p \leq q < \infty\) for nonnegative increasing functions iff \(B < \infty\), where \[ B_ 0(t)=\left[ \int_ t^ \infty (x-t)^ q x^{-q}w(x) dx \right]^{1/q} V(t)^{-1/p}, \]
\[ B_ 1(t) = \left[ \int_ t^ \infty x^{-q} w(x) dx \right]^{1/q} \left[ \int_ 0^ t (t-x)^{p'} V(x)^{ - p'} v(x) dx \right]^{1/p'}, \] where \(V(x) = \int_ 0^ x v(t) dt\), \(B_ i = \sup_{t>0} B_ i(t)\), and \(B = \max (B_ 0,B_ 1)\).
The authors also give a more complicated necessary and sufficient condition for the case \(1 < q < p < \infty\) and a necessary condition in the case \(0 < q < 1 < p < \infty\).

MSC:

26D15 Inequalities for sums, series and integrals
42B25 Maximal functions, Littlewood-Paley theory

Citations:

Zbl 0786.26015
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