Weighted weak type inequalities for modified Hardy operators and geometric means operators in dimensions one and greater

Abstract

We characterize the pairs of weights (u, v) such that the geometric mean operator G 1

Weighted inequalities for the one‐sided geometric maximal operators

“…Lemma 1.9 (7) Let −∞ < a < b < ∞, let u , v be positive locally integrable functions on ( a , b ) and let s 0 > 1. The following statements are equivalent: There exists C > 0 such that the inequality holds for all λ > 0 and all positive f . where \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\beta (x)=\exp \left(\frac{1}{b-x} \int _x^b \log \big (v^{-1} \big )\right)$\end{document}. …”

“…This result was established in 7 by performing a limiting process inspired in 10 which requires a characterization of the weighted weak type inequalities for Hardy type operators with a suitable control of the best constant. Although the result in 7 concerns only the geometric mean operator in (0, ∞), the extension to an interval ( a , b ) is straightforward.…”