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2017 Bilinear weighted Hardy inequality for nonincreasing functions
Martin Křepela
Publ. Mat. 61(1): 3-50 (2017). DOI: 10.5565/PUBLMAT_61117_01

Abstract

We characterize the validity of the bilinear Hardy inequality for nonincreasing functions \[ \|f^{**} g^{**}\|_{L^q(w)} \le C \|f\|_{\Lambda^{p_1}(v_1)}\|g\|_{\Lambda^{p_2}(v_2)}, \] in terms of the weights $v_1$, $v_2$, $w$, covering the complete range of exponents $p_1,p_2,q\in (0,\infty]$. The problem is solved by reducing it into the iterated Hardy-type inequalities \begin{align*} \left( \int\limits_0^\infty \biggl( \int\limits_0^x (g^{**}(t))^\alpha \varphi(t)\,\mathrm{d}t \biggr)^\frac{\beta}{\alpha} \psi(x)\,\mathrm{d}x \right)^\frac{1}{\beta} & \le C \biggl( \int\limits_0^\infty (g^*(x))^\gamma \omega(x) \,\mathrm{d}x \biggr)^\frac{1}{\gamma}, \\ \left( \int\limits_0^\infty \biggl( \int\limits_x^\infty (g^{**}(t))^\alpha \varphi(t)\,\mathrm{d}t \biggr)^\frac{\beta}{\alpha} \psi(x)\,\mathrm{d}x \right)^\frac{1}{\beta} & \le C \biggl( \int\limits_0^\infty (g^*(x))^\gamma \omega(x) \,\mathrm{d}x \biggr)^\frac{1}{\gamma}. \end{align*} Validity of these inequalities is characterized here for $0\lt\alpha\le\beta\lt\infty$ and $0\lt\gamma\lt\infty$.

Citation

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Martin Křepela. "Bilinear weighted Hardy inequality for nonincreasing functions." Publ. Mat. 61 (1) 3 - 50, 2017. https://doi.org/10.5565/PUBLMAT_61117_01

Information

Received: 14 January 2015; Revised: 9 September 2015; Published: 2017
First available in Project Euclid: 22 December 2016

zbMATH: 1359.26020
MathSciNet: MR3590113
Digital Object Identifier: 10.5565/PUBLMAT_61117_01

Subjects:
Primary: 26D10 , 47G10

Keywords: bilinear operators , Hardy operators , inequalities for monotone functions , weights‎

Rights: Copyright © 2017 Universitat Autònoma de Barcelona, Departament de Matemàtiques

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Vol.61 • No. 1 • 2017
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