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Bilinear weighted Hardy inequality for nonincreasing functions

Martin Křepela

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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 $0\lt\gamma\lt\infty$.

Article information

Source
Publ. Mat., Volume 61, Number 1 (2017), 3-50.

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

Permanent link to this document
https://projecteuclid.org/euclid.pm/1482375623

Digital Object Identifier
doi:10.5565/PUBLMAT_61117_01

Mathematical Reviews number (MathSciNet)
MR3590113

Zentralblatt MATH identifier
1359.26020

Subjects
Primary: 26D10: Inequalities involving derivatives and differential and integral operators 47G10: Integral operators [See also 45P05]

Keywords
Hardy operators bilinear operators weights inequalities for monotone functions

Citation

Křepela, Martin. Bilinear weighted Hardy inequality for nonincreasing functions. Publ. Mat. 61 (2017), no. 1, 3--50. doi:10.5565/PUBLMAT_61117_01. https://projecteuclid.org/euclid.pm/1482375623


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