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Weighted inequalities for iterated Copson integral operators

Volume 253 / 2020

Martin Křepela, Luboš Pick Studia Mathematica 253 (2020), 163-197 MSC: 47G10, 47B38, 26D10. DOI: 10.4064/sm181016-5-5 Published online: 24 January 2020

Abstract

We solve a long-standing open problem in the theory of weighted inequalities concerning iterated Copson operators. We use a constructive approximation method based on a new discretization principle. As a result, we characterize all weight functions $w,v,u$ on $(0,\infty )$ for which there exists a constant $C$ such that the inequality \[ \left (\int _0^{\infty }\left (\int _t^\infty \left (\int _s^{\infty } h(y)\dy \right )^{m}u(s) \ds \right )^{ {q}/{m}}w(t)\dt \right )^{ {1}/{q}} \leq C \left (\int _0^{\infty }h(t)^pv(t)\dt \right )^{ {1}/{p}} \] holds for every non-negative measurable function $h$ on $(0,\infty )$, where $p,q$ and $m$ are positive parameters. We assume that $p\geq 1$ because otherwise the inequality cannot hold for non-trivial weights, but otherwise $p,q$ and $m$ are unrestricted.

Authors

  • Martin KřepelaInstitute of Mathematics
    University of Freiburg
    Ernst-Zermelo-Straße 1
    791 04 Freiburg, Germany
    e-mail
  • Luboš PickDepartment of Mathematical Analysis
    Faculty of Mathematics and Physics
    Charles University
    Sokolovská 83
    186 75 Praha 8, Czech Republic
    e-mail

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