# Weighted inequalities for iterated Copson integral operators

Research paper by **Martin Křepela, Luboš Pick**

Indexed on: **13 Jun '18**Published on: **13 Jun '18**Published in: **arXiv - Mathematics - Functional Analysis**

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#### Abstract

We solve a long-standing open problem in theory of weighted inequalities
concerning iterated Copson operators. We use a constructive approximation
method based on a new discretization principle that is developed here. In
result, we characterize all weight functions $w,v,u$ on $(0,\infty)$ for which
there exists a constant $C$ such that the inequality $$
\left(\int_0^{\infty}\left(\int_t^\infty
\left(\int_s^{\infty}h(y)\,\text{d}y\right)^mu(s)
\,\text{d}s\right)^{\frac{q}{m}}w(t)\,\text{d}t\right)^{\frac{1}{q}} \le C
\left(\int_0^{\infty}h(t)^pv(t)\,\text{d}t\right)^{\frac{1}{p}} $$ holds for
every non-negative measurable function $h$ on $(0,\infty)$, where $p,q$ and $m$
are positive parameters. We assume that $p\geq 1$ but otherwise $p,q$ and $m$
are unrestricted.