Weighted inequalities for iterated Copson integral operators

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 $${\left({\int }_0^{\infty }{\left({\int }_t^{\infty }{\left({\int }_s^{\infty }h(y)\text{d}y\right)}^{m}u(s)\text{d}s\right)}^{\frac{q}{m}}w(t)\text{d}t\right)}^{\frac{1}{q}}\le C{\left({\int }_0^{\infty }h(t{)}^{p}v(t)\text{d}t\right)}^{\frac{1}{p}}$$ holds for every non-negative measurable function $h$ on $(0,\infty )$, where $p,q$ and $m$ are positive parameters. We assume that $p\ge 1$ but otherwise $p,q$ and $m$ are unrestricted.