Weighted inequalities for iterated Copson integral operators

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