The failure of the Hardy inequality and interpolation of intersections (Q1810332)

In this deep paper the following problem is investigated: when for parameters \(\theta \in \left( 0,1\right) \) and \(p\in \left[ 1,\infty \right] \) the following interpolation formula for weighted \(L_{p}\)-spaces is true: \[ ( N\cap L_{p}\left( \mu \right) ,N\cap L_{p}\left( \nu \right))_{\theta ,p}=N\cap \left( L_{p}\left( \mu \right) ,L_{p}\left( \nu \right) \right) _{\theta ,p}, \] where \(N\) is the linear space that consists of all functions with integral equal to \(0\). It is shown that when the weights \(\mu \), \(\nu \) are not power functions, then it is possible that the mentioned problem fails on the whole interval \( \left[ a,b\right] \), not only in a single point. As it is shown, the results are closely connected with the classical Hardy inequality.