Supremum operators and optimal Sobolev inequalities (Q2707681)

The author studies the optimality of rearrangement-invariant quasi-norms for which a Sobolev-type inequality holds. His aim is to obtain optimal pairs of rearrangement norms for such an inequality. He shows that a key role here is played by the ``supremum operator'' \(T\) given by NEWLINE\[NEWLINE (Tg)(t) = t^{-1/n} \sup _{t<s<1} s^{1/n} g^*(s), \quad t\in (0,1), NEWLINE\]NEWLINE where \(g\) is a measurable function on \((0,1)\) and \(g^*\) stands for its non-increasing rearrangement.NEWLINENEWLINEFor the entire collection see [Zbl 0933.00035].