Majorization revisited: Comparison of norms in interpolation scales

Abstract

We reformulate, modify and extend a comparison criteria of L p norms obtained by Nazarov-Podkorytov and place it in the general setting of interpolation theory and majorization theory. In particular, we give norm comparison criteria for general scales of interpolation spaces, including noncommutative L p and Lorentz spaces. As an application, we extend the classical Ball's integral inequality, which lies at the basis of his famous result on sections of the n−dimensional unit cube.

Haagerup's phase transition at polydisc slicing

“…(see also [28] and [30] for generalisations to noncentral sections). Thus, the upper bound (6) is Theorem 2 at q = −2, that is c 2 (2) = c 4,2 (2). Incidentally, the lower bound (7) follows immediately from Jensen's inequality (see, e.g.…”

“…[8,9,10,14,26,37] which heavily rely on the approach developed by Nazarov and Podkorytov in [39] to integral inequalities with oscillatory integrands. We also refer to recent papers [2] as well as [36] for connections between such integral inequalities and majorisation.…”

“…Lemma 27. With D(p) defined in (32), the function p → log D(p) is increasing, convex and positive on (2,3).…”

Transport-majorization to analytic and geometric inequalities

“…It has attracted attention and found utility as a tool for delivering relatively simple arguments for L p norm comparisons between functions that would otherwise be very challenging to compare 1 .…”

“…In fact, [16] also holds the majorization interpretation (Lemma 1.3) that was at the starting point of our investigations, even if we realized the existence of [16] only after the writing of the present article was complete 2 . We direct the reader to [1] for recent extensions of Nazarov-Podkorytov's lemma to interpolation spaces using majorization.…”