One-scale H-distributions and variants

Abstract

H-measures and semiclassical (Wigner) measures were introduced in earlyn 1990s and since then they have found numerous applications in problems involving $\mathrm{L}^2$ weakly converging sequences. Although they are similar objects, neither of them is a generalisation of the other, the fundamental difference between them being the fact that semiclassical measures have a characteristic length, while H-measures have none. Recently introduced objects, the one-scale H-measures, generalise both of them, thus encompassing properties of both. The main aim of this paper is to fully develop this theory to the $\mathrm{L}^p$ setting, $p\in(1,\infty)$, by constructing one-scale H-distributions, a generalisation of one-scale H-measures and, at the same time, of H-distributions, a generalisation of H-measures to the $\mathrm{L}^p$ setting, without any characteristic length. We also address an alternative approach to $\mathrm{L}^p$ extension of semiclassical measures via the Wigner transform, introducing new type of objects (semiclassical distributions). Furthermore, we derive a localisation principle in a rather general form, suitable for problems with a characteristic length, as well as those without a specific characteristic length, providing some applications.