One-scale H-distributions and variants. (English) Zbl 1518.35043
Summary: H-measures and semiclassical (Wigner) measures were introduced in early 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. 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 not involving a specific characteristic length, providing some applications.
MSC:
| 35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |
| 35A27 | Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs |
| 35S05 | Pseudodifferential operators as generalizations of partial differential operators |
| 42B15 | Multipliers for harmonic analysis in several variables |
| 46F05 | Topological linear spaces of test functions, distributions and ultradistributions |
Keywords:
H-measures; microlocal defect measures; one-scale H-measures; H-distributions; semiclassical measures; Wigner measures; localisation priciple; compactness by compensation; one-scale H-distributionsReferences:
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