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One-Scale H-Distributions and Variants

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Abstract

H-measures and semiclassical (Wigner) measures were introduced in early 1990 s and since then they have found numerous applications in problems involving \(\textrm{L}^2\) 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 \(\textrm{L}^p\) 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 \(\textrm{L}^p\) setting. We also address an alternative approach to \(\textrm{L}^p\) 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.

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Acknowledgements

The authors wish to thank Eduard Nigsch for numerous helpful discussions and the referees for their insightful comments that helped to improve the presentation and the quality of the paper.

Funding

This work is supported in part by the Croatian Science Foundation under the project IP–2018–01–2449 (MiTPDE) and by the Croatian-Austrian bilateral project Anisotropic distributions and H-distributions.

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  1. Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000, Zagreb, Croatia

    N. Antonić & M. Erceg

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Antonić, N., Erceg, M. One-Scale H-Distributions and Variants. Results Math 78, 165 (2023). https://doi.org/10.1007/s00025-023-01937-z

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