Microlocal defect functionals in VMO: Geometric localisation and applications to highly heterogeneous media
Abstract
We extend the concept of microlocal defect functionals to test functions belonging to the space L∞∩VMOc. Following L.~Tartar's remark that an extension of such concepts to VMO spaces should be possible, we establish a functional-analytic framework for this extension within the Lp−Lq setting. Because the topological dual of VMO is the Hardy space ℋ1, the resulting object takes the form of an H-distribution rather than a non-negative Radon measure. By assuming strict Hölder conjugate inequalities, we use the John-Nirenberg inequality over localized domains to construct these functionals. We show that the functional acts as a distribution on the inductive limit topology of the test spaces, giving geometric localisation principles for equations with rough VMO coefficients, that is, coefficients admitting sharp transitions of vanishing mean oscillation. We illustrate the framework in three settings: stratified transport, zero-order non-local cross-phase energies, and sub-critical acoustic scattering in high-contrast media. These differ in the nonlinearity generating the companion sequence but share a common geometric conclusion, since in each case the macroscopic energy defect is confined to the characteristic variety of the underlying flow, being supported where ∑j aj(bf x)ξj=0.
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BibTeX
@misc{misur2026microlocal,
title = {{Microlocal defect functionals in VMO: Geometric localisation and applications to highly heterogeneous media}},
author = {Marin Mišur},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.04294},
eprint = {2607.04294},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.04294},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Microlocal defect functionals in VMO: Geometric localisation and applications to highly heterogeneous media
AU - Marin Mišur
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2607.04294
UR - https://arxiv.org/abs/2607.04294
AB - We extend the concept of microlocal defect functionals to test functions belonging to the space L^∞∩VMO_c. Following L.~Tartar's remark that an extension of such concepts to VMO spaces should be possible, we establish a functional-analytic framework for this extension within the L^p−L^q setting. Because the topological dual of VMO is the Hardy space ℋ^1, the resulting object takes the form of an H-distribution rather than a non-negative Radon measure. By assuming strict Hölder conjugate inequalities, we use the John-Nirenberg inequality over localized domains to construct these functionals. We show that the functional acts as a distribution on the inductive limit topology of the test spaces, giving geometric localisation principles for equations with rough VMO coefficients, that is, coefficients admitting sharp transitions of vanishing mean oscillation. We illustrate the framework in three settings: stratified transport, zero-order non-local cross-phase energies, and sub-critical acoustic scattering in high-contrast media. These differ in the nonlinearity generating the companion sequence but share a common geometric conclusion, since in each case the macroscopic energy defect is confined to the characteristic variety of the underlying flow, being supported where ∑_j a_j(bf x)ξ_j=0.
ER -