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 $\mathrm{L}^\infty\cap\mathrm{VMO}_c$. Following L.~Tartar's remark that an extension of such concepts to $\mathrm{VMO}$ spaces should be possible, we establish a functional-analytic framework for this extension within the $\mathrm{L}^p-\mathrm{L}^q$ setting. Because the topological dual of $\mathrm{VMO}$ is the Hardy space $\mathcal{H}^1$, the resulting object takes the form of an H-distribution rather than a non-negative Radon measure. By assuming strict Hölder conjugate inequalities, we utilize the John-Nirenberg inequality over localized domains to construct these functionals. We demonstrate that the functional acts as a distribution on the inductive limit topology of the test spaces, enabling exact geometric localisation principles for equations with rough $\mathrm{VMO}$ coefficients, that is, coefficients admitting sharp transitions of vanishing mean oscillation. To demonstrate the versatility of this framework across diverse physical environments, we deploy it to determine the exact geometric structure and support of macroscopic energy defects in three distinct settings: the characteristic support of stratified transport, zero-order non-local cross-phase energies, and the microlocal trapping of sub-critical acoustic scattering in high-contrast media.