On generalized Dirichlet integrals in the smooth and in the o-minimal setting

TL;DR

This paper investigates conditions under which generalized Dirichlet energies on manifolds are precompact in L^2, linking geometric properties of vector fields, o-minimal tameness, and the H"ormander condition to compactness criteria.

Contribution

It provides a geometric sufficient condition for precompactness involving iterated characteristic sets, especially for tame vector fields satisfying the H"ormander condition.

Findings

Precompactness characterized by absence of characteristic submanifolds.

Tame vector fields satisfying H"ormander condition ensure precompactness.

Implications for regularity of sum-of-squares operators discussed.

Abstract

Given a compact manifold $M$ equipped with smooth vector fields $X_1,\ldots, X_r$, we consider the generalized Dirichlet energy \[\mathbf{E}(f)= \sum_{j=1}^r\int_M |X_jf|^2\, dm,\] where $dm$ is a volume form, and ask if the set \[ \mathcal{B}=\{f\in L^2(M)\colon\,\mathbf{E}(f)+\lVert f\rVert_{L^2(M)}^2\leq 1 \} \] is precompact in $L^2(M)$. We find a geometric sufficient condition in terms of "iterated characteristic sets" and use it to show that, if the vector fields are tame (in the sense of o-minimality) and satisfy the H\"ormander condition of some order $s$ on a dense set of points, then the only obstruction to precompactness is the existence of a characteristic submanifold (i.e. a nonempty submanifold of positive codimension to which each $X_j$ is tangent). Implications for global regularity of sum-of-squares operators not necessarily satisfying H\"ormander condition are discussed in an appendix.