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

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.