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

Given a compact manifold $M$ equipped with smooth vector fields $X_1,\dots ,X_r$, we consider the generalized Dirichlet energy $$\mathbf{E}(f)=\sum _{j=1}^r{\int }_M\mathrm{\mid }{X}_{j}f{\mathrm{\mid }}^{2}dm,$$ where $dm$ is a volume form, and ask if the set $$\mathcal{B}=\{f\in L^2(M)\text{\hspace{0.17em}⁣}:\mathbf{E}(f)+\parallel f{\parallel }_{L^2(M)}^2\le 1\}$$ 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 Hörmander 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 Hörmander condition are discussed in an appendix.