A Priori Estimates for Maximally Subelliptic Quadratic Forms
Brian Street
TL;DR
The work develops a rigorous, quantitative theory of subelliptic estimates for the heat operator ∂_t+ℒ tied to maximally subelliptic quadratic forms on manifolds with boundary, focusing on non-characteristic boundary points. By combining interior Rothschild–Stein-type regularity with a careful boundary analysis that reduces to tangent derivatives and uses extension/Smoothing operators, the authors obtain a priori estimates that track constant dependencies and extend to non-Dirichlet boundary conditions. The central result, a boundary-included subelliptic estimate for ∂_t+ℒ, is complemented by corollaries asserting hypoellipticity and refined control of Sobolev norms near boundary and in the interior, under the maximal subellipticity assumption. The paper also clarifies how boundary conditions emerge from the quadratic form and its boundary data, providing a framework applicable to a broad class of maximally subelliptic boundary value problems and setting the stage for sharper, adapted non-isotropic estimates in future work.
Abstract
We prove a priori subelliptic estimates, near a non-characteristic boundary point, for the heat operators associated to a wide class of maximally subelliptic quadratic forms. This is the third paper in a series devoted to studying general maximally subelliptic boundary value problems.