6 Uniform subellipticity

We establish two global subellipticity properties of positive symmetric second-order partial differential operators on L2(R ). First, if m ∈ N then we consider operators H0 with coefficients in W (R) and domain D(H0) =W (R) satisfying the subellipticity property c (φ, (I +H0)φ) ≥ ‖∆ φ‖2 for some c > 0 and γ ∈ 〈0, 1], uniformly for all φ ∈ W(R), where ∆ denotes the usual Laplacian. Then we prove that D(H) ⊆ D(∆) for all α ∈ [0, 2(m+ 1 + γ)〉. Hence there is a c > 0 such that the norm estimate c ‖(I +H)φ‖2 ≥ ‖∆ φ‖2 is valid for all φ ∈ D(H) where H denotes the self-adjoint closure of H0. In particular, if the coefficients of H0 are in C ∞ b (R ) then the conclusion is valid for all α ≥ 0. Secondly, we prove that if