An application of the almost-positivity of a class of fourth-order pseudodifferential operators

Using the almost-positivity of a class of fourth-order pseudo-differential operators, we prove the inequality |Lu|(0) + |u|(0) ≥ CK (|Xu|(1/2) + |u|(1)), CK > 0, ∀u ∈ C∞0 (K), for any compact set K ⊂ Ω, an open set of ℝn, where L = L* ∈ Ψ2phg (Ω) has principal symbol p2 ≥ 0 transversally elliptic with respect to the characteristic manifold Σ = p-12 (0), the condition ps1(ρ) + Tr+Fp2(ρ) > 0 is satisfied on Σ, and where X ∈ Ψ1phg (Ω) principal symbol vanishing on Σ. Applications to the case L = Σmj=1 X*jXj, + .X0, where X0,Xj ∈ Ψ1phg (Ω), with Xj, X complex-valued, are given.