Lower bounds for pseudo-differential operators

On the Positive Parts of Second Order Symmetric Pseudodifferential Operators

“…We end this section by recalling from [11,12] the following lower bound, which generalizes the Melin inequality (see [9]) for operators with multiple characteritics due to Mohamed [10]. In the sequel we will apply this result for m = k = 2.…”

“…Hence the spectrum of P ρ consists of a sequence of eigenvalues of finite multiplicity, diverging to +∞, and the corresponding eigenfunctions are Schwartz functions. Indeed, in [9] Melin explicitly computed the spectrum of P ρ showing that…”

“…Consider the k−th Hermite function 9) and notice that h k (|ξ 2 | 1/2 y) spans the eigenspace of the localized operator P ρ = P ρ (y, D y ) corresponding to the eigenvalues m k (x 2 , ξ 2 ). Furthermore, we observe that m 0 (x 2 , ξ 2 ) = min Spec(P ρ ).…”

Generalizations of Melin's inequality to systems

“…The proof of theorem 2.2 is in fact faithfully modelled on Melin's original proof in [12]: one can use his S 0 1/2,1/2 -partition of unity arguments to reduce the sufficiency of 2.2(i) to the following statement, where we will suppose, for simplicity (and without essential loss of generality), that A m−1 = 0, and where we write A …”

On the Generalization of Hörmander's Inequality

“…This estimate is a generalization of the classical Melin's inequality to operators with characteristics of order higher than 2 (see Melin [15], Hörmander [13], Thm. 22.3.3, and Hérau [11]).…”