On the Sharp Gårding Inequality for Operators with Polynomially Bounded and Gevrey Regular Symbols

: In this paper, we analyze the Friedrichs part of an operator with polynomially bounded symbol. Namely, we derive a precise expression of its asymptotic expansion. In the case of symbols satisfying Gevrey estimates, we also estimate precisely the regularity of the terms in the asymptotic expansion. These results allow new and reﬁned applications of the sharp Gårding inequality in the study of the Cauchy problem for p -evolution equations.


Introduction
The sharp Gårding inequality for a pseudodifferential operator was first proved by Hörmander [1] and by Lax and Nirenberg [2] for symbols in the Kohn-Nirenberg class S m (R 2n ), namely satisfying the following estimates |∂ α ξ ∂ β x p(x, ξ)| ≤ C αβ ξ m−|α| , α, β ∈ N n 0 , for some positive constant C αβ , where ξ := 1 + |ξ| 2 for every ξ ∈ R n and N n 0 stands for the set of all multi-indices of length n. In its original form, this result states that, if p ∈ S m (R 2n ), for some m ∈ R, is such that Re p(x, ξ) ≥ 0, then the corresponding operator p(x, D) satisfies the following estimate Re(p(x, D)u, u) L 2 ≥ −C u 2 for some C ∈ R, where · m−1 2 denotes the standard norm in the Sobolev space H m−1 2 (R n ). Later on, several different proofs and extensions of this result have been provided by many authors (cf. [3][4][5][6]). In particular, the inequality has been extended to symbols defined in terms of a general metric (cf. [4], Theorem 18.6.7) and to matrix valued pseudo-differential operators (cf. [4], Lemma 18.6.13, and [5], Theorem 4.4 page 134). In all the proofs of the sharp Gårding inequality, the operator p(x, D) is decomposed as the sum of a positive definite part and a remainder term providing the inequality (1). In the approach proposed in [5], this positive definite part p F is called Friedrichs part and satisfies the following conditions: (i) (p F u, v) L 2 = (u, p F v) L 2 if p(x, ξ) is real; (ii) (p F u, u) L 2 ≥ 0 if p(x, ξ) ≥ 0; and (iii) Re (p F u, u) L 2 ≥ 0 if Re p(x, ξ) ≥ 0.
Although the results in [4] are extremely general and sharp, in some applications, more detailed information on the remainder term is needed. In particular, it is important to state not only the order but also the asymptotic expansion of p − p F . This is needed in particular in the analysis of the so called p-evolution equations, namely equations of the form where p is a positive integer and a p (t) is a real valued function (cf. [7,8]). This large class of equations includes for instance strictly hyperbolic equations (p = 1) and Schrödinger-type equations (p = 2). The classical approach to study the Cauchy problem for these equations is based on a reduction to an auxiliary problem via a suitable change of variable and on a repeated application of sharp Gårding inequality which needs at every step to understand the precise form of all the remainder terms (cf. [9]). When the coefficients a j (t, x) are uniformly bounded with respect to x, this is possible using Theorem 4.2 in [5], where the asymptotic expansion of p F − p is given in the frame of classical Kohn-Nirenberg classes. In this way, under suitable assumptions on the behavior of the coefficients a j (t, x), j = 0, . . . , p − 1, for |x| → ∞, well posedness with loss of derivatives has been proved in H ∞ = ∩ m∈R H m (see [9]). In fact, for equations of the form (2), the loss of derivatives can be avoided by choosing the data of the Cauchy problem with a certain decay at infinity (cf. [10]). This motivated us to study the initial value problem for (2) in a weighted functional setting admitting also polynomially bounded coefficients, which cannot be treated in the theory of standard Kohn-Nirenberg classes but are included in the so-called SG classes (see the definition below). For this purpose we need a variant of [5] (Theorem 4.2) for SG operators with a precise information on the asymptotic expansion of p − p F .
Another challenging issue is to study Equation (2) on Gelfand-Shilov spaces of type S (cf. [11]). A first step in this direction has been done in the case p = 2, that is for Schrödinger-type equations (see [12]), and for p = 3 (see [13]). In both cases, it is sufficient to apply the sharp Gårding inequality only once. To treat p-evolution equations for p > 3, however, we need to apply the iterative procedure described above. In addition, a precise estimate of the Gevrey regularity of the terms in the asymptotic expansion of p − p F is also needed.
In this paper, we provide appropriate tools for both the aforementioned issues. This is achieved by defining in a suitable way the Friedrichs part of our operators and by studying in detail its asymptotic expansion and its regularity. With this purpose, we prove two separate results for the following classes of symbols. Fixing m = (m 1 , m 2 ) ∈ R 2 , we denote by SG m (R 2n ) the space of all functions p ∈ C ∞ (R 2n ) satisfying for any α, β ∈ N n 0 the following condition for some positive constant C α,β . These symbols have been treated by a large number of authors along the years (see [14][15][16][17][18][19][20][21]). We are moreover interested in the subclass of SG m (R 2n ) given by the SG symbols possessing a Gevrey-type regularity. Namely, for µ, ν ≥ 1, we say that a symbol p(x, ξ) belongs to the class SG m for every α, β ∈ N n 0 , x, ξ ∈ R n . This work is organized as follows. In Section 2, we recall some results concerning SG pseudodifferential operators. In Section 3, we discuss the concepts of oscillatory integrals and double symbols, which are fundamental tools in the present work. Finally, in Section 4, we study the Friedrichs part of symbols belonging to the classes SG m and SG m (µ,ν) and we prove the main results of this paper, namely Theorems 4 and 6.

SG Pseudodifferential Operators
In this section, we recall some basic facts about SG pseudodifferential operators which are used in the sequel. Although for our applications we are interested to prove the main results for the classes defined by inequalities (3) and (4), in order to prove them, we need to consider more general classes of symbols which are defined as follows.
We recall that SG m ρ,δ is a Fréchet space endowed with the seminorms for ∈ N 0 . The class SG m ρ,δ is included in the general theory by Hörmander [4]. A specific calculus for this class can be found in [22]. Pseudodifferential operators with symbols in SG m ρ,δ are linear and continuous from S (R n ) to S (R n ) and extend to linear and continuous maps from S (R n ) to S (R n ). Moreover, denoting by H s (R n ) with s = (s 1 , s 2 ) ∈ R 2 the weighted Sobolev space we know that an operator with symbol in SG m ρ,δ extends to a linear and continuous map from H s (R n ) to H s−m (R n ) for every s ∈ R 2 . Definition 2. Let, for j ∈ N 0 , p j ∈ SG (m 1,j ,m 2,j ) ρ,δ , where m 1,j , m 2,j are nonincreasing sequences and m 1,j → −∞, m 2,j → −∞, when j → ∞. We say that p ∈ C ∞ (R 2n ) has the asymptotic expansion Given p j ∈ SG (m 1,j ,m 2,j ) ρ,δ as in the previous definition, we can find p ∈ SG (m 1,0 ,m 2,0 ) ρ,δ such that [22], Theorem 2). The class SG m ρ,δ is closed under adjoints. Namely, given p ∈ SG m ρ,δ and denoting by P * the L 2 adjoint of p(x, D), we can write P * = p * (x, D) + R, where p * is a symbol in SG m ρ,δ admitting the asymptotic expansion and R : S (R n ) → S (R n ). The class SG ∞ ρ,δ := ∪ m∈R 2 SG m ρ,δ possesses algebra properties with respect to composition. Namely, given p ∈ SG m ρ,δ and q ∈ SG m ρ,δ , there exists a symbol s ∈ SG m+m ρ,δ such that p(x, D)q(x, D) = s(x, D) + R where R is a smoothing operator S (R n ) → S (R n ). Moreover, (cf. [22], Theorem 3). We now consider Gevrey regular symbols.

Definition 3.
Fixing C > 0, we denote by SG m ρ,δ;(µ,ν) (R 2n ; C) the space of all smooth functions p(x, ξ) such that Equipping SG m ρ,δ;(µ,ν) (R 2n ; C) with the norm | · | C we obtain a Banach space and we can endow SG m ρ,δ;(µ,ν) (R 2n ) with the topology of inductive limit of Banach spaces. A complete calculus for operators with symbols in this class can be found in [23]. Here, we recall only the main results. Since SG m ρ,δ;(µ,ν) ⊂ SG m ρ,δ , the previous mapping properties on the Schwartz and weighted Sobolev spaces hold true for operators with symbols in SG m ρ,δ;(µ,ν) . By the way, the most natural functional setting for these operators is given by the Gelfand-Shilov spaces of type S. We recall that, fixing For every µ ≥ µ/(1 − δ 2 ), ν ≥ ν/(1 − δ 1 ), an operator with symbol in SG m ρ,δ;(µ,ν) is linear and continuous from S ν µ (R n ) to itself and extends to a linear continuous map from the dual space (S ν µ ) (R n ) into itself (see [23], Theorem A.4). The notion of asymptotic expansion for symbols in SG m ρ,δ;(µ,ν) can be defined in terms of formal sums (cf. [23]). Here, to obtain our results, we need to use a refined notion of formal sum introduced in [13] for the case ρ = (1, 1), δ = (0, 0). All the next statements can be transferred to the case of general ρ and δ without changing the argument, thus we refer to [13] for the proofs.
and Q e t 1 ,t 2 = R 2n \ Q t 1 ,t 2 . When t 1 = t 2 = t, we simply write Q t and Q e t .

Oscillatory Integrals and Operators with Double Symbols
To define the Friedrichs part of an operator, it is necessary to extend the notion of pseudodifferential operator as in [5] by considering more general symbols called double symbols. Quantizations of these symbols are defined as oscillatory integrals.

Definition 10.
For p ∈ SG m,m ρ,δ , we define for every u ∈ S (R n ).
To obtain the same kind of result for the classes SG m,m ρ,(0,δ 2 );(µ,ν) , we need an analog of Lemma 1 with a precise estimate of the Gevrey regularity.
where r β,N is given as in (16).