On the sharp Garding inequality for operators with polynomially bounded and Gevrey regular symbols

Article

On the Sharp Gårding Inequality for Operators with

Polynomially Bounded and Gevrey Regular Symbols

Alexandre Arias Junior1 and Marco Cappiello2,*

1 _{Department of Mathematics, Federal University of Paraná, Curitiba 81531-980, Brazil; arias@ufpr.br} 2 _{Department of Mathematics, University of Turin, Via Carlo Alberto 10, 10123 Turin, Italy}

* Correspondence: marco.cappiello@unito.it

Received: 29 September 2020; Accepted: 29 October 2020; Published: 3 November 2020




 Abstract: 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.

Keywords:pseudodifferential operators; Gevrey regularity; sharp Gårding inequality; p-evolution equations

MSC:35S05; 46F05; 35S10

1. 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 Sm(R2n), namely satisfying the following estimates

|∂α_ξ∂βxp(x, ξ)| ≤Cαβhξi

m−|α|_,

α, β∈ Nn₀,

for some positive constant Cαβ, wherehξi:=p1+ |ξ|2for every ξ∈ RnandNn0stands for the set of all multi-indices of length n. In its original form, this result states that, if p∈Sm(R2n), 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)_L2 ≥ −Ckuk2m−1 2

, u∈S(Rn), (1)

for some C∈ R, wherek · km−1

2 denotes the standard norm in the Sobolev space H m−1

2 (Rn). Later on,

several different proofs and extensions of this result have been provided by many authors (cf. [3–6]). 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 pFis called Friedrichs part and satisfies the following conditions:

(i) (pFu, v)L2 = (u, pFv)L2 if p(x, ξ)is real;

(ii) (pFu, u)L2 ≥0 if p(x, ξ) ≥0; and

(iii) Re(pFu, u)L2 ≥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−pF. This is needed in particular in the analysis of the so called p-evolution equations, namely equations of the form

Dtu+ap(t)Dxp+ p−1

∑

j=0

aj(t, x)Dxju= f(t, x), t∈ [0, T], (2)

where p is a positive integer and ap(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 aj(t, x)are uniformly bounded with respect to x, this is possible using Theorem 4.2 in [5], where the asymptotic expansion of pF−p is given in the frame of classical Kohn–Nirenberg classes. In this way, under suitable assumptions on the behavior of the coefficients aj(t, x), j = 0, . . . , p−1, for|x| → ∞, well posedness with loss of derivatives has been proved in H∞= ∩_m∈RHm(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−pF.

Another challenging issue is to study Equation (2) on Gelfand–Shilov spaces of typeS (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−pFis 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= (m1, m2) ∈ R2, we denote by SGm(R2n)the space of all functions p∈C∞(R2n) satisfying for any α, β∈ Nn

0the following condition

|∂α_ξ∂βxp(x, ξ)| ≤Cαβhξi

m1−|α|_hxim2−|β|_{, x, ξ ∈ R}n_{, (3)}

for some positive constant Cα,β. These symbols have been treated by a large number of authors along the years (see [14–21]). We are moreover interested in the subclass of SGm(R2n)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 SGm_(µ,ν)(R2n)if there are constants C, C1>0 satisfying

|∂α_ξ∂βxp(x, ξ)| ≤C1C|α|+|β|α!µβ!νhξim1−|α|hxim2−|β|, (4)

for every α, β∈ Nn

0, x, ξ∈ Rn.

This work is organized as follows. In Section 2, we recall some results concerning SG pseudodifferential operators. In Section3, we discuss the concepts of oscillatory integrals and double symbols, which are fundamental tools in the present work. Finally, in Section4, we study the Friedrichs part of symbols belonging to the classes SGmand SGm_(µ,ν)and we prove the main results of this paper, namely Theorems4and6.

2. 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.

Definition 1. Given m = (m1, m2) ∈ R2, ρ = (ρ1, ρ2) ∈ (0, 1]2, δ = (δ1, δ2) ∈ [0, 1)2, with δj <ρj, j=1, 2, we denote by SGmρ,δthe space of all functions p∈C∞(R

2n_{)such that}

sup (x,ξ)∈R2n

|∂α_ξ∂βxp(x, ξ)|hξi−m1+ρ1|α|−δ1|β|hxi−m2+ρ2|β|−δ2|α|<∞. (5)

We recall that SGm_ρ,δis a Fréchet space endowed with the seminorms

|p|`:= sup (x,ξ)∈R2n |α+β|≤`

|∂α_ξ∂βxp(x, ξ)|hξi−m1+ρ1|α|−δ1|β|hxi−m2+ρ2|β|−δ2|α|,

for` ∈ N0. The class SGmρ,δis included in the general theory by Hörmander [4]. A specific calculus for this class can be found in [22]. Pseudodifferential operators with symbols in SGm_ρ,δare linear and continuous fromS(Rn)toS(Rn)and extend to linear and continuous maps fromS0(Rn)toS0(Rn). Moreover, denoting by Hs(Rn)with s= (s1, s2) ∈ R2the weighted Sobolev space

Hs(Rn):= {u∈S0

(Rn):hxis2_hDxis1_u∈L2_(Rn_)},

we know that an operator with symbol in SGm_ρ,δextends to a linear and continuous map from Hs_(Rn₎ to Hs−m(Rn)for every s∈ R2_.

Definition 2. Let, for j∈ N0, pj ∈SG

(m1,j,m2,j)

ρ,δ , where m1,j, m2,jare nonincreasing sequences and m1,j → −∞, m2,j→ −∞, when j→∞. We say that p∈C∞(R2n)has the asymptotic expansion

p(x, ξ) ∼

∑

j∈N0

pj(x, ξ)

if for any N∈ Nwe have

p(x, ξ) − N−1

∑

j=0 pj(x, ξ) ∈SG(m_ρ,δ1,N,m2,N). Given pj ∈ SG (m1,j,m2,j)

ρ,δ as in the previous definition, we can find p ∈ SG

(m1,0,m2,0)

ρ,δ such that p∼_{∑ pj}. Furthermore, if there is q such that q∼_{∑ pj}, then p−q∈SG−∞:= ∩_m∈R2SGm_ρ,δ=S(R2n)

(cf. [22], Theorem 2). The class SGm_ρ,δis closed under adjoints. Namely, given p∈SGm_ρ,δand denoting by P∗ the L2 adjoint of p(x, D), we can write P∗ = p∗(x, D) +R, where p∗ is a symbol in SGm_ρ,δ admitting the asymptotic expansion

p∗(x, ξ) ∼

∑

α∈Nn0

and R :S0(Rn) →S(Rn). The class SG∞_ρ,δ:= ∪_m∈R2SGm_ρ,δpossesses algebra properties with respect

to composition. Namely, given p∈SGm_ρ,δand q∈SGm_ρ,δ0, there exists a symbol s∈SGm+m_ρ,δ 0such that p(x, D)q(x, D) =s(x, D) +R0where R0is a smoothing operatorS0(Rn_{) →S(R}n_{). Moreover,}

s(x, ξ) ∼

∑

α∈Nn₀

α!−1∂α_ξp(x, ξ)Dα_xq(x, ξ)

(cf. [22], Theorem 3).

We now consider Gevrey regular symbols.

Definition 3. Fixing C > 0, we denote by SGm_{ρ,δ;(µ,ν)}(R2n_{; C) the space of all smooth functions p(x, ξ)} such that |p|C:= sup α,β∈Nn0 C−|α|−|β|α!−µβ!−ν sup x,ξ∈Rn hξi−m1+ρ1|α|−δ1|β|hxi−m2+ρ2|β|−δ2|α||∂α_ξ∂βxp(x, ξ)| < +∞. We set SGm_{ρ,δ;(µ,ν)}(R2n) =S C>0SGm_ρ,δ;(µ,ν)(R2n; C).

Equipping SGm_{ρ,δ;(µ,ν)}(R2n; C) with the norm | · |_C we obtain a Banach space and we can endow

SGm_{ρ,δ;(µ,ν)}(R2n) 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 SGm_{ρ,δ;(µ,ν)} ⊂ SGm_ρ,δ, the previous mapping properties on the Schwartz and weighted Sobolev spaces hold true for operators with symbols in SGm_{ρ,δ;(µ,ν)}. By the way, the most natural functional setting for these operators is given by the Gelfand–Shilov spaces of typeS. We recall that, fixing µ>0, ν>0, the Gelfand–Shilov spaceSµ

ν(Rn)is defined as the space of all functions f ∈C∞(Rn)such that for some constant C>0 sup α,β∈Nn0 C−|α+β|(α!)−ν(β!)−µ sup x∈Rn |xα ∂βf(x)| < +∞. (6)

For every µ0 ≥ µ/(1−δ2), ν0 ≥ ν/(1−δ1), an operator with symbol in SGm_ρ,δ;(µ,ν) is linear and continuous from Sν0

µ0(R

n_{) to itself and extends to a linear continuous map from the dual space}

(Sν0 µ0)

0

(Rn)into itself (see [23], Theorem A.4).

The notion of asymptotic expansion for symbols in SGm_{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

ρand δ without changing the argument, thus we refer to [13] for the proofs.

For t1, t2≥0, set

Qt1,t2 = {(x, ξ) ∈ R

2n_{:hxi <t}

1andhξi <t2} and Qe_t1,t2 = R2n\Qt1,t2. When t1=t2=t, we simply write Qtand Q

e t.

Definition 4. Let ¯σj = (kj,`j) be a sequence such that k0 = `0 = 0, kj,`j are strictly increasing, kj+N≥kj+kN,`j+N ≥ `j+ `N, for j, N ∈ N0, and kj ≥ Λ1j,`j ≥ Λ2j for j ≥1, for someΛ1,Λ2 >0. We say that ∑

j≥0

pj∈F¯σSGm_{ρ,δ;(µ,ν)}if pj∈C∞(R2n)and there are C, c, B>0 satisfying

|∂α_ξ∂βxpj(x, ξ)| ≤C|α|+|β|+2j+1α!µβ!νj!µ+ν−1hξim1−ρ1|α|+δ1|β|−kjhxim2−ρ2|β|+δ2|α|−`j

for α, β∈ Nn

0, j≥0 and(x, ξ) ∈QeB2(j),B1(j), where Bi(j) = (Bj

Definition 5. Given ∑ j≥0 pj, ∑ j≥0 qj ∈ F¯σSGµm,ν, we say that ∑ j≥0 pj ∼ ∑ j≥0 qj in F¯σSGm_{ρ,δ;(µ,ν)} if there are C, c, B>0 satisfying |∂α_ξ∂βx

∑

j<N (pj−qj)(x, ξ)| ≤C|α|+|β|+2N+1α!µβ!νN!µ+ν−1hξim1−ρ1|α|+δ1|β|−kNhxim2−ρ2|β|+δ2|α|−`N for α, β∈ Nn 0, N≥1 and(x, ξ) ∈QeB2(N),B1(N).

Remark 1. If kj = (ρ1−δ1)j,`j= (ρ2−δ2)j andΛi =ρi−δi, i=1, 2, we simply write FSG_ρm_,δ;(µ,ν)and we recover the usual definitions presented under different notation in [23]. If moreover ρ= (1, 1), δ= (0, 0), we use the notation FSG_(µ,ν)m .

Remark 2. If ∑ j≥0

pj∈F¯σSG_ρm_,δ;(µ,ν), then p0∈SG_ρm_,δ;(µ,ν). Given p∈SGm_{ρ,δ;(µ,ν)}and setting p0= p, pj =0, j≥1, we have p= _∑

j≥0

pj. Hence, we can consider SGm_ρ,δ;(µ,ν)as a subset of F¯σSGmρ,δ;(µ,ν).

Proposition 1. Given ∑ j≥0

pj ∈F¯σSG_ρm_,δ;(µ,ν), there exists p∈SGm_{ρ,δ;(µ,ν)}such that p∼ ∑ j≥0 pjin F¯σSG_ρm_,δ;(µ,ν). Proposition 2. Let p∈SG(0,0) ρ,δ;(µ,ν)such that p∼0 in F¯σSG (0,0) ρ,δ;(µ,ν). Then, p∈ Sr(R 2n_{)for r≥max{}1 ˜ Λ(µ+ ν−1), µ+ν−1}, where ˜Λ=min{Λ1,Λ2}.

Proposition 3. Let p ∈ SGm_{ρ,δ;(µ,ν)} and let P∗ be the L2 _{adjoint of p(x, D). Then, there exists a} symbol p∗∈SGm_{ρ,δ;(µ,ν)} such that P∗ = p∗(x, D) +R, where R is a Sr-regularizing operator for any

r≥ µ+ν−1

min{$1−δ1,$2−δ2}. Moreover,

p∗(x, ξ) ∼

∑

j≥0|α|=j

∑

α!−1∂α_ξDα_xp(x, ξ) in FSG_ρm_,δ;(µ,ν).

Proposition 4. Let p ∈ SGm_{ρ,δ;(µ,ν)}, q ∈ SGm_{ρ,δ;(µ,ν)}0 . Then, there exists a symbol s ∈ SGm+m_{ρ,δ;(µ,ν)}0 such that p(x, D)q(x, D) =s(x, D) +R0where R0is aSr-regularizing operator for any r≥ _min{$µ+ν−1

1−δ1,$2−δ2}. Moreover, s(x, ξ) ∼

∑

j≥0|α|=j

∑

α!−1∂α_ξp(x, ξ)Dα_xq(x, ξ) in FSGm+m 0 ρ,δ;(µ,ν).

3. 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.

3.1. Amplitudes and Oscillatory Integrals

Definition 6(Amplitudes). For m ∈ R2 _{and δ ∈ [0, 1)}2_{, we defineA}m

δ(R

2n_{)as the space of all smooth} functions a(η, y)such that

|∂α_η∂βya(η, y)| ≤C_α,βhηim1+δ1|β|hyim2+δ2|α|, η, y∈ Rn.

For` ∈ N0and a∈ Am_δ(R2n), the seminorms

|a|`= max |α+β|≤`_ηsup_,y∈Rn

turnAm

δ(R

2n_{)into a Fréchet space.}

Remark 3. In [5] (Chapter 1, Section 6), the special caseA(m,τ)_(δ,0) (Rn), where m ∈ R, τ >0 and δ∈ [0, 1), is treated.

Definition 7(Oscillatory Integral). For a∈ Am

δ,τ, we define Os− [e−iηya(η, y)] =Os− Z Z e−iηya(η, y)dyd−η :=lim ε→0 Z Z e−iηyχε(η, y)a(η, y)dyd − η,

where χε(η, y) =χ(εη, εy)and χ is a Schwartz function onR2nsuch that χ(0, 0) =1.

Theorem 1. Let a∈ Am δ(R 2n_{). If`,`}0 _{∈ N} 0satisfy −2`(1−δ1) +m1< −n, −2`0(1−δ2) +m2< −n, then|hyi−2`0<sub>hD</sub> ηi2` 0

{hηi−2`hDyi2`a(η, y)}|belongs to L1(R2n)and we have

Os− [e−iηya(η, y)] =

Z Z

e−iηyhyi−2`0hDηi2`

0

{hηi−2`hDyi2`a(η, y)}dyd−η.

Furthermore, there is C`,`0 >0 independent of a∈ Am

δ,τ(R

2n_{)such that}

|Os− [e−iηya(η, y)]| ≤C_`,`0|a|_2(`+`0₎. (7)

Proof. Integration by parts gives

Os− [e−iηya] =lim ε→0 Z Z e−iηyhyi−2`0hDηi2` 0 {hηi−2`hDyi2`(aχε)}dyd − η,

where χε(η, y) =χ(εη, εy). Since

hDηi 2`0₌

_∑

`0<sub>0</sub>+|L0<sub>|=`</sub>0 `0! `₀0!L0_!D 2L0 η , hDyi 2`₌

_∑

`0+|L|=` `! `0!L!D 2L y ,

where L0 = (`<sub>1</sub>0, . . . ,`0_n)and L= (`<sub>1</sub>, . . . ,`n), we have

hDηi2` 0 {hηi−2`hDyi2`(χεa)} =

∑

`0<sub>0</sub>+|L0|=`0 `0! `0₀!L0_!D 2L0 η {hηi −2`<sub>hD</sub> yi2`(χεa)} =

∑

`<sub>0</sub>0+|L0|=`0 `0! `0₀!L0_!

∑

α1+α2=2L0 (2L0)! α1!α2!D α1 η hηi −2`<sub>·D</sub>α2 η hDyi 2`₍ χεa) =

∑

`<sub>0</sub>0+|L0<sub>|=`</sub>0 `0! `0₀!L0_!

∑

α1+α2=2L0 (2L0)! α1!α2!D α1 η hηi −2` ×

∑

`0+|L|=` `! `0!L!D α2 η D 2L y (χεa) =

∑

`<sub>0</sub>0+|L0<sub>|=`</sub>0 `0! `0₀!L0!_α

∑

1+α2=2L0 (2L0)! α1!α2! Dα1 η hηi−2`

∑

`0+|L|=` `! `₀!L! ×

∑

α0₁+α0₂=α2

∑

β1+β2=2L α2! α0₁!α₂0! (2L)! β1!β2!D α0₁ η D β1 y χεD α₂0 η D β2 y a.

Hence, we obtain the following estimates, for ε in[0, 1], |hDηi2` 0 {hηi−2`hDyi2`(χεa)}| ≤

∑

`0<sub>0</sub>+|L0|=`0 `0! `0₀!L0_!

∑

α1+α2=2L0 (2L0)! α1!α2! ×C|α1|+1 0 α1!hηi−2`−|α1|

∑

`0+|L|=` `! `0!L!_α0

∑

1+α02=α2

∑

β1+β2=2L α2! α0₁!α0₂! (2L)! β1!β2! ×ε|α 0 1+β1|_C|α 0 1|+|β1|+1 χ α 0 1!µβ1!ν|a||α0 2+β2|hηi m1+δ1|β2|_hyim2+δ2|α02| ≤C`,`0|a|_2(`+`0₎hηim−2`(1−δ1)hyim2+2` 0_δ 2_, and hyi−2`0|hDηi2` 0 {hηi−2`hDyi2`(χεa)}| ≤C`,`0|a|_2(`+`0₎hηim1−2`(1−δ1)hyim2−2` 0_(1−δ 2)_.

Finally, by Lemma 6.3 on Page 47 of [5] and Lebesgue dominated convergence theorem, we obtain

Os− [e−iηya] = Z Z e−iηyhyi−2`0hDηi2` 0 {hηi−2`hDyi2`a}dyd−η, |Os− [e−iηya]| ≤C`,`0|a|_2(`+`0₎ Z Z hηim1−2`(1−δ1)hyim2−2` 0_(1−δ 2)_dyd−_η.

Following the ideas in the proofs of Theorems 6.7 and 6.8 of [5] (Chapter 1, Section 6), one can obtain the following result.

Proposition 5. Let a∈ Am δ(R 2n_{), α, β∈ N} 0and η0, y0∈ Rn. Then, (i) Os− [e−iηyyα_{a] =Os− [(−D} η)αe−iηya] =Os− [e−iηyDαηa]; (ii) Os− [e−iηyηβa] =Os− [(−Dy)βe−iηya] =Os− [e−iηyDyβa]; and (iii) Os− [e−iηya(η, y)] =Os− [e−i(η−η0)(y−y0)a(η−η0, y−y0)].

3.2. Operators with Double Symbols

Definition 8. Let m= (m1, m2), m0 = (m01, m20) ∈ R2and ρ= (ρ1, ρ2), δ= (δ1, δ2)such that 0≤ δj < ρj ≤ 1, j = 1, 2. We denote by SGm,m

0

ρ,δ (R4n)the space of all functions p ∈ C∞(R4n) such that for any α, α0, β, β0 ∈ Nn₀there is Cα 0_,β0 α,β >0 for which |pα,α0 β,β0(x, ξ, x 0_{, ξ}0_{)| ≤C}α0,β0 α,β hξi m1−ρ1|α|_h ξ0im 0 1−ρ1|α0|_h ξ; ξ0iδ1|β+β 0_| hxim2−ρ2|β|_hx0_im20−ρ2|β0|_{hx; x}0_iδ2|α+α0| ₍₈₎

for every x, x0, ξ, ξ0 ∈ Rn_{, where p}α,α0

β,β0=∂α_ξ∂α 0 ξ0D β xDβ 0 x0p andhz; z0i =p1+ |z|2+ |z0|2for z, z0∈ Rn. Denoting by|p|m,m0

α,α0,β,β0the supremum over x, ξ, x

0_{, ξ}0_{∈ R}n_of

|pα,α0

β,β0(x, ξ, x0, ξ0)|hξi−m1+ρ1|α|hξ0i−m 0

1+ρ1|α0|_{hξ; ξ}0_i−δ1|β+β0|_hxi−m2+ρ2|β|_hx0_i−m02+ρ2|β0|_{hx; x}0_i−δ2|α+α0|_,

the space SGm,m_ρ,δ 0is a Fréchet space whose topology is defined by the family of seminorms

|p|m,m_{`</sub> 0:= sup |α+β+α0<sub>+β</sub>0<sub>|≤`}

|p|m,m0

Definition 9. Let m = (m1, m2), m0 = (m₁0, m02) ∈ R2, ρ = (ρ1, ρ2), δ = (δ1, δ2) such that 0≤δj <ρj≤1, j=1, 2, and let µ, ν ≥ 1. We denote by SGm,m

0

ρ,δ;(µ,ν)(R

4n_{) the space of all functions} p∈C∞(R4n)such that for some C>0:

|pα,α0 β,β0(x, ξ, x0, ξ0)| ≤C|α+β+α 0_+β0_| (α!α0!)µ(β!β0!)νhξim1−ρ1|α|hξ0im 0 1−ρ1|α0|_{hξ; ξ}0_iδ1|β+β0| ₍₉₎ × hxim2−ρ2|β|_hx0_im02−ρ2|β0|_{hx; x}0_iδ2|α+α0|_.

For C>0, the space SGm,m_{ρ,δ;(µ,ν)}0 (R4n_{; C)of all smooth functions p(x, ξ, x}0_{, ξ}0_{), such that (9) holds} for a fixed C>0, is a Banach space with norm

|p|m,m_C 0 := sup α,α0,β,β0∈Nn₀

C−|α+α0+β+β0|(α!α0!)−µ(β!β0!)−ν|p|m,m 0

α,α0,β,β0.

After that, we define SGm,m_{ρ,δ;(µ,ν)}0 = S

C>0

SGm,m_{ρ,δ;(µ,ν)}0 (R4n; C)as an inductive limit of Banach spaces.

Definition 10. For p∈SGm,m_ρ,δ 0, we define

p(x, Dx, x0, Dx0)u(x):= Z eiξ(x−x0)eiξ0(x0−x00)p(x, ξ, x0, ξ0)u(x00)dx00d−ξ0dx0d−ξ = Z eiξ(x−x0)eiξ0x0p(x, ξ, x0, ξ0)u_b(ξ0)d−ξ0dx0d−ξ, for every u∈S(Rn). Lemma 1. Let p ∈ SGm,m0 ρ,(0,δ2)(R

4n_{). For any multi-indices α, α}0_{, β, β}0_{, set q = p}α,α0

β,β0, and, for θ ∈ [−1, 1], define qθ(x, ξ) =Os− Z Z e−iηyq(x, ξ+θη, x+y, ξ)dyd−η, x, ξ∈ Rn. Then, {qθ}|θ|≤1 is bounded in SGτρ,(0,δ2)(R 2n_{), where τ = (} τ1, τ2), τ1 = m1+m₁0 −ρ1|α+α0|, τ2 = m2+m02−ρ2|β+β0| +δ2|α+α0|. Furthermore, for any` ∈ N0, there are`0 := `0(`) ∈ N0and C`,`0 >0 independent of θ such that

|qθ|τ` ≤C`,`0|p|m,m 0 `0 .

Proof. First, notice that q(x, ξ+θη, x+y, ξ) ∈ A(m1,m 0

2+δ2|α+α0|)

(0,δ2) (R

2n

η,y), therefore qθ(x, ξ)is well defined for any fixed ξ, x, θ. Given γ, µ∈ Nn

0, we may write, omitting the variables(x, ξ+θη, x+y, ξ),

∂γx∂µ_ξq=

∑

µ0 ≤µ γ0≤γ µ! µ0!(µ−µ0)! γ! γ0!(γ−γ0)!p (α+µ0,α0+µ−µ0) (β+γ0_,β0_+γ−γ0₎. (10)

To prove that{q_θ}_|θ|≤1is bounded in SGτ ρ,(0,δ2)(R

2n_{), it is sufficient to show that}

|q_θ(x, ξ)| ≤C1|p|m,m 0

for some C1>0 and`1∈ N0depending on α, β, α0, β0. Indeed, if (11) holds, then we can estimate the derivatives of q_θas follows: |∂γx∂µ_ξq_θ(x, ξ)| =     Os− Z Z e−iyη∂γx∂µ_ξq(x, ξ+θη, x+y, ξ)dyd−η     ≤

∑

γ0 ≤γ µ0≤µ  µ µ0  γ γ0     Os− Z Z e−iyηp(α+µ_(β+γ00,α_,β00+µ−µ_+γ−γ00)₎(x, ξ+θη, x+y, ξ)dyd−η     ≤

∑

γ0 ≤γ µ0≤µ  µ µ0  γ γ0  C1(α, α0, β, β0, γ, µ)|p|m,m 0 `<sub>1</sub> × hξim1+m 0 1−ρ1|α+µ0+α0+µ−µ0|<sub>hxi</sub>m2+m20−ρ2|β+γ0+β0+γ−γ0|+δ2|α+µ0+α0+µ−µ0| ≤C|p|m,m<sub>`</sub> 0 1 hξi m1+m01−ρ1|α+α0+µ|_hxim2+m02−ρ2|β+β0+γ|+δ2|α+α0+µ|

where C and`1depend of α, α0, β, β0, γ, µ and does not depend of θ. Now, we show that (11) holds true. Observe that

e−iηy = (1+ hxi2δ2_|η|2₎−`<sub>(1− hxi</sub>2δ2∆<sub>y)</sub>`_e−iηy_, therefore qθ(x, ξ) =Os− Z Z e−iyηrθ(x, ξ; η, y)dyd − η, where r_θ(x, ξ; η, y) = (1+ hxi2δ2_|η|2₎−`<sub>(1− hxi</sub>2δ2∆<sub>y)</sub>`_{q(x, ξ+θη, x+y, ξ).}

If we take`satisfying 2` > |m1| +n, then rθis integrable with respect to η. Now, consider a cutoff function χ(y)such that χ(y) =1 for|y| ≤4−1hxiand χ(y) =0 for|y| ≥2−1hxi. Then, we can write

Os− Z Z e−iyηr_θ(x, ξ; η, y)dyd−η=I1+I2+I3, where I1= Z Rnη Z |y|≤4−1_hxiδ2 e −iyη_r θ(x, ξ; η, y)χ(y)dyd − η, I2= Z Rnη Z

4−1_hxiδ2≤|y|≤2−1_hxie

−iyη_r

θ(x, ξ; η, y)χ(y)dyd

− η,

I3=Os− [e−iyηrθ(x, ξ; η, y)(1−χ(y))]. Let us obtain a useful inequality when|y| ≤2−1hxi. Since

|hx+yi − hxi| ≤ Z 1 0 |d dthx+tyi|dt≤ Z 1 0 |y||x+ty| hx+tyidt≤ |y|, for|y| ≤2−1hxi, we have 1 2hxi ≤ hx+yi ≤ 3 2hxi, hx; x+yi ≤ hxi + |x+y| ≤ 5 2hxi.

Now, we proceed to estimate I1, I2, I3. We begin with I1. With aid of Petree’s inequality and using the fact that ρ2>δ2, we obtain, for|y| ≤4−1hxiδ2 and|θ| ≤1,

|rθ(x, ξ; η, y)| ≤ (1+ hxi 2δ2_|η|2₎−`

∑

`0+|L|=` `! `0!L! hxi2δ2|L|_|p(α,α0) (β,β0_+2L)| ≤ (1+ hxi2δ2_|η|2₎−`_hxi2δ2|L|

∑

`0+|L|=` `! `₀!L!|p| m,m0 |α+α0_+β+β0<sub>|+2`</sub> × hξ+θηim1−ρ1|α|hξim 0 1−ρ1|α0|<sub>hxi</sub>m2−ρ2|β|<sub>hx+yi</sub>m02−ρ2|β0+2L|<sub>hx; x+yi</sub>δ2|α+α0| ≤Ck|p|m,m<sub>k</sub> 0hξiτ1hxiτ2(1+ hxi2δ2|η|2)−`hηi|m1|+ρ1|α|,

where k= |α+β+α0+β0| +2`. Therefore, for 2` > |m1| +ρ1|α| +n,

|I1| =      Z Rnη Z |y|≤4−1_hxiδ2 e −iyη_r θ(x, ξ; η, y)χ(y)dyd − η      ≤Ck|p|m,m_k 0hξiτ1hxiτ2 Z (1+ hxi2δ2_|η|2₎−`<sub>hηi</sub>|m1|+ρ1|α|<sub>d</sub>−<sub>η</sub> Z |y|≤hxi<sub>4</sub>δ2 dy ≤Ck|p|m,m<sub>k</sub> 0hξiτ1hxiτ2 Z hηi−2`hhxi−δ2ηi|m1|+ρ1|α|d−η Z |y|≤hxi₄δ2hxi −δ2n_dy ≤Ck|p|m,m_k 0hξiτ1hxiτ2 Z hηi−2`+|m1|+ρ1|α|d−η n

∏

j=1 Z |yj|≤hxi δ2 4 hxi−δ2n_dy j ≤Ck Z hηi−nd−η|p|m,m 0 k hξiτ1hxiτ2.

To estimate I2and I3, it is useful to study|∆`η1rθ|. We have

|∆`1 ηrθ| ≤

∑

|Q|=`1 `1! Q! _Q

∑

1+Q2=2Q (2Q)! Q1!Q2! |∂Qη1(1+ hxi2δ2|η|2) −`<sub>| · |(1− hxi</sub>2δ2∆<sub>y)</sub>`_∂Q2 η q| ≤

∑

|Q|=`1 `₁! Q! _Q

∑

1+Q2=2Q (2Q)! Q1!Q2! C|Q1|+`+1<sub>hxi</sub>δ2|Q1|<sub>Q</sub> 1!(1+ hxi2δ2|η|2)−`−|Q1||θ||Q2| × |(1− hxi2δ2∆_y)`<sub>p</sub>(α+Q2,α0) (β,β0<sub>)</sub> |. Noticing that |(1− hxi2δ2∆ y)`p(α+Q2,α 0₎ (β,β0₎ | ≤

∑

`0+|L|=` `! `0!L! hxi2δ2|L|_|p(α+Q2,α0) (β,β0_+2L)| ≤

∑

`0+|L|=` `! `0!L! hxi2δ2|L|_|p|m,m0 ˜k hξ+θηim1−ρ1|α+Q2 |_h ξim 0 1−ρ1|α0| × hxim2−ρ2|β|_hx+yim20−ρ2|β0+2L|_{hx; x+yi}δ2|α+α0+Q2|_, where ˜k= |α+α0+β+β0| +2(`1+ `), we obtain |∆`1 ηrθ| ≤

∑

|Q|=`1 `₁! Q! _Q

∑

1+Q2=2Q (2Q)! Q1!Q2! C|Q1|+`+1<sub>hxi</sub>δ2|Q1|<sub>Q</sub> 1! × (1+ hxi2δ2<sub>|η|</sub>2<sub>)</sub>−`−|Q1|

∑

`0+|L|=` `! `0!L! hxi2δ2|L|_|p|m,m0 ˜k hξ+θηim1−ρ1|α+Q2 | × hξim 0 1−ρ1|α0|_hxim2−ρ2|β|_hx+yim02−ρ2|β0+2L|_{hx; x+yi}δ2|α+α0+Q2|_.

Now, we proceed with the estimate for I2. If|y| ≤2−1hxi, we get

|∆`

ηrθ| ≤C

˜k+1_|p|m,m0

˜k hξiτ1hxiτ2+2δ2`(1+ hxi2δ2|η|2)−`hηi|m1|+ρ1|α|, therefore, using integration by parts and assuming 2` > |m1| +ρ1|α| +2n,

|I2| ≤C˜k+1|p|m,m 0 ˜k hξiτ1hxiτ2+δ2(2`−n) Z hxiδ2 4 ≤|y|≤ hxi 2 |y|−2`dy.

For|y| ≥4−1hxiδ2_{, we may write}

|y|−2`≤22` n

∏

j=1 (|yj| + hxiδ2 4 ) −2` n<sub>,</sub> and then Z hxiδ2 4 ≤|y|≤ hxi 2 |y|−2`dy≤22` n

∏

j=1 Z (|yj| + hxiδ2 4 ) −2` n<sub>dyj</sub>≤<sub>C</sub>`h_xiδ2(n−2`)_. After that, |I2| ≤C˜˜k+1|p|m,m 0 ˜k hξiτ1hxiτ2.

Finally, we take care of I3. If |y| ≥ 4−1hxi, we have hx+yi ≤ 5|y| and hx; x+yi ≤ 9|y|. Hence, for|y| ≥4−1hxi, we may write

|∆`1 ηrθ| ≤C ˜k+1<sub>|p|</sub>m,m0 ˜k hξiτ1hηi|m1|+ρ1|α|hxim2−ρ2|β||y||m 0 2|+δ2|α+α0|+2δ2(`+`1)

and therefore, choosing`,`_{1∈ N0}satisfying 2` > |m1| +ρ1|α| +2n and 2`1(1−δ2) ≥ |m02| +δ2|α+ α0| +2δ2` +2n, |I3| ≤C˜k+1|p|m,m 0 ˜k hξiτ1hxim2−ρ2|β|−δ2n Z hηi|m1|+ρ1|α|−2`d−η × Z |y|≥4−1_hxi|y| |m0 2|+δ2|α+α0|+2δ2`−2`1(1−δ2)_dy. Setting r=2`1(1−δ2) − |m20| −δ2|α+α0| −2δ2`, we obtain |I3| ≤C˜k+1|p|m,m 0 ˜k hξiτ1hxim2−ρ2|β| Z hηi−2nd−ηhxin(1−δ2)−r.

Choosing`₁such that r> −m2+ρ2|β0| −δ2|α+α0| +n(1−δ2), we get

|I3| ≤C˜k+1|p|m,m 0

˜k hξiτ1hxiτ2

Z

hηi−2nd−η.

Gathering all the previous computations and choosing`,`1∈ N0satisfying 2` ≥ |m1| +ρ1|α| +2n,

2`1(1−δ2) ≥2|m02| +ρ2|β0| +δ2|α+α0| +2δ2` +2n,

we have

|q_θ(x, ξ)| ≤C˜k|p|_˜km,m0hξiτ1hxiτ2,

Remark 4. Let a∈C∞(Rn_{)such that|∂}β

xa(x)| ≤Cβhxim2−|β|, for β∈ Nn0. For each fixed x, we can look at a(x+ ·)as an amplitude inA(0,|m2|) (0,0) (R 2n_{)and, for χ∈S(R}n_{), χ(0) =1,} Os− [e−iηya(x+y)] =Os− [e−iη(y−x)a(y)] =lim ε→0 Z Z

e−iη(y−x)χ(εη)χ(εy)a(y)dyd−η

=lim ε→0 Z a(y)χ(εy)ε−nF−1(χ)(ε−1(x−y))dy =lim ε→0 Z a(x−εy)χ(ε(x−εy))F−1(χ)(y)dy =a(x) Z F−1χ(y)dy=a(x). Theorem 2. Let p(x, ξ, x0, ξ0) ∈SGm,m0 ρ,(0,δ2)and set pL(x, ξ) =Os− Z Z e−iηyp(x, ξ+η, x+y, ξ)dyd−η, x, ξ∈ Rn. Then, pL ∈SGm+m 0 ρ,(0,δ2), p(x, Dx, x 0_{, D} x0) =p_L(x, D_x)and pL(x, ξ) ∼

∑

j∈N0

∑

|α|=j 1 α!(∂ α ξD α x0p)(x, ξ, x, ξ)

Furthermore, given` ∈ N0, there is`0:= `0(`) ∈ N0such that

|pL|m+m 0

` ≤C`,`0|p|

(m,m0) `0 .

Proof. First, we notice that, repeating the ideas in the proof of [5] (Lemma 2.3, Page 65), we can conclude that pL =p as operators.

Applying Lemma1for α=α0 =β0 =β=0, we obtain that pL ∈SGm+m 0

ρ,(0,δ2).

Now, by Taylor formula, we may write

p(x, ξ+η, x+y, ξ) =

∑

|α|<N ηα α!(∂ α ξp)(x, ξ, x+y, ξ) +N

∑

|γ|=N ηγ γ! Z 1 0 (1−θ) N−1₍ ∂γ_ξp)(x, ξ+θη, x+y, ξ)dθ.

Integration by parts and Remark4give

Os− [e−iηyηα(∂α_ξp)(x, ξ, x+y, ξ)] =Os− [e−iηyD_yα(∂α_ξp)(x, ξ, x+y, ξ)] = (∂α_ξD_xα0p)(x, ξ, x, ξ) and Os−[e−iηyηγ Z 1 0 (1−θ)N−1(∂γ_ξp)(x, ξ+θη, x+y, ξ)dθ] = Os− [e−iηy Z 1 0 (1−θ)N−1(∂γ_ξDγ_x0p)(x, ξ+θη, x+y, ξ)dθ]. Hence pL(x, ξ) =

∑

|α|<N 1 α!(∂ α ξD α x0p)(x, ξ, x, ξ) +rN(x, ξ),

and Lemma1implies rN∈SGm+m

0_{−N(ρ−(0,δ}

2))

ρ,(0,δ2) .

To obtain the same kind of result for the classes SGm,m_ρ,(0,δ0

2);(µ,ν), we need an analog of Lemma1

with a precise estimate of the Gevrey regularity.

Lemma 2. Let p∈SGm,m_ρ,(0,δ0

2);(µ,ν)(R

4n_{; A)for some A >0. For any multi-indices α, α}0_{, β, β}0_{set q = p}α,α0 β,β0 and for θ∈ [−1, 1], consider qθas in Lemma1. Then,

|∂σ_ξ∂γxqθ(x, ξ)| ≤ |p| m,m0 A (CA r₎k₍ α!α0!σ!)µ˜(β!β0!γ!)˜νhxiτ2−ρ2|γ|+δ2|σ|hξiτ1−ρ1|σ| (12) where k= |α+β+α0+β0+σ+γ|,µ˜ = (1+ρ_1−δ1+δ2 2 )µ+ρ1ν, ˜ν= ρ2

1−δ2µ+ν, τ1and τ2are as in Lemma1

and C, r are positive constants depending only on ρ, δ, m, m0, µ, ν and n.

Proof. Following the ideas presented in the proof of Lemma1and using standard factorial inequalities, we obtain |qθ(x, ξ)| ≤ |p| m,m0 A (CA)˜k`! 2ν<sub>`</sub> 1!2µ(α!α0!)µ(β!β0!)νhξiτ1hxiτ2,

where C>0 depends only of µ, ν, n, m1, ˜k= |α+β+α0+β0| +2(` + `1)and`,`1are positive integers satisfying 2` ≥ |m1| +ρ1|α| +2n, and 2`1(1−δ2) ≥2|m02| +ρ2|β0| +δ2|α+α0| +2δ2` +2n. In particular, if we choose ` = _|m 1| 2 + ρ1 2 |α|  +n+1, `1=  1 1−δ2  |m0<sub>2</sub>| +δ2` + ρ2 2 |β 0_{| +}δ2 2 |α+α 0_|_+n+1,

whereb·cstands for the floor function, then we obtain

|qθ(x, ξ)| ≤ |p| m,m0 A (CAr)|α+β+α 0_+β0_| α!µ˜α0!µ(1+ δ2 1−_δ2)_β!ν_β0_!˜ν_(β!β0_!)ν_hξiτ1_hxiτ2_.

From the last estimate and (10), we get (12).

As a consequence of Lemma2, we have the following result.

Theorem 3. Let p∈SGm,m_ρ,(0,δ0 2);(µ,ν)(R 4n_{). Then, p} Lbelongs to SGm+m 0 ρ,(0,δ2);( ˜µ, ˜ν)and pL(x, ξ) ∼

∑

j∈Nn 0 pj(x, ξ) in FSGm+m 0 ρ,(0,δ2);( ˜µ, ˜ν), where pj(x, ξ) =

∑

|α|=j α!−1(∂α_ξDα_x0p)(x, ξ, x, ξ)

andµ and ˜˜ ν are as in Lemma2.

Theorem 3 states that pL has a lower Gevrey regularity than p since ˜µ > µ and ˜ν > ν.

However, we observe that, if p∈SGm,m_ρ,(0,δ0

2);(µ,ν), then∑j∈N0 pj∈ FSG m+m0 ρ,(0,δ2);(µ,ν). Thus, by Proposition 1, there exists q ∈ SGm+m_ρ,(0,δ0 2);(µ,ν)such that q ∼ ∑ pj in FSG m+m0

ρ,(0,δ2);(µ,ν). On the other hand, we have

pL∼∑ pjin FSm+m 0

ρ,(0,δ2);( ˜µ, ˜ν). Hence, pL−q∼0 in FSG

m+m0

ρ,(0,δ2);( ˜µ, ˜ν), which implies that pL =q+r, where r belongs to the Gelfand–Shilov spaceSµ+ ˜˜ ν−1(R2n). This means that we can write pLas the sum of a

symbol with the same orders and regularity as p plus a remainder term which has a lower Gevrey regularity but with orders small as we want. This is a crucial in the applications to the Cauchy problem for p-evolution equations because in the energy estimates the remainder terms can be neglected and does not affect the regularity of the solution (cf. [12]).

4. The Friedrichs Part

Fix q∈C₀∞(Rn;R)supported on Q= {σ∈ Rn:|σ| ≤1}, such that q is even,R q(σ)2dσ=1 and

|∂α_σq(σ)| ≤Cq|α|+1α!s, where 1<s≤min{µ, ν}. In this section, we consider µ, ν>1.

Lemma 3. For τ, τ0∈ (0, 1), set F :R3n→ Rgiven by

F(x, ξ, ζ) =q(hxiτ0_h ξi−τ(ζ−ξ))hξi− τn 2 h_xiτ 0_n 2 _{, (13)}

for x, ξ, ζ∈ Rn_{. Then, for any α, β∈ N}n

0, we have ∂α_ξ∂βxF(x, ξ, ζ) = hxi τ0n 2 h_ξi−τn2

∑

|γ|≤|α| γ1≤γ

∑

|δ|≤|β| ψαγγ1(ξ)φβδγγ1(x) ×(hxiτ0_hξi−τ_(ζ−ξ))γ1+δ_(∂γ+δ_q)(hxiτ0_hξi−τ_(ζ−ξ)),

where ψαγγ1 and φβδγγ1 satisfy the following estimates:

|∂µ_ξψαγγ1(ξ)| ≤Cαµhξi−|α|+(1−τ)|γ−γ1|−|µ|, (14) |∂ν_xφ_βδγγ1(x)| ≤C_βνhxi−|β|+τ 0_|γ−γ 1|−|ν|_{, (15)} for every µ, ν∈ Nn 0.

The lemma can be proved by induction on|α+β|following the same argument as in the proof

of [5] [Lemma 4.1 page 129]. Observing that|γ−γ1| ≤ |γ| ≤ |α| we have ψαγγ1(ξ)φβδγγ1(x) ∈

SG(−τ|α|,−|β|+τ0|α|)(R2n). Finally, we remark that, for α=β=0, we have ψαγγ1 ≡φβδγγ1 ≡1.

Definition 11. Let p∈ SGm. Moreover, let F(x, ξ, ζ)be defined by (13) with τ = τ0 = 1₂. We define the

Friedrichs part of p by

pF(ξ, x0, ξ0) =

Z

F(x0, ξ, ζ)p(x0, ζ)F(x0, ξ0, ζ)dζ, x0, ξ, ξ0∈ Rn_.

The following properties can be proved as in [5]. We leave the details to the reader.

Proposition 6. Let p∈SGmand let pF be its Friedrichs part. For u, v∈S(Rn), the following conditions hold:

(i) If p(x, ξ)is real, then(pFu, v)L2 = (u, pFv)L2.

(ii) If p(x, ξ) ≥0, then(pFu, u)L2 ≥0.

(iii) If p(x, ξ)is purely imaginary, then(pFu, v)L2 = −(u, pFv)L2.

(iv) If Re p(x, ξ) ≥0, then Re(pFu, u)L2 ≥0.

Theorem 4. Let p ∈SGm(R2n)and let pFbe its Friedrichs part. Then, pF,L ∈SGm(R2n)and pF,L−p ∈

SGm−(1,1)(R2n). Moreover, pF,L(x, ξ) −p(x, ξ) ∼

∑

|β|=1 q0,β(x, ξ) +

∑

|α+β|≥2 q_α,β(x, ξ),

where, for|β| =1, q0,β(x, ξ) =

∑

β1+β2+β3=β Dβ3 x p(x, ξ)

∑

|γ|≤|β||δ|≤|β

∑

1| ψ_βγγ(ξ)φ_β1δγγ(x) ×

∑

|δ0_|≤|β 2| φ_β2δ0₀₀(x) Z σγ+δ+δ 0 (∂γ+δq)(σ)(∂δ 0 q)(σ)dσ,

with ψ_βγγφ_β1δγγ∈SG(−|β|,−|β1|)(R2n), φ_β2δ0₀₀ ∈SG(0,−|β2|)(R2n)and, for|α+β| ≥2:

qα,β(x, ξ) =

∑

β1+β2+β3=β (hξi12h_xi−12₎|α| α!β1!β2!β3! ×

∑

|γ|≤|β| γ1≤γ

∑

|δ|≤|β1| ψ_βγγ1(ξ)φ_β1δγγ1(x)

∑

|δ0_|≤|β 2| φ_β2δ0₀₀(x) × Z Qσ α+γ1+δ1+δ10_(∂γ+δ_q)(σ)(∂δ0_q)(σ)dσ·_∂α ξD β3 x p(x, ξ) with ψ_βγγ1φ_β1δγγ1 ∈SG(− 1 2|β|,−|β1|+12|β|)(R2n), φ β2δ000∈SG (0,−|β2|)_(R2n_).

We need the following technical lemma whose proof follows by a compactness argument.

Lemma 4. For τ∈ (0, 1), there is C>0 such that

C−1hξi ≤ hξ+ζhξiτi ≤Chξi,

for every ξ∈ Rn_and| ζ| ≤1.

Proof of Theorem4.From Leibniz formula and Cauchy–Schwartz inequality, we get

|∂α_ξ∂α 0 ξ0∂ β0 x0pF(ξ, x0, ξ0)| ≤

∑

β0₁+β0₂+β0₃=β0 β0! β0₁!β0₂!β0₃! Z |∂α_ξ∂β 0 1 x0F(x0, ξ, ζ)|2dζ 1₂ · Z |∂β 0 2 x0p(x0, ζ)∂α 0 ξ0∂ β03 x0F(x0, ξ0, ζ)|2dζ 1₂ .

Now, by changing variables, we obtain

|∂α_ξ∂α 0 ξ0∂ β0 x0pF(ξ, x0, ξ0)| ≤ hξi n 4h_x0i−n2h_ξ0in4

∑

β0₁+β0₂+β0₃=β0 β0! β0₁!β0₂!β0₃! Z Q|(∂ α ξ∂ β0₁ x0F)(x0, ξ,hx0i− 1 2h_ξi12_σ+ξ)|2_dσ 1₂ × Z Q|(∂ α0 ξ0∂ β0₂ x0F)(x0, ξ0,hx0i− 1 2h_ξ0i12_σ+_ξ0)(_∂β 0 3 x0p)(x0,hx0i− 1 2h_ξ0i12_σ+_ξ0)|2dσ 1₂ .

Applying Lemma3, we obtain |∂α_ξ∂α 0 ξ0∂ β0 x0pF(ξ, x0, ξ0)| ≤

∑

β0₁+β0₂+β0₃=β0 β0! β0₁!β0₂!β0₃!     Z

∑

|γ|≤|α| γ1≤γ

∑

|δ|≤|β0 1|   ψαγγ1(ξ)φβ0₁δγγ1(x 0₎ σγ1+δ(∂γ+δq)(σ)    2 dσ     1 2 ×     Z

∑

|γ|≤|α0 | γ1≤γ

∑

|δ|≤|β0 2|   ψα0γγ1(ξ 0₎ φ_β0 2δγγ1(x 0₎ σγ1+δ(∂γ+δq)(σ)    2  ∂ β0₃ x0p(x0,hx0i− 1 2h_ξ0i12_σ+ξ0₎    2 dσ     1 2 .

We now observe that by Lemma4

  ∂ β03 x0p(x0,hx0i− 1 2h_ξ0i12_σ+_ξ0)   ≤Cβ0₃hξ 0_im1_hx0_im2−|β03|_. Since ψαγγ1φβ0₁δγγ1 ∈SG (−|α| 2,−|β01|+ |α| 2 )_{and ψ} α0γγ1φβ0₂δγγ1 ∈SG (−|α0 | 2 ,−|β02|+ |α0 | 2 )_{we obtain} |∂α_ξ∂α 0 ξ0∂ β0 x0pF(ξ, x0, ξ0)| ≤C_αα0_β0hξi− |α| 2 h_ξ0im1−|α20 |h_x0im2−|β0|+|α+2α0 |_,

that is pF ∈SG_{(1/2,1),(0,1/2)}(0,0),(m1,m2) . Then, by Theorem2, pF,L∈SGm(1/2,1),(0,1/2)and

pF,L(x, ξ) ∼

∑

β 1 β!(∂ β ξD β x0pF)(ξ, x, ξ) =

∑

β ˜p_β(x, ξ),

which implies that pF,L− ∑ |β|<N

˜p_β ∈ SGm−

N 2(1,1)

(1/2,1),(0,1/2) for every N∈ N. To improve this result, let us study more carefully the above asymptotic expansion. Note that

˜pβ(x, ξ) =

∑

β1+β2+β3=β 1 β1!β2!β3! Z ∂β_ξDβx1F(x, ξ, ζ)Dxβ3p(x, ζ)Dβx2F(x, ξ, ζ)dζ =

∑

β1+β2+β3=β 1 β1!β2!β3!_|γ

∑

_|≤|β| γ1≤γ

∑

|δ|≤|β1| ψ_βγγ1(ξ)φ_β1δγγ1(x)

∑

|δ0_|≤|β 2| ψ_β2δ0₀₀(x) × Z Q(D β3 x p)(x,hξi 1 2h_xi−12_σ+ξ)σγ1+δ+δ0_(∂γ+δ_q)(σ)(∂δ0_q)(σ)dσ.

By Taylor formula, we can write

Dβ3 x p(x,hξi 1 2h_xi−12_{σ+ξ) =}

∑

|α|<N (hξi 1 2hxi−12)|α|_σα α! ∂ α ξD β3 x p(x, ξ) +N

∑

|α|=N hξiN2hxi−N2σα α! Z 1 0 (1−θ) N−1₍ ∂α_ξDxβ3p)(x, θhξi 1 2h_xi−12_σ+ξ)dθ.

Then, we get ˜pβ(x, ξ) =

∑

β1+β2+β3=β

∑

|α|<N (hξi 1 2hxi−12)|α| α!β1!β2!β3! ×

∑

|γ|≤|β| γ1≤γ

∑

|δ|≤|β1| ψ_βγγ1(ξ)φ_β1δγγ1(x)

∑

|δ0_|≤|β 2| φ_β2δ0₀₀(x) × Z Qσ α+γ1+δ+δ0_∂γ+δ_q( σ)∂δ 0 q(σ)dσ·∂α_ξDxβ3p(x, ξ) +rβ,N(x, ξ) where r_β,N(x, ξ) =

∑

β1+β2+β3=β

∑

|α|=N N α!· (hξi 1 2hxi− 1 2)N β1!β2!β3! |γ|≤|β|

∑

γ1≤γ

∑

|δ|≤|β1| ψ_βγγ1(ξ)φ_β1δγγ1(x)

∑

|δ0_|≤|β 2| φ_β2δ0₀₀(x) × Z Qσ γ1+δ+δ0_+α ∂γ+δq(σ)∂δ 0 q(σ) Z1 0 (1−θ)N−1(∂_ξαDβ3x p)(x, θhξi 1 2hxi− 1 2_σ+_ξ)dθdσ. (16)

Using Lemma4, we get that rβ,Nbelongs to SG(m1−

1 2(N+|β|),m2−12(|β|+N))_{, whereas} qα,β(x, ξ) =

∑

β1+β2+β3=β (hξi12hxi−12)|α| α!β1!β2!β3! ×

∑

|γ|≤|β| γ1≤γ

∑

|δ|≤|β1| ψβγγ1(ξ)φβ1δγγ1(x)

∑

|δ0_|≤|β 2| φ_β2δ0₀₀(x) × Z Qσ α+γ1+δ1+δ10_∂γ+δ_q(σ)∂δ0_q(σ)dσ·_∂α ξD β3 x p(x, ξ) belongs to SG(m1−12(|α|+|β|),m2−12(|β|+|α|))_{. Hence,}

∑

|α+β|=j qα,β∈SGm− 1 2(j,j)_.

Then, we can find a symbol t(x, ξ)such that

t(x, ξ) ∼

∑

j∈N0

∑

|α+β|=j

qα,β(x, ξ).

Since, for every N∈ N,

˜p_β(x, ξ) −

∑

|α|<N q_α,β(x, ξ) ∈SG m−1₂(N+|β|,N+|β|) (1₂,1),(0,1₂) (R 2n_),

we obtain that pF,L−t∈S(R2n), and therefore pF,L(x, ξ) ∼

∑

j∈N0

∑

|α+β|=j

qα,β(x, ξ).

To finish the proof, let us analyze more carefully the functions q_α,β(x, ξ) when|α+β| ≤ 1.

First, we notice that if α=β=0, we have q0,0(x, ξ) =p(x, ξ). If|α| =1 and β=0,

qα,0(x, ξ) = hξi |α| 2 h_xi− |α| 2 α! Z σαq(σ)2dσ=0,

because σα_q2_{(σ)is an odd function. In the case|α| =0 and|β| =1, we have} q0,β(x, ξ) =

∑

β1+β2+β3=β Dβ3 x p(x, ξ)

∑

|γ|≤|β| |γ1|≤|γ|

∑

|δ|≤|β1| ψ_βγγ1(ξ)φ_β1δγγ1(x) ×

∑

|δ0_|≤|β 2| φ_β2δ0₀₀(x) Z σγ1+δ+δ 0 (∂γ+δq)(σ)(∂δ 0 q)(σ)dσ.

If|γ1| < |γ|in the above formula, we have γ1=0 and|γ| =1, and, since q is even,

Z σδ+δ 0 (∂ej+δq)(σ)(∂δ 0 q)(σ)dσ=0, j=1, . . . , n. Therefore, q0,β(x, ξ) =

∑

β1+β2+β3=β Dβ3 x p(x, ξ)

∑

|γ|≤|β||δ|≤|β

∑

1| ψ_βγγ(ξ)φ_β1δγγ(x) ×

∑

|δ0_|≤|β 2| φ_β2δ0₀₀(x) Z σγ+δ+δ 0 (∂γ+δq)(σ)(∂δ 0 q)(σ)dσ,

and by Lemma3q0,β∈SGm−(1,1)(R2n). Hence pF,L(x, ξ) −p(x, ξ) ∼

∑

|β|=1

q0,β(x, ξ) +

∑

|α+β|≥2

qα,β(x, ξ)

and in particular that pF,L−p∈SGm−(1,1)(R2n).

Proposition6and Theorem4imply the well known sharp Gårding inequality.

Theorem 5. Let p∈SGm(R2n_{). If Re p(x, ξ) ≥0, then} Re(p(x, D)u, u)_L2 ≥ −Ckuk2

H12(m−(1,1)) u∈S(R

n_),

for some positive constant C.

Proof. Setting q= p−pF,L∈SGm−(1,1)(R2n)and recalling that pFand pF,Ldefine the same operator, we may write, by (iv) of Proposition6:

Re(p(x, D)u, u)_L2 =Re(q(x, D)u, u)_L2 +Re(pFu, u)L2 ≥Re(q(x, D)u, u)_L2.

Now, observe that for any s= (s1, s2) ∈ R2

|(q(x, D)u, u)_L2| = |(hxis2hDxis1q(x, D)u,hxi−s2hDxi−s1u)L2|

≤ kq(x, D)ukHskuk_H−s ≤Ckuk_Hs+m−(1,1)kuk_H−s.

Choosing s= 1₂[(1, 1) −m], we conclude that

Re(p(x, D)u, u)_L2 ≥ −Ckuk2

To study the Friedrichs part of symbols satisfying Gevrey estimates, we need the Faà di Bruno formula. Given smooth functions g : Rn → Rp, g = (g1, . . . , gp), f : Rp → Rand γ ∈ Nn0− {0}, we have ∂γ(f◦g)(x) =

∑

γ! k1! . . . k`! (∂k1+···+knf)(g(x)) `

∏

j=1 p

∏

i=1 " 1 δj! ∂δjgi(x) #kji , (17)

where the sum is taken over all` ∈ N, all sets{δ1, . . . , δ`}of`distinct elements ofNn0− {0}and all

(k1, . . . , k`) ∈ (N p 0− {0})`, such that γ= `

∑

s=1 |ks|δs.

It is possible to show that there is a constant C>0 such that

∑

(k1+. . .+k`)! k1! . . . k`!

≤C|γ|+1, γ∈ N − {0},

and|k1+. . .+k`|! ≤ |γ|!, where the summation and k1, . . . , k` are as in (17). For a proof of these assertions, we refer to Proposition 4.3 (Page 9), Corollary 4.5 (Page 11) and Lemma 4.8 (Page 12) of [24].

Let p ∈SGm_(µ,ν). We already know that pF ∈ SG(0,0),(m₍1 1,m2) 2,1),(0,12)

. Now, we want to obtain a precise information about the Gevrey regularity of pF. By Faà di Bruno formula,

∂βx∂α_ξq(hξi− 1 2h_xi12_(ζ−_{ξ)) =∂}β_x

∑

`,k1,...,k` α! k1! . . . k`! × (∂k1+...+k`q)(hξi− 1 2h_xi12_(ζ−_ξ)) `

∏

j=1 hxi |_kj| 2 n

∏

i=1 " 1 δj! ∂δ_ξj{hξi− 1 2_(ζi−_ξi)} #kji =

∑

`,k1,...,k` α! k1! . . . k`!_β1+β

∑

_2=β β! β1!β2! ∂βx1(∂k1+...+k`<sub>q)(hξi</sub>−12h<sub>x</sub>i12<sub>(ζ</sub>−<sub>ξ))</sub> ×∂βx2 `

∏

j=1 hxi |_kj| 2 n

∏

i=1 " 1 δj! ∂δ_ξj{hξi− 1 2_(ζi−_ξi)} #k_ji =

∑

`,k<sub>1</sub>,...,k` α! k1! . . . k`!_β1+β

∑

_2=β β! β1!β2!_`0_,k

∑

0 1,...,k0`0 β1! k0<sub>1</sub>! . . . k0<sub>`</sub>0! × (∂(k1+...+k`)+(k 0 1+...+k0`0)_q)(hξi−21h_xi12_(ζ−_ξ)) × `0

∏

j0₌₁ n

∏

i0₌₁ " 1 δ0_j0! ∂ δ0_j xhξi−12h_xi12_(ζi0−ξ_i0) #k0 j0i0 ×

∑

σ1+...+σ`=β2 β2! σ1! . . . σ`! `

∏

j=1 ∂σxjhxi |_kj| 2 n

∏

i=1 " 1 δj! ∂δ_ξj{hξi− 1 2_(ζi−_ξi)} #kji ,

hence ∂βx∂α_ξq(hξi− 1 2hxi12(_ζ−_ξ)) =

∑

`,k<sub>1</sub>,...,k` α! k1! . . . k`!_β1+β

∑

_2=β β! β1!β2! ×

∑

`0<sub>,k</sub>0 1,...,k0`0 β1! k0₁! . . . k0<sub>`</sub>0! (∂(k1+...+k`)+(k 0 1+...+k0`0)q)(h<sub>ξ</sub>i−21h<sub>x</sub>i12<sub>(ζ</sub>−<sub>ξ))</sub> × `0

∏

j0₌₁ n

∏

i0₌₁ " 1 δ0_j0! ∂ δ_j00 x hxi 1 2h_xi−12 #k0 j0i0_h hξi− 1 2h_xi12_(ζi0−ξi0) ik0 j0i0 ×

∑

σ1+...+σ`=β2 β2! σ1! . . . σ`! `

∏

j=1 ∂σxjhxi 1 2|kj| n

∏

i=1 " 1 δj! {∂δ_ξjhξi−12_(ζi−_{ξi) −δji∂}δj−ei ξ hξi −1 2} #k_ji .

Noticing thathxi12h_ξi−12|_ζ−_ξ| ≤_{1 on the support of q, we have}

|∂βx∂α_ξq(hξi− 1 2h_xi12_(ζ−_{ξ))| ≤C}˜_q,s|α+β|+1_(α!β!)sh_xi |α| 2 −|β|h_ξi− |α| 2 _.

We now apply the above inequality to estimate the derivatives of F. We have

|∂α_ξ∂βxF(x, ξ, ζ)| ≤

∑

α1+_α2=α β1+β1=β α!β! α1!β1!α2!β2! ∂α_ξ1hξi− n 4_∂β_x1h_xin4|_∂α2 ξ ∂ β2 x q(hξi− 1 2h_xi12_(ζ−_ξ))| ≤Cq,s|α+β|+1(α!β!)shξi− n 4− |α| 2 hxin4+ |α| 2−|β|_{. (18)}

Finally we proceed with the estimates for pF. Denoting

Qx,ξ = {ζ∈ Rn: hxi 1 2h_ξi−12|_ζ−_ξ| <₁}_{, x, ξ}∈ Rn_, we obtain |∂α_ξ∂α 0 ξ0∂ β0 x0pF(ξ, x0, ξ0)| ≤

∑

β1+β2+β2=β β! β1!β2!β3! " Z Qx,ξ |∂α_ξ∂β 0 1 x F(ξ, x0, ζ)|2dζ #1₂ × " Z Q_x0_,ξ0 |∂β 0 3 x p(x0, ζ)∂α 0 ξ ∂ β0₂ x F(x0, ξ0, ζ)|2dζ #1₂ ≤

∑

β1+β2+β2=β β! β1!β2!β3! C|α+α0+β01+β02|+2 q,s (α!α0!β0₁!β0₂!)shξi− |α| 2 h_ξ0i− |α0 | 2 × hxi12|α+α0|−|β01+β02| " Z Q_x0_,ξ hξi− n 2h_x0in2_dζ #1₂ · " Z Q_x0_,ξ0 hξ0i− n 2h_x0in2|_∂β 0 3 x p(x0, ζ)|2dζ #1₂ ≤

∑

β1+β2+β2=β β! β1!β2!β3! C|α+α 0_+β0 1+β02|+2 q,s (α!α0!β0₁!β0₂!)shξi− |α| 2 h_ξ0i− |α0 | 2 × hxi|α+2α0 |−|β01+β02| Z |ζ|<1dζ 1₂ Z |ζ|<1|∂ β0₃ x p(x0,hx0i− 1 2h_ξi12_ζ+ξ0_)|2_dζ 1₂ .

Using Lemma4and recalling that s≤min{µ, ν},

|∂α_ξ∂α 0 ξ0∂ β0 x0pF| ≤C|α+α 0_+β0_|+1 (α!α0!)µβ0!νhxim2−|β 0_|+1 2|α+α0|h_ξi− |α| 2 h_ξ0im1−|α20 |_, which means pF ∈SG(0,0),(m₍1 1,m2) 2,1),(0,12);(µ,ν) (R4n_).