STUDIA MATHEMATICA 167 (2) (2005)

Lemma 1.3. (1) For every λ, s, t > 0 we have 2λϕ∗ s + t 2λ  ≤λϕ∗  s λ  + λϕ∗  t λ  ≤λϕ∗ s + t

Lemma 1.4. For every n, k ∈N and t ≥1 we have (1) tk ≤enϕ∗(k/n)enω(t), (2) infj∈N0 t−jekϕ∗(j/k) ≤e−kω(t)+log t. The following result permits us to split R into intervals in which the infimum in 1.4 is attained in a finite set.

Lemma 1.5. Fix k, N ∈N and assume k N ϕ∗ N k  ≤log t < k N + 1 ϕ∗ N + 1 k  . Then (1) min0≤j≤N t−jekϕ∗(j/k) ≤e−kω(t)+log t,

Lemma 1.4. (2) We already know that t−(N−l)ekϕ∗((N−l)/k) ≤e−kω(t)+log t for some l = 0, 1, . . . , N (see 1.4). Then, using the inequality (k/l)ϕ∗(l/k) ≤log t, we obtain t−Ne2kϕ∗(N/2k) ≤t−(N−l)t−lekϕ∗((N−l)/k)ekϕ∗(l/k) ≤e−kω(t)+log t. The definition of symbol in [13, 23, 25] motivates our next definition. As we will check in 2.11, for the limit case ω(t) = log(1 + t) we recover the symbols in [13], whereas for the Gevrey weights ω(t) = td, 0 < d < 1, our definition is what can be reasonably expecte

Lemma 1.8. Let a(x, y, ξ) be an amplitude in Sm,ω ϱ,δ (Ω). Then for every compact set Q ⊂Ω× Ωthere exists a sequence Cn > 0, n ∈N, such that |Dα xDγ ya(x, y, ξ)| ≤Cne(ϱ−δ)nϕ∗(|α|/n)e(ϱ−δ)nϕ∗(|γ|/n)|ξ|δ|α+γ|emω(ξ) for every (x, y) ∈Q and |ξ| ≥1.

Proposition 1.9. Let a(x, y, ξ) be an amplitude in Sm,ω ϱ,δ (Ω) and let f ∈D(ω)(Ω). For every compact set K ⊂Ωand n, λ ∈N there is a constant

Lemma 2.1. Let a(x, y, ξ) be an amplitude in Sm,ω ϱ,δ (Ω) and let Ψ ∈ D(ω)(Rp). Then (1) K(x, y) := a(x, y, ξ)ei(x−y)ξΨ(ξ) dξ belongs to E(ω)(Ω× Ω), (2) B : D(ω)(Ω) →E(ω)(Ω), B(f)(x) := K(x, y)f(y) dy, is a continu- ous linear operator. Let Ψ ∈D(ω)(Rp) be a test function such that Ψ(ξ) = 1 for |ξ| ≤1 and Ψ(ξ) = 0 for |ξ| ≥2. We put (Aδf)(x) := a(x, y, ξ)ei(x−y)ξf(y)Ψ(δξ) dy dξ.

Theorem 2.2. Let a(x, y, ξ) be an amplitude in Sm,ω ϱ,δ (Ω). Then (1) For every f ∈D(ω)(Ω) the limit A(f) := E(ω)(Ω)-limδ→0+ Aδ(f) exists and A : D(ω)(Ω) →E(ω)(Ω) is a continuous linear operator, (2) (Af)(x) = ( a(x, y, ξ)ei(x−y)ξf(y) dy) dξ.

Theorem 2.2 is called a pseudodifferential operator of (ω)-class associated to the amplitude a(x, y, ξ). In the case a(x, y, ξ) = p(x, ξ) the pseudodifferential operator A is de- noted by P(x, D) and we have P(x, D)f = p(x, ξ)eixξ bf(ξ) dξ for every f ∈D(ω)(Ω). It is clear that the expression above makes sense for f ∈D(ω)(Rp), and even for a wider class of functions.

Proposition 2.4. The operator P(x, D) associated to a symbol p(x, ξ) in Sm,ω ϱ,δ (Ω) can be extended to DL1,(ω)(Rp) and the extension is linear and continuous taking values in E(ω)(Ω).

Theorem 2.5. The pseudodifferential operator A associated to an am- plitude a(x, y, ξ) in Sm,ω ϱ,δ (Ω) admits a continuous linear extension E′ (ω)(Ω) → D′ (ω)(Ω).

Theorem 2.2(i) gives the conclusion.

Corollary 2.6. Let A : D(ω)(Ω) →E(ω)(Ω) be the pseudodifferential operator with amplitude a(x, y, ξ). Then At|D(ω)(Ω) : D(ω)(Ω) →E(ω)(Ω) is a pseudodifferential operator with amplitude a(y, x, −ξ).

Theorem 2.7. The extension P(x, D) : E′ (ω)(Ω) →D′ (ω)(Ω) of the pseu- dodifferential operator P(x, D) is given by ⟨P(x, D)µ, ψ⟩= bµ(ξ)  eixξp(x, ξ)ψ(x) dx  dξ.

Corollary 2.8. Let p(x, ξ) and q(x, ξ) be symbols in Sm,ω ϱ,δ (Rp) defining the same pseudodifferential operator. Then p(x, ξ) = q(x, ξ).

Proposition 2.9. Let p(x, ξ) a symbol in Sm,ω ϱ,δ (Rp) and assume that P(x, D) admits a continuous linear extension A : E(ω)(Rp) →E(ω)(Rp). Then p(x, ξ) = 1 (2π)p e−ixξA(ei(·)ξ)(x).

Proposition 2.12. Let ω be a weight with ω(t) = o(td), d < 1, and let G(D) be an (ω)-ultradifferential operator with constant coefficients such that G(ξ) does not vanish on Rp. If one of the following two conditions is satisfied: (1) G(D) is elliptic, or (2) G(D) : E(ω)(Rp) →E(ω)(Rp) is surjective and {td}-hypoelliptic, then there exists a pseudodifferential operator of (ω)-class P : D(ω)(Rp) → E(ω)(Rp) such that G(D) ◦P is the identity on D(ω)(Rp).

Proposition 2.14. Let P(x, D) be the pseudodifferential operator as- sociated to p(x, ξ) ∈ASm,ω ϱ,δ (Ω). Then there is an ultradifferential oper- ator G(D) of (ω)-class and a symbol q(x, ξ) ∈ASm,ω ϱ,δ (Ω) of finite order such that if Q(x, D) is the corresponding pseudodifferential operator, then P(x, D) = Q(x, D) ◦G(D).

Proposition 2.16. Let b(y, ξ) be an amplitude in Sm,ω ϱ,δ (Ω), and let B be the associated pseudodifferential operator. Then Bf ∈DL1,(ω)(Rp) for every f ∈D(ω)(Ω), and B : D(ω)(Ω) →DL1,(ω)(Rp) is continuous.

Theorem 2.17. The (ω)-singular support of the kernel K of a pseudo- differential operator A is contained in ∆:= {(x, y) ∈Ω× Ω: x = y}.

Theorem 2.18. Let A : E′ (ω)(Ω) →D′ (ω)(Ω) be the pseudodifferential operator associated to an amplitude a(x, y, ξ) in Sm,ω ϱ,δ (Ω). Then sing(ω) supp(Aµ) ⊂sing(ω) supp(µ) for every µ ∈E′ (ω)(Ω). If a convolution operator ψ 7→ψ ∗S, ψ ∈D(ω)(Rp), S ∈D′ (ω)(Rp), is a pseudodifferential operator, by 2.18 the (ω)-singular support of S reduces

Proposition 3.5. Let A be the pseudodifferential operator defined by an amplitude a ∈ASm,ω ϱ,δ (Ω) which is equivalent to zero. Then A is an (ω)-smoothing operator.

Lemma 3.6 ([24, p. 241]). There is a sequence (Φl)l≥1 and constants C, D > 0 such that Φl ∈D(ω)(Rp), |Φl(ξ)| ≤1, Φl(ξ) = 1 for |ξ| ≤2, Φl(ξ) = 0 for |ξ| ≥3 and |Dα ξ Φl(ξ)| ≤C(D/3)|α|l|α|+1 whenever |α| ≤l. We now fix a positive constant R ≥1 and we put Ψj,n(ξ) := 1 −Φj  ξ Re(n/j)ϕ∗(j/n)  .

Theorem 3.7. Let P aj ∈FASm,ω ϱ,δ (U) and let Ωbe a relatively compact open subset of U. Then there is an amplitude a ∈ASm,ω ϱ,δ (Ω) such that a ∼P aj on Ω.

Lemma 3.8. Let a ∈Sm,ω ϱ,δ (Ω) and let A be the pseudodifferential opera- tor defined by a. Then, for every u ∈D(ω)(Ω), A(u) = ∞ X j=0 Aj(u), where Aj is the pseudodifferential operator with amplitude aj(x, y, ξ) := (ϕj −ϕj+1)(ξ)a(x, y, ξ).

Lemma 3.9. Let P∞ j=0 pj(x, ξ) be a formal sum in FASm,ω ϱ,δ (U), Ωa rela- tively compact open subset of U, and (jn) as in the proof of 3.7 and satisfying the additional assumption (n/j)ϕ∗(j/n) ≥max(n, log Cn) for j ≥jn, (Cn) being the constants of Definition 3.1 relative to the closure of Ω. Let p(x, ξ) := ∞ X j=0 ϕj(ξ)pj(x, ξ), which is a symbol in ASm,ω ϱ,δ (Ω). Then the corresponding pseudodifferential operator P(x, D) is the limit in L(D(ω)(Ω), D′ (ω)(Ω)) of the sequence of op-

Lemma 3.10. For every n ∈N we have lim N→∞ N e(n/N)ϕ∗(N/n) = 0.

Lemma 3.11. Let m ≥n and 1 e e(m/j)ϕ∗(j/m) ≤t ≤e(n/j)ϕ∗(j/n).

Lemma 3.12. Let σ(t) = td, 0 < d < 1, and let ω be a weight function such that ω(t) = o(σ(t)). Then there are λ > 0 and a sequence (jn) of natural numbers such that λσ(e(n/j)ϕ∗(j/n)) ≥j for every j ≥jn.

Theorem 3.13. Let ω be a weight such that ω(t) = o(td), d ≤ϱ −δ, d < 1. Let a ∈ASm,ω ϱ,δ (U) with associated pseudodifferential operator A and let Ωbe a relatively compact open subset of U. Then there are a pseudo- differential operator P(x, D) : D(ω)(Ω) →E(ω)(Ω) and an (ω)-smoothing operator R : E′ (ω)(Ω) →E(ω)(Ω) such that Aϕ = P(x, D)ϕ + Rϕ for every ϕ ∈D(ω)(Ω). Moreover p(x, ξ) ∼ ∞ X j=0 pj(x, ξ), where pj(x, ξ) =

Proposition 3.14. Let P(x, D) be the operator associated to p(x, ξ) ∈ ASm,ω ϱ,δ (U) and let Ωbe a relatively compact open subset of U. Then the transposed operator (restricted to D(ω)(Ω)) can be decomposed as P(x, D)t = Q(x, D) + R, where R is (ω)-smoothing and Q(x, D) is defined by a symbol q(x, ξ) ∼P qj, and we have qj(x, ξ) := P |α|=j(1/α!)∂α ξ Dα xp(x, −ξ).

Proposition 3.16. (1) ((P pj)t)t ∼P pj. (2) If P pj ∼P p′ j and P qj ∼P q′ j, then (P pj) ◦(P qj) ∼(P p′ j) ◦ (P q′ j).

Lemma 3.17. Let Ω⊂Rp be an open bounded set, and let p(x, ξ), q(x, ξ) ∈ASm,ω ϱ,δ (Ω). Assume b(x, ξ) ∈ASm,ω ϱ,δ (Ω) satisfies b(x, ξ) ∼qt(x, −ξ) and r(x, ξ) ∈AS2m,ω ϱ,δ (Ω) is equivalent to P j P |α|=j 1 α!∂α ξ Dα y (p(x, ξ)b(y, ξ))|y=x. Then r(x, ξ) ∼p(x, ξ) ◦q(x, ξ).

Theorem 3.18. Let p(x, ξ), q(x, ξ) ∈ASm,ω ϱ,δ (U) and let Ωbe an open set which is relatively compact in U. Denote by P and Q the corresponding pseu- dodifferential operators and assume that either P or Q is properly supported. Then P ◦Q : D(ω)(Ω) →E(ω)(Ω) coincides, modulo an (ω)-smoothing opera- tor, with the pseudodiferential operator associated to (2π)p(p(x, ξ) ◦q(x, ξ)).

Definition 1.1 ([8]). A weight function is an increasing continuous function ω : [0, ∞[ →[0, ∞[ with the following properties: (α) there exists L ≥0 with ω(2t) ≤L(ω(t) + 1) for all t ≥0, (β) ∞ 1 (ω(t)/t2) dt < ∞, (γ) log(t) = o(ω(t)) as t tends to ∞, (δ) ϕ : t 7→ω(et) is convex.

Definition 1.2 ([8]). Let ω be a weight function. For an open set Ω⊂ Rp we let E(ω)(Ω) := {f ∈C∞(Ω) : |f|K,λ < ∞for every λ > 0, and every K ⊂Ωcompact}, where |f|K,λ := sup x∈K sup α∈Np 0

Definition 1.6. Let Ωbe an open set in Rp, 0 ≤δ < ϱ ≤1, d := ϱ −δ and assume that ω(t) = o(td) as t →∞. An amplitude in Sm,ω ϱ,δ (Ω) is a

Definition 2.3. The operator A : D(ω)(Ω) →E(ω)(Ω) introduced in

Definition 2.10. Let Ωbe an open set in Rp, 0 ≤δ < ϱ ≤1, d := ϱ−δ and assume ω(t) = o(td) as t →∞. An amplitude in ASm,ω ϱ,δ (Ω) is a function

Definition 3.1. We denote by FASm,ω ϱ,δ (Ω) the set of all formal sums X j∈N0 aj(x, y, ξ) such that aj(x, y, ξ) ∈C∞(Ω×Ω×Rp) and for every compact set Q ⊂Ω×Ω there are R, B ≥1 and a sequence Cn > 0, n ∈N, with |Dα xDγ yDβ

Definition 3.3. Two formal sums P aj and P bj in FASm,ω ϱ,δ (Ω) are said to be equivalent if for every compact set Q ⊂Ω×Ωthere are R, B ≥1 and two sequences Cn > 0 and Nn (n ∈N) with Dα xDγ yDβ ξ X j<N

Definition 3.15. (1) For P pj ∈FASm,ω ϱ,δ (Ω) we define (P pj)t as the formal sum P j qj, where qj is as before. (2) For P pj ∈FASm1,ω ϱ,δ (Ω) and P qj ∈FASm2 ϱ,δ we define (P pj) ◦ (P qj) = P rj, where rj is as above. The following two results are straightforward, therefore we omit their