Regularity of partial differential operators in ultradifferentiable spaces and Wigner type transforms

Lemma 2.1. Let P(Dx, Dy) be a linear partial differential operator with constant coefficients. Then, for every w ∈S(R2), P(D1, D2) Wig[w](x, y) = Wig[P(D1 + D2, M2 −M1)w](x, y). (2.10)

Proposition 2.2. Let P(x, y, Dx, Dy) be a linear partial differential operator with polynomial coefficients. Then, for all w ∈S(R2), the following formula holds: P(M1, M2, D1, D2) Wig[w] = Wig  P 1 2(M2 + M1), 1 2(D1 −D2), D1 + D2, M2 −M1  w  .

Theorem 3.1. Let B(x, y, Dx, Dy) be a linear partial differential operator with polynomial coefficients and let σ = F −1(e−iP (ξ,η)) ∈S′(R2) for some P ∈R[ξ, η]. Then, for every w ∈ S(R2), the time-frequency representation Q[w] = σ ∗Wig[w] satisfies: B(M1, M2, D1, D2)Q[w] = Q[ ¯B(M1, M2, D1, D2)w], (3.13) where ¯B is the linear partial differential operator with polynomial coefficients defined by ¯B(M1, M2, D1, D2) := B M2 + M1 2 + P ∗ 1 , D1 −D2 2 + P ∗ 2 , D1 + D2, M2 −M1 

Theorem 3.2. Let B(x, y, Dx, Dy) be a linear partial differential operator with polynomial coefficients and let σ = F −1(e−iP (ξ,η)) ∈S′(R2) for some P ∈R[ξ, η]. Then, for every w ∈ S(R2), the time-frequency representation Q[w] = σ ∗Wig[w] satisfies: Q[B(M1, M2, D1, D2)w] = ˜B(M1, M2, D1, D2)Q[w], (3.18) where ˜B is the linear partial differential operator with polynomial coefficients defined by ˜B(M1, M2, D1, D2) (3.19) = B  M1 −1 2D2 −P1, M1 + 1 2D2 −P1, 1 2D1 + M2 −P2, 1 2D1 −M2 + P2

Lemma 3.3. Let ϕ ∈C∞ p (Rn). If u ∈S(Rn), then ϕu ∈S(Rn); if w ∈S′(Rn), then ϕw ∈ S′(Rn). We recall the notion of regularity from [23]: Definition 3.4. A linear operator A on S′(Rn) is regular if Au ∈S(Rn) ⇒ u ∈S(Rn), ∀u ∈S′(Rn). We have the following:

Lemma 3.5. For σ = F −1(e−iP (ξ,η)) ∈S′(R2) with P ∈R[ξ, η] and Q[w] = σ ∗Wig[w], we have that: (i) Q : S′ →S′ is invertible; (ii) Q is regular; (iii) Q : S →S.

Theorem 3.6. Let B(x, y, Dx, Dy) be a linear partial differential operator with polynomial coefficients and let σ = F −1(e−iP (ξ,η)) ∈S′(R2) for some P ∈R[ξ, η]. If B is regular and ¯B is defined by (3.14), then also ¯B is regular.

Theorem 3.7. Let B(x, y, Dx, Dy) be a linear partial differential operator with polynomial coefficients and let σ = F −1(e−iP (ξ,η)) ∈S′(R2) for some P ∈R[ξ, η]. If B is regular and ˜B is defined by (3.19), then also ˜B is regular.

Proposition 3.8. Let σ = F −1(e−iP (ξ,η)) ∈S′(R2) with P ∈R[ξ, η], σ1 = q(D1, D2)σ for a polynomial q(ξ, η) that never vanishes on R2, and set Q(σ1)[w] = σ1 ∗Wig[w]. Then: (i) Q(σ1) : S′ →S′ is invertible; (ii) Q(σ1) is regular; (iii) Q(σ1) : S →S.

Theorem 3.9. Let B(x, y, Dx, Dy) be a linear partial differential operator with polynomial coefficients. Let σ = F −1(e−iP (ξ,η)) ∈S′(R2) for some P ∈R[ξ, η] and σ1 = q(D1, D2)σ for some q ∈C[ξ, η] never vanishing on R2. Then Q(σ)[w] = σ ∗Wig[w] and Q(σ1)[w] = σ1 ∗Wig[w] satisfy, for w ∈S′(R2), Q(σ1)[Bw] = g AB Q(σ)[w], (3.28) where A is the operator defined by A(M1, M2, D1, D2) = q(D1 + D2, M2 −M1), and g AB is obtained from AB as in (3.19). Moreover, B is regular if and only if g AB is regular.

Lemma 4.5. Let ω be a weight function and D be a constant such that ω(et) ≤D (ω(t) + 1) for every t ≥0 (such constant exists from condition (α)). Fix λ, ρ > 0; then for every 0 < λ′ ≤

Lemma 4.7. Let ω be a weight function as in Definition 4.1. Then, for every λ > 0, k ∈N and t ≥1 we have: (i) tke−λω(t) ≤eλϕ∗( k λ), (ii) inf j∈N0 t−jeλϕ∗( j λ) ≤e− λ−1 b
ω(t)−a/b, where a, b are the constants of condition (γ) of Defi- nition 4.1.

Theorem 4.8. Let u ∈S(Rn) and ω a non-quasianalytic weight function. Then u ∈Sω if and only if one of the following equivalent conditions is satisfied: (1) u satisfies the conditions: (i) ∀λ > 0, α ∈Nn 0 : sup x∈Rn eλω(x)|Dαu(x)| < +∞; (ii) ∀λ > 0, α ∈Nn 0 : sup ξ∈Rn eλω(ξ)|Dαbu(ξ)| < +∞. (2) u satisfies the conditions: (i)′ ∀λ > 0, α ∈Nn 0 : sup x∈Rn eλω(x)|xαu(x)| < +∞;

Lemma 4.7(i), e−λω(ξ)+|α−γ| log |ξ| ≤e−λω(ξ)+|α| log |ξ| ≤eλϕ∗( |α| λ ). Therefore, substituting in (4.1): |xβDαu(x)| ≤C′ β,λ2|α|eλϕ∗( |α| λ ) Z e−λω(ξ)dξ (4.2) for some C′ β,λ > 0. But from Lemma 4.5 we have that for all 0 < λ′ ≤λ/D, there exists Cλ′ > 0 such that

Proposition 4.9. Let ω be as in Definition 4.1. (a) The Fourier transform is a continuous automorphism F : Sω(Rn) →Sω(Rn). It can be extended to S′ ω(Rn) in the standard way, by the formula ⟨bu, ϕ⟩= ⟨u, bϕ⟩ ∀ϕ ∈Sω. (b) Sω(Rn) is an algebra under multiplication and convolution. (c) The differentiation Dα, the multiplication by xα, for α ∈Nn 0, the multiplication by ei⟨·,a⟩ and the translation τa acting as τau(x) := u(x−a), for a ∈Rn, are continuous on Sω(Rn). (d) The following inclusions hold: D(ω)

Proposition 4.9 (e) and [8, Thm. 8.1] (cf. also [3], since we assume condition (γ) of Definition 4.1 instead of log(t) = o(ω(t)) as t →∞). Furthermore, the linear change of variable T : Sω →Sω defined in (2.3) is invertible and therefore from (2.2) we deduce that also the Wigner transform Wig : Sω −→Sω S′ ω −→S′ ω is invertible. The following lemma can be deduced as Lemma 3.3 above.

Lemma 4.11. If ϕ ∈C∞ p (Rn) and u ∈Sω then ϕu ∈Sω. If w ∈S′ ω then ϕw ∈S′ ω.

Proposition 4.12. For every non-quasianalytic weight function ω we have S′(Rn) ⊂S′ ω(Rn).

Proposition 4.14. Let σ = q(D1, D2)F −1(e−iP (ξ,η)) for some P(ξ, η) ∈R[ξ, η] and q(ξ, η) ∈ C[ξ, η] with q(ξ, η) ̸= 0 for all ξ, η ∈R. Let u ∈S′ ω for a non-quasianalytic weight function ω. Then Q[u] = σ ∗Wig[u] is well defined and satisfies: (i) Q : S′ ω →S′ ω is invertible; (ii) Q is ω-regular; (iii) Q : Sω →Sω.

Theorem 4.15. Let ω be a non-quasianalytic weight function, P(ξ, η) ∈R[ξ, η] and q(ξ, η) ∈ C[ξ, η] with q(ξ, η) ̸= 0 for all ξ, η ∈R. Let σ = F −1(e−iP (ξ,η)) ∈S′ ⊂S′ ω, σ1 = q(D1, D2)σ, Q(σ)[w] = σ ∗Wig[w] and Q(σ1)[w] = σ1 ∗Wig[w] for w ∈S′ ω. Then, if B(x, y, Dx, Dy) is a linear partial differential operator with polynomial coefficients, we have that Q(σ1)[Bw] = g ABQ(σ)[w], (4.16) where A is the operator defined by A(M1, M2, D1, D1) = q(D1 + D2, M2 −M1) and g AB is obtained from AB as in (3.19).

Proposition 4.14 (iii), we have that u = Q(σ)[w] ∈Sω and we have proved that g AB is ω- regular. Reciprocally, assuming that g AB is ω-regular, if Bu ∈Sω for some u ∈S′ ω, then Q(σ1)[Bu] ∈Sω by Proposition 4.14 (iii) and therefore g ABQ(σ)[u] = Q(σ1)[Bu] ∈Sω. By the ω-regularity of g AB we have that Q(σ)[u] ∈Sω and hence u ∈Sω by Proposition 4.14 (ii). This proves that B is ω-regular. □

Proposition 5.2. Let ω be a non-quasianalytic weight function and let B be a continuous linear operator on S′ ω(R) such that B(Sω(R)) ⊆Sω(R). Let I be the indentity operator on S′(R) and consider the operator B ˆ⊗I, interpreted as the “extension of B from one variable in R to two variables in R2”. If B ˆ⊗I is ω-regular in S′ ω(R2), then B is ω-regular and injective in S′ ω(R).

Proposition 5.2 has already been proved in [9] in the Schwartz case, i.e. ω(t) = log(1 + t). Under suitable assumptions also the converse is true in S′, as it was proved in [9, Thm. 3]. Example 5.3. As first example consider the simple cases of a multiplication operator B(x, y, Dx, Dy) = b(x, y), where b is a polynomial. Then it is easy to prove that B is regular if and only if b never vanishes. We then have from Theorems 3.7 and 4.15 (cf. also Remark 4.16) that the operator ˜B = b  M1 −1 2D2 −P

Theorem 3.7 we have that the operator ˜B =  M1 −1 2D2 −P1(D1, D2) 2 +  M2 + 1 2D1 −P2(D1, D2) 2 (5.4) is regular in Schwartz spaces, where P1 = (iD1P)(D1, D2), P2 = (iD2P)(D1, D2)

Proposition 5.2, from the ω-regularity of B = B1 ˆ⊗I.