Gevrey Vectors (2)

Gevrey vectors in locally integrable structures

Paulo Domingos CordaroJoint work with J.E. Castellanos Ramos e G. Petronilho

University of Sao Paulo

May 7, 2010

P.D.Cordaro (University of Sao Paulo) Gevrey vectors May 7, 2010 1 / 16

Introduction

P = P(x ,D) =∑|α|≤m aα(x)Dα: analytic LPDO in Ω ⊂ RN open.

Definition.

u ∈ D′(Ω) is a s-Gevrey vector for P (s ≥ 1) if Pku ∈ L1loc(Ω) for every

k = 0, 1, . . . and for every K ⊂⊂ Ω there is C = C (K ) > 0 such that

‖Pku‖L1(K) ≤ C k+1k!ms , k = 0, 1, 2, ...

G s(Ω;P).

= u ∈ D′(Ω) : u is a s-Gevrey vector for P

u ∈ L1loc(Ω) : Pu

.= f ∈ G s(Ω) ⊂ G s(Ω;P).

Introduction

Definition.

‖Pku‖L1(K) ≤ C k+1k!ms , k = 0, 1, 2, ...

G s(Ω;P).

.= f ∈ G s(Ω) ⊂ G s(Ω;P).

Introduction

Definition.

‖Pku‖L1(K) ≤ C k+1k!ms , k = 0, 1, 2, ...

G s(Ω;P).

.= f ∈ G s(Ω) ⊂ G s(Ω;P).

A bit of history

If P is elliptic in Ω and if s ≥ 1 then G s(Ω;P) ⊂ G s(Ω).Kotake–Narasimhan (1962), Lions–Magenes (1968). Microlocalversion: Bolley–Camus–Mattera (1978)

If ∃s > 1 : G s(Ω;P) ⊂ G s(Ω) then P is elliptic. Metivier (1978)

Assume P of principal type and hypoelliptic. Then P is subellipticand analytic-hypoelliptic (Treves, 1971).

If U ⊂⊂ Ω and if s ≥ 1 then G s(U;P) ⊂ G s′(U), where

s ′ = (sm − δ)/(m − δ);0 ≤ δ < 1: subellipticity index of P over U.

Baouendi–Metivier (1982) •

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A bit of history

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A bit of history

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A bit of history

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A bit of history

For M = ∂2t + t2∂2

x we have G 1(R2;M) 6⊂ Gσ(R2) if 1 ≤ σ < 2,although M is analytic-hypoelliptic in R2. Goualouic (1969)

X1, . . . ,Xp: real-valued, analytic vector fields on Ω ⊂ RN opensatisfying Hormander’s condition: there is r <∞ such that theirbrackets of length ≤ r span TxΩ, ∀x ∈ Ω.

If u ∈ C∞(Ω) is such that, for every K ⊂⊂ Ω,

‖X jνu‖L2(K) ≤ C (K )j+1j!, 1 ≤ ν ≤ p, j = 1, 2, . . .

then u ∈ Cω(Ω). Helffer-Mattera (1980)

More results in this direction: Damlakhi-Helffer (1980)

A bit of history

For M = ∂2t + t2∂2

‖X jνu‖L2(K) ≤ C (K )j+1j!, 1 ≤ ν ≤ p, j = 1, 2, . . .

A bit of history

For M = ∂2t + t2∂2

‖X jνu‖L2(K) ≤ C (K )j+1j!, 1 ≤ ν ≤ p, j = 1, 2, . . .

A bit of history

For M = ∂2t + t2∂2

‖X jνu‖L2(K) ≤ C (K )j+1j!, 1 ≤ ν ≤ p, j = 1, 2, . . .

Joint work with J.E.Castellanos Ramos and G. Petronilho

Θ: open ball centered at the origin in Rn;

Φ = (Φ1, . . . ,Φm) : Θ→ Rm, analytic near Θ, Φ(0) = 0;

Z = (Z1, . . . ,Zm) : Θ→ Cm, Z (x , t) = x + iΦ(t), defines anintegrable structure (of “tube type”) in Rm ×Θ.

L = L1, . . . , Ln, where

Lj =∂

∂tj− i

m∑k=1

∂Φk

∂tj(t)

∂xk,

spans a complex vector subbundle of C⊗ T(Rm ×Θ) of rank n,which is the orthogonal of spandZ1, . . . , dZm.[Lj , Lj ′

]= 0, j , j ′ = 1, . . . , n.

L = L1, . . . , Ln, where

Lj =∂

∂tj− i

m∑k=1

∂Φk

∂tj(t)

∂xk,

]= 0, j , j ′ = 1, . . . , n.

L = L1, . . . , Ln, where

Lj =∂

∂tj− i

m∑k=1

∂Φk

∂tj(t)

∂xk,

spans a complex vector subbundle of C⊗ T(Rm ×Θ) of rank n,which is the orthogonal of spandZ1, . . . , dZm.

[Lj , Lj ′

]= 0, j , j ′ = 1, . . . , n.

L = L1, . . . , Ln, where

Lj =∂

∂tj− i

m∑k=1

∂Φk

∂tj(t)

∂xk,

]= 0, j , j ′ = 1, . . . , n.

Gevrey vectors for L = L1, . . . , Ln

Ω ⊂ Rm ×Θ open.

Definition.

We say that u ∈ D′(Ω) is a s-Gevrey vector for L (s ≥ 1) if Lαu ∈ L∞loc(Ω)for every α ∈ Zn

+ and, for every K ⊂ Ω compact, there is C > 0 such that

‖Lαu‖L∞(K) ≤ C |α|+1α!s , α ∈ Zn+.

G s(Ω;L) = u ∈ D′(Ω) : u is a s-Gevrey vector for L

G 1(Ω;L) = analytic vectors for L on Ω

Main question: Regularity for the elements of G s(Ω;L) when L isanalytic-hypoelliptic (AHE).

Definition.

‖Lαu‖L∞(K) ≤ C |α|+1α!s , α ∈ Zn+.

Definition.

‖Lαu‖L∞(K) ≤ C |α|+1α!s , α ∈ Zn+.

Definition.

‖Lαu‖L∞(K) ≤ C |α|+1α!s , α ∈ Zn+.

First remarks

If u ∈ L∞loc(Ω) and Lju ∈ G s(Ω), j = 1, . . . , n, then u ∈ G s(Ω;L).

If u ∈ C 1(Ω) is such that

Lju = gj(x , t, u), j = 1, . . . , n,

where gj ∈ G s(Ω,O(C)), j = 1, . . . , n, then u ∈ G s(Ω;L).

If u ∈ G s(Ω;L) then WFs(u) is contained in the characteristic set ofL. In particular, G s(Ω;L) ⊂ G s(Ω) when L is elliptic and s ≥ 1.

First remarks

Lju = gj(x , t, u), j = 1, . . . , n,

First remarks

Lju = gj(x , t, u), j = 1, . . . , n,

Proposition.

If L is not elliptic and if 1 < s ≤ s ′ < 2s − 1 then there is a s-Gevreyvector for L which is not a Gevrey function of order s ′.

Can assume (Φ(t) · ξ0) = O(|t|2) for some ξ0 ∈ Rm \ 0.

If s ′ < 1/α < 2s − 1 then σ.

= s − (1− α)/(2α) > 1.

Let ζ ∈ Gσc (|t| < ρ) satisfy ζ(0) = 1 (ρ small).

u(x , t) =

∫ ∞1

e iλZ(x ,t)·ξ0−λαζ(λ(1−α)/2t

)dλ ∈ C∞(Rm+n)

(ξ0 · Dx)ku

(0, 0) =

∫ ∞1

λke−λα

dλ =⇒ u 6∈ G s′

Lγu(x , t) =

∫ ∞1

λ(1−α)|γ|/2e iλZ(x ,t)·ξ0−λα ζ(γ)(λ(1−α)/2 t) dλ.

|Lγu(x , t)| ≤ C|γ|+11 γ!σ

∫ ∞1

λ(1−α)|γ|/2e−cλα

dλ ≤ C|γ|+12 γ!σ+(1−α)/(2α)

Proposition.

If s ′ < 1/α < 2s − 1 then σ.

= s − (1− α)/(2α) > 1.

u(x , t) =

∫ ∞1

)dλ ∈ C∞(Rm+n)

(ξ0 · Dx)ku

(0, 0) =

∫ ∞1

λke−λα

dλ =⇒ u 6∈ G s′

Lγu(x , t) =

∫ ∞1

|Lγu(x , t)| ≤ C|γ|+11 γ!σ

∫ ∞1

dλ ≤ C|γ|+12 γ!σ+(1−α)/(2α)

Proposition.

If s ′ < 1/α < 2s − 1 then σ.

= s − (1− α)/(2α) > 1.

u(x , t) =

∫ ∞1

)dλ ∈ C∞(Rm+n)

(ξ0 · Dx)ku

(0, 0) =

∫ ∞1

λke−λα

dλ =⇒ u 6∈ G s′

Lγu(x , t) =

∫ ∞1

|Lγu(x , t)| ≤ C|γ|+11 γ!σ

∫ ∞1

dλ ≤ C|γ|+12 γ!σ+(1−α)/(2α)

Proposition.

If s ′ < 1/α < 2s − 1 then σ.

= s − (1− α)/(2α) > 1.

u(x , t) =

∫ ∞1

)dλ ∈ C∞(Rm+n)

(ξ0 · Dx)ku

(0, 0) =

∫ ∞1

λke−λα

dλ =⇒ u 6∈ G s′

Lγu(x , t) =

∫ ∞1

|Lγu(x , t)| ≤ C|γ|+11 γ!σ

∫ ∞1

dλ ≤ C|γ|+12 γ!σ+(1−α)/(2α)

Proposition.

If s ′ < 1/α < 2s − 1 then σ.

= s − (1− α)/(2α) > 1.

u(x , t) =

∫ ∞1

)dλ ∈ C∞(Rm+n)

(ξ0 · Dx)ku

(0, 0) =

∫ ∞1

λke−λα

dλ =⇒ u 6∈ G s′

Lγu(x , t) =

∫ ∞1

|Lγu(x , t)| ≤ C|γ|+11 γ!σ

∫ ∞1

dλ ≤ C|γ|+12 γ!σ+(1−α)/(2α)

Proposition.

If s ′ < 1/α < 2s − 1 then σ.

= s − (1− α)/(2α) > 1.

u(x , t) =

∫ ∞1

)dλ ∈ C∞(Rm+n)

(ξ0 · Dx)ku

(0, 0) =

∫ ∞1

λke−λα

dλ =⇒ u 6∈ G s′

Lγu(x , t) =

∫ ∞1

|Lγu(x , t)| ≤ C|γ|+11 γ!σ

∫ ∞1

dλ ≤ C|γ|+12 γ!σ+(1−α)/(2α)

Proposition.

If s ′ < 1/α < 2s − 1 then σ.

= s − (1− α)/(2α) > 1.

u(x , t) =

∫ ∞1

)dλ ∈ C∞(Rm+n)

(ξ0 · Dx)ku

(0, 0) =

∫ ∞1

λke−λα

dλ =⇒ u 6∈ G s′

Lγu(x , t) =

∫ ∞1

|Lγu(x , t)| ≤ C|γ|+11 γ!σ

∫ ∞1

dλ ≤ C|γ|+12 γ!σ+(1−α)/(2α)

Proposition.

If s ′ < 1/α < 2s − 1 then σ.

= s − (1− α)/(2α) > 1.

u(x , t) =

∫ ∞1

)dλ ∈ C∞(Rm+n)

(ξ0 · Dx)ku

(0, 0) =

∫ ∞1

λke−λα

dλ =⇒ u 6∈ G s′

Lγu(x , t) =

∫ ∞1

|Lγu(x , t)| ≤ C|γ|+11 γ!σ

∫ ∞1

dλ ≤ C|γ|+12 γ!σ+(1−α)/(2α)

Regularity of the analytic vectors for L

Theorem (Baouendi-Treves, 1982)

L is (AHE) if and only if the following condition holds:

(?) For each ξ ∈ Rm, ξ 6= 0, the map t 7→ Φ(t) · ξ is open

Theorem 1.

If (?) holds then G 1(Ω;L) ⊂ Cω(Ω).

Proof. If u ∈ G 1(Ω;L) then v(x , y , t) =∑

α(−i)|α|Lαu(x , t)yα/α!satisfies

(Lj − i

)v = 0 .

L1, . . . , Ln

↔ Φ(t) = (Φ(t), t1, . . . , tn).

Theorem 1.

(Lj − i

)v = 0 .

L1, . . . , Ln

↔ Φ(t) = (Φ(t), t1, . . . , tn).

Theorem 1.

(Lj − i

)v = 0 .

L1, . . . , Ln

↔ Φ(t) = (Φ(t), t1, . . . , tn).

Case s > 1: Maire’s example

m = n = 2, Φ(t1, t2) =(−3t1, t

31 (1 + t1t2)

)satisfies (?).

L1 =∂

∂t1+ i

∂x1− (3 + 4t1t2)t2

L2 =∂

∂t2− it4

L = L1, L2 is (AHE) but not C∞-hypoelliptic (Baouendi-Treves, 1981)

Theorem 2.

There is Ω ⊂ R2 open, 0 ∈ Ω, such that for every s ≥ 4 and every Ω′ ⊂ Ωopen, 0 ∈ Ω′, there is u ∈ C (Ω) such that Lju ∈ G s(Ω), j = 1, 2, butu|Ω′ 6∈ C 1(Ω′).

m = n = 2, Φ(t1, t2) =(−3t1, t

31 (1 + t1t2)

)satisfies (?).

L1 =∂

∂t1+ i

∂x1− (3 + 4t1t2)t2

L2 =∂

∂t2− it4

Theorem 2.

m = n = 2, Φ(t1, t2) =(−3t1, t

31 (1 + t1t2)

)satisfies (?).

L1 =∂

∂t1+ i

∂x1− (3 + 4t1t2)t2

L2 =∂

∂t2− it4

Theorem 2.

The case m = 1

Φ ∈ Cω(Θ;R), Φ(0) = 0, coordinates (x , t1, . . . , tn)(?) means: t 7→ Φ(t) is an open map.

Condition (?)⇒

L is analytic hypoelliptic (Baouendi-Treves, 1982)

L is C∞-hypoelliptic (Maire, 1980)

Return

We shall assume dΦ(0) = 0 (non-elliptic case)

Lojasiewicz: ∃ θ ∈]0, 1[, C > 0: |Φ(t)|θ ≤ C |dΦ(t)| near the origin;

The infimum of such values of θ is denoted by θφ and is called theLojasiewicz exponent of Φ;

We have θφ ∈ [1/2, 1[ and |Φ(t)|θΦ ≤ C |dΦ(t)| near the origin, forsome constant C > 0.

The case m = 1

Condition (?)⇒

Return

The case m = 1

Condition (?)⇒

Return

The case m = 1

Condition (?)⇒

Return

The main result

Main Theorem.

Assume that m = 1 and that L satisfies property (?).

1 The system L is G s -hypoelliptic for every s ≥ 1; Hypoellipticity

2 If s > 1 and if Ω ⊂ R×Θ is open then G s(Ω;L) ⊂ G s′(Ω), wheres ′ = s/(1− θΦ).

The main result

Main Theorem.

The main result

Main Theorem.

Proof of Main Theorem - part (2)

Ω ⊂ Θ× R open, N ∈ Z+.

C∞(Ω)(N): space of all smooth tensor fields of type (0,N) on Ω ofthe form

f =n∑

i1,...,iN=1

fi1,...,iN (x , t) dti1 ⊗ · · · ⊗ dtiN ,

where fi1,...,iN ∈ C∞(Ω).

Define IL : C∞(Ω)(N) −→ C∞(Ω)(N+1) by

ILf =n∑

i1,...,iN=1

n∑j=1

(Lj fi1,...,iN )(x , t) dti1 ⊗ · · · ⊗ dtiN ⊗ dtj .

By integrating the corresponding tensors over the integral curves of±~∇Φ/|~∇Φ| (Lojasiewicz argument) one can construct parametricesfor ILN .

f =n∑

i1,...,iN=1

fi1,...,iN (x , t) dti1 ⊗ · · · ⊗ dtiN ,

ILf =n∑

i1,...,iN=1

n∑j=1

f =n∑

i1,...,iN=1

fi1,...,iN (x , t) dti1 ⊗ · · · ⊗ dtiN ,

ILf =n∑

i1,...,iN=1

n∑j=1

Assume (?). Given

Θ′ ⊂⊂ Θ open ball centered at 0 ∈ Rn;

J ⊂ R open interval centered at 0 ∈ R;

1 < τ ≤ 1/(1− θφ) and χ ∈ G τc (J), χ ≡ 1 near the origin,

there is C > 0 such that

∣∣∣(χu)(ξ, t)∣∣∣ ≤ CN+1

|ξ|N(1−θΦ)

∑|α|=N

‖Lαu‖L∞(Θ,L1(J))+

+N!2τ−1

|ξ|N∑

|γ|≤N−1

γ!‖Lγu‖L∞(Θ,L1(J))

holds for all ξ 6= 0, N ∈ N, t ∈ Θ′ and u ∈ D′(J ×Θ) such thatLαu ∈ L∞(Θ, L1(J)) for all α.

Assume (?). Given

Θ′ ⊂⊂ Θ open ball centered at 0 ∈ Rn;

J ⊂ R open interval centered at 0 ∈ R;

1 < τ ≤ 1/(1− θφ) and χ ∈ G τc (J), χ ≡ 1 near the origin,

there is C > 0 such that

∣∣∣(χu)(ξ, t)∣∣∣ ≤ CN+1

|ξ|N(1−θΦ)

∑|α|=N

‖Lαu‖L∞(Θ,L1(J))+

+N!2τ−1

|ξ|N∑

|γ|≤N−1

γ!‖Lγu‖L∞(Θ,L1(J))

holds for all ξ 6= 0, N ∈ N, t ∈ Θ′ and u ∈ D′(J ×Θ) such thatLαu ∈ L∞(Θ, L1(J)) for all α.

Final comments (m = 1)

When n = 1 then θΦ = (k − 1)/k , where k is the order of thevanishing of the function Φ at the origin.

s ′ = s/(1− θΦ) = ks

(Baouendi-Metivier, 1982, BM ) When k is odd then δ = (k − 1)/k ,and we can take s ′ = ks − (k − 1), this result being sharp.

When n ≥ 2 condition (?) 6=⇒ L subelliptic.

Journe-Trepreau (2007):

If L satisfies (?) then given s ∈ R and u ∈ D′0 such that Lju ∈ Hs0 ,

j = 1, . . . , n, it follows that u ∈ Hs−n/20 .

Given ρ > −(n − 1)/4 there is L satisfying (?) for which there ares ∈ R and u ∈ D′0 such that Lju ∈ Hs

0 , j = 1, . . . , n, but u 6∈ Hs+ρ0 .

Derridj–Helffer (2008): classes of subelliptic systems L (n = 2)

s ′ = s/(1− θΦ) = ks

0 , j = 1, . . . , n, but u 6∈ Hs+ρ0 .

s ′ = s/(1− θΦ) = ks

0 , j = 1, . . . , n, but u 6∈ Hs+ρ0 .

s ′ = s/(1− θΦ) = ks

0 , j = 1, . . . , n, but u 6∈ Hs+ρ0 .

Derridj’s criterium =⇒ subellipticity for the system L with loss ofδ ≥ 0 derivatives (based on a geometric escape condition for thefunction Φ),

Conjecture: Derridj’s criterium ⇒ every s-Gevrey vector for L belongsto G s′ , s ′ = (s − δ)/(1− δ). In particular we would recover, whenn = 1, the Baouendi-Metivier and also obtain sharper regularity resultsfor the classes of systems L discussed in [Derridj-Helffer (2008)].

What is the best Gevrey regularity for the solutions u ∈ C 1(Ω) of

Lju = gj(x , t, u), j = 1, . . . , n,

where gj ∈ G s(Ω,O(C)), j = 1, . . . , n, s > 1, and L satisfies (?)?

Every such solution u belongs to G s′ , s ′ = s/(1− θΦ);If gj = gj(t, ζ), and the corresponding hamiltonian system is ininvolution, then every solution u belongs to G s .

Lju = gj(x , t, u), j = 1, . . . , n,

Every such solution u belongs to G s′ , s ′ = s/(1− θΦ);

If gj = gj(t, ζ), and the corresponding hamiltonian system is ininvolution, then every solution u belongs to G s .

Lju = gj(x , t, u), j = 1, . . . , n,

Gevrey Vectors (2)

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