# Gevrey Vectors (2)

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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) =||m a(x)D

: analytic LPDO in RN open.

Definition.

u D() is a s-Gevrey vector for P (s 1) if Pku L1loc() for everyk = 0, 1, . . . and for every K there is C = C (K ) > 0 such that

PkuL1(K) C k+1k!ms , k = 0, 1, 2, ...

G s(;P).

= {u D() : u is a s-Gevrey vector for P}

{u L1loc() : Pu.

= f G s()} G s(;P).

P.D.Cordaro (University of Sao Paulo) Gevrey vectors May 7, 2010 2 / 16

Introduction

P = P(x ,D) =||m a(x)D

: analytic LPDO in RN open.

Definition.

u D() is a s-Gevrey vector for P (s 1) if Pku L1loc() for everyk = 0, 1, . . . and for every K there is C = C (K ) > 0 such that

PkuL1(K) C k+1k!ms , k = 0, 1, 2, ...

G s(;P).

= {u D() : u is a s-Gevrey vector for P}

{u L1loc() : Pu.

= f G s()} G s(;P).

P.D.Cordaro (University of Sao Paulo) Gevrey vectors May 7, 2010 2 / 16

Introduction

P = P(x ,D) =||m a(x)D

: analytic LPDO in RN open.

Definition.

u D() is a s-Gevrey vector for P (s 1) if Pku L1loc() for everyk = 0, 1, . . . and for every K there is C = C (K ) > 0 such that

PkuL1(K) C k+1k!ms , k = 0, 1, 2, ...

G s(;P).

= {u D() : u is a s-Gevrey vector for P}

{u L1loc() : Pu.

= f G s()} G s(;P).

P.D.Cordaro (University of Sao Paulo) Gevrey vectors May 7, 2010 2 / 16

A bit of history

If P is elliptic in and if s 1 then G s(;P) G s().KotakeNarasimhan (1962), LionsMagenes (1968). Microlocalversion: BolleyCamusMattera (1978)

If s > 1 : G s(;P) G s() then P is elliptic. Metivier (1978)Assume P of principal type and hypoelliptic. Then P is subellipticand analytic-hypoelliptic (Treves, 1971).

If U and if s 1 then G s(U;P) G s(U), wheres = (sm )/(m );0 < 1: subellipticity index of P over U.

BaouendiMetivier (1982)

- o - o - o -

P.D.Cordaro (University of Sao Paulo) Gevrey vectors May 7, 2010 3 / 16

A bit of history

If P is elliptic in and if s 1 then G s(;P) G s().KotakeNarasimhan (1962), LionsMagenes (1968). Microlocalversion: BolleyCamusMattera (1978)

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

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

If U and if s 1 then G s(U;P) G s(U), wheres = (sm )/(m );0 < 1: subellipticity index of P over U.

BaouendiMetivier (1982)

- o - o - o -

P.D.Cordaro (University of Sao Paulo) Gevrey vectors May 7, 2010 3 / 16

A bit of history

If P is elliptic in and if s 1 then G s(;P) G s().KotakeNarasimhan (1962), LionsMagenes (1968). Microlocalversion: BolleyCamusMattera (1978)

If s > 1 : G s(;P) G s() then P is elliptic. Metivier (1978)Assume P of principal type and hypoelliptic. Then P is subellipticand analytic-hypoelliptic (Treves, 1971).

If U and if s 1 then G s(U;P) G s(U), wheres = (sm )/(m );0 < 1: subellipticity index of P over U.

BaouendiMetivier (1982)

- o - o - o -

P.D.Cordaro (University of Sao Paulo) Gevrey vectors May 7, 2010 3 / 16

A bit of history

If s > 1 : G s(;P) G s() then P is elliptic. Metivier (1978)Assume P of principal type and hypoelliptic. Then P is subellipticand analytic-hypoelliptic (Treves, 1971).

If U and if s 1 then G s(U;P) G s(U), wheres = (sm )/(m );0 < 1: subellipticity index of P over U.

BaouendiMetivier (1982)

- o - o - o -

P.D.Cordaro (University of Sao Paulo) Gevrey vectors May 7, 2010 3 / 16

A bit of history

For M = 2t + t22x we have G

1(R2;M) 6 G(R2) if 1 < 2,although M is analytic-hypoelliptic in R2. Goualouic (1969)

X1, . . . ,Xp: real-valued, analytic vector fields on RN opensatisfying Hormanders condition: there is r

A bit of history

For M = 2t + t22x we have G

1(R2;M) 6 G(R2) if 1 < 2,although M is analytic-hypoelliptic in R2. Goualouic (1969)X1, . . . ,Xp: real-valued, analytic vector fields on RN opensatisfying Hormanders condition: there is r

A bit of history

For M = 2t + t22x we have G

1(R2;M) 6 G(R2) if 1 < 2,although M is analytic-hypoelliptic in R2. Goualouic (1969)X1, . . . ,Xp: real-valued, analytic vector fields on RN opensatisfying Hormanders condition: there is r

A bit of history

For M = 2t + t22x we have G

1(R2;M) 6 G(R2) if 1 < 2,although M is analytic-hypoelliptic in R2. Goualouic (1969)X1, . . . ,Xp: real-valued, analytic vector fields on RN opensatisfying Hormanders condition: there is r

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

: open ball centered at the origin in Rn; = (1, . . . ,m) : Rm, analytic near , (0) = 0;

Z = (Z1, . . . ,Zm) : Cm, Z (x , t) = x + i(t), defines anintegrable structure (of tube type) in Rm .L = {L1, . . . , Ln}, where

Lj =

tj i

mk=1

ktj

(t)

xk,

spans a complex vector subbundle of C T(Rm ) of rank n,which is the orthogonal of span{dZ1, . . . , dZm}.[Lj , Lj

]= 0, j , j = 1, . . . , n.

P.D.Cordaro (University of Sao Paulo) Gevrey vectors May 7, 2010 5 / 16

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

: open ball centered at the origin in Rn; = (1, . . . ,m) : Rm, analytic near , (0) = 0;Z = (Z1, . . . ,Zm) : Cm, Z (x , t) = x + i(t), defines anintegrable structure (of tube type) in Rm .

L = {L1, . . . , Ln}, where

Lj =

tj i

mk=1

ktj

(t)

xk,

spans a complex vector subbundle of C T(Rm ) of rank n,which is the orthogonal of span{dZ1, . . . , dZm}.[Lj , Lj

]= 0, j , j = 1, . . . , n.

P.D.Cordaro (University of Sao Paulo) Gevrey vectors May 7, 2010 5 / 16

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

: open ball centered at the origin in Rn; = (1, . . . ,m) : Rm, analytic near , (0) = 0;Z = (Z1, . . . ,Zm) : Cm, Z (x , t) = x + i(t), defines anintegrable structure (of tube type) in Rm .L = {L1, . . . , Ln}, where

Lj =

tj i

mk=1

ktj

(t)

xk,

spans a complex vector subbundle of C T(Rm ) of rank n,which is the orthogonal of span{dZ1, . . . , dZm}.

[Lj , Lj

]= 0, j , j = 1, . . . , n.

P.D.Cordaro (University of Sao Paulo) Gevrey vectors May 7, 2010 5 / 16

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

: open ball centered at the origin in Rn; = (1, . . . ,m) : Rm, analytic near , (0) = 0;Z = (Z1, . . . ,Zm) : Cm, Z (x , t) = x + i(t), defines anintegrable structure (of tube type) in Rm .L = {L1, . . . , Ln}, where

Lj =

tj i

mk=1

ktj

(t)

xk,

spans a complex vector subbundle of C T(Rm ) of rank n,which is the orthogonal of span{dZ1, . . . , dZm}.[Lj , Lj

]= 0, j , j = 1, . . . , n.

P.D.Cordaro (University of Sao Paulo) Gevrey vectors May 7, 2010 5 / 16

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

Rm open.

Definition.

We say that u D() is a s-Gevrey vector for L (s 1) if Lu Lloc()for every Zn+ and, for every K compact, there is C > 0 such that

LuL(K) C ||+1!s , Zn+.

G s(;L) = {u D() : u is a s-Gevrey vector for L}

G 1(;L) = {analytic vectors for L on }

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

P.D.Cordaro (University of Sao Paulo) Gevrey vectors May 7, 2010 6 / 16

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

Rm open.

Definition.

We say that u D() is a s-Gevrey vector for L (s 1) if Lu Lloc()for every Zn+ and, for every K compact, there is C > 0 such that

LuL(K) C ||+1!s , Zn+.

G s(;L) = {u D() : u is a s-Gevrey vector for L}

G 1(;L) = {analytic vectors for L on }

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

P.D.Cordaro (University of Sao Paulo) Gevrey vectors May 7, 2010 6 / 16

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

Rm open.

Definition.

We say that u D() is a s-Gevrey vector for L (s 1) if Lu Lloc()for every Zn+ and, for every K compact, there is C > 0 such that

LuL(K) C ||+1!s , Zn+.

G s(;L) = {u D() : u is a s-Gevrey vector for L}

G 1(;L) = {analytic vectors for L on }

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

P.D.Cordaro (University of Sao Paulo) Gevrey vectors May 7, 2010 6 / 16

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

Rm open.

Definition.

LuL(K) C ||+1!s , Zn+.

G s(;L) = {u D() : u is a s-Gevrey vector for L}

G 1(;L) = {analytic vectors for L on }

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

P.D.Cordaro (University of Sao Paulo) Gevrey vectors May 7, 2010 6 / 16

First remarks

If u Lloc() and Lju Gs(), j = 1, . . . , n, then u G s(;L).

If u C 1() is such that

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

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