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 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

    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.

    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

  • 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,