Intro The A Priori Estimate Standard Forms Microlocal Regularity Examples

Intro The A Priori Estimate Standard Forms Microlocal Regularity Examples
Wave front set of solutions to sums of
squares of vector elds
Paolo Albano
Universita di Bologna
Paris, November 2010Intro The A Priori Estimate Standard Forms Microlocal Regularity Examples
Preliminaries
Hypoellipticity
!Let’s consider N vector elds with real analytic ( C ) coe cients
nX (x; D); j = 1;:::; N; x2 UR :j
Let
NX
2P(x; D) = X (x; D) :j
j=1
1 @pHere D = D = .xj j @x1 j
1 0We say that P is analytic [C ]-hypoelliptic in U if for every U
open subset of U, whenever
! 0 1 0Pu2 C (U )[C (U )]
! 0 1 0 0we have u2 C (U )[C (U )]. Here u2D (U).Intro The A Priori Estimate Standard Forms Microlocal Regularity Examples
Preliminaries
H ormander’s Theorem
We state H ormander’s theorem
Theorem
Assume that the Lie algebra generated by the elds X as well asj
by their commutators of length up to r has dimension n. Then L is
1C hypoelliptic.
Moreover the a priori estimate holds:
dX
2 2 2kuk + kX uk C jhPu; ui +kukj1=r
j=1
(subelliptic estimate)Intro The A Priori Estimate Standard Forms Microlocal Regularity Examples
Preliminaries
H ormander’s Theorem
We state H ormander’s theorem
Theorem
Assume that the Lie algebra generated by the elds X as well asj
by their commutators of length up to r has dimension n. Then L is
1C hypoelliptic.
Moreover the a priori estimate holds:
dX
2 2 2kuk + kX uk C jhPu; ui +kukj1=r
j=1
(subelliptic estimate)Intro The A Priori Estimate Standard Forms Microlocal Regularity Examples
The Strati cation
Poisson-Treves Strati cation
Theorem (F. Treves, 1999)
Let = Char(X ), the characteristic set of the system of the above
!vector elds. Then there is a strati cation of in C manifolds
such that
Each stratum is a real analytic manifold
The symplectic form has constant rank on each stratum
Let be one of the strata. Then all the brackets of the;kj
vector elds of length vanish on , but there is atj ;kj
least one bracket of length + 1 which is non zeroj
De nition
The length of a Poisson bracket is the number of elds forming it.Intro The A Priori Estimate Standard Forms Microlocal Regularity Examples
The Strati cation
Poisson-Treves Strati cation
Theorem (F. Treves, 1999)
Let = Char(X ), the characteristic set of the system of the above
!vector elds. Then there is a strati cation of in C manifolds
such that
Each stratum is a real analytic manifold
The symplectic form has constant rank on each stratum
Let be one of the strata. Then all the brackets of the;kj
vector elds of length vanish on , but there is atj ;kj
least one bracket of length + 1 which is non zeroj
De nition
The length of a Poisson bracket is the number of elds forming it.Intro The A Priori Estimate Standard Forms Microlocal Regularity Examples
The Strati cation
Poisson-Treves Strati cation
Theorem (F. Treves, 1999)
Let = Char(X ), the characteristic set of the system of the above
!vector elds. Then there is a strati cation of in C manifolds
such that
Each stratum is a real analytic manifold
The symplectic form has constant rank on each stratum
Let be one of the strata. Then all the brackets of the;kj
vector elds of length vanish on , but there is atj ;kj
least one bracket of length + 1 which is non zeroj
De nition
The length of a Poisson bracket is the number of elds forming it.Intro The A Priori Estimate Standard Forms Microlocal Regularity Examples
The Strati cation
Poisson-Treves Strati cation
Theorem (F. Treves, 1999)
Let = Char(X ), the characteristic set of the system of the above
!vector elds. Then there is a strati cation of in C manifolds
such that
Each stratum is a real analytic manifold
The symplectic form has constant rank on each stratum
Let be one of the strata. Then all the brackets of the;kj
vector elds of length vanish on , but there is atj ;kj
least one bracket of length + 1 which is non zeroj
De nition
The length of a Poisson bracket is the number of elds forming it.Intro The A Priori Estimate Standard Forms Microlocal Regularity Examples
FBI
The FBI Transform
We de ne the FBI transform of a temperate distribution u as
Z
2(x y) n
2Tu(x;) = e u(y)dy; x2C
nR
where 1 is a large parameter.
We remark that
T (@ u)(x;)) =@ (Tu)(x;) =@ (Tu)(x;),y x Rexj j j

@xjT (y u)(x;) = x + (Tu)(x;),j j
j = 1;:::; n, i.e. associated to T we have the (complex) canonical
transformation
(y;)7! (y i;):Intro The A Priori Estimate Standard Forms Microlocal Regularity Examples
FBI
The FBI Transform
We de ne the FBI transform of a temperate distribution u as
Z
2(x y) n
2Tu(x;) = e u(y)dy; x2C
nR
where 1 is a large parameter.
We remark that
T (@ u)(x;)) =@ (Tu)(x;) =@ (Tu)(x;),y x Rexj j j

@xjT (y u)(x;) = x + (Tu)(x;),j j
j = 1;:::; n, i.e. associated to T we have the (complex) canonical
transformation
(y;)7! (y i;):