aa r X i v : . [ m a t h . A P ] M a r SOME REMARKS ON TR`EVES’ CONJECTURE
T. DAHN
Abstract.
I will give a discussion of the conditions involved in Treves’ conjec-ture on analytic hypoellipticity. I will discuss some microlocally characteristicsets and introduce a topology of monotropic functionals as suitable for solvingthe conjecture. The pseudodifferential operator representation is inspired byCousin (cf. [5]) Introduction
Treves’ conjecture is existence of an involutive stratification equivalent with hy-poellipticity. The concept of hypoellipticity is very sensitive to change of topologybut there are geometric sets that are characteristic. We will discuss lineality and aset that relates to orthogonality. We will also consider three sets that occur in liter-ature and that we consider as not characteristic. The first is relative representationof spectral function to a hypoelliptic operator (6.6). The third (section 11) relatesto hypoelliptic operators as limits of operators dependent on a parameter . In thesecond we consider continuation of the contact transform to ( T ), which is consid-ered as a B¨acklund transform. For this continuation, algebraicity is considered tobe characteristic for hypoellipticity (10.4). It is necessary for hypoellipticity thatthe singularities have measure zero and in this study we assume parabolic singular-ities. The regular approximations are transversals and we only briefly discuss somepossible generalizations..The set of lineality is defined for a polynomial over a real (or complex) vectorspace E R (or E C ) is∆( P ) = { η ∈ E R P ( ξ + itη ) − P ( ξ ) = 0 ∀ ξ ∈ E R ∀ t ∈ R } It can be proved that ∆ is standard complexified in the topology for
Exp (cf. [12]), why it is sufficient to consider purely imaginary translations as above. The setcan be generalized to symbol classes where ∆ has a locally algebraic definition orwhere the definition is locally algebraic modulo monotropy. The pseudo differentialoperators are realized from the symbol ideals using a representation derived fromCousin.For constant coefficients polynomial differential operators, we note that the classof operators hypoelliptic in D ′ is not radical. We can prove for the radical tothe class, that the lineality is decreasing for iteration. For variable coefficientspolynomial differential operators, we consider formally hypoelliptic operators, thatis where the symbol is equivalent in strength with a constant coefficients polynomialoperators, as the variable varies. We also assume that the real part of the symbol Tove Dahn, Lund University, Sweden. is unbounded and does not change sign in the infinity, as the variable varies on aconnected set.The generalization to more general symbols will be using a lifting operator actingon a dynamical system, that maps into analytic symbols f ( ζ ) ∈ ( I )(Ω), where ( I )is an ideal over a pseudo convex domain. We will mainly discuss operators A λ onthe form A λ = P λ + H λ , where P λ is a polynomial for finite parameter values and H λ is regularizing. Proposition 1.1.
Assume S a pseudo differential operator, self-adjoint andof exponentially finite type. Assume the symbol in ( I )(Ω) , where ( I ) is finitelygenerated and Ω is pseudo convex. Assume the lineality to S , Ω is decreasing foriteration. Assume singular points are mapped on to singular points in the dynamicalsystem, with tangent determined (global pseudo base). Then, for u ∈ D ′ W F a ( Su ) = ˜Ω ∪ W F a ( u )Here ˜Ω is a set only dependent on Ω and the symbol.If Ω = lim j Ω j , where Ω j ⊂ a pseudo convex set (and Ω j algebraic), thenΩ must be an analytic set. Given that the level surfaces are of order 1, Ω has alocally algebraic definition through transversality. Conversely, if Ω has an algebraicdefinition and if we have a global pseudo base for ( I ), then regular approximationsare transversals and Ω is given by regular approximations. If Ω j are given by thelineality locally to A λ = P λ + H λ and P λ ∼ P λ , then Ω is a set of lineality for thelimit of P λ .We will use monotropic functionals to study both the symbols to hypoellipticoperators and the equations in operator space. For the representations we consider,monotropy is microlocally indifferent, that is does not influence the geometry in amicrolocally significant way. We will use the notation f ∼ m B ( R n ) and B ( R n ), we consider over an ǫ − neighborhood of thereal space, the space B m of C ∞ − functions bounded in the real infinity by a smallconstant with all derivatives. Thus, consider D α φ − µ α → α and µ α constants. Obviously, the space of monotropic functionals B ′ m ⊂ D ′ L ,why T ∈ B ′ m has representation P | α |≥ k D α f α with f α ∈ L . If T ∈ D ′ L and φ ∈ B m , there is a S ∈ B ′ m such that S = T over B m . We have that R n = ∪ ∞ j =0 K j ,for compact sets K j . Let Φ j, = ( S − T ) | K j ∈ E ′ ⊂ B ′ m and Φ j, = Φ − Φ j, .We chose S such that Φ j, = 0 for all j and Φ j, ∈ B ′ m . This gives existence ofa functional S such that S ( φ ) = P α f α ( x ) dx = lim j →∞ T j ( φ ), where the limit istaken in D ′ L Assume Ω ⊂ U ⊂ V , where U is an open set. Assume U quasi-porteur for S ∈ H ′ ( V ), that is T = t i ( u S ) for u S ∈ H ′ ( U ), where i is the restriction homomorphism.Assume r ′ T the transposed ramifier. Algebraicity for r ′ T means that we can provethat the wave front-set is defined by b Γ ([20]) in H ′ . Assume for the vorticityto the dynamical system b w changes sign finitely many times locally. On regionswhere b w has constant sign, we have isolated singularities in a sufficiently smallneighborhood. The lift function F in f ( ζ ) = F ( γ )( ζ ), can be represented by Q p F p relative a division in contingent regions. Proposition 1.2.
Assume F reduced and F T algebraically dependent on T . Then F T is not regularizing. OME REMARKS ON TR`EVES’ CONJECTURE 3
Proof:We have assumed conditions on the ramifier r ′ T such that we have existence ofconstants c , c such that c | γ |≤| r ′ T γ |≤ c | γ | as | γ |→ ∞ , that is the type | F | I = lim sup r →∞ r log | F ( γ ) | is not dependent on T in the | γ | − infinity and | F T | I = | F | I . If for this reason F is not of type −∞ , then the same holds for F T . (cid:3) . Proposition 1.3.
The condition that F ( γ T ) is analytically hypoelliptic does notimply that Re F ( γ T ) or Im F ( γ T ) is analytically hypoelliptic. Assume P T the pseudo-differential operator that corresponds to F T and that P T u = f T in H ′ ( V ), for an open set V , where we are assuming f T holomorphic,that lim T → f T = f in H ′ ( V ) and lim T → < u, P T ϕ > = < f, ϕ > , for ϕ ∈ H .We are assuming that P T maps H → H and that D ( P T ), the domain for P T , has D ( P T ) ⊂ H ( V ).1.1. Paradoxal arguments.
First note that among parametrices to partially hy-poelliptic differential operators, considered as Fredholm operators on L , there areexamples of operators with non-trivial kernels. These can be proved to be hypoel-liptic outside the kernel. If they are defined as regularizing on the kernel, they willnot be hypoelliptic there. The class of partially hypoelliptic differential operatorscan be shown to be different from the class of hypoelliptic differential operators on L . The following argument for C ∞ -hypoellipticity is based on two fairly trivialobservations,i) The Dirac measure δ is not a ( C ∞ − )hypoelliptic operator.ii) If E is a parametrix to a differential operator P such that P E − δ x ≡ V , an open set in the real space (a neighborhood of x ), then P is not a( C ∞ − )hypoelliptic operator.Proof of the observations:For the first proposition, define a convolution operator on E ′ , H ( ϕ ) = E ∗ ϕ ,where E is a fundamental solution with singularities in 0 to P ( D ) and where P ( D ) is a ( C ∞ − )hypoelliptic differential operator with constant coefficients. If δ were ( C ∞ − ) hypoelliptic over E ′ , then sing supp δ ∗ ϕ = sing supp ϕ andalso sing supp H ( t P ( D ) ϕ ) = sing supp ϕ , but since E is regularizing outside theorigin, ϕ can have singularities that H ( ϕ ) does not have.For the second proposition, we use the notation I E ( ϕ ) = R E ( x, y ) ϕ ( y ) dy and I denotes the identity operator, that is an operator such that sing supp Iu =sing supp u for all u ∈ D ′ . If P ( D ) were ( C ∞ − ) hypoelliptic, then I E − I wouldbe locally regularizing. If locally I P E = I , we also have that locally u − P u ∈ C ∞ for u ∈ D . But if P ( D ) is ( C ∞ − ) hypoelliptic, then the same must hold for P − (cid:3) The first observation can immediately be adapted to analytic hypoellipticity.For the second we note that if P is a differential operator, then P − f is the symbol to an analytically hypoellipticpseudo differential operator in the class that we are studying, we have that allapproximations f T can be chosen regular. The condition that the dependence of T is algebraic for f T is necessary to avoid a paradox in the analogue to Weyl’slemma. It is necessary to have symplecticity on each stratum. The involution is T. DAHN used to guarantee existence of an inverse lifting function, since in this case F T canbe chosen regular in T .For a symbol in B m over the real space (modulo regularizing operators), weagain consider (modulo monotropy) locally algebraic symbols. For an constantcoefficients polynomial operator a condition equivalent with hypoellipticity is thatevery distributional parametrix is very regular. These parametrices map D ′ → D ′ F ,why it is necessary for the pseudo differential operator to be hypoelliptic, that thesymbol is of real dominant type (orthogonal real and imaginary parts). The analysisis focused on the microlocal contribution from the lineality. The singular supportis considered as a formal support in a ball of ǫ − radius..Assume temporarily that the operator is not self-adjoint. Consider E , a parametrixto a homogeneously hypoelliptic, constant coefficients operator P ( D ). Then, I E − I is regularizing in D ′ F and thus is represented by a kernel in C ∞ , which has a regu-larizing action in E ′ . However, it is not trivial to extend this action to D ′ . Considerinstead C I E = I E ϕ − ϕI E , for some suitable real function in C ∞ acting on D ′ .Since C I E f = ϕf − ϕI E f + I E ϕf − ϕf f ∈ D ′ This operator will be regularizing in D ′ .2. Lineality
The lineality and the wavefront set.
The lineality Ω can be consideredas the ”boundary” to the frequency component. More precisely, assume Γ a simplecone in Ω and B Γ = lim t → A Γ , where A Γ = F − τ Γ F : H ( E R ) → H ′ ( E R ), for areal vector space E R . Assume h F the growth indicator to B Γ and that g is growthindicator for the frequency component to W F a ( u ) (cf. [12] Ch.2, Theorem 4.3).As h F = g on ∆ = Ω \
0, we see that cones in ∆ have indicator ≥
0. Let W be the convex closure of the real support to B Γ , that is W = { y < y, η > ≤ h F ( η ) | η | = 1 } . Let W + = { y ∈ W < y, η > ≥ | η | = 1 } and let W − be the complementary set. Let V + = { η < y, η > ≥ y ∈ W + } , then∆ ∩ V + = { η < y, η > = 0 y ∈ W + } . Further, since g = h F = 0 on ∆ , wemust have Σ ∩ ∆ ∩ V − = ∅ .2.2. The lineality is standard complexified.
We can show that the linealityto a polynomial, is standard complexified in
Exp , why it is sufficient to considercompletely imaginary translations of the real space. We shall now see that if wehave lineality and if the lineality is locally algebraic, there is lineality in a completedisk (cf. [8]). Assume 0 an essential singularity and ∆ simply connected and closed ∋ ′ ∪ Γ ′′ . If for a holomorphic function f , | f | is bounded on ∆and f ( z ) → w as Γ ′ ∋ z → ′′ ∋ z →
0, then f ( z ) → w uniformly as z → ′ , Γ ′′ are different, then f can not be boundedon ∆. Assume ∆ with a algebraic definition locally, then given a sector A B where f is assumed holomorphic, if f → w as z → L in this sector, the sameholds for any sector inner to A B . Thus, if we have lineality on a line OL , we havelineality on the disk. The conclusion also holds for the several dimensional set oflineality, but since hypoellipticity can be derived from one dimensional translations,we do not prove this here.2.3. Remarks on hypoellipticity and symmetry.
An operator is consideredas hypoelliptic, if its symbol is reduced in a neighborhood of the infinity, but fora holomorphic symbol it is not simultaneously reduced in a neighborhood of the
OME REMARKS ON TR`EVES’ CONJECTURE 5 origin. Note also that if f ( z ) is reduced as z → ∞ , then f ( z ) is not necessarilyreduced as z → ∞ . If f ( z ) = f ( z ), we have that f ( z ) is not necessarily reduced,as z → ∞ . This property is consequently not symmetric with respect to the realaxes. A necessary condition on a mapping c to preserve reducedness, when f ( c ( z )) = c ( f ( z )), is that it is bijective.In this context we consider the property ( P ) for a continuous function d , that is d ( T ) = d ( T ) as T → ∞ . For instance if d is the distance function to the boundary,if there is no essential singularity in the infinity and if all singularities are isolatedin the finite plane, then d is globally reduced and d has the property ( P ).2.4. The property (P).
Assume again that f = e ϕ with ϕ = e α and L ( e ϕ ) = b L ( ϕ ) = e ˜ L ( ϕ ) and if ee L ( − α ) = − ee L ( α ), we have L ( e ϕ ) = e L ( ϕ ) , which we denoteproperty log( P ). If L is algebraic, we have that it has the property log( P ). Theproperty (P) means that ˜ L ( ϕ ) + ˜ L ( ϕ ) ∼ L − = − ˜ L . If we assume ˜ L ∼ m ˜ W , where W is algebraic in the infinity, then e ˜ W ( − ϕ ) = W − ( e ϕ ). We will consider ˜ L →
0, such that we have existence of an al-gebraic morphism ˜ W such that ˜ L ∼ m ˜ W where ˜ W has the property (P). We assumeexistence of L − over an involutive set where we have a regular approximation. If ϕ is a holomorphic function with ϕ ( ζ T ) = ϕ T ( ζ ) and ζ T → ζ , as T → ∞ , then usingWeierstrass theorem, we have existence of s continuous, such that s ( ϕ T + a ) = ζ T ,for a constant a and s ( a ) = ζ . Further, s can be approximated by polynomials of1 / ( ϕ T + a ).2.5. Lineality and the characteristic set.
Treves’ conjecture is given for thecharacteristic set Σ and our argument is given for the set of lineality. We will nowargue that the conjecture can be derived from our result. Assume Σ = { ζ f ( ζ ) =0 } and Σ = Σ ∩ Σ , where Σ = { Re f ( ζ ) = 0 } . Thus, if Im f is algebraic, wehave that Σ is removable. The condition Re f ⊥ Im f is considered as necessaryfor hypoellipticity. We note in this connection the well-known Weyl’s lemma (cf.[1]), if w ∈ L ( | z | <
1) and for all V ∈ C ( | z | < w, dV ) = ( w, dV ♦ ) =0, (harmonic conjugation) then w is equivalent with a C form.Assume ( I ) = ( ker h ), where h is a homomorphism and assume existence of ahomomorphism g , such that dh ( f ) = g ( f ) dz . If g is algebraic and g − (0) = const ,we can define ∆ as semi-algebraic. Note however that existence for a global basefor g , does not imply existence of a global base for h . Let C = { f = c } and assume∆ = V \ C , where V = { f = 0 } and f = τ f − f . Let ∆ be ∆ \{ x } , where x is the intersection point. We can choose g ( f ) = 0 on C and g ( f ) = 0 on ∆.Note that if ∆ ∪ C = V , then I ( V ) = I (∆) I ( C ). Assume ∆ ∩ C = ∅ , then g ∈ I ( V ) implies g = pq , where p ∈ I (∆) and q ∈ I ( C ). Assuming C oriented,we can choose ∆ as locally algebraic p + p − q = g , where p ± , have one-sided zero-sets. If we assume C = { f = c ddT f = c ′ } and I the ideal of non-constantfunctions and N I = V ∪ V , then IN I ∼ rad I . If V ∩ V = ∅ , we canwrite g = g g ∈ rad I . Assume ρ a measure such that ρ ( I (Σ)) = ρ ( I (Σ ))and correspondingly for ρ .If Σ ∩ Σ = ∅ , then the measures can not be absolutecontinuous with respect to each other. If we instead consider two ideals of analyticfunctions I = { f dh ( f ) = 0 } and I = { f f = const. } and the correspondingmeasures ρ ( I ) = ρ ( I ), ρ ( I ) = ρ ( I ). Then if 0 = ρ ( I c ) implies ρ ( I c ) = 0, we have ρ is absolute continuous with respect to ρ . Thus, we have existence of f Baire(cf. [6]), such that ρ ( f f ) = ρ ( f ) and f ∈ L ( ρ ). T. DAHN
Proposition 2.1.
Given an analytic symbol with first surfaces C , the linealitycan be studied locally as transversals. Conversely, given the lineality and a normalmodel, the lineality approximates the first surfaces to the symbol. Existence of lineality can be seen as a proposition of possibility to continue thesymbol on a set of infinite order, that is the symbol is not reduced with respect toanalytic continuation. Assume for a measure ρ , ρ ( T ϕ ) = ρ ( ϕ ∗ ), for ϕ ∈ L , on analgebraic set and ρ ( T ϕ ) = 0 implies ρ ( ϕ ∗ ) = 0, then we have existence of ϕ Bairesuch that ρ ( T ϕ ) = ρ ( ϕ ϕ ∗ )We know that (cf. [19]) every form P j B j dx j invariant relative closed contours,has the representation R P j B j dx j = R dW + R P j c j dx j , where dW is exact andthe last integral is an absolute invariant. The argument can be repeated for ourramifier and R dV = R B ( dx T , dy ) − B ( dx, dy ) with V ( x, y ) = W ( r ′ T x, y ) − W ( x, y )and dW exact. We have assumed that the ramifier is close to translation and wehave the following explanation of this. Assume P j F j dx j invariant in the sensethat P j R F j ( r ′ T x ) dx j = R F j ( x ) dx j and assume that τ Γ is translation. Let dK T = X j (cid:2) F j ( x )( r T ζ ) − F j ( x )( ζ ) (cid:3) dx j dL T = X j (cid:2) F j ( x )( τ Γ ζ ) − F j ( x )( ζ ) (cid:3) dx j We can prove that over regular approximations, we have that R dK T + R P j C T,j dx j ∼ R dL T + R C ′ T,j dx j , for constants C T,j , C ′ T,j .3.
Involution
Introduction.
Given a multivalued surface, a canonical approximation is thespiral Puiseux approximation, but some results require a tangent determined, whywe prefer transversal approximations. Sufficient conditions for existence of transver-sals are discussed in connection with the lifting principle.We note that assuming polynomial right hand sides, for the associated dynamicalsystem, monotropy is microlocally invariant. That is since a bounded set can notcontribute as lineality, obviously ǫ translation does not affect this proposition. Inthis case monotropy (cf. [5]) correponds exactly to adding a small constant (thevalue in the origin to a polynomial) to the symbol in the infinity. For analyticright hand sides, the two monotropy concepts are no longer equivalent, but themicrolocal invariance can be proved for both separately.3.2. Tr`eves curves.
Assume <, > = Re < y ∗ T , y T > − γ T ∈ Γ. If Γ T describes a line, we have that γ ∗ T describes a line. Let Σ = { γ T ddT F T = F T } and Σ = { γ T ddT F T = F T = 0 } . Let F ′ T be the transposed operator to F T withrespect to ddT , that is F ′ T ddT = ddT F T , why on Σ, F ( ddT γ T − γ T ) = 0 ⇔ ddT ( F ′ T − F T ) = 0. Let A = T Σ = { γ T d dT F T = ddT F T } . If for every θ T ∈ T Σ, we have < ddT γ T , θ T > = 0, we have that ddT γ T ∈ bd A . Further, dF T dγ T dγ T dT = F ′ T dγ T dT over A ,why F ′ T = dF T /dγ T . Thus, for instance F ′ T ( γ ) = 0, where F T ( γ ) is constant. If F ′ T maps A ⊥ → A ⊥ over Σ, we have that < ddT F T ( γ ) , θ T > = 0, so ddT F T ( γ ) ∈ A ⊥ and < F T ( γ ) , θ T > = 0. A sufficient condition for F T to map A ⊥ → A ⊥ is that < F T ( γ ) , θ T > = ρ T < γ T , θ T > , where ρ T is a function, not involving anydifferentials (a multiplier). The proposition, is that γ ∗ T ⊥ ( bd A ) ⇒ γ ∗ T ⊥ A , which OME REMARKS ON TR`EVES’ CONJECTURE 7 can be fulfilled if A is on one side locally of a hyperplane. If the symbol ideal issymmetric and finitely generated over a pseudo convex domain, this can be assumed.Assume Φ ⊥ bd Σ = { F T ( η ) = c } , for a constant c , implies Φ ⊥{ η ≥ c } (a semialgebraic characteristic set = Σ). If < F T ( η ) , φ > = C T < η, φ > = 0. Assumefurther that F T ( η ) = c ⇔ η = c , for constants c, c , why F T maps Σ → Σ. Weknow that if η ∗ T = y ∗ T /x ∗ with x ∗ , y ∗ T polynomials and R Σ η ∗ T dxdx ∗ = 0, then Σhas measure zero. In the same manner for η T . Assume bd Σ = { the set where η changes sign } and where Σ has measure >
0. If we have existence of γ ⊥ Σholomorphic, we must γ ≡ γ ⊥ Σ with these conditions. The conclusion is thatif γ ⊥ bd Σ, we can not have, for γ algebraic, that Σ stays locally on one side of ahyperplane. More precisely, if there are 2 m characteristics through a singular point(cf. [3]), where m is referring to the order of X, Y in the associated dynamicalsystem, and if the sign is changed passing the characteristics, then the set of forinstance positive sign is not separated by a hyperplane. By giving the characteristicsa direction however, the problem can be handled. Assume Σ + = Σ ∪ Σ , the domainfor positive sign and that η is an algebraic characteristic with R Σ + ηdxdy = 0, then η = 0 either on Σ or Σ , depending on which direction η has. We can thus havehalf-characteristics η ⊥ Σ ∪ Σ with algebraic definition. F T is said to be reduced for involution, if given existence of a regular approxi-mation G T in ( I ) with ( I ) = ker H V , we have { F T , G T } = 0 on S T implies T = 0.In this case there are no level surfaces to F T on S T . Over reduced x , we havethat r ′ T x = x implies r ′ T − id is only locally algebraic. If r ′ T is algebraic in T withminimally defined singularities, then r ′ T − id ∼ a polynomial. Boundary condition 3.1.
The boundary is characterized by the condition that F T is holomorphic in T , for T / ∈ Σ or dF T dT holomorphic in T , for T / ∈ Σ , where Σ is given by locally isolated points and the regularity is close to the boundary. More precisely, let Σ = { ζ T F T = const. dF T dT = const. } and as previously( I ) = { γ T F ( γ T ) is not constant } , where F T is assumed holomorphically depen-dent on the one dimensional parameter T . Let ( I ) = (( I ) ∩ ( ddT ( I )) and N ( I ) = V ∪ V , where V = { ζ T F T is not constant } and V = { ζ T ddT F T is not constant } .Using the Nullstellensatz, we can form IN ( I ) ∼ rad ( I ) and we claim that ( I ) isradical. The condition can be generalized to higher order derivatives. Lemma 3.2.
The condition that F T is not reduced for involution means thatthere exist Tr`eves curves in S T . Proof:Assume for this reason that T = 0 and that there exist γ T ⊂ S T such that { G T , F T } = 0 over γ T , where F T is a lifting function and G T is a regular ap-proximation of a singular point. Assume G T ( γ ) = G ( γ T ) and ddT G ( γ T ) = G ( dγ T dT )with G invertible over dγ T dT . Assume existence of v T , a regular approximation of T Σ with dv T dT = G dγ T dT , then < dv T dT , θ T > = 0 and if G : A ⊥ → A ⊥ (independenceof T at the boundary), we see that there exist Tr`eves curves for F T in S T . (cid:3) First surfaces.
Consider the system dxX = dyY = dt and the correspond-ing variation equations dx ∗ dt = dXdx x ∗ + dXdy y ∗ and dy ∗ dt = dYdx x ∗ + dYdy y ∗ . Assume F T ( x, y, x ∗ , y ∗ ) a first integral to the variation equations, algebraic in x, y and ho-mogeneous of order 1 in x ∗ , y ∗ . It is well known that R F T ( dx, dy ) is invariantintegral to the given system. Conversely, if R F T ( dx, dy ) is invariant integral to the T. DAHN system, then F T ( x ∗ , y ∗ ) is integral to the variation equation. Assume V the Hamil-ton function to the system, that is dxdt = dVdx ∗ , dx ∗ dt = − dVdx and dydt = dVdy ∗ , dy ∗ dt = − dVdy .Then given that { V, V } = 0, also V is a Hamilton function. If V is a Hamiltonfunction, we have that { V, V } = 0 and { V , V } = 0. We will consider an involutiveset S T such that for F T a lifting function and V a Hamilton function, { V, F T } = 0over S T . One of the most important problems in this approach seems to be exis-tence of an inverse for F T . A sufficient condition is reducedness, but this is notsuitable in connection with invariant integrals.Assume V a Hamilton function and F T a lifting function to the system { γ T } corresponding to the symbol. Further that G T is a regular approximation (withrespect to ddT ) to the singularity in { γ T } , not necessarily a lifting function. As { V, ·} = H V defines an ideal ( I ), we note that if F T ∈ ( I ) = ( I )( S T ) with(1) dF T dx ∗ = dVdx ∗ and dF T dy ∗ = dVdy ∗ and if G T ∈ ( I ), we have that dF T dT = dG T dT . We see that F T is regular under theseconditions. The proposition is that existence of G T regular in ( I )( S T ) and S T involutive means that the lifting function with (1) is regular.3.4. Continuation of the representation.
Assume W ⊂ V ⊂ V ′ and Λ complexvarieties and consider the mapping r ⊥ : V ⊥ → Λ ⊥ with ker r ⊥ = W ⊥ . We thenhave, given T ∈ H ′ ( Λ ), there is a U ∈ H ′ ( V ′ ) with F ( U ) = F ( T ) if and only if F ( U ) is constant on W ⊥ . Particularly, if F ( T ) has isolated singularities in theinfinity, there is a continuation principle through the projection method.Given a finitely generated system with polynomial right hand sides P, Q . If theconstant surface corresponding to
P/Q is = { } , then L T is reduced with respectto contraction, that is − dL T dy / dL T dx = dxdy ⇒ T = 0. Particularly, consider F over γ with right hand sides P, Q and df T = dF ( ζ + T ) − dF ( ζ ) for ζ ∈ Ω. Then, over the lineality for
Q/P , df T dx / df T dy = dy/dx = Q/P . Inthe same manner, if G is a different form to the same system and dg T as above, if { f T , g T } = 0 over V , then df T = dg T over V . If further { f T , g T } ∈ I ( V ), we have df T = dg T + c T on V , for a constant c T . Note that there is an ideal J such that radJ ∼ m I ( V ). If f Γ = F ( x + i Γ , y ) − F ( x, y + i Γ) and g Γ = f ∗ Γ , we have for aninvolutive set, that df Γ = dg Γ . If we have f Γ = g Γ in L ∩ H we know that g Γ = t f Γ .Obviously, H P defines a functional in H ′ . If R Σ H P ( f T ) dxdy = 0, we have either H P ( f T ) ≡ R Σ dxdy = 0.3.5. Continuous ramification.
We are assuming the ramifier defines a regularcovering ([2]), that is we are assuming Ψ : ( I )(Ω) → ( I T )(Ω), where the first is aHausdorff space, Ψ is continuous, proper and almost injective (singular points aremapped on to a discrete set (subset of transversals) in Ω). We write r ′ T for Ψ, N :( I )(Ω) → Ω and the ramifier is the lift Ω → ( I T )(Ω), such that r ′ T I (Ω) = I ( r T Ω).Denote the critical points to r T with A and assume that they are parabolic. Weassume Ψ such that I ( r T A ) is nowhere dense in ( I )(Ω) and so that Ψ is locally ahomeomorphism outside critical points. Finally, we are assuming that for all γ ∈ ( I )(Ω), there is a small neighborhood U γ , open and arc-wise connected, such thatthe U γ − I ( r T A ) is arc-wise connected. Wherever Ψ is holomorphically dependenton the parameter, the inverse r T will be assumed continuous outside a discrete set. OME REMARKS ON TR`EVES’ CONJECTURE 9
If for instance A = { ζ d ζ f ( ζ ) = f ( ζ ) = 0 } , we are studying points ζ T that can beused to reach A from { f = c } , for a constant c .Assume Ω ⊂ Ω, where Ω is assumed a pseudo convex domain. Assume U an open set such that Ω ⊂ U ⊂ Ω. Assume T (= B Γ ) an analytic functional, T ∈ H ′ (Ω), quasi portable by Ω , that is we have existence of u ∈ H ′ ( U ) with T = i U,V ( u ) (restriction homomorphism). Let Ω ⊂ ∪ Nj =1 U j , for open sets U j ⊂ Ωand T = P Nj =1 T j , that is we can write T j = i U j , Ω ( u j ) with u j ∈ H ′ ( U j ). Assumenow the restriction homomorphism algebraic, then we have if Ω is complex analyticin a real analytic vector space, that T is portable by Ω (cf. [12] Ch.2, Section 2).Assume h algebraic and let v T ( x ) = h ( r ′ T x ) − h ( x ), where r ′ T is a continuouslinear mapping d ( r ′ T f ) /dx = df /dx and we write f ( r T ζ ) = r ′ T f ( ζ ). Let ∆ = { T f ( r T ζ ) = f ( ζ ) ∀ ζ } . Over an ideal, finitely generated and of Schwartz typetopology with (weakly) compact translation (cf. [12]), there are given T ∈ ∆, T j regular such that T j → T and f ( r T j ) ζ ) = f ( ζ ) + C T j , for constants 0 = C T j → j → ∞ . The sets { v T = 0 } will not contribute micro locally, however the sets { v T = const } contributes to invariance in the tangent space and gives a micro localcontribution.Assume L an analytic line, transversal in a first surface S through p and con-sider a neighborhood Γ of p on L . Denote Σ Γ the set of points that can be joinedwith a point in Γ, through a first surface to f . We assume L transversal to everyfirst surface through Γ of order 1. Transversality means existence of regular ap-proximations. We will in this approach not assume minimally defined singularities.If for a first surface S , we have S ∩ Σ Γ = ∅ , we have S ′ ⊂ Σ Γ , for all S ′ ∼ S (conjugated in the sense of [15]). Thus for a generalization of the inhomogeneousHange’s result, it is sufficient to consider the normal tube. That is if Γ gives amicro local contribution in p , then if S ⊥ (transversal) has S ⊥ ∩ Σ ( u ) = ∅ , wehave S ∩ Σ Γ = ∅ implies S ⊂ Σ Γ , so that Γ ∈ S ⊥ . Note however that it is necessaryfor micro local contribution, that the set is not bounded globally.3.6. The condition on involution.
First a few notes on the lifting principle.Assume γ ∈ P , an analytic polyeder. It is not true that the lifting principle holdsover every P , but by constructing a normal model Σ ([16]) to P , we have always(modulo monotropy) a lifting function. Let Ω = { ζ r ′ T γ ( ζ ) ∈ Σ γ ∈ P} , bythe definition of the ramifier r T , Ω = { r T ζ γ ( r T ζ ) ∈ Σ γ ∈ P} . We assume γ T (real-) analytic on V × Ω ∋ ( T, ζ ). For ζ fix in a neighborhood defined by T , F can be chosen holomorphic. Let P = { γ ( ζ ) γ holomorphic in ζ ∈ Ω } . Then,if we assume P finitely generated over Ω and r ′ T P = Σ, we get a corresponding˜Ω = { r T Ω } and f ( r T ζ ) = F ( γ T )( ζ ) for ζ fix, can be extended to the domain for f , in a neighborhood of a first surface. Thus, the construction is such that Ω is aneighborhood of { T = 0 } and ζ in a first surface, why we have existence locally ofa lifting function for a normal model.The condition on involution gives existence of the inverse lifting function G T = F − T . We are now interested in determining the domain where G T is constant,algebraic, holomorphic etc. Note that if G T ( f ) is algebraic in f and f is the symbolto a hypoelliptic operator, then in the real space, G T ( f ) ∈ B m . Assume existenceof G ′ , derivative with respect to argument, then from the regularity conditions forthe dynamical system, G T ( f ) has isolated singularities and if G ′ holomorphic orconstant, we must have isolated singularities for the symbol f T . Consider ( I ) + = { γ T F ( γ T ) = F ( γ ) Im T > } and correspondingly ( I ) − . Assume F T algebraicin T , then the signs will give an orientation to the first surfaces. Thus, ( I ) + will correspond to conjugate classes of first surfaces ([14]). For instance in case F ( γ T ) = F ( γ T ), we have the same first surface in ( I ) ± but different orientations.We will assume the number of classes constant, when Im T is small (comparewith the regularity conditions ([3])). The regularity for G T will now determine thecharacter of the first surfaces. Regular first surfaces, for instance have only trivialconjugates, which will be the case if G T is reduced. We have noted that all normalapproximations can be chosen regular.Consider the symbol F = P F with P a polynomial, F = b f and F holomorphicor monotropic with a holomorphic function. Let( I ) Ω = { f ∈ ( B m ) ′ P b f = 0 on Ω } If Λ = Z P (zero-set), we have that F ⊥ Λ implies f ∈ ( I ) Λ . Conversely, if thepolynomial P is reduced and | P F | < ǫ at the boundary for a small number ǫ ,then the Nullstellensatz ([17]) gives that F is bounded by a small number at theboundary.3.7. The lifting principle.
Assume the right hand sides to the associated dynam-ical system
X, Y are polynomials in ζ , then according to the lifting principle (cf.[17]), we have on | X |≤ | Y |≤
1, existence of a function F holomorphic in x, y ,such that f ( ζ ) = F ( x, y )( ζ ). If ζ is in a polynomially convex and compact set, f canbe represented as a polynomial. Assume ϑ = Y /X and η = y/x , for polynomials X, Y . Further, for constants, c, c ′ , | ϑ − η | > c and | η | < c ′ | ηϑ − | locally. We candetermine w algebraic and locally maximal, such that | wϑ − | <
1. For ηϑ ∼ m wϑ ,we have existence of a holomorphic function F , such that F ( η )( ζ ) ∼ m f ( ζ ) and F ( η ) = const. ⇔ η = const. If F is invariant for monotropy, the result F ( η ) = f follows directly from the lifting principle. Assume P an analytic polyeder withseparation condition (cf. [17]), P = ( x, h ( x )). Assume ∆ = {| z j |≤ } and∆ ǫ = {| z j |≤ ǫ } , close to ∆, for j = 1 ,
2. Assume Σ = Φ( P ), such thatΦ( δ P ) ⊂ δ P (conformal) and that Σ is an analytic set with continuation in ∆ ǫ .Then, Σ is a (normal) model for P . Assume f analytic on P , then we have existenceof F holomorphic on ∆, such that f ( ζ ) = F (Φ( P ))( ζ ). Note that we are assuming ζ in a symmetric neighborhood of { T = 0 } . We can, according to Rouch´e’s principleassume, | z j − w j | < ǫ | z j | and | w j |≤
1, for z j ∼ m w j . For w j polynomials, thisis a proposition on F being invariant for monotropy.3.8. Exactness and involution.
We will use the Poisson bracket
U, V = P i δUδx i δVδy i − δUδy i δVδx i . Assume V defined through δVδy i = dx i dt , δVδx i = − dy i dt . Concerning the two pos-sibilities for ( x i , y i ) where i = 1 ,
2, A) ( x, y, x ∗ , y ∗ ) B) ( x, x ∗ , y, y ∗ ) ,it does notappear to be important what representation we use.Consider the sets Φ ϑ = { e ϑ M = W } and analogously for Φ ∗ . Thus, F ♦ ( M ) = F ( e ϑ M ). Assume over an involutive set that ∃ F − and let G = F − F ♦ over M . Then, G ( M ) /M = e ± ϕ . We will study the parabolic sets ± ϕ <
0, so that G ( M ) = const.M . The spectrum is { e ϑ M = W } , then for a lifting operator F ,invertible and over ϑ <
0, we have F ♦ ( M ) = const.F ( M ), if the constant is real,we have real eigenvectors. There will be a boundary in this approach, given bythe set where ϑ changes sign. Finally, we consider the sets where ϑ > OME REMARKS ON TR`EVES’ CONJECTURE 11 constitute neighborhoods of the constant surfaces. If we consider F as an analyticfunctional, we have that F has the closure of { ϑ < α } as semi-porteur if and onlyif the type for b F is ≤ α , which particularly means that it is portable by any convexneighborhood of the semi-porteur.3.9. Dependence of parameter.
Given a closed trajectory, that does not end ina singular point P , that is the point P stays inner to the trajectory. The point P is called a center, if there are infinitely many closed trajectories, arbitrarily closeto P , that circumscribes the point. We could say that the trajectory γ T → γ = P ,but does not reach it. We will assuming the boundary not C , but holomorphicand with only parabolic singularities, consider the problem of removing the centerpoint as a Dirichlet problem.There are certain conclusions on the singularities in Ω ζ , given the dependenceof the parameter in L . We have the following weak form of minimally definedsingularities. For F = w ∈ B ′ m , if x, y ∈ B m and R I w T ( x, y ) dσ → R I w ( x, y ) dσ ,through a normal and regular approximation. Assume that the dependence of T isholomorphic and w algebraic in ( x, y ). We have that { ( x, y ) w T ( x, y ) = w ( x, y ) } has σ − measure zero. Assume that R | w | dσ < ∞ and ww ∗ = w ∗ w and that( x, y ) is in the normal tube. Then we have normal and regular approximations,say g T of { x = const., y = const. } . Assume f T →
0, as T → df T dT is holomorphic in T (that is not a non-zero constant).If df T dT = C dg T dT , for a constant C on a domain of positive measure, we still havea regular approximation. If γ T = ( f T , h ( f T ) is the regular approximation andif the dependence of T is algebraic in dγ T dT , then according to Hurwitz theorem,since polynomials never have zero-sets of infinite order, then the zero-set must havemeasure zero. Thus, given existence of regular and normal approximations, wherewe assume algebraic dependence of the parameter T , in the tangent space, then allnormal approximations, algebraically dependent on the parameter in the tangentspace, can be assumed regular (at least after adding a regular approximation). Proposition 3.3.
Assume F T with L − dependence in the parameter and exis-tence of a normal and regular approximation algebraically dependent of the param-eter in the tangent space, then all normal approximations, algebraically dependentof the parameter in the tangent space, can be chosen as regular. Note that when the parameter is with respect to the ramifier, we assume alge-braic dependence over transversals and tangents. There are numerous exampleswhere ( dI ) has a global (pseudo-)base, but not ( I ). Finally, note that of Ω ( dI ) = { T r ′ T F x = F x F x ∈ ( dI ) } and Ω ( dI ) = { T r ′ T F x = F x F x ∈ ( dI ) } , where F = 0 we have that T ∈ Ω ⇒ T ∈ Ω iff r ′ T is algebraic in the sense that it isgeometrically equivalent with a polynomial. Assume all approximations of a par-abolic singular point are on the form η T = α T e ϕ T , since we know that all normalapproximations are regular, we can assume the singularities for α T simple. Assume η T ( x j ) ∼ m η jT ( x ), then it is sufficient to consider the case where η j,T = d j dx j η T hasisolated singularities. Since | e − ϕ ∗ η ∗ j,T | < M as | x ∗ |→ ∞ implies | e − ϕ ∗ η ∗ T | < M ,as | x ∗ |→ ∞ . We will see that monotropy is a micro local invariant, this meansthat it is sufficient to consider parabolic approximations for η ∗ .Note that presence of lineality for the symbol, may result in Im F in the spaceof hyperfunctions. We now note that if F is symmetric, entire and of finite typein Exp , then the condition that f represents a hypoelliptic operator, means thatfor some λ , ( Im) λ = P A j F j on a domain of holomorphy, for constant coefficients and a global pseudo-base representing the ideal of hypoelliptic operators. Thus,symbols to hypoelliptic operators do not have imaginary part outside the spaceof distributions and if hyperfunction representation is necessary, we must havecontribution of lineality in the infinity.3.10. A generalized Cousin integral.
We denote with ˜ M = − Y dx + Xdy andcorrespondingly for ˜ W Assume ˜ M exact and ˜ W closed, then the form correspondingto c M is exact after analytic continuation and in the same manner for c W . Notehowever that the forms corresponding to c M and c W are not locally holomorphic,that is we do not have locally isolated singularities and the center case could appear.Assume µ is a positively definite measure and considerΦ µ ( dγ ) = Z P µ =0 dµ ( γ )where P µ is a polynomial and gives a local definition of ∆. Approximating asingular point through dγ →
0, then either Φ µ ( dγ ) → µ ′ such that (cid:2) Φ µ + Φ µ ′ (cid:3) ( dγ ) →
0. Thus, for the measurecorresponding to a hypoelliptic operator, we can choose µ with point support.Assume Φ µ ( d ( γ T − γ )) = R γ T − γ dµ . If dµ is a reduced measure, we must have γ T = γ . We know that if dµ is holomorphic (that is holomorphic coefficients), then dµ will be reduced, for T close to 0. Assume dµ continuous and locally bounded,for all T and that f dµ = dµ + dµ , where dµ is assumed with point support and f dµ is holomorphic. Then R γ T − γ f dµ = 0 implies γ T = γ . Assume γ T a closed contourand γ a point, then for T not close to 0, we have R γ T − γ dµ = 0, implies γ T = γ .This case includes the case with a center (cf. [3], Theorem 4).4. Stratification
Introduction.
If we consider a hypoelliptic analytic symbol f as locally re-duced, it is naturally necessary to use a stratification to define a globally hypoellipticsymbol. The model is centered around the set of lineality and we are always assum-ing the lineality locally is a subset of a domain of holomorphy, which means thatits local complement set is analytic, We consider it to be necessary for the conceptof hypoellipticity to have an approximation property for log f . We will discuss aninterpolation property. Further, it is necessary to have a concept of orthogonalitybetween the real and imaginary parts of the symbol.4.2. The arithmetic mean.
For the arithmetic mean, we have thatlim ǫ → Z C ǫ M V dz ( T ) = M V (0)given that
M V is holomorphic, regularly that is without a porteur (cf. [12]). If forall closed contours R C ǫ M V dz ( T ) = M V (0) implies C ǫ = { } , then M V is reducedfor analytic continuation. If R C ǫ − M V dz ( T ) = 0 for all closed contours in a leaf L ,then the form M V dz ( T ) is closed in L and we have a mean value property abovefor the arithmetic mean in L . Further, the closed contour C ǫ ∼ L .4.3. The concept of stratification.
Assume X ⊂ Y are separable topologiclalvectorspaces. We say that Y is a stratifiable space if it has the property that toany open set U we associate a sequence { U j } ∞ j =1 of open sets in X , such thati) U n ⊂ U for all n ii) U = ∪ ∞ j =1 U j iii) U ⊂ V implies U n ⊂ V n for all n OME REMARKS ON TR`EVES’ CONJECTURE 13
Further, (cf. [4]) given a topological vector space X and with Y as above, we canassociate a topological vector space Z ( X ), such that X is closed in Z ( X ). We saythat X is locally RA (retractible), if X has a local extension property with respectto the stratification. Particularly, if Γ is closed in Y and f is a continuous mappingΓ → X , we have existence of ˜ f that maps Y → Z ( X ).4.4. A stratification using averages.
A topological vector space X is stratifi-able, if for any open set U , there is a continuous mapping f U X → nbhd
0, suchthat f − U (0) = X − U and if U, V are open sets with U ⊂ V , we have f U ≤ f V .We will for this reason study the averages M ≥ M ≥ . . . ≥ ϕ , where the bound-ary M j = ϕ is common for all the averages and where M j →
0, as ϕ →
0. Let F = {M ≥ ϕ } and let f be a continuous function such that ker f = bd F and f = M ( ϕ ) − ϕ . If M is holomorphic and M ( x ) = lim ρ → R C ρ M ( x ) dx andif C ⊂ bd F and ϕ ( x ) = lim ρ → R C ρ ϕ ( x ) dx . Further, f ( ϕ ) = M ( ϕ ) − ϕ withker f = bd F , where F = { f ≥ } why F ⊂ F and f ≤ f , and so on.A stratification of ˙ B can be mapped into a stratification of B m , through i a :˙ B → B m and i a ( ϕ ) = ϕ + a , for a constant a . This is a compact mapping with i a ( ϕ j − a ) = i a ( ϕ j ) − a . ( B m ) stratified in this manner with topology induced ofSchwartz type is FS , why the dual space ( B m ) ′ is DS (cf. [12])4.5. The arithmetic mean and duality in L . Assume F = {M ( φ ) ≥ φ } and f = M ( φ ) − φ and Γ = { f = 0 } . If we assume M ( φ ) holomorphic, we musthave that F lux ( M ( φ )) = 0. Note that M ( φ ♦ ) = M ( φ ) ♦ in L and M ( dφ ) = d M ( φ ). Thus, given a dynamical system with right hand sides harmonic conjugates,satisfying the regularity conditions, we see that the arithmetic mean satisfies acondition on vanishing flux. If Γ is always reduced to a dynamical system consideredin L , the boundary problem is solvable in L .Assume now that the boundary value problem is solvable for M V ( φ ), that isassume ∆ M V ( φ ) = 0 on an open set Ω. Using duality with respect to the scalarproduct in L , we consider0 → φ → M V ( φ ) → ∆ M V ( φ ) → ← M V − (∆ σ ) ← ∆ σ ← σ ← σ ∈ L with σ ∈ L . Let E = ∆ L and X = { φ t M V ( φ ) ∈ E } , that is for a f ∈ L , we have t M V ( φ ) = ∆ f in L . More precisely, we candescribe ∆ L = E through the closure of ( M, W ) with respect to L . AssumeΦ ⊥ M V ( W ) ♦ and Φ ⊥ M V ( W ) in L . Assume Φ with support in a bounded neigh-borhood of the boundary (restriction to strata). The relations will then also holdin L and we can apply Weyl’s lemma to conclude Φ ∈ C locally. Assume in aneighborhood of the boundary that 0 = | ( ϕ, M V ( W )) |≥| ( ϕ, W ) | . Thus, if theproblem is solvable for M V , it is solvable for (
M, W ), given the inequality above.We now have ϕ ∈ C . The parametrix to the problem then has a trivial kernel andthe problem is solvable. Proposition 4.1.
The arithmetic means applied to f (and log f ) form a stratifi-cation over ( B m ) ′ associated to f in a finitely generated symmetric ideal of analyticfunctions over a pseudoconvex domain with transversals given by a locally algebraicramifier. We have assumed parabolic singularities and no essential singularity inthe infinity. Reduction to tangent space.
Assume F ∼ V + iV , and consider thecondition(2) ddx log V reduced and ddy log V reducedGiven that V + iV is hypoelliptic with ϑ = log V , we have that if the property (2)holds for M N ( ϑ ), then it also holds for ϑ . Note also that if M ⊥ W with T W = 0,then we can not conclude that T V has vanishing flux. However, if the condition (2)is satisfied for V and M ⊥ W , we can conclude that V ⊥ V . Let ˜ M = Xdx + Y dy and ˜ W = dF . Then we can consider F ♦ defined as dFdx = dF ♦ dy and dFdy = − dF ♦ dy ,so that ˜ M = dF ♦ . If the involution is taken over F, V, G , where V is the Hamiltonfunction, F is the lifting function and G is a regular approximation, then we canrelate the involutive set to a condition R C dF = 0. Assume F N corresponds to M N ( F ) and C N is the corresponding contour, such that R C dF ∼ R C N dF N and C N ⊂ C ⊂ C − N . Then, the conclusion is that the stratification of negative orderis a covering of the involutive set.4.7. Example.
Assume for instance that V = V + iV + ∆, such that δδx ∆ = δδy ∆ = 0 and where ∆ is defined through involution and through the conditions δδx ( V ⊥ V ) = δδy ( V ⊥ V ). Hypoellipticity means that supp ∆ = { } .4.8. The lineality as closed contours.
The lineality has a pre-image in thecontour C T in the following manner. Let f T = e ϑ T and assume ϑ T − ϑ ≡ ζ T ∈ ∆ locally (lineality). Assume C T = F − { ϑ T − ϑ } describes a simple contourwith an analytic parametrization, then on C T , M − N ( ϑ T − ϑ ) ≤ ϑ T − ϑ . Assume∆ = { ϑ T − ϑ ≡ } locally analytic, then we have locally I (∆) = { ϑ T ϑ T − ϑ ≡ ζ T ∈ ∆ } and N I (∆) = ∆. This means that C T :0 ≡ M − N ( ϑ T − ϑ ) has a pointin common with I (∆), that is { ζ T F − ( ϑ T − ϑ ) ∈ C T } and ∆ have points incommon. The contour C T gives a micro-local contribution, if M − N ( ϑ T − ϑ ) ≡ F , M − N maps locally the geometric ideal I (∆) on to the closedcontour C T .Consider again the problem if the zero set has points in common with C T . If M − N ∈ D ( k ′ ) L , we can assume that the restriction of a complex operator P ( δ T ) tothe real space, is such that P ( δ T ) σ | R ∼ M − N in D ( k ′ ) L , where σ ∈ L ( R ). Extendthe definition of σ (standard complexify) to L ( R ). We can then, in a neighborhoodof the boundary corresponding to the symbol, assume that the parametrix to P ( δ T )is injective, E ( P ( δ T ) σ ( φ )) = σ ( φ ) − r T , where r T is regularizing and r T → T →
0. We must assume that σ is not identically 0, but that σ T ≡ C T . Theproposition is now that φ T has a zero on C T . Assume for this reason that { σ s } isa family of measures, depending on a parameter as above, such that σ s → δ γ as s → C T has an analyticparametrization as a closed and simple contour. If σ s ( φ ) ≡ C T as s →
0, thenwe have existence of γ on C T such that φ ( γ ) = 0.4.9. Further remarks on the stratification.
Assume a global pseudo base inthe tangent space and that F ( dz ) = f ( z ) dz , where f is given by a locally reducedfunction. We are assuming F has no lineality in the tangent space and that ∆ canbe given as a semi-algebraic set. If F ∈ L in the parameter, then dFdz = f ( z ) a.e.A sufficient (and necessary) condition for equality, is that F is absolute continuous.For example, if df = f dg , where f corresponds to continuation. If f is reducedwith respect to analytic continuation (over strata) then f is locally reduced. If OME REMARKS ON TR`EVES’ CONJECTURE 15 df, dg are of type 0, then the same must hold for f . If g = hf , then over dh = 0,for f to be reduced, we must have that h is minimally defined. The first relationparticularly means that F preserves order of zero’s if f is regular, particularly F maps exponentials onto exponentials. If f is absolute continuous, then zerosets aremapped onto zerosets. When F ( e x ) = e ˜ F ( x ) , if we assume ˜ F ( x ) = ˜ F ( x ). If we onlyhave F ( e x ) = βe φ , where φ ( x ) = φ ( x ), then e − φ ( x ) f ( x ) = (cid:2) β ( x ) − β ( x ) (cid:3) e − i Im φ .Note that reducedness for β is not necessarily symmetric.4.10. Condition ( M ) relative the stratification. We are assuming T : x dηdx → x ∗ dη △ dx ∗ and x dηdx = 0 and that the systems ( M, W )... are regular. In particular, weassume r ′ T dFdx = dFdx r ′ T and in the same manner for y . Further, we are assuming that F ( X, Y ) ∼ R ( M, W ) and F :0 → ∆ and F − :∆ → B m ) ′ and is relative T W = e φ △ , where ± φ △ > M ). Let L N ( ω T ) = | ρ | R C N M N ( ω T ) dz ( T ),where C N is a closed contour, parameterized through T , such that T → C N ↓ { } as N ↑ ∞ and T → C − N ↑ bd C , as − N ↑ ∞ and where ρ is the radius for C N , where ρ = ρ ( N, T ). The condition ( M ) is that lim N →∞ L N ( ω T ) is regular,that is that the functional corresponding to M N ( ω T ) is of real type. Note that if σ △ N ∈ L , with k σ △ N k = 1, we have the same argument for M − N and L − N . Wewill now argue that if ω △ T ≡
0, for T small, then M − N ( ω △ T ) ≡ C − N . Let X N = { ω T ≤ M N ( ω T ) } , for N ≥ N , where M N ( ω T ) ∈ L andalgebraic in T , for T small. If we extend the definition of σ △ N , such that σ △ N is theevaluation functional on the boundary { ω T = M N ( ω T ) } , with ω T ∈ L , then if < M − N ( ω △ T ) , M N ( ω T ) > ≡
0, we can chose M − N ( ω △ T ) = σ △ N on the inner of X N ,why ω △ T ⊥M N ( ω T ), for T small. If M N − I is locally algebraic, then ω T ∈ Γ = Γ ⊥ .We are assuming the Lagrange condition Γ = I (∆) = I (∆) ⊥ = Γ ⊥ Assume ω △ T ≡ ω △ ∞ ( ∼ ω /T ), for T small, on a set with complex dimension,then we must have existence of M − N as described above, such that for N large, < M − N ( ω △ T ) , M N ( ω T ) > ≡ < ω △ T , M N ( ω T ) > ≡
0, according tothe conditions, we have that M − N ( ω △ T ) ≡ C − N , for N large. We are assumingthat C − N includes the real infinity, as N → ∞ . Conversely, if L − N ( ω △ T ) = 0, as N > N implies ω T is not ≡ ω ∞ , in the real infinity. The conclusion is that if thestratification has condition ( M ) in the infinity, it is not possible to have lineality.Consider the limit L N ( ω T ) = R C δ M N ( ω T − ω ) dz , where z ( T ) ∈ C δ a closedcontour of radius δ and let A N be the porteur set to this limit considered in H ′ .Obviously, we have A N ⊂ ∆, for N ≥
0. Consider the stratification of ( B m )with { X △ N } , that is a stratification using the means M N . If L N are not regular,that is A N = { } , then we have on a connected set that ω T − ω ≡
0. (We areassuming Schwartz type topology for the symbol space). Conversely, consider thestratification of ( B m ) ′ and the contours { C T } that contribute to ∆ through commonpoints. In this case, if M − N are of real type, there is no possibility of lineality.Thus, given an operator with lineality, we do not have condition ( M ) for ( B m ) ′ inthe stratification using M − N . Proposition 4.1.
If the stratification that we are considering has condition ( M ) ,that is if all the means are of real type, then the symbol ideal is locally reduced andconversely. We will discuss two other similar topological conditions in a later section. Sinceit is topological, we prefer the set of lineality to characterize hypoellipticity. The condition ( M ) at the boundary, means that the boundary behavior does not in-fluence the microlocal behaviour in the infinity. A globally hypoelliptic operatoris in this context a globally defined operator that is hypoelliptic in the infinityand for which the topology for the symbol space has condition ( M ) (or a similartopological condition) at the boundary.4.11. Reduction to real type.
Assume F holomorphic and of finite exponentialtype. Further that F has finitely many zero’s on X \ U , where X is assumed abounded domain and U is a neighborhood of the infinity. Further, we assume thatthe zero’s P , . . . , P ν are isolated and of finite order. Assume U is a neighborhoodof P that does not contain any other zero’s. Then we have on X a holomorphicfunction F , such that F − F is of type 0 on X and F is of type −∞ on U . Theremaining P ′ j s are treated in the same way. Thus, F − P j F j is of type 0 on X andeach F j is of type −∞ on the corresponding U j .4.12. Remarks on a spectral mapping problem.
The definition of the mapping T starts with − Y dx + Xdy → − b Y dx ∗ + b Xdy ∗ and we are requiring { W = 0 } →{ c W = 0 } → { cc W = 0 } . We consider the multipliers χX = Y , χ △ b X = b Y , λH = G and λ △ b H = b G . We assume T : χ → χ △ and { η = χ } → { η △ = χ △ } . We have that T preserves parabolic points, but is usually not a contact transform. If T has theproperty that it maps constants on constants and exponentials on exponentials,we know that T preserves parabolic approximations. Through the condition onvanishing flux, we can assume ( w, T w ) pure and that T preserves analyticity.Consider ( J ) = { f R ( I ) f dσ ( t ) = 0 ˜ V } , where ˜ V is a geometric set. One ofthe more difficult problems in our approach is to see that the spectral mappingresult we use respects the stratification, that is if starting with a stratification of( B m ) ′ and c W , { X ∗ j } , we have that the sets { X △ j } where T X j = X △ j constitute astratification. Consider Φ △ = { ϑ △ e − ϑ △ χ △ = const ∃ ϑ △ } and Φ = { ϑ e − ϑ χ = const ∃ ϑ } . Consider the Legendre transform R , according to RE < R ( χ ) , χ > − R ( e ϑ ) = b R ( ϑ ) = b IR ( ϑ ) = e ϑ ∗ and we note that (cid:2) b R, I (cid:3) = (cid:2)b I, R (cid:3) implies that R is algebraic in H ′ over Φ △ → Φ ∗ → Φ and over a regular parabolic approximation,we can argue as in the spectral mapping theorem. For a hypoelliptic system, thecontinuation to T is algebraic and the stratification of X ∗ gives a stratification of X △ . We can conversely argue that if these stratifications are equivalent, the systemhas no lineality. 5. Topology
Introduction.
The concept of hypoellipticity is dependent on topology andwe will use the monotropic functionals both for limits in the symbol space and forthe equations in the operator space. The topological arguments are comparativeand we compare with the more familiar hyperfunctions. However there are geomet-ric sets that are characteristic for hypoellipticity, such as lineality and the set oforthogonality, for all topologies that we consider. Several parameters are necessaryto define the class of hypoelliptic symbols. We give the approximation of the op-erator using operators dependent on a parameter and a second parameter is usedto trace the transversal in determining microlocal contribution. Since this is ananalytical study and not a geometrical, we do not attempt to minimize the numberof parameters.
OME REMARKS ON TR`EVES’ CONJECTURE 17
Topological fundamentals.
The space H ( V ), where V is a complex analyticvariety, countable in the infinity, is the space of holomorphic functions with topologyof uniform convergence on compact sets. This is a Frechet type of space (FS) andthe dual space is denoted (DS). Given F-spaces { E i } , if i : E i +1 → E i the projectivespace is (FS). If t i : E i → E i +1 compact, the inductive limit is compact. We startwith a topology of Schwartz type, that is given a separated space E , if V is aconvex disc neighborhood of the origin in E , then we have existence of a convexdisc in E that is a neighborhood of the origin such that U ⊂ V and such that E ˜ U → E ˜ V is compact, where E ˜ U is the completion of the normalized set E U .The topological arguments in this study are comparative. The symbols moduloregularizing action are considered in i neighborhood of the real space where wecompare with monotropic functionals and the D ′ Lp spaces ( p = 1 , Proposition 5.1. If ( I ) is an ideal of holomorphy with topology of Schwartztype and a compact translation, consider rad ( I ) with Schwartz type topology anda weakly compact translation, if ψ ∼ m in the | ζ | − infinity, is in rad ( I ) , then { d ζ ψ = ψ = 0 } is nowhere dense in N ( I ) . Proposition 5.2.
Assume ( J ) = ker h a finitely generated ideal with topol-ogy of Schwartz type and r ′ T weakly compact. Assume for all ψ ∈ ( J ) , we have h ( r ′ T ψ ) /ψ ∼ m in the ζ − infinity. If η = const. , we have that ψ is in a boundedset with respect to the origin. Proof:We can prove an estimate | η − c R | < / | ζ | | ζ | > R for a constant c R and R sufficiently large. Thus, for φ ∈ ( J ) | φ | < c / | ζ | + c | η ( φ ) | | ζ | > R for constants c , c . Symmetry follows from the conditions on r ′ T . (cid:3) Monotropic functionals.
Assume B m test functions, that is C ∞ − functions,bounded by a small constant in the infinity, such that ˙ B ⊂ B m ⊂ E and ( B m ) ′ ⊂D ′ L . The Fourier transform over the real space is P f , where P is a polynomialand f is a continuous function. We will modify f to an ǫ − neighborhood of thereal space as followsi) F is continuous on the real space and locally bounded on an ǫ − neighbor-hood of the real space.ii) we have existence of lim Γ → F ( ξ + Γ), for any line Γiii) any line Γ ⊂ ∆( F ) is such that Γ ⊂ Ω, where Ω is a domain of holomorphy.Note that the difference τ Γ F − F , even when it is not holomorphic, will preserveconstant value over the lineality corresponding to F . Finally, assumeiv) F ∼ m W , where W is holomorphic and in Exp of finite type.We then have existence of B Γ (modulo monotropy). Assume further that the trans-lation is algebraic over P F and for W , that the lineality is quasi-porteur (cf. [12])for B Γ .The first observation is that if f T ∈ L then M N ( d N T N f T ) = σ T ∈ L . Par-ticularly, we have f ∈ D ′ mL implies M m ( f ) ∈ L . As df = αdx , we have if d M ( f ) = βdx , then we must have β = M ( α ), thus in L , ddT M ( α ) = M ( ddT α ) Wenow argue that M − N is surjective in B ′ m . Consider for this reason M N in B m andassume that ( I ) is defined by f ∈ ( I ) ⇔ f ∈ B m and f ≤ M ( f ). We then have M ( f − f ) = 0 implies f − f = 0. Thus, we have that M − N is surjective over ( I ) ′ .As ( I ) ⊂ B m , we must have B ′ m ⊂ ( I ) ′ , why the surjectivity follows for B ′ m . Notethat f = e ϕ with ϕ subharmonic, if we let ˜ M ( e ϕ ) = e M ( ϕ ) and M ( ϕ − ϕ ) = 0,then ϕ − ϕ = 0 implies f = f .We have studied regular approximations according to F ( ζ + T j ) = F ( ζ ) + c j as0 = c j → T j → T . Note that more generally, for dF ( ζ + T ) − dF ( ζ ) = dL T ( ζ )with for instance dL T ∼ m | ζ |→ ∞ . The projection method gives that f ∼ m | ζ |→ ∞ , means existence of g holomorphic such that g → | ζ |→ ∞ . Thuswe have existence of a polynomial P such that | g ( ζ ) − P ( ζ ) | < ǫ as | ζ |→ ∞ , wherewe have assumed f = τ ǫ g . Note that 1 /ζ = (1 /ζ , . . . , /ζ n ) and we can assume thecondition in some variabels and assume the others fixed and in the finite plane.5.4. Algebraicity for exponential representations.
Consider the following prob-lem, when for a continuous homomorphism L and L ′ = {∃ ! η L ( e ψ ) = e <η,ψ> } ,do we have L ∈ L ′ . Let L = {∃ ! η L ( ψ ) = < η, ψ > } , where in this case L is assumed continuous and linear. If X = H (Ω), for an open set Ω, we assume b X = { e ψ ψ ∈ H (Ω) } . If we have existence of η x , for x fix, such that L ∈ L ( b X ),we have L ∈ ( b X ) ′ , . Assume for M ∈ ( X ) ′ , that M ( ψ ) = < b L, ψ > = < L, b ψ > ,then L = F − M . When L ∈ L ′ and if L is algebraic in e ψ , that is linear in ψ ,we have L ( e ψ + ψ ) = L ( e ψ ) L ( e ψ ) = e <η,ψ > e <η,ψ > . Further if L , L ∈ L ′ ,we have L L ( e ψ ) = e <η + η ,ψ> = e <η ,ψ> e <η ,ψ> . If (cid:2)b I, N (cid:3) ( ψ ) = e <η,ψ> and (cid:2) b N, I (cid:3) ( ψ ) = < η, ψ > , we assume the commutator C such that C (cid:2)b I, N (cid:3) = (cid:2) b N , I (cid:3) ,then F − C F (cid:2) I, N (cid:3) = (cid:2) N, I (cid:3) . If (cid:2)b
I, N (cid:3) = (cid:2) b N , I (cid:3) , we say that N is algebraic. Let < N ( ψ ) , θ > = < ψ, t N ( θ ) > . Then N ( ψ ) ∈ ( b X ) ′ implies t N ( e θ ) ∈ X ′ , for x fix and t N b I = (cid:2) F N (cid:3) t . We have < b I ( ψ ) , θ > = < ψ, I t F θ > iff < e ψ , θ > = < ψ, t F θ > .Assume Σ ∋ γ →| γ |∈ R is on the form ( e ϕ , h ( e ϕ )). Assume L within a constantis algebraic over R (does not imply algebraic over γ ). Note that if π is the projectionΣ → R and π − R = ˜Σ, then ˜Σ may have points in the edge, even when Σ doesnot. Thus. there may be points in common for L T ∈ ( I ˜Σ ), that are not present for L T ∈ ( I Σ ).5.5. Some generalizations.
Assume V ⊂ V ⊂ V , where V , V are semi-algebraicand V analytic. Assume V ⊂ Ω a domain of holomorphy, such that the limit B Γ isindependent of starting-point, then V is quasi-porteur to B Γ and the same followsfor V . Further, since V is analytic, V is porteur to B Γ . Assume ˜ V the extensionto full lines. Assume g algebraic, such that if V = { p ( γ ) ≥ } , g p ( γ ) = γ locally. Then g (0) = 0 and V → γ → ˜ γ , where the last mapping is into the wavefront set, but regular approximations are assumed in V ⊂ V . If V is porteur toa functional T ∈ H ′ , we can chose V as a cone which through the topology canbe assumed compact. Note that V = { γ ∗ T Re < γ T , γ ∗ T > ≤ γ T ∈ V } and bdV corresponds to the orthogonal complement to bdV . If B Γ has indicator h ,we have for Γ a compact, convex cone ∋ B Γ is portableby Γ ⇔ h ≤ bd Γ .If γ is defined by a homomorphism h such that h N = 1, we have that all regularapproximations of a singular point P , γ → P as t → ∞ , can be seen as on oneside of a hyperplane { ( x, h ( x )) dh ( x ) ≥ µdx } , for a constant µ . More precisely,for a curve that reaches Σ as t → ∞ , if the part of the curve that is situatedoutside Σ is finitely generated, we claim that γ ⊥ can be chosen locally on sideof a hyperplane (cf. section on paradoxal arguments). We are assuming in the OME REMARKS ON TR`EVES’ CONJECTURE 19 following that x is reduced. We have { p ( x, y ) = 0 } ∼ { ˜ p ( η ) = 0 } , for a polynomial˜ p and η = y/x . Further, there are polynomials in x, y , r , s such that p ( γ T ) = ddT r ( γ T ) = s T ( ddT γ T ). If p is reduced, there is a polynomial q such that p = q .Assume F T → ddT F T > { p ( γ T ) > } . If we let ddT F T ∼ p , we areassuming F T conformal in T and algebraic in x, y . Assume ˜ V , ˜ V are extensionsto full lines with indicators h and h . It is then sufficient to prove that h ≤ ch ,for a constant c , to have ˜ V ⊂ ˜ V . Further, c h ≤ h ≤ c h , for constants c , c ,gives ˜ V ∼ ˜ V . If V is a semi-algebraic quasi-porteur to a functional for instancedefining the first surface to the symbol with V ⊂ V , where V is analytic, then V isporteur. Assume that p , p have the same micro-local properties, in the sense thattheir sets of constant sign coincides. Assume V i = { p i ( γ T ) ≥ } and V ⊂ V ⊂ V and V = { g = 0 } with g analytic, then { p < } ⊂ supp g ⊂ { p < } , Thus, ifsingular points are in { g = 0 } , then regular points will be in an ”octant”. We couldsay that the micro-local contribution from the symbol, is given by this ”octant”.5.6. A comparison of hyperfunctions and monotropic functionals.
If thesymbol F T ( γ ) preserves a constant value in the γ - infinity, then F T ∈ B m , that isit is C ∞ and bounded by a small constant in the infinity. In this case the Cauchyinequalities can be satisfied for a monotropic function, that is there is a ϕ T ∈ A (real-analytic functions) such that F T ∼ m ϕ T . If for F T ( γ ) = P α F α,T /α ! γ α thereis a number ρ with ρ < A , for a constant A , such that ρ α sup | F α,T |→ | α |→ ∞ , then we have that F T is entire in γ and of exponential type A ([12]).Note that a sufficient condition for existence of a global pseudo-base for the symbolideal, is that it has an induced topology with Oka’s property.If γ T is in A (Ω ζ ), then F T ∈ B K (Ω ζ ) that is hyperfunctions with compact support([11]). In the case where F is real-analytic, then so is f . If we assume instead that G T,k = d k dT k F T has isolated singularities at the boundary and preserves constantvalue in the infinity, then intuitively we would have at worst algebraic singularitiesin the infinity. Assume f ∈ B m in x, y , then we have f ( x, y ) = g ( x, y ) + P ( x , y ),where g is radial and bounded by a small number in the infinity and P is polynomial.Further, if f ∈ B m there is a ϕ ∈ A such that | f − ϕ | < ǫ at the boundary, for asmall number ǫ .An important difference between B m and A is the algebraic properties. A func-tion f is in A if both its real and imaginary parts satisfy the Cauchy’s inequalities.More precisely, assume L ∈ B (Ω) where L is defined by composition, such that φ ∈ A (Ω) and L ( φ ) = L ( J ( D ) ϕ ), where J ( D ) is a local elliptic operator (cf. [9])and where L ∈ E ′ (Ω) such that J ( D ) ϕ ∈ E . Then ϕ ∈ A and also J ( D ) ϕ ∈ A , fromthe properties of J ( D ). Any element in A has a representation through J ( D ) ϕ asabove, why L is defined on A and L ∈ B K . However we can have 1 /f N → N without having 1 /f ∈ B m , for instance if f is thesymbol to a self-adjoint operator partially hypoelliptic in D with Re f ≺≺ Im f .Thus we do not expect a radical behavior in the case of monotropic functionals.Another important difference is the global property of the hyperfunctions ([11]),which is not present with the monotropic functionals. However, we can give thefollowing argument. Let F ( T, γ ) = F ( γ T ), where γ = ( x, y ) and T ∈ V = ∪ Nj =1 V j ,the parameter space. Assume F algebraic in the parameter in the sense that F ( T .T , γ ) = F ( T , γ ) .F ( T , γ ) (the dot signifies concatenation of curve segments). We are assuming γ T real-analytic as T ∈ V , but we are not assuming T → γ T algebraic in T . With theseconditions we do not necessarily have that F T ( γ ) is algebraic in x and y . For thesymbol now holds that if f ( r T ζ ) = 0 for all T j ∈ V j and ζ fixed, then we have f ( r T ζ ) = 0 for T = T . . . T N ∈ V .5.7. The operator space.
Given an open set Ω ⊂ R n and a differential operator P with constant coefficients, we have that P ( D ) A (Ω) = A (Ω). If P a differentialoperator hypoelliptic (in the sense of A ), we have P ( D ) B (Ω) = B (Ω)The global property is particularly interesting in connection with the solvabilityproblem. It is known for an elliptic d.o P ( D ), that P A (Ω) = A (Ω) ([10]). For anelliptic d.o P ( D ), (hypoelliptic in the sense of A ), we have that P ( D ) B (Ω) = B (Ω)(Harvey [11]), where Ω is an open set in R n . Note also that through majorants anoperator with coefficients ≤| c α | and such that lim sup | c α | | α | →
0, as | α |→ ∞ (compare exponentially finite type) maps B (Ω) → B (Ω) (cf. [9])We have not discussed hypoellipticity in the sense of monotropic functionals, butwe can relate to A -hypoellipticity using monotropy. We assume the proposition(1.1). We can now use that monotropy is micro-locally invariant in the symbol-space. Assume for this reason P a ps.d.o hypoelliptic in A − sense. Further that P u ∈ B m and φ ∈ A such that φ = P v in A and | P u − φ | < ǫ at the boundary Γ.The problem is now to prove existence of a symbol P such that ∆( P ) = ∆( P ) and P u ∈ A with | P u − φ | < ǫ at the boundary. Define P such that c P ∼ m b P , that is τ ǫ c P ∼ b P . The conclusion is that given a ps.d.o P , hypoelliptic in A − sense, thereis a ps.d.o P with the same lineality (= { } ), such that W F a ( P u ) = W F a ( u ),why using the claim (1.1), we have that P is hypoelliptic in B m .For the discussion of the symbol space, we will use a topological argument.Assume the topology of Schwartz type and with (weakly) compact translation.Let Ω ( j )0 = { Γ F ( γ j )( ζ + Γ) = F ( γ j )( ζ ) ∃ γ j } , where F Γ are assumed to sat-isfy the regularity conditions for the dynamical system. We can now prove for asequence of γ j , that approximate a singularity, Ω ( j )0 ↓ { } as j ↑ ∞ . Let J bedefined by N ( J ) = Ω ( j )0 , for some j and let ( I ) = { γ F ( γ )( ζ + Γ) = F ( γ )( ζ ) + C Γ for some constant C Γ } . Then we have existence of J as above with rad ( J ) ∼ I .Further, there is a regular sequence of Γ j , such that F ( ζ + Γ j ) = F ( ζ ) + C Γ j where 0 = C Γ j → j → Ω . In the same manner, if the dependence of T is holomorphic for r ′ T , we have existence of a regular sequence of T outsideΩ ′ = { T F ( r T ζ ) = F ( ζ ) + C T C T = 0 } , such that 0 = C T → T → Ω ′ . Thismotivates why there is no loss of generality in assuming that r T behaves locally astranslation, in a regular approximation of Ω ′ .6. The mapping T Introduction.
We have seen that certain trace sets (clustersets) are charac-teristic for hypoellipticity, more precisely the absence of these sets is necessary. Themapping T which is derived from dynamical systems theory ([3]) will in this studybe used to define and describe these sets. Characteristic for hypoellipticity, assum-ing the real and imaginary parts of the symbol are orthogonal, is that T , given asa continuation of the contact transform (Legendre), is (topologically) algebraic.6.2. Systems of multipliers.
Consider the system with right hand sides (
X, Y )and η ∗ b X = b Y . In the same manner, to the system ( M, W ), γH = G . This can be OME REMARKS ON TR`EVES’ CONJECTURE 21 seen as a multiplier problem. Note that if η is a polynomial and the correspondingconvolution equation is seen over E ′ , then ηδ b ∗ X = η ∗ b X . Thus, if we assume X, Y are holomorphic and of type 0, then η ∗ = b η . Assume using the Fourier-Boreltransform, that M = X x + Y y → H and W = Y x − X y → G , then the condition c W = 0 is the condition that x ∗ dη ∗ dx ∗ = 0, that is η ∗ b X = b Y . Further, M = H x + G y and W = G x − H y , why the condition c W = 0 is a condition γ ∗ b H = b G , that is x ∗ dγ ∗ dx ∗ = 0. For M = ( b X ) x ∗ + ( b Y ) y ∗ and W = ( b Y ) x ∗ − ( b X ) y ∗ , if we assume b X, b Y are holomorphic of type 0, then the condition c W = 0 is a condition x dηdx = 0, thatis ηX = Y . Finally, M = ( b H ) x ∗ + ( b G ) y ∗ → C dx ′ dt and W = ( b G ) x ∗ − ( b H ) y ∗ →− C dy ′ dt , give the condition c W = 0 is a condition x dγdx = 0, assuming b H, b G areholomorphic. Let S denote the mapping ( X, Y ) → ( M, W ) and T the mapping( M, W ) → ( M , W ). If we assume that all elements are holomorphic over regularapproximations, we can prove that the mapping T preserves order of zero.Consider the following scheme( M, W ) → ( H, G ) → ( M , W ) ↓ T ↓ ↓ T ( M , W ) → ( b H, b G ) → ( M , W )and the corresponding characteristic sets b Σ = { ( H, G ) = 0 } , b Σ = { ( c M , c W ) = 0 } , b Σ = { ( c M , c W ) = 0 } , b Σ = { ( c M , c W ) = 0 } . Let Σ → T Σ ,Σ → T Σ . Let e S ( M , W ) = ( M , W ), The sets Σ → Σ are connected through S F S = e SS F over ( X, Y ).Assume g = b Y / b X − η ∗ >
0, then we know that there exists a measure v ,non-negative and slowly growing such that g = b v . The condition on positivityimplies exactness over the tangent space (global pseudo-base). We can say that T maps contingent regions on contingent regions, in the sense that the order ofthe regions are preserved, that is the number of defining functions is preserved.Let ϑ = W /M and ϑ = W /M . Then, we have that ϑ changes sign as( ϑ + η ) / (1 + ηϑ ).6.3. Degenerated points for the method.
The problem of determining x ∈ L ∩ H , so that f = c x + c h ( x ) is trivial over { η = const. } . Consider a neighborhoodwhere η is quasi conformal, | η ( x ) − x | < c , locally for a constant c . Assume − c M c W = xX + yYxY − yX , where M = 0 is the equation for mass conservation and W is thevorticity. The Poincar´e index counts the number of changes of sign −∞ to ∞ andback, for this quotient, why if for instance { η + ϑ ≥ µ } , for a positive constant µ ,the index is zero. For η conformal we have 0 < | ϑ − η | . Assume η algebraic over ϑ = η . In a neighborhood of c W = 0, if η is constant, then − c M c W changes sign as1 / ( ϑ + η ). In the same manner in a neighborhood of c M = 0, the correspondingtest is for 1 / ( η − + ϑ ). Note that conformal mappings do not preserve continuumor reducedness, unless they are bijective.Consider the mapping dx → dv ( x ) → dh ( x ), then F ( x, v ( x )) have isolatedsingularities. To describe the singularities to F ( x, h ( x )), we start with ( M, W )and consider W = const. (not necessarily non-zero) or W regular. We will use c W = G = P f , where P is a polynomial and f is a continuous function and uni-formly bounded (in the real space). When have c W = 0, equivalently x ∗ dη ∗ dx ∗ = 0 ⇔ η ∗ b X = b Y . In the same manner x ∗ dγ ∗ dx ∗ = 0 ⇔ γ ∗ b H = b G and we can map η ∗ → γ ∗ by a contact transform (in the case where T is a contact transform). In thecase where the multipliers are polynomials, there is a simple connection ηX = Y , γH = G and η ∗ = b η , γ ∗ = b γ . Locally, where H = 0, we have | γ |≤ C | P | , for aconstant C . Consider the set Σ γ ∗ = { γ ∗ = dγ ∗ } and outside this set, G is regular ifand only if H is regular ( = 0). In the same manner for Σ η ∗ . Consider for G = P f ,the set Σ f = { f = 0 } and Σ f ⊂ { x ∗ dγ ∗ dx ∗ = 0 } . Assume Ω = { γ = P = 0 } algebraic.Further, either Σ γ ∗ has isolated singularities locally or Σ γ ∗ = { f = 0 }\ Ω. We areassuming H = 0 why γ ∗ = 0 implies f = 0 while γ ∗ = H = 0 implies ( x, y ) ∈ Ω.Where H = 0, we have f = 0 outside Ω. Further, that H = 0 corresponds tothe proposition on lineality in the tangent space and if H = 0, we can only havelineality in the symbol space.A neighborhood of a singular point P , is divided into contingent regions where c W has constant sign. Let ( J ′ p ) = { ( x, y ) Y ≥ µ p X } , for a constant µ p . Thenwe have ( J ′ p ) = { ( x, y ) γ ∗ ≥ µ p } which, under the conditions above is a semi-algebraic set and on this set, every boundary curve has a local maximum. In thecase where γ ∗ is not a polynomial, we can find semi-algebraic sets V , V , such that V ⊂ ( J ′ p ) ⊂ V ( H = 0). Assume the boundary is given by a quasi conformalmapping c such that we have existence of a conformal mapping c on the boundarywith c ∼ m c , c ( x , y ) = 0 and either dc dy = 0 or dc dx = 0 in ( x , y ). We canassume that c has isolated singularities on the boundary. If we assume c algebraicmodulo monotropy at the boundary with w − c conformal and w − c ( x ) = 0 and w algebraic. Thus, c has isolated singularities at the boundary. Assume c satisfiesPfaff’s equation, dc − pdx − qdy = 0 with p = dcdx and q = dcdy . If p = 0 at theboundary dy/dx = − q/p and the condition | p |≤ k | q | gives in the context | dy |≥ c | dx | at the boundary.6.4. The spectrum to T . Assuming y = h ( x ) and dr ′ T xdx = 1 (we can assume x reduced) and Y T = χ T X T and if dhdx is locally bounded, the system is well defined.Let T : ( X, Y ) → ( b X, b Y ) and T ( Xdx + Y dy ) = b Xdx ∗ + b Y dy ∗ , where dx → dx ∗ is the Legendre. The discussion has to do about when T is analytic. A sufficientcondition for this is that T is pure.Given a formally self-adjoint and hypoelliptic differential operator L , the spec-trum to a self-adjoint realization in L ( R n ), σ ( A ) = { L ( ξ ) ξ ∈ R n } , is left semi-bounded. If we let ω = χ ( −∞ ,λ ) ◦ L and F ( E λ u ) = ω F u , for the spectral projection E λ and F − w ∈ D ′ L . For Baire functions we have the spectral mapping theorem,that is if A is harmonic conjugation, σ ( T ( A )) ⊂ T ( σ ( A )) and when T is algebraic,we have equality.6.5. The spectrum to multipliers.
The spectral theory is set over χX = Y and χ △ b X = b Y , that is dh ( x ) dt = χ dxdt and λ △ H = G , λ b H = b G . Let Φ = { λ = const. } andΦ △ = { λ △ = const } . Let Ψ and Ψ △ be the constant sets to χ and χ △ . Let V be theHamilton function corresponding to ∆ V = − W . Assume that T : λ → λ △ is analyticon Φ and Ψ. Assume A the linear operator corresponding to A ( M ) = M ♦ , that isharmonic conjugation. Thus, if Y = ηX , then A ( X ) = χX and A ( b X ) = χ △ b X . If Sp ( T ( A )) ⊂ R , then Sp ( T ( A )) is constant ⊂ Φ △ .We will also consider the sets { e − ϑ △ χ △ = const. } and { e − ϑ χ = const. } as para-bolic Riemann surfaces. Over these sets T , when it acts as a Legendre transform, OME REMARKS ON TR`EVES’ CONJECTURE 23 we can consider it as algebraic and we can apply the spectral mapping theorem.Note that through symplecticity we have that if χ △ /χ = const. , then this holds ina point. If χ is algebraic in T , then χ △ is algebraic 1 /T . We add the condition T (0) = 0 to the definition of T , corresponding to the condition that P = P △ exists, which is necessary for analytic continuation.6.6. Conclusions concerning the trace formula.
In the representation e λ ( x, y ) = R ∆( ξ ) <λ e i ( x − y ) · ξ dξ ,(cf. [13]) a trace in ( x − y ) corresponds to a trace in ξ , consideredas a functional in H ′ . Through Iversen’s result, the correspondent to r ′ T in ξ , is afunction, however multi-valued in most cases. If we reduce the situation to a realparameter, the continuity of r T means that there can be no trace in ξ correspondingto the leafs, since the operator is elliptic. The only possibility is through change ofleafs, but through the conditions that the covering is regular, we know that thesetrace sets do not have a measure .6.7. Conjugation.
Assume χ △ H = G and consider the two FBI-transforms F G and F H , where the kernels are harmonically conjugated. We are assuming H = e Φ ,where Φ ≥ < const. If χ △ = 1, we have F G = F H and we assume F H/G = F = δ , the evaluationfunctional. Consider the problem of geometric equivalence. The mapping T : Ψ = x dχdx → x ∗ dχ △ dx ∗ , maps the set χ △ = 0 (or χ = const ) on to T Ψ = 0. Further − χ △ = − HG = T Ψ T Ψ ( quotient of polynomials ) and we have T Ψ = − Ψ. We can showthat T is not algebraic but if T maps 0 onto 0, it has the property that T Ψ = 0implies Ψ = 0. In the case where χ △ is constant, we have not excluded the casewhere G = 0 ( and H = 0). These points are singular and parabolic. Thus, if χ △ = 1, we have T Ψ + T Ψ = 0. In the case where χ △ is algebraic, we have that − H/G changes sign as T Ψ / T Ψ, a quotient of polynomials.We are considering two types of conjugates (still referring to kernels), F ♦ H = F G and F △ Ψ = F T Ψ . We assume as above F = F T Ψ / T Ψ corresponds to the proposition F T Ψ = F T Ψ , where as before T Ψ = ϑ △ − χ △ and T Ψ = ϑ △ + χ △ . Particularly, if T Ψ = ρ Ψ and T ( ρ Ψ) = ρ T Ψ, we have T Ψ = − /ρ Ψ. Further, x ∗ x dχ △ dη = ρ dx ∗ dx .Now let χ △ = iρ △ , for a real ρ △ and assume ϑ △ = ϑ △ . Thus, T Ψ = S Ψ, why S = T T . A differential Ψ = w + iw ♦ is said to be pure, if Ψ ♦ = − i Ψ. We have T ( w + i T w ) = − i ( w + i T w ), T ( w + iSw ) = − i ( − w + i T w ) and S ( w + i T w ) = − i ( − w + iSw ). Finally, S ( w + iSw ) = − i ( − w + iSw ). Thus, w + i T w is pure andif w is symmetric with respect to the origin, also w + iSw is pure. If T Ψ = g and g = αg , we have g = αg which is pure, if α = i . Let p = ( w, w ♦ ) and consider r = ( w, T w ). Through the definition of T , we have T w ♦ = ( T w ) ♦ . Thus, if( w, w ♦ ) is pure, the same holds for ( w + iw ♦ ) + i ( w ♦ + iT w ♦ ) and it follows fromthe condition T = − I over ( w, w ♦ ) that r is pure.Assuming w is pure, we have that ( w, w ♦♦ ) is pure. This follows since ( w + iw ♦♦ ) = w − w ♦ , which is pure if w ⊥ w ♦ , where we have used that iw ♦♦ = − w ♦ and if w ♦ = − iw , then ( w + iw ♦♦ ) = w + iw . If ( iw ♦ ) ♦ = − iw , then the form( w + iw ♦♦ ) is pure.6.8. Symplecticity and forms.
Assume u = adx + bdy and let F ( u ) = F ( a ) dx + F ( b ) dy . Then F ( u ♦ ) = − F ( b ) dx + F ( a ) dy = F ( u ) ♦ , assuming that F ( − b ) = − F ( b ). Further, if ddx F ♦ = F ♦ ddy , then ∆ F ♦ = F ♦ ∆. Assume F ♦ a homo-morphism and that it maps 0 →
0. We can as before write, < F ( M ) , θ > = ρ T ( x, y ) < M, θ > and as F ( M ) = − F ♦ ( W ) = ∆ F ♦ ( V ), such that if ∆ V = 0,then F ♦ ( W ) = 0 and F ( M ) = 0.Consider the mapping T : Xdx + Y dy → b Xdx ∗ + b Y dy ∗ . We are assuming γ and γ △ in duality and write T ˜ M ( dγ ) = ˜ M ( dγ △ ). T is first assumed an extension ofa contact transform in the sense that T ˜ M = ρ L ˜ M , where ρ is at least a Bairefunction. Assume T f τ /f τ = const. ⇒ τ = 0 and d T f τ /df τ = const. ⇒ τ = 0.Assume α ( w, w ♦ ) a symplectic form over E = V × V ♦ and consider E ×T E . Assume S an involutive set with respect to the bracket dVdx ddy − dVdy ddx + dVdx ∗ ddy ∗ − dVdy ∗ ddx ∗ .Assume T equivalent with a form symplectic with respect to α , in the sense thatthe sets { χ = ddx χ = 0 } and { χ △ = ddx ∗ χ △ = 0 } are both minimally defined andequivalent. That is formally, α ( T w, T w ♦ ) = ̺α ( w, w ♦ ) and ̺ ∼ constant .Consider the form ( p, q ) σ = − ( q, p ) σ , where p = ( w, w ♦ ) and q = T p , where weare assuming ( p, q ) σ = ( T p, T p ) σ = − ( q, p ) σ , that is skew-symmetric and bilinear,assuming the double transform in Exp is equivalent with − I (after analytic contin-uation). Through the conditions ( w τ ) ♦ = ( w ♦ ) τ and the quotient T ( w τ ) / ( T w ) τ is never algebraic. We conclude that T under these conditions is symplectic for() σ and that the involutive set S has a corresponding extended involutive set withrespect to () σ . Proposition 6.1.
The mapping T when planar (pure) preserves analyticity. The reflection principle.
Consider Φ ∗ = { ϑ ∗ e − ϑ ∗ χ ∗ = const ∃ ϑ ∗ } andΦ = { ϑ e − ϑ χ = const ∃ ϑ } . Consider the Legendre transform R , according toRE < R ( χ ) , χ > − χ ∗ /χ = f ( χ ), where f has slow growth like e − ϕ as 0 ≤ ϕ → ∞ . Thus, R ( e ϑ ) = e − ϕ + ϑ and ϑ ∗ = − ϕ + ϑ . Note that if T is considered as acontinuous morphism on a Banach algebra A with T ( e ϑ ) = e ϑ △ , then ϑ △ ∈ A . Forinstance if ϑ T is algebraic in T or a quotient of algebraic functions, for T close to 0,then the same holds for ϑ △ T .The spectrum for R ( f ) /f contains under the conditionson f both 0 and ∞ . If R preserves first surfaces, then we can extend the definitionof T to T (0) = 0, that is a part of the boundary. Since T is pure if we assume thecorresponding form closed, we have an analytic mapping.Assume that the segments γ, γ ∗ have a point in common P = P ∗ . As γ = R Γ dγ = R Γ ⊥ dγ ∗ = γ ∗ , where Γ is a closed contour Γ ∼ Γ . We thus have areflection principle for T expressed in preservation of the condition on flux. Thatis F lux ( T W ) = R Γ d T W . Staring with the condition − H/G ∼ T W/ T W , weare assuming F lux ( T W ) = F lux ( T W ) = 0, reflecting symmetry with respect toharmonic conjugates. Through the condition on parabolic singularities, there mustbe a point at the boundary, where T W = 0 such that W = 0. If the point is singularfor the associated dynamical system ( M, W ), the condition must be symmetric, thatis must have M = W = 0. Thus the condition on flux is necessary for regularityfor this dynamical system. Consider T ( χ ) = χ △ such that γ . . . γ N ∼ T ( γ . . . γ N ) ∼
1, that is preserves the closed property. The proposition is thusthat T preserves flux, but through symplecticity, that is T f = f implies f a point,it is not considered to be a normal mapping on a non-trivial set at the boundary.6.10. Codimension one singularities.
To determine if the symbol correspondsto a hypoelliptic operator, we must prove that every complex line, transversal tofirst surfaces, does not contribute to lineality. A complex line is considered as
OME REMARKS ON TR`EVES’ CONJECTURE 25 transversal, if it intersects a first surface and the origin. We are assuming that F is the lifting function with ∆ F = − W on a domain Ω (and the associated equation∆ F ♦ = − M on Ω). The boundary condition is assumed parabolic, that is Γ is suchthat P ( δ T ) F T = 0 implies T = 0 for a polynomial P and F T = F in T = 0. Weare thus assuming that the parametrix is invariant at the boundary, E T F = F on Γ. Let φ T = E T F . We are assuming in this approach that W T is definedby ∆ φ T = E T W + R T . Assume the operator T is continuous at the boundary,with T W = T W ♦ , we then have T W ♦ = ζ T W , for ζ ∈ C and | ζ | = 1. In aPuiseux-expansion, we have that the coefficient for t , χ ∗ = a + a t + . . . is = 0 and t = √ ζ . Thus, the order of the critical surface is one and we have singularities ofcodimension 1. When T preserves the order of contact, we have the same conclusionfor the multiplier χ .A co-dimensional one variety S ( p ), is such that S ( p, x ) = 0 and s x = 0, where p = s x is a characteristic variety, if g ( x, s x ) = 0, p.dx = 0 and g x dx + g p dp = 0, dS = pdx . As before, we have r ′ T ds = ( r ′ T p ) dx . S ( p ) is involutive for g, γ , if H g ( γ ) = 0 (the Poisson bracket). For parabolic singularities at the boundary, weare considering isolated singularities in higher order derivatives ψ N − = d N − ψdT N − = 0and ψ N = 0. Assume g is defined through dS = dgdx dx + dgdy dy + dgdx ∗ dx ∗ + dgdy ∗ dy ∗ = − Y dx + Xdy − b Y dx ∗ + b Xdy ∗ , that is dS = W ♦ + T W ♦ . Symplecticity givesthat dγ ∗ T /dγ T = const. ⇒ T = 0. We have a canonical symplectic form, d
= 0 on S ( p ), where p = dϕ/dx and d ( dϕ ) = 0. Note that we are assuming T : x dχdx → x ∗ dχ ∗ dx ∗ and we must assume x dχdx > x ∗ dχ ∗ dx ∗ > T is algebraic, the spectral theorem can be applied with advantage and wemust assume that T does not have zero’s on the boundary, that is x ∗ = dχdx and < dχdx , x > = < x ∗ , dχ ∗ dx ∗ > = 0. These sections correspond to T : < x ∗ , x > → < x, x ∗ > (normal sections). We are not assuming the trajectories in a reflexive space, but itis sufficient to consider the Lagrange case Γ ⊥ = Γ.6.11. The mapping T and parabolicity. Consider T over a set where it isalgebraic in H ′ , for instance Legendre. We have seen that there can be no closedcontour in the infinity. On the other hand, if T e ϑ = e ϑ ∗ + ϕ △ , where ϑ ∗ is related to ϑ through a Legendre transform, we have that e ϕ △ can define a circle in the infinity.Further, if ϑ △ T = P ( T ) ϑ ∗ and ϑ T = P ( T ) ϑ , taking the closure of the domain meansthat ϑ △ T = e α ( T ) ϑ ∗ and that ϑ △ T /ϑ T = e α ( T ) − α ( T ) ϑ ∗ /ϑ and we may well have that α ( T ) − α ( T ) → T → ∞ simultaneously as T → T preserves the parabolic property for the stratification,that is it maps exponentials on exponentials. Note that we have a parabolic approx-imation if and only if for all functions u harmonic in a neighborhood of the idealboundary, with finite Dirichlet integral, we have vanishing flux (cf. [1]). Assume T maps finite Dirichlet integrals on finite Dirichlet integrals. We have defined T such that { W = 0 } → { c W = 0 } , why ∆ F = − W → ∆ △ T F = −T W and wesee that T preserves parabolic approximations. Consider T algebraic, in the sensethat T ( e − v χ ) = T ( e − v ) T ( χ ). We then have that T maps constants on constants,why if χ = e v , we have that T ( e − v χ ) = const. and T ( χ ) = T ( e v ) = e v ∗ . As-sume that T ( e − v χ ) = e ϕ △ , that is T maps constants on to exponentials and that T ( e − v χ ) = e ϕ △ . Thus, if χ △ = T ( χ ), we have that χ △ = e v ∗ + ϕ △ . If T = T , thereis no room for a closed contour, in the infinity. If we assume T ∼ T , in the sensethat e ϕ △ = T ( e − v χ ) = T ( e − v χ ) along two different paths, the result depends on the property of monodromy for the stratification. That is if T is locally injectivewith respect to path, we can write T T ∼ I .6.12. The vanishing flux condition in phase space.
Consider the linear func-tional L ( φ ) = R β dφ and consider the difference L ( e φ ) − e L ( φ ) = (cid:2) b I, L (cid:3) − (cid:2)b L, I (cid:3) .We note that the vanishing flux condition L ( ϕ ♦ ) = 0 does not imply L ( e φ ♦ ) = 0.Assume that ( β ) is a neighborhood of the origin and note(3) Z β ˜ W ♦ = Z ( β ) W dxdy where ˜ W ♦ = P dx + Qdy . Immediately, we note that b L ( φ ) = L ( e φ ) = R β de φ bounded in the infinity implies that b L ∼ P ( T ), as T → ∞ . Note that if theindicator for L is α , then the functional L has support on a ball of radius α . Startingwith (3), if ˜ W T is algebraic in T , then the measure for ( β ) is zero, that is L ( e φ )is of real type. Note also that R β ∗ d T W = 0 does not imply R β dW = 0, however R β ∗ d T W = 0. Particularly, if e φ = e e α , we can consider (cid:2) L, (cid:2)b I, b I (cid:3)(cid:3) = (cid:2)b I, (cid:2)b I, L (cid:3) .For instance if R β dφ = −∞ then L ( e φ ) = e L ( φ ) implies L ( e φ ) = 0. Assume ˜ φ T algebraic in T , then F lux ( φ T ) = 0. In the same manner if φ T is harmonic, then F luxφ T = 0, further if ˜ φ T is algebraic in T , then R β dφ ♦ T = 0 implies ( β ) hasmeasure zero. Thus, if L ( e φ T ) = 0 implies that the measure of ( β ) is zero. We notethe following result. Assume ˜ φ T algebraic in T , then d ˜ φ T , d ˜ φ ♦ T are closed, whichimplies that φ T is harmonic, why we have a real type operator. Note that if φ T isharmonic on a disc, the mean is constant ( ≡ −∞ ) and the measure for the idealboundary ( β ) is zero. 7. Boundary conditions
Introduction.
In the model, the singularities on the first surfaces to the sym-bol are mapped on to the boundary of the stratification, which is parabolic or moregenerally very regular. Hypoellipticity is a condition on behavior for the symbolin the infinity but the method using T (basically the projection method) requiresa discussion on the simultaneous behavior at the boundary. The boundary to thestrata is defined by {M N ( f ) = f } but we also discuss the phase correspondent {M N (log f ) = log f } .7.2. The δ -Neumannproblem. We will deal with the following problem, given aregular approximation of a singular point in the boundary, Γ, determine F T suchthat i) F T ( γ )( ζ ) = f ( r T ζ )ii) F is holomorphic in γ and algebraic in T .We assume here that the boundary is finitely generated and in semi-algebraic neigh-borhood. The first part of ii ) is the lifting principle for a semi-algebraic domain.For i ) we note that as f ∈ ( I ), a finitely generated ideal, there is no problem todetermine γ , such that the formal series for F T converges. Consider now the secondpart of ii ). Given a regular approximation U T with algebraic dependence of theparameter T , we can use the δ − Neumann problem to determine a lifting function F T such that δ T F T = δ T U T and F T = U T + L T . We assume the boundary finitelygenerated and we can in a suitable topology assume that given F T , there is a do-main of γ such that the dependence of T is as prescribed and F T ( γ ) = f ( r T ζ ). The OME REMARKS ON TR`EVES’ CONJECTURE 27 domain in ζ is a neighborhood of the first surface generated by one variable T andit is pseudo convex.Note the symmetry condition that if for T ∈ V , (cid:2) Lr ′ T (cid:3) ∗ = r ′∗ T L ∗ is algebraicin T corresponding to a real coefficients polynomial in T , we have that r ′ T − c T I holomorphic, that is r ′ T is holomorphic modulo monotropy. This means that for analgebraic dependence of T , δ T F T can have level surfaces.7.3. Multipliers.
As the mapping x → x ∗ preserves order of zero, we see thatif x is locally reduced, then the same holds for x ∗ . Consider the system of in-variant curves { C j } = { ( x, h j ( x )) } . Assume η j ( x ) = h ( x j ) x such that η j ( x ) = x j − η ( x ) = xη j − ( x ). Assume also that we identify using monotropy, the curves { C j } ∼ m { ( x, h ( x j )) } . Assume < ηX, b ϕ > = < η ∗ b X, ϕ > = C < b X, ϕ > . Thus, { η = const } → { η ∗ = const } . The condition η ∗ = const. ⇔ η ∗ = const./x ∗ and soon. Assume existence of an algebraic homomorphism w , such that | w − η ∗ − x ∗ | < ǫ as | x ∗ |→ ∞ . Then η ∗ ( x ∗ ) ∼ m w ( x ∗ ) as | x ∗ |→ ∞ and that η ∗ preserves constantvalue in the x ∗ − infinity, why the projection method can be applied to η ∗ . It is ofno significance what level surface we start with, that is η ∗ j = const. ⇒ η ∗ preservesa constant value in the infinity. Lemma 7.1.
Given a system of invariant curves { ( x ) , h j ( x ) } such that h ( x j ) ∼ m h j ( x ) , we have that η ∗ preserves a constant value in the infinity and the projectionmethod can be applied. Assume existence of a finite j such that for a lifting function F T , d j dT j F T isalgebraic in T , that is F jT ∼ α j ( T ) F , in a neighborhood of T , where α j is alocally defined polynomial. Assume F jT ( x, y ) = G ( e x , y ) → | x |→ ∞ for a G ∈ H ′ and y = h ( x ) finite in modulus. Thus, G ( e x , h ( e x )) = G ( e x , e ˜ h ( x ) ) is therepresentation we prefer. We have also assumed that F jT preserves a constant valuein x as | x |→ ∞ and y finite. If ddT F ( γ T ) = F ( dγ T dT ) and θ T ∈ T Σ, we see thatif F is algebraic, then θ T = F − ddT F ( γ T ) for a γ T ∈ Σ. Further, < γ T , θ T > = 0implies γ T ∈ { B ( γ T ) = µ } , for a constant µ and B algebraic. Assume B self-transposed and such that B ( F ( γ T )) = t F B ( γ T ). A sufficient condition for F tomap Γ ⊥ → Γ ⊥ , given that Γ is symmetric, is that it maps constants on constants.For a finitely generated boundary, we have the following result. Assume thesingularity at the boundary, described by T , that is F T − µ →
0, for a smallconstant µ , such that F /T is close to a polynomial as T →
0. We are assuming γ fix at the boundary and γ T ∈ a neighborhood of γ with F T ( γ ) = F ( γ T )and where | w − F T − /T | < ǫ , for an algebraic homomorphism w , as T → F T invertible over regular approximations, such that dγ T dT = F − φ T = 0and d dT γ T = ddT F − φ T = 0. Thus, if F maps constants on constant, we have that F maps singular points on singular points and regular points on regular points.7.4. The orthogonal to the boundary.
Assume F preserves a constant valueas | x |→ ∞ and | y |→ ∞ . Assume F ( ηX ) = η ∗ F X . Degenerate points are thenon the form y ∗ dx ∗ − x ∗ dy ∗ = 0. If F is not algebraic, we are at least assumingthat F − maps constants on constants or that F ∼ m an algebraic function closeto the boundary. Let < y ∗ , y > = Re < y ∗ , y > −
1, for y ∈ Γ and y ∗ ∈ Γ .Then if < y ∗ , y > = 0, Γ is a line if Γ is a line. We write y ∗ ⊥ y for < y ∗ , y > = 0and we define bdΓ as the set where this relation holds over Γ. If Γ = Γ, thenΓ ⊂ Γ. If < y ∗ , · > is reduced, we have isolated singularities at the boundary of Γ. If < y, y > = 0 for all y ∈ Γ, we have that Γ ⊂ bd Γ . Since y → y ∗ isa contact transform, we have N (Γ) ⊂ N (Γ ). Further, assuming y ∗ → y ∗∗ is acontact transform with y ∗∗ ∼ y , we get rad (Γ) ∼ rad (Γ ) (equivalence in senseof ideals). Consider with these conditions Γ = { y < y ∗ , y > = 0 y ∗ ∈ Γ } ,then also rad (Γ ) ∼ rad (Γ ⊥ ). Note that ( T Σ ⊥ ) does not completely describe themicro-local contribution.We have a few immediate results. Let Γ T = { γ T γ T = r ′ T γ γ ∈ Γ } andassume the boundary condition (3.1) Proposition 7.1. r ′ T − is locally algebraic if and only if Γ T ∼ Γ ⊥ T . Proof:We are assuming r ′ T γ ⊥ γ for γ ∈ Γ, that is < r ′ T γ, γ > = 0 for γ ∈ Γ impliesΓ T ⊂ Γ ⊥ . Assume further the ramifier symmetric, in the sense that < r ′ T x, y > =
We define < γ T , θ T > = Re < γ T , θ T > −
1. Thus, < F ( γ T ) , i Im θ T > = < Re F ( γ T ) , θ T > and if θ T ∈ T Σ has the property that θ T ∈ T Σ ⇔ θ T ∈ T Σ, we have that Re F ( γ T ) ⊥ Re θ T implies iF ( γ T ) ⊥ θ T . Con-versely, if T Σ is symmetric with respect to the origin, we have existence of γ T such that Re F ( γ T ) ⊥ θ T implies F ( γ T ) ⊥ i Im θ T and analyticity for Im F ( γ T )only means that < F ( γ T ) , θ T > = < F ( γ T ) , θ T > . Assume dFdγ :Γ ⊥ → Γ ⊥ , suchthat < dFdT ( γ T ) , θ T > = 0 implies < dγ T dT , θ T > = 0. Over { F T = ddT F T } , we thushave that if F T ⊥ θ T , we have existence of Tr`eves curves. Conversely, given exis-tence of γ T such that < dγ T dT , θ T > = 0 implies < ddT Im F ( γ T ) , θ T > = 0, why < ddT F ( γ T ) − ddT F ( γ T ) , θ T > = 0 which is always true for real T . We conclude ashas been noted before that the condition that F T is analytically hypoelliptic doesnot imply that the real and imaginary parts are analytically hypoelliptic.Note that it is possible to have ( dI ) has a global pseudo-base, when the pseudo-base for ( I ) is only local. However, Proposition 7.1. If ( J ) is a finitely generated ideal of Schwartz type topology andwith a compact ramification, such that r ′ T φ/φ ∼ m in the ζ − infinity. Then, ( I ) ∼ ( r ′ T J ) has a global pseudo base. Proof:We are considering ( I ) ∼ ( r ′ T J ), where ( J ) = ker h and as before η ( φ ) = h ( φ ) /φ .Through the conditions, we can satisfy | r ′ T h ( φ ) | < c + | h ( φ ) | , for a constant c andfor φ ∈ ( J ), so r ′ T is quasi conformal and ( I ) is finitely generated, if ( J ) is. Giventhat h is algebraic and such that h N = 1, we can show that h ( r ′ T φ ) ∼ m φ reduced, we have that η ∼ m I ). If h is analytic, we assumelocally h is monotropic to an algebraic homomorphism. Thus, we can find an entirefunction γ such that γ ∼ m η . (cid:3) Analytic set theory.
Starting with the boundary condition in a higher, finiteorder derivative F ( j ) T = const. implies T = 0. Consider the sets Σ = { ζ T F T = const } , Σ = { ζ T F T = const., F (1) T = const. } and so on. This gives a finite OME REMARKS ON TR`EVES’ CONJECTURE 29 sequence Σ j ↓ { T = 0 } . We can form the corresponding ideals in γ , such that N ( I j ) = V ∪ . . . ∪ V j , where V = { ζ T F T not constant } . If we assume algebraicdependence of the parameter for F T − const.I and that we have a neighborhoodof ζ T that is a domain of holomorphy. Then the sets V j as geometric complementsof algebraic sets, are analytic. We have noted examples where V * V . We alsonote the following example, assume F T = α T /β T such that α ′ T = γ T β ′ T , where γ T is assumed non-constant and regular holomorphic (not-Fuchs equation), then if V , V are analytic, then since V * V , the inclusion V ⊂ V ∪ V is strict and wehave for the corresponding ideals I ⊂ I . For a parabolic approximation, the set { ζ T ϑ T ( ζ ) > } is the geometric complement to a first surface, which is with theconditions above an analytic set. Note that we may still have that the sets V ⊥ arefirst surfaces.A different argument can be given using Tr`eves-curves. Assume Ω a domainof holomorphy with ddT γ T = 0 on Ω. If R Ω ( ddT γ T ) θ T dT = 0 and if we assume theintegrand holomorphic, we must have θ T = 0 on Ω, assuming it of positive measure.Assume bdΩ on one side locally of a hyperplane, then we have that γ T = 0 on Ω.Assume now ∆ ⊂ Ω where ∆ is algebraic. Then, γ T has no zero’s on ∆, but is notconstant. We have assumed that constant functions are not holomorphic and wemust also assume that they are not algebraic. For instance the complement to a firstsurface in a domain of holomorphy is not necessarily an analytic set. Note furtherthat if ∆ is algebraic, we can assume ∆ ⊥ is not algebraic and with the conditionsunder hand, it must be a first surface. Thus, we have that Ω \ ∆ is analytic andsimultaneously ∆ ⊥ is a first surface.7.7. A Tauberian problem.
Let V N = { ζ T e M N ( η T ) M N ( η T ) subharmonic } ,which is a subset to V N +1 . Assume V the set corresponding to η subharmonic. Con-sider the complement set in V N , V c = { ζ T η > M ( η ) } , then V ⊂ V N , throughthe conditions and V c is analytic, if V is analytic. We have that log X ∼ I = { η ≥ } , In the same manner we consider I = { η ≥ } = { η + M ( η ) ≥ } , . . . , I N = { η N ≥ } = { P N − j =0 M ( j ) ( η ) ≥ } . Associated to these ideals, we con-sider J = { e η η ≥ } and so on and we have if e η N is analytic, that N ( J N )contains a path ζ T that is continuous. The proposition is thus that given a Taube-rian condition, we have existence of a continuous approximation of a singular point.For instance if we have that J N is defined by an analytic function and the set V c above is analytic, then we have existence of a continuous path in N ( J ).7.8. The distance function. If e g represents the distance to isolated (essential)singularities, all situated on a finite distance from the origin, then this distancefunction is globally reduced. For a holomorphic function u , bounded in the infinity,we must have that the distance to essential singularities is finite. It is sufficientto consider points P in a punctuated neighborhood of the origin. For a harmonicfunction u , we have that it is bounded in the finite plane, and we only have to applyPhragm´en-Lindel¨of’s theorem. Consider the representation u = e g + m harmonic,where m is symmetric. Over a parabolic approximation where − g − m is subhar-monic, we assume m → d globally reducedand that d →
0, as P → P ∈ Γ. Further, d ( P ) → ∞ , as P → Γ and d ( P ) → ∞ ,as P → Γ. Then we can find ǫ small such that | d ( P ) − d ( P ) | < ǫ , as P → Γ. If allsingularities for u are at a finite distance from the infinity, we have that d ( P ) → P → Γ over the set where d ( P ) > P = P . The singularities at the boundary are assumed removable. Assume P = 0and that, for instance d ( P ) ∼| P − P | , then d ( P ) ∼| P − P |→ P = P , using reducedness for d and we can conclude that the Dirichlet problem ∆ u = 0 on a set (it is sufficient to assume parabolic) with boundary value d issolvable, modulo monotropy. Proposition 7.1.
Assume the boundary holomorphic and only with parabolic sin-gularities, then there is a regular approximation of a singular point that will reachthe point.
Localization at the boundary.
Assume P ( δ T ) is the operator used to definethe boundary condition, such that P ( δ T ) is hypoelliptic. C T ( φ ) = P ( δ T ) φ − φP ( δ T ), where φ is a real test function and where P is assumed such that P ishypoelliptic. Thus, P ( δ T )( φF T ) = φP ( δ T ) F T + C T ( φ ). If P is hypoelliptic, we havethat C T ≺≺ Re P T . Otherwise, we will assume that P ( δ T )( φ ) − P ( δ T ) φP ( δ T )+ P ( δ T ) φP ( δ T ) − φP ( δ T ) ∼ Im P ( δ T ) ≺≺ Re P ( δ T ). Thus, if P ( δ T ) C T ( φ ) + C T ( φ ) P ( δ T ) ∼ Im P ( δ T ). As T →
0, we have that C T ( φ ) → φ = 1at the boundary). Using Nullstellensatz, that is P C T + C T PP →
0, as 1 /T → | P C T + C T P | < ǫ , for large T ( and real). Let ( P C T ) ∗ = C ∗ T P ∗ and if P ∗ = P implies C ∗ T ∼ − C T . Symmetry with respect to ∗ gives C T ≺≺ Re P in the infinity,for P such that the square is hypoelliptic.Assume now that the boundary condition is given by a differential operator(reduced) P ( x, ddT ) such that there is a function g N ∈ L in the parameter close tothe boundary, ( g N = M N ( f )) with P ( x, ddT ) g N = I − r N , where r N is regularizingas a pseudo differential operator. If we regard g N as an operator L → D ′ L , wecan construct g N as an operator with kernel G N ∈ D ′ L , that is a parametrix, g N ( φ ) = R G N ( x, y ) φ ( y ) dy . Given a parabolic boundary condition, we can assume P ( x, ddT ) g N ∈ D ′ L for x in a neighborhood of a point x at the boundary andwith sing supp g N = { x } . Assume φ a regular approximation of the singularpoint such that P ( x, ddT ) φ = 0 implies x = x . For the parametrix, we thenhave < G N , P ( x, ddT ) φ > = 0 implies x = x (modulo regularizing action), that is P ( x, ddT ) G N ⊥ φ implies x = x Further remarks on the boundary. If r ′ T is an algebraic homomorphism,then r ′ T e φ = e r ′ T φ , and consequently R ( I ) r ′ T e φ dz ( T ) = R ( I ) e r ′ T φ dz ( T ) = 0, implies m ( I ) = 0 (measure zero set), using a result by Hurwitz. If (cid:2) e v △ T − (cid:3) is holomorphic,we have either that e v △ T ≡ m ( I ) = 0. Lemma 7.2.
Assume e v △ T − is holomorphic in the parameter T with v △ T algebraicin T . We then have that R ( I ) ( e v △ T − dz ( T ) = 0 implies m ( I ) = 0 . We also note the following consequence of the condition on vanishing flux, R ( C ) M dxdy = R C ˜ M ♦ = 0, means that there is a trajectory γ such that M ( γ ) = 0 in( C ). Particularly, if f is such that R S M N ( d ˜ f ) ♦ = 0 that is we have R ( S ) ∆ M N ( f ) dxdy =0, we must have that M N ( f ) changes sign in points inner to ( S ).7.11. A very regular boundary.
The boundary is said to be very regular, ifthe singularities are located in a locally finite set of isolated points or segmentsof analytic curves (cf. [18]). Thus, we are assuming that if f is a boundaryelement, then a very regular representation of the boundary preserves the localityof singularities, but not necessarily the order. Assume Γ = { Γ j } is a locally finiteset of analytic curves, where the set of common points is a discrete set. Given anelement in ( B m ) ′ ⊂ D ′ L , we know for the real Fourier transform, that b f = P ( ξ ) f ,where f is a continuous function in the real space and P a polynomial. Extend f to OME REMARKS ON TR`EVES’ CONJECTURE 31 a continuous function in a complex neighborhood of the real space and denote ˜ f thefunction such that b ˜ f is the extended function. More precisely b ˜ f | R n = f . Assume˜ f has a very regular representation at the boundary, with isolated singularities.Then b f = 0 from P ( ξ ) = 0, gives an extension of singularities to Γ j , locally algebraicsegments. At the boundary, in a complex neighborhood of the real space, we areconsidering the symbol as F ( γ ) = P ( D ) ˜ f .Consider in d D ′ L , R ( ζ ) f ( ζ ), where f is the Fourier transform of a very regularoperator, that is F ( γ ) ∼ R ( D ) ˜ f , where b ˜ f | R n = f and ˜ f is very regular. ConsiderΓ → Γ ∗ through a simple Legendre transform. If we assume Σ discrete and thatall approximations of Σ through Γ are regular (transversal intersection), then wecan assume existence of a norm ρ , such that ρ ( z ) ≤ ⇔ z ∈ Γ. In conclusion, weare assuming a very regular boundary continued to δ Γ − γ δ , that corresponds toa normal tube in Ω, thus that all singularities are situated on first surfaces. Theparabolic singularities can be given by a very regular boundary, that d k dT K F T = 0implies T = 0 and it corresponds to a very regular representation in the right handside. If the differential operator is given as hypoelliptic, then F T is very regular.7.12. A very regular representation.
For a boundary operator L , a very regularrepresentation is given by L ( f ) = f + γ δ ( f ), where γ δ is regularizing, for δ >
0. Notethat for a finite N , the term M − N γ δ is still regularizing, for δ >
0. We note that inthis representation, the locality of singularities is not affected by the means, but theorder of singularity is decreased by the mean and increased by the mean of negativeorder. Thus, given singularities of finite order, we must have that application ofa finite order mean, decreases the set of singularities. Through the result fromIversen, we see that the set of singularities in a very regular boundary, must beassumed of measure zero. Note however, that if M N ( f ) is locally injective, thenthe corresponding M − N ( f ) is locally surjective. We are assuming the neighborhoodof Γ one-sided, that is γ δ ( f ) ≥ f ≥ R bd ( dLf ) △ = R bd df △ = 0, since dγ δ = 0 for δ = 0. That is if wehave a ”planar” reflection through the boundary, this is preserved by the boundaryrepresentation.7.13. The extended Dirac distribution.
Assume Σ a set of common pointsfor finitely many analytic curve segments, a discrete set without positive measure.The boundary condition corresponding to a very regular boundary can now beformulated, f T is regular outside Γ ∪ . . . ∪ Γ N , that is at least one of the seg-ments Γ j is singular for f T . This means that if the boundary element is δ Γ − γ δ ,then (cid:2) δ Γ , δ Γ (cid:3) = (cid:2) δ Γ , δ Γ (cid:3) . Only points in Σ give raise to a commutative sys-tem. Further, the system will be finitely generated in the sense that finitely many(sufficiently many) iterations of δ Γ j , will for different Γ j produce regular points.More precisely, assume Γ is a singular analytic curve, for φ and Γ is a regularanalytic curve for φ except for a point in Σ and in Γ ∩ Γ . Then φ | Γ | Γ is theresult of a regular approximation of a singular point, but φ | Γ | Γ gives a singularapproximation of a singular point. Compare Nishino’s concept of a normal tube(cf. [15]).Consider δ x → δ Γ , where Γ has an analytic parametrization. This meanslim Γ ∋ x → x φ ( x ) = φ ( x ). Assume the set Σ = ∩ j Γ j discrete (compact), that isalgebraic. If r Γ , Σ is the restriction homomorphism H Γ ( V ) → H Σ ( V ), for a do-main V and if T ∈ H ′ I, Σ ( V ) and we have existence of U T ∈ H ′ I, Γ ( V ) such that T = r Γ , Σ ( U T ), then we say that T has a continuation to Γ. Thus, we can see δ Γ as a continuation of δ Σ to a very regular boundary. If V is a domain of holomorphyand if the definition of a normal operator L at the boundary, is not dependent onchoice of Γ, we say that { Γ } is a quasi porteur for δ Γ . When Γ is analytic, we saythat it is a porteur. 8. A monodromy condition
Introduction.
Since we are discussing the symbol in ( B m ) ′ , there is no ob-vious monodromy concept that can be assumed. Assume condition ( M ) is thecondition that all means are of real type. For f ∈ ( B m ) ′ we have a local represen-tation f ∼ P ( D ) ˜ f , where close to the boundary ˜ f is very regular. Assume thelocal condition ( M ) is the condition that P ( D ) is hypoelliptic. Finally,Assume pr : L → R ( L λ ), for a formally self-adjoint and hypo-elliptic differ-ential operator. Assume L λ ∈ φ ( D ′ L , L ) (unbounded Fredholm-operator) then D ′ L = X N N ( L λ ), L = Y N R ( L λ ) and the inverse B λ is bounded in ( L , D ′ L ).Through a fundamental theorem in Fredholm theory, we have existence of B λ ∈ Φ( L , D ′ L ) such that B λ L λ = I on X and L λ B λ = I on R ( L λ ). Proposition 8.1.
Assume E λ a parametrix to L λ a hypoelliptic operator in L ,then ( δ x − E λ ) is regularizing. Conversely, if E λ is regularizing, then δ x − E λ ishypoelliptic. We give a short proof, if R is regularizing then || ψRu || s ≤ C , for a constant C = C ( ψ, s ) and ψ ∈ C ∞ and a real s and u ∈ D ′ . We can write || ψu || s ≤ C || ψ ( A − R ) u || s , where A = E − I is hypoelliptic. If E is a parametrix P Eu − u ∈ C ∞ ,for a u ∈ D ′ and if P hypoelliptic in L , P Eu − P u ∈ C ∞ implies Eu − u ∈ C ∞ ,that is ( δ x − E ) is regularizing. The result can e extended to D F ′ , then I E − I isregularizing in the space of D F ′ , but the result can not be extended to D ′ .The condition ( M ) is the condition that the symbol considered as a parametrixto a boundary operator has a trivial kernel.8.2. The (M)-conditions and orthogonality.
Assume T γ corresponds to ana-lytic continuation. We will assume that T γ can be divided into translational move-ment and rotational movement, not necessarily independent. For V = ( V , V )the vorticity is given as w = δV δx − δV δy . Given that dydx is bounded in the infin-ity, we have that dV dV dydx = 1 in the infinity iff w = 0 in the infinity, that is ifthe limit V V = e − ϕ →
0, in the infinity, is a “simple zero“. If e − ϕ T ∼ P (1 /T ) ∼ c + c /T + . . . , then we must have c = 0 and c = 0. If we compare with theglobal problem, the condition r ′ T f − f = 0 implies T = 0 is locally a condition ( M )and r ′ T log f − log f = 0 implies T = 0 is related to local parabolicity. The conditionthat r ′ T log f = log r ′ T f , means that the ramifier only contributes to the phase. Ifwe assume V ⊥ V , we can find a polynomial P such that V V ∼ m P , in the infinity.Note in connection with the conditions ( M ), if we assume V ⊥ V , then for ahypoelliptic symbol both V and V will be unbounded in the infinity and thus therespective inverse is bounded in ∞ . In presence of lineality, we will later arguethat the imaginary part can be assumed bounded, where we assume Im F /T → / Im F T preserves conditions ( M ). Thus, in order to discuss the conditions ( M )using only bounded symbols for orthogonal parts, it is necessary to assume F adjT − F T bounded, where F adjT is the adjoint symbol in ( B m ) ′ . OME REMARKS ON TR`EVES’ CONJECTURE 33
A condition ( M ) operator at the boundary. We now wish to define aformal condition ( M ) operator on ( B m ) ′ . This operator can be used to define aglobal condition ( M ). For such an operator, we must have that t T γ has representa-tion with a point support measure µ . For a condition ( M ) with respect to paths,the limit must be independent of starting point, why we must assume µ ∈ E ′ (0) ,that is of order 0. If we have the parabolic property, we must further assume themeasure is positive in phase space. For a formal ( M )-operator, we only assume F ∈ L implies t T γ F ∈ ( B m ) ′ . Consider now t T γ modulo regularizing action, thatis t T γ ∼ δ + ν γ , where ν γ has support outside a point. If we assume t T γ F ∈ B m , wehave [ t T γ F = Qf . We write t T γ F = Q ( D ) ˜ f , where ˜ f is a very regular operator.Assume F ∈ B m , then we have F ∼ Q λ ˜ f . We can in this context consider ˜ f as aglobal representation. Given that Q λ is hypoelliptic and such that Q λt T γ = t T γ Q λ we see that in the case with condition ( M ), t T γ preserves parametrices to Q λ .Conversely, given that t T γ preserves very regular parametrices to Q λ (that is wehave condition ( M )) then we can derive that Q λ is hypoelliptic. The conclusionis that with the conditions above, we have that hypoellipticity for Q λ means that t T γ defines condition ( M ) and conversely. Note also that t T γ can be globally rep-resented in ( B m ) ′ . Given Q , we can define the possible continuations that preservecondition ( M ), as t T γ such that t T γ Q = Q t T γ . For instance if t T γ F = F + F , wemust assume QF = 0, why F is regularizing. If on the other hand t T γ F = cF ,for a c in the ∞ and t Q = Q ∗ , then for all real c , we have cQ − Qc ≺≺ Q , that is t T γ Q − Q t T γ = 0 modulo regularizing action. Thus, localizing with a real c is pos-sible and corresponds to localizing ˜ f . Given that F is hypoelliptic and representedas Q ˜ f at the boundary, this property can be continued using t T γ , given that thisis an algebraic action. Assume for this reason, ( t T γ F )( ϕ ) = R Ω k γ ϕdσ = 0 implies Qϕ = 0 on Ω or σ (Ω) = 0, that is in the case where t T γ is algebraic, we havecondition ( M ). Note that R Ω ˜ f ϕdσ = 0 implies σ (Ω) = 0, which corresponds to”isolated zeros“ to ˜ f . In the case where t T γ ( ˜ f )( ϕ ) = 0, when t T γ ˜ f is algebraic,we see that σ (Ω) = 0, that is we have condition ( M ).Note that if t T γ ˜ f = 0 alonga line in Ω, this can be compared with the extension δ to δ Γ , where condition ( M )is not longer with respect to a point. Lemma 8.2.
Assuming the symbol has a representation in ( B m ) ′ satisfying con-dition ( M ) , then this property is preserved if the continuation t T γ as above isalgebraic. Condition ( M ) does not however imply that t T γ is algebraic.8.4. The operator T and the conditions (M). We have that ( w µ ) ♦ = ( w ♦ ) µ but ( T w ) µ = T ( w µ ). Assume w = W/M ∼ e ϕ → T e ϕ = e ϕ △ .The problem iswhen e ϕ △ respects the conditions (M). Assume H ( w ) = w ♦ and T Legendre andthat we have algebraic dependence of the parameter T , then T Hτ µ − τ µ T H ∼ e P ( T ) − e P ( T ) , for a polynomial P . The condition ( M ) means that e P ( T ) − P ( T ) isnot ∼
1. Thus, the operator T does not necessarily preserve the condition ( M ).The same holds for the conditions ( M ),( M ).If the operator T is studied using t T γ acting on the Legendre transform, we seethat an algebraic continuation in the infinity implies that we do not have linealityand further the conditions ( M ) and ( M ) in parameter infinity. If we only assumethe continuation very regular on all strata, we still have conditions ( M ) , ( M ) and( M ), but not necessarily an algebraic continuation. We can consider it to be alge-braic modulo monotropy locally. Finally note that an algebraic continuation doesnot necessarily preserve condition ( M ). However, by considering symbols modulo regularizing action, we can restrict the representation to real type symbols. Forthis representation the corresponding functional is an infinite sequence of constantcoefficients polynomials acting on a point support measure.9. Further remarks on algebraicity
Introduction.
An algebraic set is geometrically equivalent to a zero set ofpolynomials. Characteristic for an algebraic mapping L is that L ( e ϕ ) = e L ( ϕ ) and the zero set to an algebraic mapping is locally an algebraic set. The identity(evaluation) is considered as algebraic and we consider any operator that commuteswith the identity in H ′ as (topologically) algebraic.9.2. Clustersets for multipliers.
We will prove that given that M ⊥ W implies V ⊥ V , we have that T V − N algebraic in ∞ if we have hypoellipticity. Assumefor this reason V ⊥ V with c V c V = e − ϑ ∗ →
0, as | ξ |→ ∞ If T V = e ϑ △ T V , thenwe must have e ϑ △ → ∞ , as | ξ |→ ∞ . For instance if ϑ △ = ϑ ∗ + p , where weassume p is harmonic, in the sense that p → ∞ , as | ξ |→ ∞ . Otherwise, if weassume e ϑ △ bounded as | ξ |→ ∞ , we have unbounded sublevel sets (cluster sets) forthe multipliers. When T is the Legendre transform, this will disrupt the conceptof monodromy in the infinity. We have noted that if V T is algebraic in T and F lux ( V ) = 0, we have that Im V T ≡
0. Note that if T V − N is taken as the limitover strata, in case the symbol is not hypoelliptic, we have distributional limits insymbol space for the representation of V .9.3. Example on algebraic mappings.
Consider the mapping A (cid:2) a, b (cid:3) = b ba . If E is a (topologically) algebraic mapping, and if E ( e φ ) = e ˜ E ( φ ) , then ˜ E is oddand also ˜ A . The proposition that ˜ A is odd, corresponds to E ( e b I ( φ ) ) = E ( e b I ( φ ) ) ,thus if A is (topologically) algebraic, then A is (topologically) algebraic. Notethat E ( e e − φ ) = E ( e eφ ) . If E ∼ m E and E is (topologically) algebraic, then | E ( e f ) − E ( e /f ) | < ǫ in ∞ . If E ∈ L , then b E ( f ) →
0, as f → ∞ . This meansthat E ∼ m E , where E is (topologically) algebraic, then | b E ( f ) − b E ( f ) |→ M N ( E ) ∈ L , which means that M N ( E ) ∼ m W ,where ˜ W is odd.Further, we have A (cid:2) I, E (cid:3) ( φ ) = E ( e φ ) and in the same manner A (cid:2) E, I (cid:3) ( φ ) = e bb E ( φ ) . Thus, if E ( e φ ) = e e ( φ ) , that is v ( φ ) = (cid:2) b I, e (cid:3) ( φ ), then bb E ∼ e . If E is assumed(topologically) algebraic and bb E ∼ E , then we should have that e is (topologically)algebraic (modulo algebraic sets). Further, if e is linear and bb E → I at the boundaryand if bb E ∼ E , then we must have E − I ∼ W , where W is algebraic at the boundary(with respect to concatenation of curve segments). More precisely ( E − I )( E + I ) = E − I + R , where R is assumed to vanish. If E − I → I at the boundary andthe same condition holds for E + I , we must have E − I →
0, which is seenas E being (topologically) algebraic at the boundary, that is E − I ∼ W ( e − φ ),where W is algebraic in T (compare the Lagrange condition). We are assuming W ( e φ ) W ( e − φ ) = W (1) Thus, if W (1) = 1, we have W ( e − φ ) = e ˜ W − ( φ ) and theodd condition means that ˜ W − ( φ ) = − ˜ W ( φ ).9.4. The Legendre transform has removable singularities.
Assume ϑ △ T /ϑ T →
0, as T → ∞ and further that < ϑ △ ± ϑ, ϑ △ ± ϑ > = 0 or formally ϑ △ /ϑ + ϑ/ϑ △ ∼ F △ T .F T ∼
1, as T → ∞ and the first condition, then the condition v/v △ ∼ OME REMARKS ON TR`EVES’ CONJECTURE 35 is impossible. If ϑ △ − ϑ real, then the condition contradicts symplecticity. Other-wise, we are assuming Im ( ϑ △ − ϑ ) ⊥ Re ( ϑ △ − ϑ ). The conclusion is that if ϑ △ and ϑ are related through a simple Legendre transform, then F △ T .F ∼ T
1, as T → ∞ isnot possible, that is the singularities are removable.9.5. Sufficient conditions for orthogonality.
Assume now M ⊥ W and considerthe lift V ( M, W ) = V + iV . The problem is now under what conditions wehave that V ⊥ V (that is V ≺≺ V ). Assume V is the Hamilton function, that is M = δV δy = δV δx and W = δV δy = − δV δx . We write formally the condition that M ⊥ W as δδx ( V ⊥ V ) and δδy ( V ⊥ V ). The vanishing flux condition is F lux ( T W ) = 0.The corresponding condition for T V is V ≺≺ V . Note that the lifting func-tion F ( X, Y ) and the Hamilton function V ( M, W ) may have quite different alge-braic properties, but are considered as related by involution and the condition onequal derivatives in the first order x ∗ , y ∗ -variables, with respect to ( X, Y ) − argu-ments. For V algebraic, we always have that δδx V ⊥ V , that is δδx log V →
0, as | T |→ ∞ . The condition δδx V ⊥ V is ( δδx log V ) V V →
0, as | T |→ ∞ and if δδx log V is bounded in the infinity, then we could write the condition δδx V ⊥ V as( δδx log V ) V V →
0, as | T |→ ∞ .Let W/M = e ϕ and consider the iterated mean M N ( ϕ ), for N large. We can thenassume ddx M N ( ϕ ) reduced at the boundary. Let V ( N )1 , V ( N )2 be the correspondingHamilton function (locally). Consider the condition ( δδx log V ( N )1 ) V ( N )1 V ( N )2 →
0, as T →
0. This is obviously true for large N . If δδx log V ( N )1 is reduced at the boundary,then we have that V ( N )2 /V ( N )1 →
0, as T → ∞ . Further, δδx V ( N )1 ⊥ V ( N )2 . A sufficientcondition to conclude that V ⊥ V is that V corresponds to a reduced symbol. Notethat we have assumed that V | Γ ∼ V | ∞ and V | Γ ∼ V | ∞ .9.6. The orthogonal condition and the Dirichlet problem.
Assume insteadof the condition Im F T ≺≺ Re F T , that(4) Re F T Im F T → T → ∞ The condition is to be understood using T = (Re T, Im T ) ∈ R n , n = 2 and | Im F T || Re F T |→
0, as | T |→ ∞ , where we assume that the factors are not withoutsupport in a neighborhood of the infinity. Given that F T is holomorphic in T , weknow that F T can not be reduced in the origin and in the infinity simultaneously.Assume Re F T reduced in the T − infinity, we then have that Im F T is bounded inthe infinity. Assume F △ T = t F T , so that Im F T ∼ F △ T − F T . The condition that F T is algebraic in T close to the boundary, does not imply that Im F T is algebraic in T , close to the boundary. As we are assuming real dominance for the operator, wewill assume the real part algebraically dependent on the parameter.In the same manner, given that F T ∈ D ′ L , we do not necessarily have that F △ T − F T ∈ D ′ L . Note that for the restriction to the real space, the Fouriertransform to F T is on the form P ( ξ ) f ( ξ ), why on the support of f , we necessarilyhave finite order of zero. We are assuming 0 ∈ supp f . We have that Re F T cannot be reduced in a neighborhood of parameter origin. If Im F T were bounded in aneighborhood of the origin, the condition (4) could not be possible. The conclusionis that Im F T must be unbounded at the boundary.Conversely, if we assume Re F T is reduced in a neighborhood of the parameterorigin, then we have that Im F T is bounded in a neighborhood of the boundary and we have that Im F T is necessarily unbounded in the infinity. This, does notcontradict the condition (4). Proposition 9.1.
Assume condition (4). For a hypoelliptic operator, both the realpart and the imaginary part are unbounded in the infinity. In presence of linealityfor the real part the imaginary part is bounded in the infinity.
The conclusion is that given the condition (4), if we further assume that Re F T isreduced in the infinity, then we have that Im F T can be assumed to be bounded atthe boundary. The proposition that Im F T is bounded in the infinity, means that itcan not be represented be a polynomial operator, unless the set of unboundednessis of measure zero. In this case if the set is normal (finite Dirichlet integral),we can give an algebraic representation for this part of the symbol. Sato gives awell known example of a (hyper-) function defined at the boundary, that is not adistribution in 0. Assume ϑ T algebraic in T and that ϑ △ T − ϑ T ∼ Im ϑ T , such thatIm ϑ T ∼ P ( T ) ϑ △ − P ( T ) ϑ .9.7. A necessary condition.
Assume(5) e α ( T ) − e α ( T ) e α ( T ) → T → ∞ We then have, e α ( T ) − α ( T ) − e − i Im α ( T ) →
0, as T → ∞ . A sufficient conditionfor this is that α ( T ) − α ( T ) − i Im α ( T ) →
0, where α ( T ) ∼ α ( T ). Thus, therelationship between α △ ( T ) = α ( T ) and α ( T ) = α ( T ), as a simple Legendre typecondition, means that if e α is locally algebraic, then the same must hold for e α ( T ) .Formally T (Im e ϑ T ) ∼ e α ( T ) Im ϑ where e ϑ T is locally algebraic. Note that without the condition (5), it is necessaryto consider T as acting on hyper-functions.9.8. The complement sets to the first surfaces.
Solvability in this contextcorresponds to the regularity conditions in dynamical systems and the correspond-ing conditions for first surfaces. The neighborhood { ϕ > } ∼ { ϑ = 0 } , that is P ϕ j > P ϑ j ≡
0, gives an analytic parametrization. If the set isnot analytic, we do not have local solvability. (The Fuchs condition). If we do nothave regular approximations of a first surface, the representation of the operator isnot defined. Thus a local complement { f = c } c analytic, is necessary for solvabil-ity. A locally algebraic transversal is necessary for hypoellipticity. For example, if e − φ b f T →
0, as T → ∞ , given an essential singularity in the infinity, we may havelocal solutions that are not analytic.Assume { f = c } a first surface to a holomorphic function. Given minimallydefined singularities to f , we know that the first surfaces are also analytic. Thismeans that if f ∈ L , the complement in a domain of holomorphy is analytic. If weonly assume f ∈ D ′ L , we do not have this strong result. Given that ∆ is analyticand ∆ → ∆ ⊥ is a Legendre transform, that is a contact transform, we should beable to prove Ω \ ∆ ∼ Ω \ ∆ ⊥ , that is if ∆ ⊥ is a first surface, it must be analytic.If f is reduced, we have that ddT log f → ddT f →
0. Otherwise, if 1 /f isbounded, then the converse implication holds. Algebraic dependence implies that ddT log f →
0. Since { f = c } ⊂ { ddT f = 0 } , we must have Ω \{ f = c } analyticimplies { ddT f = 0 } analytic. Obviously ddT log f = 0 implies ddT f = 0 (assumingboth not zero). Thus, { log f = 0 } is analytic and { f = c } is analytic. In thiscase, if { f = c } is analytic, then { log f = c } is analytic. If the complement to the OME REMARKS ON TR`EVES’ CONJECTURE 37 second set is analytic, the complement to the first set is analytic. The last conditionimplies solvability. Counterexamples can be given by f = βe ϕ .9.9. A counterexample to solvability.
The problem that we start with is whenthe complement to a fist surface in a domain of holomorphy, is analytic? As-suming parabolicity, we make the approach e φ , with φ subharmonic. Assumethe neighborhood of the first surface is given by { φ ≥ } . We are now dis-cussing φ ( x − y ) = − φ ( y ) + ψ ( x, y ). Through Tarski-Seidenberg’s result, we haveif x, y in a semi-algebraic { P ( x, y ) ≥ } and related through a Legendre trans-form, we have y in a semi-algebraic set { Q ( y ) ≥ } . The problem is now, if { Q ( y ) ≥ } = { F Q ( y ) = 0 } locally, where F Q is analytic. For instance R analyticwith R ( x, y ) F P ( x − y ) = F Q ( y ). Assume now x = 0, so φ ( − y ) = C − φ ( y ). If φ ( − y ) ≤
0, then φ (0) − φ ( y ) ≤ φ ( − y ) = C , a constant,as y → ∞ . Thus, φ ( y ) = C − C constant. If φ ( − y ) >
0, then C − φ ( y ) >
0, that is φ ( y ) is bounded, as y → ∞ . In this case { y φ ( y ) < C } is unbounded. The integral R φ
Introduction.
To determine the class of hypoelliptic pseudo differential op-erators, we will first assume the operator has representation as an unbounded Fred-holm operator with symbol in the radical to the ideal of hypoelliptic operators. Inthe last section we consider a different representation based on Cousin (cf. [5]).10.2.
Pseudodifferential operators as Fredholm operators.
From the theoryof linear Fredholm-operators, we know that any Fredholm operator A : E → E between Banach spaces, has a twosided Fredholm inverse, that is there is a B : E → E , such that BA = I − P , AB = I − P , with P , P finite rank operators, P isthe projection E → Ker A and ( I − P ) is the projection E → Im A . Conversely,given an operator A , continuous and linear E → E , such that we have operators B , B with B A = I + R , AB = I + R , R , R compact operators, then A isFredholm. Finally, the class of Fredholm operators is invariant to addition of acompact operator.For our pseudodifferential operator A , given existence of left- and right para-metrices, the operator A is Fredholm and we have a Fredholm inverse. We haveearlier noted that our pseudodifferential operator A can be compared with polyno-mial operators according to H λ = A λ − P λ , with H λ regularizing. Any parametrixto P λ can be considered as a parametrix to A λ . Assume now B λ the Fredholminverse to P λ , modified as in the preliminaries. We then know that B λ − I isregularizing outside Ker B λ as | λ |→ ∞ . We shall see below, that this is a ”radical” property, which means that for ϕ / ∈ Ker B λ we have sing supp L ( ϕ ) =sing supp L ( B λ P λ ϕ ) = sing supp L ( P λ ϕ ). Naturally, A λ has the same domainfor hypoellipticity as P λ . Assume E λ a left- and F λ a right-parametrix to A λ and B λ the modified Fredholm parametrix to P λ . Then A λ (cid:0) E λ − B λ (cid:1) = R + P , R regularizing and P : E → Im P λ ⊥ continued with regularizing action outsideKer B λ . As A λ is hypoelliptic outside Ker B λ , E λ = B λ + Γ , with Γ regular-izing. In the same manner F λ = B λ + Γ , with Γ regularizing. The constructiongives that Ker B λ is a finite dimensional space and on this space any parametrixto A λ is either regularizing or 0. We can assume Γ j = 0 on Ker B λ . Proposition 10.1.
Assume B λ the modified Fredholm inverse to P λ as above, thatis B λ P λ = I − P , where P is regularizing and P λ B λ = I − P with P regularizingoutside Ker B λ . Further A λ a pseudodifferential operator so that H λ = A λ − P λ with H λ regularizing, then A λ is hypoelliptic in L if and only if Ker B λ = { } The proposition can be read as follows, for a hypoelliptic pseudodifferential op-erator in our class A λ , we have that for ϕ ∈ L , B λ ϕ = 0 implies ϕ = 0. Thefollowing Lemma is trivial Lemma 10.2. If P Nλ is hypoelliptic, then B Nλ − B Nλ ∈ C ∞ . By choosing λ appropriately, we can assume N = 2. Lemma 10.3.
For u ∈ L sing supp L (cid:0) B λ u − u (cid:1) = sing supp L (cid:0) B λ u + u (cid:1) Proof: Assume N = 2. We obviously have P λ − I ∼ P λ + I , which means thatthe singular supports for P λ ( B λ + I ) u and P λ ( I − B λ ) u coincide. Thus, thelemma holds for B λ . Finally, the singular support for ( I + B λ + B λ B λ ) u is thesame as the one for ( I + B λ B λ − B λ ) u . (cid:3) According to Lemma 10.2 ( B λ − I )( B λ + I )+ I − B λ is regularizing and accordingto Lemma 10.3, this implies ( B λ − I ) + I − B λ is regularizing. So, if I − B λ isregularizing we have I − B λ is regularizing outside Ker B λ Proposition 10.4. If P Nλ is hypoelliptic, then I − P N − j =1 B jλ is regularizing outsideKer B λ Proof: We have ( B λ − I ) P ∞ j =0 B jλ = I . Thus, ( B Nλ − I ) P ∞ j =0 B jλ + + P N − j =1 B jλ ∼ I , where ( B Nλ − I ) is regularizing outside Ker B λ . (cid:3) Assume one more time N = 2, then N ( B λ ) ⊂ N ( B λ ) and if ϕ ∈ N ( B λ ),according to Lemma 10.2, we have B λ ϕ ∈ C ∞ . This gives good estimates usingHadamard’s lemma as we have, k B Nλ k c ≤ C N | λ | − N N ≥
2. Finally, since B λ is a L -kernel of finite rank, we can find a canonical kernel K λ = P p k µν e µ ⊗ e ν such that k B λ − K λ k = 0 and e µ some orthonormal system. That is K λ ϕ = P k µν ( ϕ, e ν ) e µ . If ϕ ∈ L is such that B λ ϕ = 0, using the theory of integralequations (cf. [21]) we can make an orthogonal decomposition of N ( B λ ) accordingto S Nλ = R ( B Nλ ) ∩ R ( B ( N +1) λ ) ⊥ Thus, ϕ = B Nλ ϕ + N λB ( N +1) λ ϕ on S Nλ . If N >
2, say N , then S Mλ ⊂ C ∞ for M ≥ N . On S Nλ and N < N we have ϕ − B Nλ ϕ ∈ C ∞ with K Nλ ϕ = P µ,ν k Nµν ( ϕ, e ν ) e µ . These kernels B Nλ can thus be considered as hypoelliptic.10.3. Asymptotically hypoelliptic operators. If B λ is a parametrix to an op-erator L λ in L and B λ L − hypoelliptic on N ( B N λ ) ⊥ , where N is the minimalinteger such that the zero-space is stable. Then, if N = 1, we have that L λ is hy-poelliptic on L . Assume B λ,N a L − parametrix in L to the iterated operator L Nλ , OME REMARKS ON TR`EVES’ CONJECTURE 39 then N ( B N λ,N ) = N ( B B λ ) implies N = 1. and B λ,N L N λ = L N λ B λ,N = δ x − γ forsome γ ∈ C ∞ where γ = 0 on N ( E N λ ). Further, we have that L N λ is hypoellipticon L . If B λ,N is parametrix to a L − operator L N λ , by adding a solution tothe homogeneous equation H = 0 on N ( B λ,N ), we get N ( E λ,N + H ) = { } and B λ,N − B N λ ∈ C ∞ , that is the parametrix to L N λ is regularizing on N ( E N λ ).10.4. Iversen’s condition and hypoellipticity.
We have seen that when T = L (Legendre), we do not have micro local contribution. Consider for this reason thecondition: If the phase space sequence v µ in(6) 0 < | T v µ − L v µ | < ǫ µ → e v µ of a singular point that can be continued ana-lytically over the origin, then the corresponding operator does not have micro localcontribution from this sequence. More precisely, if V ( M, W ) = V + iV is thelifting function, let V N be the localization to X N . For d V N = P N ( T ) f , for f corresponding to a very regular operator, we assume b V − N = (1 /P N ) g , where g is in the same class as f . We then have that for V corresponding to a hypoellipticoperator, we must have that the continuation from b V − N , ( T − L ) V − N is algebraicas T → ∞ . Conversely, given that ( T − L ) V − N is algebraic, there is no roomfor lineality. In the terminology of Iversen (cf. [7]), if we can find a Jordan arcemanating from the origin, on which the limes inferior of the modulus of the closedcontours corresponding to the strata have a limit, then there is a subsequence of µ n such that v µ n → v on this arc. Assuming existence of a point in (6) where v µ is fi-nite, we can find the arc ̺ such that lim µ ∈ ̺ v µ = v . The conclusion is that we haveanalytic continuation in this case. Thus, assuming the conditions on V as aboveand that M ⊥ W , we can use the path in Iversen’s proof to derive hypoellipticity. Proposition 10.5.
Given the representation f = V ( M, W ) with V ⊥ V , assume V − N is the restriction to strata X − N and that the stratification of ( B m ) has property ( M ) . Then we have that if ( T − L ) V − N is algebraic, as T → ∞ , there is no closedcontour contributing microlocally to the symbol. Algebraic continuation and orthogonality.
The information on the closedcontour giving microlocal contribution is in T V , in the manner that if ( T − L ) V − N is algebraic, there is no possibility for presence of a closed contour that would con-tribute and we can conclude that V ≺≺ V . Assume now that V ( N ) ∈ ( B m ) ′ ⊂ D ′ L ,such that b V ( N ) = P N ( T ) f , with f continuous (very regular). We can now assume b V − ( N ) ∼ P N g , where g has the same property as f . Our proposition is that if T ( ϑ ) ∼ α L ϑ , with α holomorphic, then for a hypoelliptic operator, α/P N will bealgebraic, as N → ∞ . Presence of a contributing closed contour means that α willbe exponential. We will argue that for V corresponding to a hypoelliptic operator,we have that ( T − L ) V − ( N ) is algebraic in T − infinity. Assume α = q N + r N , where q N is a polynomial. The condition on involution means that r N /P N behaves likethe symbol, in the tangent space. If | r N | < ǫ , then according to Nullstellensatz, wehave that | α/P N | < ǫ as T → ∞ . Thus, the condition that P N is reduced in the T -infinity and r N small, means that α/P N is algebraic in the T -infinity, as N → ∞ .If we assume α/P N bounded in the T -infinity, then we have that also V − ( N ) hastype 0. The conclusion is given that M ⊥ W and M = e ϕ W , if F is algebraic, thenwe must have V = e Φ V and that e − Φ → e − ϕ →
0, as T → ∞ . This meansfor the continuation of F using T , that the orthogonality is preserved. Final remarks on hypoellipticity.
We have seen that under the conditionthat M ⊥ W implies V ⊥ V , then we have that absence of closed contour that con-tributes microlocally is equivalent with the proposition ( T − L ) V − N is algebraic.Through the condition we have that V preserves parabolic approximations.It is known that for a symbol such that f N ∈ ( I HE ), for some integer N , we have∆( f ) ⊂ ∆( f ). We can give an interpolation problem for the iterates, ψ j + ϕ jN = M N ( ψ j ) and e ψ j = f j . As the kernel to the parametrix gets smaller as j increases,why Im f ≡ V jN , for all N when j is large. More precisely, if we assume f ( ξ )( Im F ) → ∞ , as | ξ |→ ∞ . Further, if we have f λ ∈ ( I HE ), as | ξ |→ ∞ .Assume for simplicity, | ξ | δ ≤ C | f ( ξ ) N | , for N positive and | ξ |→ ∞ . Theproblem is now if we can find δ such that | ξ | δ ≤ C ′ | f ( ξ ) | , as | ξ |→ ∞ ?. In thiscase we can choose δ = 1 /N . In this manner we can prove that given that the realpart has lower bound with exponent σ , then we can select δ = σ/N as exponentfor the lower bound to the imaginary part.We have noted that presence of lineality for the symbol, may result in Im F inthe space of hyperfunctions. We now note Proposition 10.6. If F is symmetric, entire and of finite type in Exp , thenthe condition that f represents a hypoelliptic operator, means that for some λ , ( Im ) λ = P A j F j on a domain of holomorphy, for constant coefficients and aglobal pseudo-base F j representing the ideal of hypoelliptic operators. Thus, symbols to hypoelliptic operators do not have imaginary part outside thespace of distributions and if hyper-function representation is necessary, we musthave contribution of lineality in the infinity.11.
Examples
Introduction.
There is a big number of examples published in the literature(cf. [22]) and we will deal with only some of them briefly here. We are assumingthe pseudo differential operator P is defined as lim λ → P λ , where the dependenceof λ is locally algebraic in the symbol space. We are assuming a dependence ξ ( x )through reciprocal polars, in this context, that is x T → ξ /T . We are assumingthe limit unique, in the sense that dP T dT = 0 implies T = 0, for small T (regularapproximations). However, we may have d dT P T = 0, even locally. An operatorthat is the regular limit of analytically hypoelliptic operators, with the conditionsthat we have, is analytically hypoelliptic. We can use Proposition (3.3) to constructan approximating sequence of symbols. The L − dependence for parameter, meansthat the limit of P λ in operator space is continuous.11.2. Example 1.
Consider P ( x, y, ξ, η ) = ξ + x m η . This corresponds to anoperator analytically hypoelliptic in R n with n = 2 , m ≥
1. If, for a constant c T , ξ /T = x mT η − c T , then P T is not analytically hypoelliptic, when c T = 0. But,when c T = 0, we have that P T is analytically hypoelliptic and the limit P is a limitof analytically hypoelliptic operators.11.3. Example 2.
Consider P ( x, y, z, ξ, η, ν ) = ξ + x η + ν . This correspondsto an operator not analytically hypoelliptic in 0 in R n , for n = 3. For this reason weconsider ξ /T + x T η + ν /T = c T , for a (possibly zero) constant c T such that c T → T → c ′ T , as T → x T η = c ′ T ( ξ /T + ν /T ). Wenow have constant surfaces through a suitable choice of η , which means that theoperator is not analytically hypoelliptic in 0. OME REMARKS ON TR`EVES’ CONJECTURE 41
Example 3.
Consider P ( x, y, z, ξ, η, ν ) = ξ + (cid:2) η + ( x + xy ) v (cid:3) = ξ + (cid:2) η + x ( x + y ) v (cid:3) . As we have assumed real arguments, it can be proved that P is a regular limit of analytically hypoelliptic operators in ( x, ξ ),( y, η ),( z, ν ) and incombinations of these. 12. The mapping ( I ) → Op ( I )Assume T = { y = F } a transversal manifold, that is for a submersion p , ker p = T x L L x , for a manifold L . For instance, assume h such that dh ( f ) = 0 implies f = const. , then if dh is locally algebraic (in the parameter), we must have where f = const. , that dh ( f ) = const. . Assuming on an irreducible component in { f = const } , there is at least one point where dh ( f ) = 0 we can conclude that f = const implies dh ( f ) = 0. A sufficient condition for this is that the tangent set ( I µ ) = { ζ h ( f ) = µf } exists and has irreducible.components. Note that under theseconditions, the first surfaces have a locally algebraic definition and the complementsets are assumed locally analytic. We can thus assume that in a neighborhood of df = 0, we have that dh ( f ) = 0 gives a regular set. If we consider transversals onthe form dh ( f ) = ρdf , for a locally regular function ρ , we can form the extendedtransversal as a Baire function.Assume Γ A is the boundary given by dF = 0 and Γ µ is given by { ζ log dF =0 } . If γ is transversal to Γ UA (= Γ A ∩ U ), we haveΦ dF = a γ ( z − , ζ k ) = Z Γ UA d z F ( z − + ζ ) ζ − ζ k dζ We then have, if g Φ dF is the analytic continuation over transversals, that g Φ dF − Φ dF → dF = 0 on Γ A . Further, d ζ ( g Φ dF − Φ dF ) = 0 at isolated points.Now consider F ( ζ, z ) analytic for ζ bounded and z large, such that F ( ζ, z ) → z → ∞ . We assume F ( ζ, z ) = d z F ( z + ζ ) d z F ( z ) − Q in 1 /t such that | F ( ζ, z ) | <ǫ + Q (1 /t ) on ∆ t,ǫ , a conical neighborhood of the lineality. We say that ζ preservesa constant value for F ( ζ, z ). Thus, | d z F ( ζ + z ) − d z F ( z ) | < ( ǫ t + Q (1 /t )) | d z F ( z ) | with Q (1 /t ) → t → ∞ . Assume further | dF | < | dF + dL | , where ker p = { dF + dL } . Then formally d z F ( ζ + z ) ∼ m ( dF + dL ) and dL ∼ m
0. Withinmonotropy, we thus have slow oscillation in the limit z → ∞ . We have that d z F is holomorphic with respect to { d z F = 0 } according to d z F = tr g Φ dF , where g Φ dF corresponds to to d z L = L z in the transversal decomposition. If F is a minimallydefining function of N ( J h ), we have dF = 0 on this set. If µ is such that h ( F ) = µF ,we have that f Φ F − Φ F → N ( J h ), why Φ F is analytic over the characteristicset { F = 0 } . Note that | d z F ( z + ζ ) dF ( z ) + dL ( ζ ) − | < ǫ ⇒| dL ( z ) dF ( z ) | < ǫ as z → ∞ along a transversal γ emanating from the origin. Consider first L z and thecorresponding g Φ dF . This corresponds through the inverse Fourier-Borel transform,to an analytic functional that allows real support, why it is sufficient to considerthe real space and b Γ . Assume u an entire function on a univalent domain andzero on points for multi-valentness and on Γ µ . We shall see that the corresponding form defines a good contour for the associated pseudo differential operator. Thisis according to (cf. [20]) given by a realization with regularizing action on D ′ whywe have no loss of generality from the conditions on the zero-set in the approach.Consider now the form θ : i dϕdx + iR ( x − y ) with | x − y |≤ r . Assume y = x − ζ and | ζ |≤ r . We then have e λ ( ϕ ( x + ζ ) − ϕ ( x )) − λ Re dϕdx · ζ e −| ζ | = (cid:2) u λ ( x + ζ ) u λ ( x ) e − Re (cid:2)(cid:0) P j δjuλuλ · ζ (cid:1)(cid:3)(cid:3) e − R | ζ | The slow oscillation that we already established (within monotropy) implies thatparticularly | δ j u λ /u λ | bounded, as | x |→ ∞ We conclude that for | ζ | bounded, as | x |→ ∞ , have that the bracket tendsto 1, as | x |→ ∞ and we have a good contour Γ( x ) for the form θ . The pseudodifferential operator can be realized through f H µ u λ ( x ) = C λ Z Γ( x ) e iλ ( x − y ) · ξ L µz ( x, y, ξ ) u λ ( y ) dydξ where L µz has compact support and the operator acts D ′ → C ∞ . Finally, we havethe case with T z . Using Weierstrass theorem, we can find a polynomial P µ,c whichinclude the foliation in its zero’s. For any polynomial, we have that δ j P µ ≺ P µ ,why the second term is bounded, for λ finite. The first term is bounded by slowoscillation as before. We have apparently a good contour also in this case.We conclude that given the foliation for the symbol { f = c } and the tangentspaces ( I µ ), we can realize the pseudo differential operator as a locally polynomialoperator where the polynomial part of the operator has zero’s on the foliation. Thisis a Levi decomposition of the operator Au λ = X j,µ ( P µ,c j + ^ H µ,c j ) u λ References
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