because ËPV (ξ,t)=(ËPξ)V (0,t) it follows immediately from the definition that ...... that Md+1 = 0 and that H0 is a supporting hyperplane of Î. Moreover, Md+1 =.
Surjectivity of augmented linear partial dierential operators with constant coe cients and a conjecture of Trèves
Habilitationsschrift
Dem Fachbereich IV der Universität Trier vorgelegt von
Dr. Thomas Kalmes
Trier, im Januar 2012
Für Inga
Contents Introduction
1
1 The problem of surjectivity of augmented operators
5
2 Conditions for P-convexity
9
2.1
Continuation of dierentiability . . . . . . . . . . . . . . . . . . .
2.2
Su cient conditions for P-convexity
. . . . . . . . . . . . . . . .
17
2.3
Complements of closed proper convex cones . . . . . . . . . . . .
30
3 Surjectivity of augmented operators
9
35
3.1
Some positive results . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.2
A surjective operator with non-surjective augmented operator . .
48
4 Dierential operators in two variables
51
4.1
On a conjecture of Trèves
4.2
Augmented operators in two variables
. . . . . . . . . . . . . . .
57
4.3
Ultradistributions of Beurling type . . . . . . . . . . . . . . . . .
60
Bibliography
. . . . . . . . . . . . . . . . . . . . . .
52
67
Introduction The question of solvability of a linear partial dierential equation with constant coe cients in some open set
X ⊆ Rd P (D)u = f
is a classical problem.
Depending on the properties of the right hand side
this problem leads in a natural way to the question of surjectivity of
P (D)
f
on
various spaces of functions and distributions. Malgrange [25] proved in 1955 that if the above equation has a distributional solution for every there is always a solution
∞
u ∈ C (X).
f ∈ C ∞ (X)
then
Moreover, Malgrange was able to give a
P (D) on C ∞ (X) by a certain kind of geometric condition involving the adjoint P (−D) of P (D). P (D) is surjective ∞ on C (X) if and only if X is P -convex for supports. Roughly speaking this ∞ means that for compactly supported u ∈ C (X) the location of their support is determined by the location of the support of P (−D)u. As is well-known, in ∞ general surjectivity of the dierential operator P (D) on C (X) does not imply 0 surjectivity on D (X) (see e.g. [13, Section 6]). In order to have surjectivity 0 on D (X) a second condition apart from P -convexity for supports has to be 0 satis ed. As proved by Hörmander [13] in 1962, P (D) is surjective on D (X) if and only if X is P -convex for supports as well as P -convex for singular supports. The later means, roughly, that for compactly supported distributions u in X the complete characterization of surjectivity of
location of their singular support is determined by the location of the singular support of
P (−D)u.
Despite the ingenious elegance of these characterizations,
in concrete examples it can be rather involved to verify their validity. Although
P -convexity
for supports does not imply
P -convexity
for singular
supports in general, Trèves conjectured in [35, Problem 2, page 389] that in the special case of a dierential operator in two independent variables, i.e. when
X ⊆ R2 , P -convexity for supports of X ⊆ R2 already implies P -convexity for ∞ singular supports of X , or equivalently surjectivity of P (D) on C (X) implies 0 surjectivity on D (X). Another problem concerning the notion of P -convexity for singular supports considered in this treatise arises from the work of Bonet and Doma«ski.
In
[5], Bonet and Doma«ski investigated the parameter dependence problem for solutions of partial dierential equations with constant coe cients, that is the problem whether for a given family on a (real) parameter
λ
(fλ ) of distributions on X
depending nicely
it is possible to solve
P (D)uλ = fλ in such a way that the solution family
(uλ )
depends on the parameter in the
same way. Their results lead to the following problem [5, Problem 9.1]. Does
P (D) on D 0 (X) imply surjectivity of the augmented operator D (X × R), where P + (x1 , . . . , xd+1 ) = P (x1 , . . . , xd )? The moti-
surjectivity of
+
P (D)
on
0
vation of this problem will also be discussed in chapter 1. Clearly, one has to
P -convexity for supports and singular supports of X implies X × R. It will be shown in chapter 1 that convexity for supports does not cause any problem. P -convexity + for supports of X always implies P -convexity for supports of X × R. So, as investigate whether
P + -convexity
for supports and singular supports of
for Trèves' conjecture, the problem of Bonet and Doma«ski leads again to the
question if some open set,
X × R in P +.
this case, is convex for singular supports
with respect to the polynomial
Motivated by these two problems, we will prove su cient conditions for convexity for singular supports in chapter 2.
P-
Our main tool will be a deep
result about the continuation of dierentiability for distributional solutions of the dierential equation
P (D)u = 0
due to Hörmander [18, Section 11.3]. This
result involves the zeros of a certain function
Rd .
subspaces of
σP
de ned on the non-trivial
After a careful analysis of this function in section 2.1 we give
su cient conditions for
P -convexity
for singular supports in section 2.2. The
most important one of these conditions for our purposes is an exterior cone condition for every boundary point of the set
X
under consideration. Although
this su cient condition is far from being necessary in general, we show in section 2.3 that for certain sets
P -convexity
X
the exterior cone condition in fact characterizes
for singular supports.
Moreover, it will be shown in section 4.1
that in the two dimensional case the exterior cone condition characterizes
P-
convexity for singular supports no matter the geometry of the open (connected) set
X ⊆ R2 .
Analogous conditions for
P -convexity
for supports will also be
proved in chapter 2. In chapter 3 we turn our attention to the problem of Bonet and Doma«ski. In section 3.1 we rst present some results showing that for special classes of dierential operators (namely semi-elliptic dierential operators and operators of principal type) the problem has a positive solution if the underlying set satis es certain geometrical properties.
X
Furthermore, we give an alternative
proof of a result due to Vogt [39] stating that the problem of Bonet and Doma«ski has a positive solution for every elliptic operator. However, in section 3.2 we present an example of a surjective dierential operator
P (D)
on
D 0 (X)
d
X ⊆ R , for arbitrary d ≥ 3, such that the augmented operator P + (D) is not surjective on D 0 (X ×R). Thus, we solve the problem of Bonet and for some open
Doma«ski in the negative. Moreover, the dierential operator in this example is even hypoelliptic so that it also answers a problem posed by Varol in [38, Section 3]. Additionally, it will be shown in section 4.2 that in the two dimensional case the problem of Bonet and Doma«ski has a positive solution. But before we do so, we prove in the a rmative Trèves' conjecture in section 4.1. Moreover, using results due to Langenbruch [23] about the continuation of ultradierentiability of ultradistributional solutions
P (D)u = 0
u
of Beurling type of the dierential equation
we prove an analogue result of Trèves' conjecture in the setting of
ultradistributions of Beurling type section 4.3.
0 D(ω) (X)
for non-quasianalytic weights
ω
in
These spaces of distributions generalize classical distributions by
allowing more exible growth conditions for the Fourier transforms of the corresponding test functions than the classical Paley-Wiener weights. In particular, we prove that contrary to
0 D(ω) (X)
d≥3
in the case of
d=2
surjectivity of
does not depend on the speci c weight function
ω.
P (D)
on
These results of
chapter 4 complement results of Zampieri [44, 45] who proved that, again in con-
d ≥ 3, in two dimensions surjectivity of a dierential operator C ∞ (X) is equivalent to surjectivity on the space of real analytic functions A (X) on X . Contrary to d ≥ 3, there is a geometrical characterization ∞ 2 of surjectivity on C (X) due to Hörmander for X ⊆ R [18, Theorem 10.8.3] trast to the case
P (D)
on
which can easily be applied to concrete examples. Due to the equivalences of
the surjectivity of a dierential operator on the various spaces of functions and distributions mentioned above this characterization is now also applicable in all the previously mentioned settings. Throughout this treatise we will use standard notation from the theory of linear partial dierential operators and functional analysis. denote by
E (X)
the space
C ∞ (X)
In particular, we
equipped with its standard Fréchet space
topology. For any notion not explained explicitly, see [17, 18] and/or [30]. It is a great pleasure for me to express my deep gratitude to Prof. Dr. Leonhard Frerick and to Prof. Dr. Jochen Wengenroth not only for turning my attention to the subject of this treatise but also for constant encouragement and helpful discussions for quite a long time. Moreover, I want to thank Prof. Dr. Michael Langenbruch for valueable hints concerning the literature and Prof. Dr. Dietmar Vogt for interesting and stimulating discussions.
The problem of surjectivity of augmented operators
5
1 The problem of surjectivity of augmented linear partial dierential operators with constant coe cients In this short chapter we will introduce one of the problems considered in this treatise.
X ⊆ Rd1
Let
P1 ∈ C[X1 . . . , Xd1 ], P2 ∈ C[X1 , . . . , Xd2 ] be polynomials and Y ⊆ Rd2 be open sets such that the dierential operators
as well as
P1 (D) : D 0 (X) → D 0 (X), P2 (D) : D 0 (Y ) → D 0 (Y ) are both surjective. It is a natural question whether
P1 ⊗ P2 : D 0 (X × Y ) → D 0 (X × Y ) P1 ⊗ P2 is the polynomial in d1 + d2 variables de ned P1 ⊗ P2 (x, y) = P1 (x)P2 (y). ˆ ε D 0 (Y ), the complete ε-tensor product, are canonD 0 (X × Y ) and D 0 (X)⊗
is surjective again, where as
ically isomorphic as locally convex spaces (see e.g. [36, Theorem 51.7]) and for this isomorphism we have
P1 ⊗ P2 (D) = (P1 (D) ⊗ idY ) ◦ (idX ⊗ P2 (D)), where idX and idY denote the identity operator on
P1 (D) ⊗ idY
tively. Obviously,
and idX
⊗ P2 (D)
D 0 (X)
and
D 0 (Y ),
respec-
commute so that the following
proposition is trivial.
Proposition 1.1. P1 (D) ⊗ idY
and
P1 ⊗ P2 (D) is surjective on D 0 (X × Y ) if idX ⊗ P2 (D) are surjective on D 0 (X × Y ).
and only if both
It turns out that as a model space for the above, it su ces to investigate the special case when
Y =R
which is a consequence of the next theorem proved
independently by Valdivia [37] and Vogt [40].
Theorem 1.2. For every open X ⊆ Rd we have D 0 (X) ∼ = (s0 )N as locally convex spaces, where s0 denotes the strong dual of the nuclear Fréchet space of rapidly decreasing sequences. In particular, D 0 (X) ∼ = D 0 (R).
As the tensor product is commutative modulo canonical isomorphism, by
P1 ⊗ P2 (D) is surjective on D 0 (X × Y ) if and only if both operators P1 (D) ⊗ idR and P2 (D) ⊗ idR are surjective on D 0 (X × R) and D 0 (Y × R), + respectively. Moreover, if we denote for a polynomial P ∈ C[X1 , . . . , Xd ] by P
the above
the polynomial de ned as
P + (x1 , . . . , xd+1 ) := P (x1 , . . . , xd ) we have
P (D) ⊗ idR = P + (D).
augmented operator of P (D).
We call the dierential operator
P + (D)
the
With this notion, the original question leads in a
natural way to the following problem posed by Bonet and Doma«ski [5, Problem 9.1]:
D 0 (X) imply D 0 (X × R)?
Does surjectivity of a dierential operator on augmented operator on
surjectivity of the
6
The problem of surjectivity of augmented operators
This question is also connected with the parameter dependence of solutions of the dierential equation
P (D)uλ = fλ , see [5]. By the above considerations a positive answer to the problem of Bonet and Doma«ski also implies the surjectivity of the vector valued operators
P (D) : D 0 (X, D 0 (Y )) → D 0 (X, D 0 (Y )), where
0
Y ⊆ Rd
is any non-empty open set.
There are various results about surjectivity of vector valued dierential operators. Just to mention a few, surjectivity of where
X
P (D) : D 0 (X, F ) → D 0 (X, F ),
F is a nuclear Fréchet space with the linear topological (Ω) is considered by Bonet and Doma«ski in [6, Section 5]. Examples F are nuclear power series spaces Λr (α), see [30, Section 29]. The case is convex and
invariant for such of
F
being a Banach space is treated in [4, Theorem 36]. Moreover, Doma«ski
P (D) : D 0 (X, A (U )) → D 0 (X, A (U )),
investigates in [9] surjectivity of
A (U ) U.
where
denotes the space of real analytic functions on a real analytic manifold
Clearly, the problem of Bonet and Doma«ski has a positive solution when the operator
R⊗
P (D)
has a continuous linear right inverse
idR is a continuous linear right inverse of
+
P (D)
R on
D 0 (X) for D (X × R).
on
0
then The
existence of such a right inverse has been characterized by Meise, Taylor, and Vogt in [28] and [29] via the existence of shifted fundamental solutions with
P (D) has a continuous linear
additional properties. Moreover, as shown in [29], right inverse on on
E (X).
D 0 (X)
if and only if
P (D)
has a continuous linear right inverse
It was already shown by Grothendieck (see e.g. [36, Theorem 52.4])
that elliptic operators
P (D)
for
d≥2
never possess such a right inverse. More
generally, the same holds for hypoelliptic operators, as shown by Vogt [39], [41]. On the other hand, Bonet and Doma«ski proved in [5, Proposition 8.3] that a positive solution of their problem is equivalent to the kernel of having
P (D) in D 0 (X)
(P Ω), a linear topological invariant introduced by them in [5]. As is wellP (D) in D(X) and in E (X)
known, for hypoelliptic polynomials the kernel of
coincide as locally convex spaces (this follows for example from [17, Theorem 4.4.2]), so that it is a nuclear Fréchet space and hence has property only if it has the linear topological invariant
(Ω).
by Vogt [39] that the kernel of any elliptic operator on
X
is an arbitrary open subset of
Rd .
(P Ω)
if and
It has already been shown
E (X)
has
(Ω),
where
Thus, in case of an elliptic operator the
problem of Bonet and Doma«ski has always a positive solution. Moreover, it is
X every dierential operator which is not D 0 (X). Because X × R is convex if this holds for
well-known that for convex open sets identically zero is surjective on
X
the problem of Bonet and Doma«ski also has a positive solution for convex
sets. By a classical result due to Hörmander [13] if and only if
X
is
P (D)-convex
P (D)
is surjective on
D 0 (X)
for supports as well as for singular supports.
Since these notions will be important throughout the whole text we recall their de nition.
De nition 1.3. mial.
Let
X ⊆ Rd
be open and let
P ∈ C[X1 , . . . , Xd ]
be a polyno-
The problem of surjectivity of augmented operators
i)
X
7
P (D)-convex for supports if to every compact set K ⊆ X there L ⊆ X such that u ∈ E 0 (X) and supp P (−D)u ⊆ K supp u ⊆ L.
is called
is another compact set implies ii)
X is called P (D)-convex for singular supports if to every compact set K ⊆ X there is another compact set L ⊆ X such that u ∈ E 0 (X) and sing supp P (−D)u ⊆ K implies sing supp u ⊆ L.
It is well-known that for P (D)-convexity for supports it is enough to consider u ∈ D(X) and that X is P (D)-convex for (singular) supports if and only if for 0 c every u ∈ E (X) the distance of X to (sing)supp u coincides with the distance to (sing)supp P (−D)u (see e.g. [18, Theorem 10.6.3 and Theorem 10.7.3]). A dierent characterization without the notions of convexity for (singular) supports for the surjectivity of
P (D)
on
D 0 (X)
in the spirit of [28], [29] via the
existence of shifted fundamental solutions with additional properties was given only recently by Wengenroth [43]. Because his characterization of surjectivity seems di cult to apply in concrete situations, we will treat the problem of Bonet and Doma«ski by using Hörmanders classical approach.
Thus we are
P + -convex for supports as well as P + -convex for singular supports in case of X being P -convex for supports and P -convex for singular supports. We will see immediately that P -convexity for supports + of X is passed on to P -convexity for supports of X × R. Recall, that P (D) is surjective on E (X) if and only if X is P -convex for supports, as was proved by Malgrange [25]. For two locally convex spaces E and F we denote their ˆ πF . complete π -tensor product by E ⊗ interested in whether
X×R
is
Proposition 1.4. Let E and F be locally convex spaces, E complete, F 6= {0}. Moreover let T : E → F be continuous and linear. ˆ π idF : E ⊗ ˆ π F → E⊗ ˆ π F is surjective the same holds for T . i) If T ⊗
ii) If E and F are Fréchet spaces and T is surjective the same holds for ˆ π idF . T⊗ Proof.
i) Let y0 ∈ F \{0}. We denote by [y0 ] the subspace of F generated by y0 and by P : F → [y0 ] the corresponding continuous projection. Clearly, via Ψ(x) := x ⊗ y0 a topological isomorphism from E onto E ⊗π [y0 ] is de ned, ˆ π F by the completeness of E . so that E ⊗π [y0 ] is a closed subspace of E ⊗ ˆ π F with continuous Moreover, E ⊗π [y0 ] is a complemented subspace of E ⊗ ˆ ˆ π idF projection idE ⊗π P . Because the later commutes with the surjection T ⊗ we obtain
ˆ π idF (E ⊗π [y0 ]) = E ⊗π [y0 ]. T⊗ As
ˆ π idF ) ◦ Ψ = T Ψ−1 ◦ (T ⊗
T follows. π -tensor product (see e.g. [36, Proposition
the surjectivity of
ii) is a well-known result about the 43.9]). Using the nuclearity of
E (X)
(see e.g. [36, Corollary of Theorem 51.5], [30,
ˆ ε E (R) ∼ E (X)⊗ = E (X ×R) 0 (see e.g. [36, Theorem 51.6] or [12, Section 5]) as well as the nuclearity of D (X)
Example 28.9], or [12, Corollaire of Théorème 3]) and
(see e.g. [36, Corollary of Theorem 51.5] or [12, Corollaire of Théorème 3]) and
ˆ ε D 0 (R) ∼ D 0 (X)⊗ = D 0 (X × R) result.
proposition 1.4 immediately yields the following
8
The problem of surjectivity of augmented operators
Theorem 1.5. Let X ⊆ Rd be open and P
∈ C[X1 , . . . , Xd ].
i) P (D) is surjective on E (X) if and only if P + (D) is surjective on E (X×R). ii) P (D) is surjective on D 0 (X) if P + (D) is surjective on D 0 (X × R).
Remark 1.6. that and
It should be mentioned that proposition 1.4 immediately implies
P1 ⊗ P2 (D) is surjective on E (X × Y ) if and only if both P2 (D) are surjective on E (X) and E (Y ), respectively.
operators
P1 (D)
In order to solve the problem of Bonet and Doma«ski, some preparations have to be made, which will be presented in the following chapters. It will be shown in section 3.2 that contrary to theorem 1.5 i) in general surjectivity of
P (D) on D 0 (X) does not imply surjectivity of the augmented operator P + (D) 0 on D (X × R) (see also example 3.5). So in general the problem has a negative solution. However, we will provide some special cases when the answer is positive (see theorem 3.16 and theorem A at the beginning of chapter 4).
Conditions for P-convexity
9
2 Conditions for P-convexity The main purpose of this chapter is to prove some su cient conditions for
P -convexity
X
for singular supports of an open subset
of
P -convexity
Moreover, we give some su cient conditions for
Rd
in section 2.2.
for supports of
X
as well. In the third section of this chapter we characterize these properties for arbitrary polynomials and certain open subsets of
Rd
having a rather special
geometric form. As a starting point for all this, in the rst section we consider a
P (D)u u for distributions u ∈ D 0 (X) with P (D)u = 0. This result involves a certain d function σP de ned on the subspaces of R which we carefully analyse. result of Hörmander about the the continuation of dierentiability from to
Apart from standard notation we use the following. For an a ne subspace
V
Rd
V⊥
V. H = {x ∈ Rd ; hx, N i = α} in Rd , where N ∈ Rd \{0} and α ∈ R we have that H ⊥ is the one-dimensional subspace spanned d+1 0 d by N . Moreover, for x = (x1 , . . . , xd+1 ) ∈ R we set x = (x1 , . . . , xd ) ∈ R 0 0 d+1 and more generally, we write M = {x ; x ∈ M } for a subset M of R . This of
we denote by
the orthogonal space of the subspace parallel to
In particular, for a hyperplane
notation will be used throughout the whole text. Furthermore, a cone is always assumed to be non-empty.
2.1 Continuation of dierentiability X
As is well-known (see e.g. [18, Theorem 10.7.3]) supports if and only if for every and sing supp u to
X c coincide.
u ∈ E 0 (X)
is
P -convex
for singular
the distance of sing supp P (−D)u
Obviously sing supp P (−D)u is always contained
in sing supp u so the problem we consider is related to deriving bounds for sing supp u by knowledge of sing supp P (−D)u. If of
Pˇ
Pˇ (x) = P (−x)) W F (u) of u one has
(where as usual
wave front set
we have
E is a fundamental solution u = P (−D)u ∗ E so that for the
W F (u) ⊆ {(x + y, ξ); (x, ξ) ∈ W F (P (−D)u)
and
(y, ξ) ∈ W F (E)}
(1)
(cf. [17, p. 270, Formula (8.2.16)]), where the wave front set of a distribution
v
is a subset of
Rd × S d−1
whose projection onto
Therefore, knowledge about
W F (P (−D)u)
Rd
is precisely sing supp v .
as well as
W F (E)
will allow to
obtain bounds for sing supp u. For every polynomial
P
there is a speci c fundamental solution
E(P )
for
which the location of its wave front set is well understood. This speci c fundamental solution is given by
∀ ϕ ∈ D(X) : hE(P ), ϕi = (2π)−d
Z Rd
where
ϕˆ
Z dζ ϕ(−ξ ˆ − ζ)
dξ Cd
ϕ, Pξ (x) = P (ξ + x) and Φ is ζ ∈ Cd such that Φ(Q, ζ) vanishes "in
denotes the Fourier transform of
certain function of polynomials controlled manner" if
Q(ζ) = 0
Q
and
Φ(Pξ , ζ) , Pξ (ζ) a a
(see [17, Section 7.3]).
E(P ) is described by means of the so P whose de nition we want to recall. For a Pξ (η) = P (η + ξ). We denote by L(P ) the set
The location of the wave front set of called localizations at in nity of polynomial
P
and
ξ ∈ Rd
we set
of limits (in the unique Hausdor linear topology on the space of polynomials
10
Conditions for P-convexity
of degree not exceeding
deg P ,
η 7→
as
ξ
tends to in nity, where
d
R \{0}
P)
the degree of
P˜ξ (0) =
of the normalized polynomials
Pξ (η) P˜ξ (0)
qP
α
(α)
|Pξ (0)|2 .
More precisely, if
N ∈
ξ tends to in nity and ξ/|ξ| → N/|N | is LN (P ) = LαN (P ) for every N ∈ Rd \{0}, α > 0, so without loss of generality that |N | = 1. Obviously, L(P )
then the set of limits where
denoted by
LN (P ).
that we can assume
Hence,
LN (P ) are closed subsets of the unit sphere of all polynomials in d variables of degree not exceeding the degree of P , equipped with the norm ˜ . The non-zero multiples of elements of L(P ) (resp. of LN (P )) are Q 7→ Q(0) called localizations of P at in nity (resp. localizations of P at in nity in direction ˇ ∈ L−N (P ). N ). Clearly, Q ∈ LN (Pˇ ) if and only if Q We de ne for a polynomial Q as well as
Λ(Q) = {η ∈ Rd ; ∀ ξ ∈ Rd , ∀ t ∈ R : Q(ξ + tη) = Q(ξ)}, Λ0 (Q) the orthogonal space of Λ(Q). Clearly, by an appropriate linear change of variables Q does 0 only depend on k variables where k = dim Λ (Q). In particular, Q is constant 0 if and only if Λ (Q) = {0} and by an application of the Tarski-Seidenberg Theorem Hörmander proved N ∈ Λ(Q) if Q ∈ LN (P ) (see [18, Theorem 10.2.8]). 0 d Therefore, Λ(Q) 6= {0} and hence Λ (Q) 6= R for every Q ∈ L(P ).
which is obviously a subspace of
Rd .
Moreover, denote by
By a result due to Hörmander (cf. [18, Theorem 10.2.11]) the wave front set
W F (E(Pˇ ))
of the above mentioned fundamental solution
E(Pˇ )
is contained in
the closure of the set
{(x, N ) ∈ Rd × S d−1 ; x ∈ Λ0 (Q)
for some
Q ∈ LN (Pˇ )}.
u ∈ E 0 (X) the nonˇ ), or better the non-trivial subspaces Λ0 (Q), are the constant elements of L(P ones which may cause sing supp u to be much larger than sing supp P (−D)u. d De ne for a polynomial Q, a subspace V of R , and t ≥ 1 From this and equation (1) above it clearly follows that for
˜ V (ξ, t) = sup{|Q(ξ + η)|; η ∈ V, |η| ≤ t} Q and
˜ t) = Q ˜ Rd (ξ, t). Q(ξ, ξ ∈ Rd
Clearly, for every
˜ t) Q(ξ,
and
˜ V (ξ, t) t≥1Q
is a continuous seminorm, while
is a continuous norm depending continuously on
non-constant then
ξ
and
t.
If
Q ∈ L(Pˇ )
is
˜ Λ(Q) (0, t) Q . ˜ t) t≥1 Q(0,
0 = inf Moreover, since
Q ∈ L(Pˇ )
tending to in nity such that
it follows that there is a sequence
Q = limn→∞ Pˇξn /P˜ˇξn (0),
hence
˜ Λ(Q) (0, t) Q P˜ˇΛ(Q) (ξn , t) . = inf lim ˜ t) t≥1 n→∞ t≥1 Q(0, P˜ˇ (ξ , t)
0 = inf
n
(ξn )n∈N
in
Rd
Conditions for P-convexity
11
De ning for an arbitrary subspace
V
Rd
of
σPˇ (V ) = inf lim inf t≥1 ξ→∞
P˜ˇV (ξ, t) , P˜ˇ (ξ, t)
σPˇ (V ) = σP (V ). of σP (span{y}).
it follows immediately that simply write
σP (y)
Remark 2.1.
instead
Moreover, for
V1 ⊆ V2 are subspaces σP (V1 ) ≤ σP (V2 ).
a) Clearly, if
de nition that we have
P
b) Recall that a polynomial
of
Rd
y ∈ Rd
we shall
it follows from the
is hypoelliptic if and only if all of its lo-
calizations at in nity are constant (cf. proof of [18, Theorem 11.1.11]).
σP (V ) = 1
Therefore it follows that
for every subspace
is hypoelliptic. Moreover, observe that a polynomial and only if the polynomial
Pˇ (ξ) = P (−ξ)
P
V
of
Rd
if
P
is hypoelliptic if
is hypoelliptic (this follows e.g.
from [18, Theorem 11.1.11]) which together with [18, Corollary 11.3.3] and the above observations gives the equivalence of the following properties of
P.
a polynomial
i) Every open set ii)
P
is
P -convex
for singular supports.
is hypoelliptic.
iii)
σP (V ) 6= 0
iv)
σP (y) 6= 0
c) Let
X ⊆ Rd
V, W ⊆ Rd
for every subspace for every
V
of
Rd .
y ∈ Rd \{0}.
be two subspaces and
d(V, W ) =
sup
(
|x − y|).
inf
x∈V,|x|=1 y∈W,|y|=1 Then
|σP (V ) − σP (W )| ≤ C max{d(V, W ), d(W, V )}, where
C >0
is a constant depending only on
Since for unit vectors
N1 , N2 ∈ S d−1
P
(cf. [18, Section 11.3]).
we have
d(span{N1 }, span{N2 }) ≤ |N1 − N2 | it follows in particular that unit sphere The function
σP
is a continuous function on the
d-dimensional
S d−1 .
σP
is much more powerful than simply indicating the existence
of non-constant elements of
L(Pˇ ).
The values of
continue dierentiability of zero solutions of
P (D)
σP
govern the possibility to
across a hyperplane
H = {x ∈ Rd ; hx, N i = α}, N ∈ S d−1 , α ∈ R. x0 ∈ X and N ∈ S d−1 be such that σP (N ) 6= 0. Then 0 there is a neighborhood U ⊆ X of x0 such that u ∈ E (U ) whenever u ∈ D (X) with P (D)u = 0 as well as u|X− ∈ E (X− ), where X− = {x ∈ X; hx, N i < hx0 , N i}. This is only a very special case of the following deep theorem due to Let
X ⊆ Rd
be open,
Hörmander (cf. [18, Theorem 11.3.6]).
12
Conditions for P-convexity
Theorem 2.2. Let
X be an open subset of Rd , x0 ∈ X and let φ1 , . . . , φk ∈ be real valued functions such that ∇φ1 (x0 ), . . . , ∇φk (x0 ) are linearly independent. Assume that σP (W ) 6= 0 for the linear space W spanned by the gradients ∇φ1 (x0 ), . . . , ∇φk (x0 ) and set
C 1 (X)
X− = {x ∈ X; φj (x) < φj (x0 )
for some j = 1, . . . , k}.
If u ∈ D 0 (X), P (D)u ∈ E (X) and u ∈ E (X− ) then u ∈ E (U ) in a neighborhood U of x0 which is independent of u. Some kind of converse of the above theorem is also true.
Again it is due to
Hörmander (cf. [18, Theorem 11.3.1]).
Theorem 2.3. Let
be a linear subspace of Rd such that σP (V ⊥ ) = 0. For every non-negative integer k one can nd u ∈ C k (Rd ) with P (D)u = 0 and sing supp u = V . More precisely, we can nd u so that u 6= C k+1 (U ) for any open set U intersecting V . V
As a consequence of the previous two deep results one obtains the following corollary (see [18, Corollary 11.3.7]). This will be the starting point of deriving su cient conditions for
P -convexity
for singular supports.
Corollary 2.4. Let X1 ⊆ X2 be open convex sets in Rd and P (D) a dierential operator with constant coe cients. Then the following conditions are equivalent: i) Every solution u ∈ D 0 (X2 ) of the equation P (D)u = 0 with u|X1 ∈ E (X1 ) already belongs to E (X2 ). ii) Every hyperplane H with σP (H ⊥ ) = 0 which intersects X2 already intersects X1 . One way we use
σP (V )
is given by the following result which is a reformu-
lation of Corollary 2.4 from [10] more suitable for our aims.
Proposition 2.5. Let X1 ⊆ X2 be open and convex, and let P be a non-constant polynomial. Then the following are equivalent: i) Every u ∈ D 0 (X2 ) satisfying P (D)u ∈ E (X2 ) as well as u|X1 ∈ E (X1 ) already belongs to E (X2 ). ii) Every hyperplane H with σP (H ⊥ ) = 0 which intersects X2 already intersects X1 . Proof.
u ∈ D 0 (X2 ) satisfy P (D)u ∈ E (X2 ) as well as u|X1 ∈ E (X1 ). By the convexity of X2 we nd v ∈ E (X2 ) such that P (D)v = P (D)u. Therefore w := u − v ∈ D 0 (X2 ) satis es P (D)w = 0 and w|X1 ∈ E (X1 ). Now if ii) holds it follows from Corollary 2.4 that w ∈ E (X2 ), thus u ∈ E (X2 ). That i) implies ii) is just a special case of Corollary 2.4. Let
σP σP (V ) = 0.
In order to apply the above proposition it is crucial to know the zeros of on
S d−1 ,
or more generally, to identify the subspaces
Since the very de nition of
σP
V ⊆ Rd
with
seems not very appropriate to investigate this
question we give a dierent representation of
σP .
At the beginning of this
section we have already indicated the connection between the localizations of at in nity and the function
σP .
P
The next lemma strengthens this connection.
Conditions for P-convexity
13
Its usefulness will be shown in chapter 3 when dealing with a problem posed by Bonet and Doma«ski as well as in chapter 4 where we will prove a conjecture by Trèves.
Lemma 2.6. Let P be of degree m, P
j=0 Pj with Pj being a homogeneous polynomial of degree j , Pm the principal part of P . i) For every subspace V of Rd and t ≥ 1 we have
lim inf ξ→∞
=
Pm
˜ V (0, t) Q P˜V (ξ, t) = inf . ˜ t) Q∈L(P ) Q(0, P˜ (ξ, t)
ii) Let N ∈ S d−1 and Q ∈ LN (P ). If Pm (N ) 6= 0 then Q is constant. iii) If P is non-elliptic then for every subspace V of Rd and t ≥ 1 we have P˜V (ξ, t) = inf N ∈S d−1 ,Pm (N )=0 P˜ (ξ, t)
lim inf ξ→∞
˜ V (0, t) Q . ˜ t) Q∈LN (P ) Q(0, inf
iv) With the convention that the in mum taken over an empty subset of [0, 1] equals 1 we have σP (V ) = inf
inf
˜ V (0, t) Q . ˜ t) Q∈LN (P ) Q(0,
and each
t ≥ 1
t≥1 N ∈S d−1 ,Pm (N )=0
Proof.
V
i) Since for every subspace
inf
because
P˜V (ξ, t) = (P˜ξ )V (0, t)
˜ V (0, t) R 7→ R d variables and
the maps
are continuous seminorms on the space of all polynomials
R
in
it follows immediately from the de nition that
˜ V (0, t) P˜V (ξ, t) Q ≥ lim inf ˜ ˜ (ξ, t) ξ→∞ P Q(0, t) for every
Q ∈ L(P ). (ξn )n∈N
Moreover, if
lim inf ξ→∞
tends to in nity such that
P˜V (ξ, t) P˜V (ξn , t) (P˜ξn )V (0, t) = lim = lim ˜ ˜ n→∞ n→∞ P (ξ, t) P (ξn , t) P˜ξn (0, t)
we can extract a subsequence of
(ξn )n∈N
which we again denote by
such that the sequence of normalized polynomials compact unit sphere of all polynomials in limit belongs to
L(P )
Pξn /P˜ξn (0)
d variables
(ξn )n∈N
converges in the
of degree at most
m.
This
and we get
lim inf ξ→∞
˜ V (0, t) Q P˜V (ξ, t) ≥ inf ˜ ˜ t) Q∈L(P ) Q(0, P (ξ, t)
completing the proof of i). The proof of ii) is a consequence of Taylor's formula. For
ξ n ∈ Rd
we have
(α)
Pξn (x)
=
Pj
X 0≤|α|≤j≤m
(ξn ) α x α!
=
|ξn |m
X 0≤j≤m
j
|ξn | ξn Pj ( )+ |ξn |m |ξn |
X 00 |(ξ,η)|>r
r>0 |(ξ,η)|>r
ξ∈Rd
P˜W 0 (ξ, t) P˜ (ξ, t)
as well as
+ P˜W 0 ×{0} ((ξ, η), t) P˜W 0 (ξ, t) lim inf = inf ˜ (ξ, t) (ξ,η)→∞ ξ∈Rd P P˜ + ((ξ, η), t) which gives
+ P˜W 0 ×R ((ξ, η), t) = σP0 (W 0 ), ˜ + ((ξ, η), t) t>1 (ξ,η)→∞ P
σP + (W 0 × R) = inf lim inf as well as
σP + (W 0 × {0}) = σP0 (W 0 ). Thus i) is proved.
W is contained in the kernel of πd , (ξ, η) ∈ Rd × R
In order to prove ii) assume rst that
W ⊆ {0} × R.
Then we have for
i.e.
+ P˜W ((ξ, η), t) = sup{|P (ξ)|; (0, xd+1 ) ∈ W, |xd+1 | ≤ t} = |P (ξ)| = P˜W 0 (ξ, t). As in the proof of i) it then follows that
σP + (W ) =
inf
t>1,ξ∈Rd
Hence, without loss of generality, let we get
p1 > 0
P˜W 0 (ξ, t) = σP0 (W 0 ). P˜ (ξ, t)
W * {0}×R.
Then, by setting
p1 := kΠ|W k
as well as
+ P˜W ((ξ, η), t)
=
sup{|P (ξ + x0 )|; (x0 , xd+1 ) ∈ W, |(x0 , xd+1 )| ≤ t}
≤
sup{|P (ξ + x0 )|; x0 ∈ W 0 , |x0 | ≤ tp1 } P˜W 0 (ξ, tp1 ).
=
πd|W : W → W 0 is not injective we clearly have {(0, y); y ∈ R} ⊆ W . Therefore, recalling that πd as an orthogonal projection satis es p1 = kπd|W k ≤ kπd k ≤ 1 Now we distinguish two cases. If
sup{|P (ξ+x0 )|; x0 ∈ W 0 , |x0 | ≤ tp1 } = sup{|P (ξ+x0 )|; (x0 , xd+1 ) ∈ W, |(x0 , xd+1 )| ≤ t} because if
x00 ∈ W 0
|x00 | ≤ tp1 is a point where the 0 0 then (x0 , 0) ∈ W with |(x0 , 0)| ≤ t.
with
hand side is attained
+ P˜W 0 (ξ, tp1 ) = P˜W ((ξ, η), t).
supremum on the left Therefore
16
Conditions for P-convexity
In case of
πd|W : W → W 0 being injective (πd|W )−1 : W 0 → W
is well-de ned
and continuous and we get
P˜W 0 (ξ, t k(Π|W )−1 k−1 )
=
sup{|P (ξ + x0 )|; x0 ∈ W 0 , |x0 | ≤ t k(πd|W )−1 k−1 }
≤ sup{|P (ξ + x0 )|; (x0 , xd+1 ) ∈ W, |(x0 , xd+1 )| ≤ t} = P˜ + ((ξ, η), t). W
Hence, in both cases there are
p1 , p2 > 0
such that
+ P˜W 0 (ξ, tp2 ) ≤ P˜W ((ξ, η), t) ≤ P˜W 0 (ξ, tp1 ) for all
ξ ∈ Rd , η ∈ R, t ≥ 1.
Altogether this yields
P˜ + ((ξ, η), t) P˜W 0 (ξ, tp1 ) P˜W 0 (ξ, tp2 ) ≤ lim inf W ≤ inf , d + ˜ ˜ (ξ,η)→∞ P ((ξ, η), t) ξ∈R P (ξ, t) P˜ (ξ, t)
inf
ξ∈Rd so that
inf
t≥1,ξ∈Rd
P˜W 0 (ξ, tp2 ) P˜W 0 (ξ, tp1 ) ≤ σP + (W ) ≤ inf . d ˜ t≥1,ξ∈R P (ξ, t) P˜ (ξ, t)
(2)
Now, as on the nite dimensional vector space
{Q|W 0 ; Q ∈ C[X1 , . . . , Xd ], deg Q ≤ deg P } all norms are equivalent, there are
C[X1 , . . . , Xd ] 1/Cj
with
deg Q ≤ deg P |Q(x0 )| ≤
sup x0 ∈W 0 ,|x0 |≤p
Cj > 0, j = 1, 2, such j = 1, 2 |Q(x0 )| ≤ Cj
sup x0 ∈W 0 ,|x0 |≤1
j
that for every
Q ∈
we have for
|Q(x0 )|.
sup x0 ∈W 0 ,|x0 |≤p
j
ξ ∈ Rd , and t > 1 the degree of the polynomial y 7→ P (ξ +ty) P it follows that for j = 1, 2
Since for arbitrary equals that of
1/Cj for all
ξ ∈ Rd
and
P˜W 0 (ξ, tpj ) P˜W 0 (ξ, t) P˜W 0 (ξ, tpj ) ≤ ≤ Cj P˜ (ξ, t) P˜ (ξ, t) P˜ (ξ, t) t > 1.
(3)
Now ii) follows from the inequalities (2) and (3)
completing the proof of the lemma. Lemma 2.7 will be used in the next two sections to give su cient conditions, respectively to give a characterization for certain sets for singular supports of
X ×R
in terms of
P
and
X.
X,
of the
P + -convexity
As an application of these
results it will be shown in example 3.5 that for the hypoelliptic polynomial in two variables inducing the heat operator
P (ξ1 , ξ2 ) = iξ1 + ξ22
+
there are open
X × R is not P -convex for singular supports. As P is hypoelliptic this set X is P -convex for singular supports. However, the set X in this example is not P -convex for supports. subsets
X⊆R
2
such that
Conditions for P-convexity
17
2.2 Su cient conditions for P-convexity P -convexity for sinP -convexity condition for P -convexity for
In this section we will mainly give su cient conditions for
gular supports. Moreover, we also present su cient conditions for for supports. Part i) of theorem 2.9, a su cient
supports, is originally due to Tervo [34, Theorem 4.3]. We give a simpli ed and more transparent proof here. Moreover, our proof has the advantage that it can easily be modi ed in such a way as to give a similar su cient conditions for
P -convexity
for singular supports. We denote by
the origin with radius
r > 0.
B(0, r)
the open ball about
Recall that a hyperplane
H = {x ∈ Rd ; hx, N i = α} N ∈ S d−1 , α ∈ R is called characteristic for a polynomial P if Pm (N ) = 0, where Pm is the principal part of P . The next well-known theorem will be used with
several times in this section so we state it here for the reader's convenience (see e.g. [17, Theorem 8.6.8]).
Theorem 2.8. Let X1 and X2 be open convex sets in Rd such that X1 ⊆ X2 and let P ∈ C[X1 , . . . , Xd ] be non-constant. Then the following are equivalent. i) Every u ∈ D 0 (X2 ) with P (D)u = 0 and u|X1 = 0 vanishes in X2 . ii) Every characteristic hyperplane for P which intersects X2 already intersects X1 . P + -convexity P and X rather
It should be noted that in part iii) of the following theorem the for singular supports of than from properties of
X × R is derived P + and X × R.
from properties of
Theorem 2.9. Let X be an open, connected subset of Rd and let P be a non-constant polynomial with principal part Pm .
∈ C[X1 , . . . , Xd ]
i) X is P -convex for supports if for every x ∈ ∂X and every r > 0 there are convex sets C1 ⊆ C2 ⊆ Rd \X such that x ∈ C2 , C1 ⊆ Rd \B(0, r), and every characteristic hyperplane for P which intersects C2 also intersects C1 . ii) X is P -convex for singular supports if for every x ∈ ∂X and every r > 0 there are convex sets C1 ⊆ C2 ⊆ Rd \X such that x ∈ C2 , C1 ⊆ Rd \B(0, r), and every hyperplane H with σP (H ⊥ ) = 0 intersecting C2 already intersects C1 . iii) X × R is P + -convex for singular supports if for every x ∈ ∂X and every r > 0 there are convex sets C1 ⊆ C2 ⊆ Rd \X such that x ∈ C2 , C1 ⊆ Rd \B(0, r), and every hyperplane H with σP0 (H ⊥ ) = 0 intersecting C2 already intersects C1 . Proof.
u ∈ E 0 (X) and set K := supp P (−D)u and δ := dist(K, X ). Moreover, set L := supp u and let r > δ be such that L ⊆ B(0, r − δ). Let x ∈ ∂X be arbitrary and choose C1 and C2 according to i) for x and r . Then Xj := B(0, δ) + Cj , j = 1, 2, are open convex sets. In order to prove i) we x
c
Recall that by extending any compactly supported distribution by zero to all of
Rd
E 0 (X) ⊆ D 0 (Rd ) and thus E 0 (X) ⊆ D 0 (Rd ) ⊆ D 0 (Y ) for every Y ⊆ Rd .
we have
open subset
18
Conditions for P-convexity
r we have u|X1 = 0 and X2 ∩ K = ∅ because of C2 ⊆ Rd \X P (−D)u|X2 = 0. Hence, by the hypothesis and theorem 2.8 we u|X2 = 0, in particular u vanishes in B(x, δ) and as x ∈ ∂X was
By choice of so that also conclude
chosen arbitrarily dist(supp u, X follows. Since supp P (−D)u is
P -convex
c
) ≥ δ = dist(supp P (−D)u, X c )
⊆ supp u we have equality of the distances,
thus
X
for supports by [18, Theorem 10.6.3].
Referring to proposition 2.5 instead of theorem 2.8 and replacing supports by singular supports in the proof of part i) immediately gives ii). In order to proof iii) we observe that with
C2 × R are convex, open, and non-empty. d+1 from R N = (N 0 , Nd+1 ) ∈ S d and α ∈ R
C1 , C2 ⊆ Rd
trivially
C1 × R
and
Assume that for some unit vector the hyperplane
H = {x ∈ Rd+1 ; hx, N i = α} satis es
σP + (N ) = 0, H ∩ (C2 × R) 6= ∅ and H ∩ (C1 × R) = ∅. Without loss C1 × R ⊆ {x ∈ Rd+1 ; hx, N i > α}. But this implies Nd+1 = 0
of generality let
because otherwise we had
(x,
1 (α − hx, N 0 i) ∈ (C1 × R) ∩ H Nd+1
x ∈ C1 . Thus, Nd+1 = 0 so that H = {x ∈ Rd ; hx, N 0 i = α} × R 0 0 implying H ∩ C2 6= ∅ as well as H ∩ C1 = ∅. By σP + (N ) = 0 and lemma 2.7 0 0 ii) we conclude σP (N ) = 0. Now, if (x, t) ∈ ∂X × R = ∂(X × R) and r > 0 are arbitrary let C1 and C2 be as in iii) for x and r. From the hypothesis and what we have observed d+1 above C1 × R ⊆ C2 × R are convex, (x, t) ∈ C2 × R ⊆ R \(X × R), C1 × R ⊆ Rd+1 \B(0, R) and every hyperplane H ∈ Rd+1 with σP + (H ⊥ ) = 0 intersecting C2 × R also intersects C1 × R. By ii) X × R is therefore P + -convex for singular for any
supports.
C1 ⊆ Rd \B(0, r) together with the intersecting C2 should also intersect C1 may
In the above theorem, the condition that condition that certain hyperplanes
sometimes be hard to verify in concrete examples. Therefore we consider in the sequel convex cones as special cases for
C2
and we will see that in this case one
can give a su cient condition for the convexity properties solely in terms of a single convex set. Recall that a cone
C
is called
proper if it does not contain any a ne subspace Γ ⊆ Rd its dual
of dimension one. Moreover, recall that for an open convex cone
cone is de ned as
Γ◦ := {ξ ∈ Rd ; ∀ y ∈ Γ : hy, ξi ≥ 0}. It is a closed proper convex cone in convex cone
C
in
Rd
Rd .
On the other hand, every closed proper
is the dual cone of a unique open convex cone which is
given by
Γ := {y ∈ Rd ; ∀ξ ∈ C\{0} : hy, ξi > 0} = {y ∈ Rd ; ∀ξ ∈ C ∩ S d−1 : hy, ξi > 0}.
Conditions for P-convexity
Γ
19
Rd \Γ is closed one uses the compactC = Γ◦ can be done with the Hahn-Banach ◦ Therefore, we use the notation Γ also for arbitrary
is obviously a convex cone. To see that
ness of
C ∩ S d−1
while the proof of
Theorem (cf. [17, p. 257]).
closed convex proper cones. Before we continue to prove su cient conditions for some more preparations.
P -convexity
we need
We begin with the following proposition containing
some elementary geometric results which will be useful in the sequel.
Proposition 2.10.
a) If C ⊆ Rd is closed, convex, and unbounded, then for every x ∈ C there is ω ∈ S d−1 such that x + tω ∈ C for every t ≥ 0.
b) Let Γ◦ 6= {0} be a closed proper convex cone in Rd and N ∈ S d−1 . For c ∈ R let Hc := {x ∈ Rd ; hx, N i = c}. Then the following are equivalent. i) ii) iii) iv) Proof.
H0 ∩ Γ◦ = {0}.
or −N ∈ Γ. If x ∈ Rd and Hc ∩ (x + Γ◦ ) 6= ∅ then Hc ∩ (x + Γ◦ ) is bounded. If x ∈ Hc then Hc ∩ (x + Γ◦ ) = {x}. N ∈Γ
x ∈ C . Replacing C by C − x we may assume without loss of x = 0. Let (xn )n∈N be a sequence in C with |xn | ≥ n for all n ∈ N. Because 0 ∈ C we have xn /|xn | ∈ C for every n ∈ N. Passing to a subsequence if necessary, we can assume that (xn /|xn |)n∈N converges to ω ∈ S d−1 . For every t ≥ 0 we have t/|xn | < 1 for n su ciently large, hence txn /|xn | ∈ C . Since C is closed it follows that tω ∈ C . b) By translating and changing c appropriately, we can assume throughout the proof that x = 0. Obviously, i) is then equivalent to iv). a) Let
generality that
To show that i) implies ii) let
H + := {x ∈ Rd ; hx, N i > 0}
and
H − := {x ∈ Rd ; hx, N i < 0}.
H + ∩ Γ◦ 6= ∅ then H − ∩ Γ◦ = ∅. Indeed, assume there are x 6= y in Γ◦ such hx, N i > 0 and hy, N i < 0. Convexity of Γ◦ together with H0 ∩ Γ◦ = {0} imply the existence of λ ∈ (0, 1) such that λx + (1 − λ)y = 0, hence −x = (1 − λ)/λ y . Since Γ◦ is a cone and (1 − λ)/λ > 0 it follows that −x ∈ Γ◦ . Hence {0} = 6 span{x} ⊆ Γ◦ contradicting that Γ◦ is proper. − ◦ + ◦ Analogously one shows that H ∩ Γ 6= ∅ implies H ∩ Γ = ∅. Moreover, + assuming H ∩ Γ◦ = ∅ as well as H − ∩ Γ◦ = ∅ implies Γ◦ ⊆ H0 . This yields Γ◦ = {0} because of Γ◦ ∩ H0 = {0}, contradicting Γ◦ 6= {0}. + ◦ Without loss of generality we therefore may assume that H ∩ Γ 6= ∅. From ◦ d ◦ the above we obtain Γ ⊆ {x ∈ R ; hx, N i ≥ 0}. Since H0 ∩ Γ = {0} it follows ◦ that for all x ∈ Γ \{0} we have hx, N i > 0 which shows ii). If
that
That ii) implies i) is trivial.
H0 ∩ Γ◦ 6= {0}. Then, there t ≥ 0. If x ∈ Hc ∩ Γ◦ it follows
In order to show that iii) implies i) assume that is
◦
d−1
ω∈S such that tω ∈ H0 ∩ Γ for every x + tω ∈ Hc . Moreover, because of x ∈ Γ◦
that
we have
∀ y ∈ Γ, t ≥ 0 : hy, x + tωi = hy, xi + thy, ωi ≥ 0, hence
x + tω ∈ Hc ∩ Γ◦
for all
t≥0
contradicting the boundedness of
Hc ∩ Γ◦ .
20
Conditions for P-convexity
Hc ∩ Γ◦ 6= ∅ is unbounded. It follows ω ∈ S d−1 such that x + tω ∈ Hc ∩ Γ◦
To show that i) implies iii) assume that
a) that all t ≥ 0.
x ∈ Hc ∩ Γ◦ \{0}
from
for
for
Thus
there is
c = hx, N i = hx, N i + thω, N i, i.e.
ω ∈ H0 ,
and
∀y ∈ Γ, t ≥ 0 : 0 ≤ hy, x + tωi. Since
Γ
is a cone, this implies
∀y ∈ Γ, t ≥ 0, ε > 0 : 0 ≤ hεy, x + t/ε ωi = εhy, xi + thy, ωi. The special case
t := hy, xi
gives
∀y ∈ Γ, ε > 0 : 0 ≤ (ε + hy, ωi)hy, xi. x ∈ Γ◦ \{0} we have hy, xi > 0 for every y ∈ Γ, ◦ inequality yields hy, ωi ≥ 0 for all y ∈ Γ, thus ω ∈ Γ . ◦ d−1 ω ∈ H0 ∩ Γ ∩ S contradicting i). Because
so that the above We conclude that
With the aid of the above proposition and theorem 2.9 we can now prove the next theorem.
Theorem 2.11. Let
be an open, connected subset of Rd and let P be a non-constant polynomial with principal part Pm . X
i) X is P -convex for supports if for every x ∈ ∂X there is an open convex cone Γ 6= Rd such that (x + Γ◦ ) ∩ X = ∅ and Pm (y) 6= 0 for all y ∈ Γ. ii) X is P -convex for singular supports if for every x ∈ ∂X there is an open convex cone Γ 6= Rd such that (x + Γ◦ ) ∩ X = ∅ and σP (y) 6= 0 for all y ∈ Γ. iii) X × R is P + -convex for singular supports if for every x ∈ ∂X there is an open convex cone Γ 6= Rd such that (x + Γ◦ ) ∩ X = ∅ and σP0 (y) 6= 0 for all y ∈ Γ. Proof. ◦
We begin with a general observation. Let
d
x ∈ Rd
be arbitrary and let
Γ 6= {0} be a closed proper convex cone in R . Moreover, let π be a supporting ◦ ◦ hyperplane of C2 := x + Γ with π ∩ (x + Γ ) = {x0 }. If r > 0 let π ˜ be a ◦ halfspace with boundary parallel to π such that C1 := (x + Γ ) ∩ π ˜ ⊆ Rd \B(0, r) is unbounded. From the choice of π it follows that C2 \C1 is bounded. d Now, if H = {ξ ∈ R ; hξ, N i = α} is a hyperplane intersecting C2 then by proposition 2.10 b) H ∩ C2 is unbounded if and only if {N, −N } ∩ Γ = ∅. As C2 \C1 is bounded we obtain that the hyperplane H intersecting C2 also intersects C1 if and only if {N, −N } ∩ Γ = ∅. In order to prove i) let x ∈ ∂X be arbitrary and let Γ be as in the hypothesis d of i). Setting C2 as above we have x ∈ C2 ⊆ R \X . For r > 0 arbitary let C1 d be as above, too. Then C1 ⊆ R \B(0, r). Moreover, C1 ⊆ C2 are convex sets and by hypothesis and the fact that Pm (N ) 6= 0 if and only if Pm (−N ) 6= 0 it follows from the above observation that every characteristic hyperplane for P which intersects C2 also intersects C1 . From theorem 2.9 i) it follows that X is P -convex for supports. 0 0 Taking into account that σP (y) = σP (−y) and σP (y) = σP (−y) for all d y ∈ R the proofs of ii) and iii) are obvious modi cations of the above.
Conditions for P-convexity
21
We provide an alternative way to proof theorem 2.11 as was originally done in [22]. This proof will involve two results which are interesting in their own right, see propositions 2.12 and 2.15 below.
Proposition 2.12. Let Γ be an open proper convex cone in Rd , x0 ∈ Rd , and
let P be a non-constant polynomial. If for X := x0 + Γ no hyperplane H = {x ∈ Rd ; hx, N i = α}
with σP (H ⊥ ) = 0 intersects X only in x0 , the following holds. Each u ∈ D 0 (X) with P (D)u ∈ E (X) which is C ∞ outside a bounded subset of X already belongs to E (X). Proof.
P (D)u ∈ E (X) and assume that u is C ∞ outside a bounded subset of X . Since Γ is a proper cone, there is a hyperplane π intersecting X only in x0 . Let Hπ be a halfspace with boundary parallel to π such that X1 := X ∩ Hπ 6= ∅ is unbounded and u|X1 ∈ E (X1 ). Denoting X2 := X we have convex sets X1 ⊆ X2 and by the hypothesis and proposition ⊥ 2.10, each hyperplane H with σP (H ) = 0 and H ∩ X2 6= ∅ already intersects X1 . Proposition 2.5 now gives u ∈ E (X). Let
u ∈ D 0 (X)
satisfy
In order to obtain an analogous result of the above for
P and X M ∈ Rd+1
P+
and
X ×R
only
involving properties of
we continue with some geometrical considera-
tions. Recall that for
and
A ⊆ Rd+1
M 0 = (M1 , . . . , Md ) ∈ Rd
and
we write
A0 = {ξ 0 ; ξ ∈ A}.
Proposition 2.13. Let Γ be an open proper convex cone in Rd , x0 ∈ Rd , and
N ∈ S d−1 such that π := {x ∈ Rd ; hx, N i = α} is a supporting hyperplane of x0 + Γ intersecting x0 + Γ only in x0 and x0 + Γ ⊆ {x ∈ Rd ; hx, N i > α}. For β > α set X˜1 := {x ∈ x0 +Γ; hx, N i > β}, X1 := X˜1 ×R, and X2 := (x0 +Γ)×R. If H = {x ∈ Rd+1 ; hx, M i = c} is a hyperplane with X2 ∩ H 6= ∅ as well as X1 ∩ H = ∅ then the hyperplane Hx0 := {x ∈ Rd+1 ; hx, M i = hx0 , M 0 i} is a supporting hyperplane of X2 with Hx0 ∩ X2 = {x0 } × R and Md+1 = 0. Moreover, Hx0 0 = {x ∈ Rd ; hx, M 0 i = hx0 , M 0 i} is a supporting hyperplane of x0 + Γ such that Hx0 0 ∩ (x0 + Γ) = {x0 }.
Proof. 0.
Without loss of generality, let
Suppose
H0
x0 = 0.
α = 0 and H0 contains 0 ∈ H0 ∩ X2 that hv, M i < 0 < hw, M i,
In this case,
is not a supporting hyperplane of
X2 .
v, w ∈ X2 = Γ × R such hx, M i < 0 < hy, M i for some x, y ∈ Γ × R. d+1 Set R := (N, 0) ∈ R . Then |R| = 1 and because
this means that there are
Because of
hence
of
Γ ⊆ {v ∈ Rd ; hv, N i > 0} we have
X2 ⊆ {v ∈ Rd+1 ; hv, Ri > 0}. λ1 := hx, Ri > 0 as well as λ2 := hy, Ri > 0. Since X2 is a cone we β+1 x1 := β+1 λ1 x, y1 := λ2 y ∈ X2 and from X1 = {v ∈ X2 ; hv, Ri > β} we get x1 , y1 ∈ X1 . Because hx1 , M i < 0 < hy1 , M i it follows for some t > 1
Therefore, have
htx1 , M i < c < hty1 , M i.
22
Conditions for P-convexity
Hence there is
λ ∈ (0, 1)
with
hλtx1 + (1 − λ)ty1 , M i = c, i.e. λtx1 + (1 − λ)ty1 ∈ H . Obviously, X1 is convex and for every x ∈ X1 and t > 1 we have tx ∈ X1 . Therefore we have λtx1 + (1 − λ)ty1 ∈ H ∩ X1 which contradicts our hypothesis.
H0 is a supporting hyperplane of X2 = Γ × R. This immediately implies Md+1 = 0 and that H00 is a supporting hyperplane of Γ. Moreover, Md+1 = ˜1. 0 implies that H 0 = {x ∈ Rd ; hx, M 0 i = c} intersects Γ but not X10 = X 0 Because Γ is a proper cone and Γ\X1 = {x ∈ Γ; hx, N i ≤ β} this implies that H 0 ∩Γ is bounded. Since H00 is a supporting hyperplane of Γ this yields H00 ∩Γ = {0} by proposition 2.10 b), hence H0 ∩ X2 = (H00 × R) ∩ (Γ × R) = {0} × R. So,
that
Proposition 2.14. Let Γ be an open proper convex cone in Rd , x0 ∈ Rd , and
let X1 and X2 be as in proposition 2.13. Moreover, let P be a non-constant polynomial. Assume that no hyperplane H in Rd with σP0 (H ⊥ ) = 0 intersects x0 + Γ only in x0 . Then for every hyperplane H in Rd+1 with H ∩ X2 6= ∅ and σP + (H ⊥ ) = 0 it follows that H ∩ X1 6= ∅. Proof.
H = {x ∈ Rd+1 ; hx, M i = β} be a hyperplane with H ∩ X2 6= ∅ H ∩ X1 = ∅. We have to show that σP + (M ) 6= 0. 0 From proposition 2.13 it follows that M = (M , 0) and that Let
but
Hx0 0 = {x ∈ Rd ; hx, M 0 i = hx0 , M 0 i} is a supporting hyperplane of
x0 + Γ
with
Hx0 0 ∩ (x0 + Γ) = {x0 }. In particular, the hypothesis gives
σP0 (M 0 ) 6= 0.
With lemma 2.7 we get
0 6= σP0 (M 0 ) = σP + (span{M 0 } × {0}) = σP + (M ), proving the proposition. Now, we can prove an analogous result to proposition 2.12 for which only relies on properties of
P
and
P+
and
X ×R
X.
Proposition 2.15. Let Γ be an open proper convex cone in Rd , x0 ∈ Rd , and let P ∈ C[X1 , . . . , Xd ] be a non-constant polynomial. Assume that no hyperplane H in Rd with σP0 (H ⊥ ) = 0 intersects x0 + Γ only in x0 . Then, every u ∈ D 0 ((x0 + Γ) × R) with P + (D)u ∈ E ((x0 + Γ) × R) for which there is a bounded subset B of x0 + Γ such that u is C ∞ outside B × R already satis es u ∈ E ((x0 + Γ) × R). Proof.
u ∈ D 0 (Γ × R) with P + (D)u ∈ E (Γ × R) and let B ⊆ Γ be bounded such that u|Γ\B×R ∈ E (Γ\B × R). Because Γ is a proper cone in Rd there is a hyperplane H1 intersecting Γ only in 0. Let X˜1 be the intersection of Γ with a halfspace whose boundary is parallel to H1 ˜1 is unbounded and B ⊆ Γ\X˜1 . such that X Without restriction, assume
x0 = 0.
Let
Conditions for P-convexity
23
X1 := X˜1 × R, and X2 := Γ × R. Then X1 ⊆ X2 are open convex Rd+1 and it follows from proposition 2.14 that for every hyperplane H in Rd+1 with σP + (H ⊥ ) = 0 and H ∩ X2 6= ∅ already H ∩ X1 6= ∅. Since u ∈ D 0 (X2 ), P + (D)u ∈ E (X2 ) and u|X1 ∈ E (X1 ) it follows from proposition 2.5 that u ∈ E (X2 ). Let
subsets of
We now give an alternative proof of theorem 2.11.
Alternative proof of theorem 2.11.
Again, the alternative proofs of all three
parts are very similar, so we give the proof of part iii) and only sketch the proofs of i) and ii).
u ∈ E 0 (X × R). We set K := sing supp P + (−D)u and δ := dist(K, X × R). Let x0 ∈ ∂(X × R) = ∂X × R and let Γ be as 0 ◦ in the hypothesis for x0 ∈ ∂X . Then (x0 + (Γ × R)) ∩ (X × R) = ∅, thus ◦ d+1 (x0 + y + (Γ × R)) ∩ K = ∅ for all y ∈ R with |y| < δ . Therefore, for xed y ˜ in Rd with Γ ˜ ⊃ Γ◦ \{0} such with |y| < δ , there is an open proper convex cone Γ 0 0 ˜ × R)) ∩ K = ∅. Hence, u ∈ E (X × R) ⊆ D (x0 + y + (Γ ˜ × R)) that (x0 + y + (Γ + ˜ × R)). We show that u ∈ E (x0 + y + (Γ ˜ × R)) satis es P (−D)u ∈ E (x0 + y + (Γ In order to prove iii), let
c
by applying proposition 2.15. Let
H = {v ∈ Rd ; hv, N i = α}
be a hyperplane with
σP0 (N ) = 0.
As
˜ Γ
is a closed proper convex cone with non-empty interior, it is the dual cone of
˜ ⊃ Γ◦ that Γ1 ⊆ Γ. Γ1 . It follows from Γ◦1 = Γ σP0 (N ) = 0 it follows from the hypothesis on Γ that {N, −N } ∩ Γ = ∅, hence {N, −N } ∩ Γ1 = ∅, so that by proposition 2.10 b) H does not intersect ˜ only in x0 + y 0 . Now sing supp u is compact since u ∈ E 0 (X × R). x00 + y 0 + Γ 0 some open proper convex cone
Because
Because
˜ × R)), P + (−D)u ∈ E (x0 + y + (Γ we have
˜ × R)) u ∈ E (x0 + y + (Γ by proposition 2.15.
Since
x0 ∈ ∂X × R
and
y
with
|y| < δ
were chosen
arbitrarily, it follows that dist(sing supp u, X
c
× R) ≥ δ = dist(sing supp P (−D)u, X c × R),
which proves iii).
u ∈ E 0 (X).
We set again K := sing supp P (−D)u and ). Again, we have to show that dist(sing supp u, X c ) ≥ δ in order to prove P -convexity for singular supports of X . Let x0 ∈ ∂X and let Γ be as in the hypothesis for x0 ∈ ∂X . As above, d ˜ such that for y ∈ R with |y| < δ we nd an open proper convex cone Γ 0 0 ˜ ˜ u ∈ E (X) ⊆ D (x0 +y + Γ) satis es P (−D)u ∈ E (x0 +y + Γ). Using proposition To prove ii) let
δ :=
dist(K, X
c
2.12 instead of proposition 2.15 the proof of ii) is now completed exactly as the one of iii).
u ∈ E 0 (X), K := supp P (−D)u and δ := dist(K, X c ). c one has to show dist(supp u, X ) ≥ δ which is done as
In order to prove i), let By [18, Theorem 10.6.3]
in the proof of iii) and ii), respectively, by using [17, Corollary 8.6.11] instead of proposition 2.12.
24
Conditions for P-convexity
P-
We close this section with yet another pair of su cient conditions for convexity for supports and singular supports.
These will be used to give an
alternative proof of a result due to Vogt [39] stating that for elliptic operator
P + (D)
is always surjective on
D 0 (X × R).
For
x, y ∈ Rd
P
the
we de ne
[x, y] = {γx + (1 − γ)y; γ ∈ [0, 1]}. X ⊆ Rd
Moreover, for
open,
x ∈ X , r ∈ Rd \{0},
we de ne
λX (x, r) := sup{λ > 0; ∀ 0 ≤ µ < λ : [x, x + µr] ⊆ X}. In case of
λX (x, r) = ∞
we write
[x, x + λX (x, r)r]
instead of
∪0 0. Therefore (D1 (ξn ))n∈N tends to zero. From the
we immediately obtain
0 = lim |∇Pm ( n→∞
ξn )| = |∇Pm (N )|, |ξn |
Surjectivity of augmented operators
45
contradicting ∇Pm (N ) 6= 0. Hence, if Q ∈ LN (P ) is not constant, we have h∇Pm (N ), xi = 6 0 for every x with Q(x) 6= Q(0). For x with Q(x) 6= Q(0) we therefore obtain that the numerator in (9) converges to h∇Pm (N ), xi = 6 0, so that again by (9) and the de nition of D1 (ξn ) we conclude that (D1 (ξn ))n∈N does not have any unbounded subsequence. Therefore, by passing to a subsequence if necessary, we may assume that (D1 (ξn ))n∈N converges in [0, ∞) and that (|ξn |Pm (ξn /|ξn |))n∈N converges to some β ∈ C, hence
lim D1 (ξn ) =
n→∞
p |β + Pm−1 (N )|2 + |∇Pm (N )|2 .
From this and equation (8)
Q(x) = lim
n→∞
nally follows. desired form.
β + Pm−1 (N ) + h∇Pm (N ), xi Pξn (x) =p P˜ξn (0) |β + Pm−1 (N )|2 + |∇Pm (N )|2
Thus, every non-constant polynomial
Q ∈ LN (P ) is ξn = nN
of the
To nish the proof we observe that choosing
yields
0 ∈ λ1 (N ).
Remark 3.13.
λ1 (N ). q 1 1 For example, let P (x1 , x2 ) = x1 x2 and consider ηn := ( 1 − 2 , ). Then n n |ηn | = 1 and (ηn ) converges to the simple zero (1, 0) of P . For β > 0 arbitrary and ξn := nβ ηn it follows r 1 ξn ) = β 1 − 2 −→n→∞ β. (ξn )n∈N ∈ ω (1, 0) and |ξn |P ( |ξn | n In general, nothing speci c can be said about the set
Moreover, considering
ξn := en ηn
we obtain
limn→∞ |ξn |P ( |ξξnn | ) = ∞.
One
easily checks that then the corresponding localization at in nity is constant. Lemma 3.12 will now be used to derive the following result. Part ii) corrects a notational inaccuracy in [15, Theorem 6.9].
Theorem 3.14. Let P be a polynomial with principal part Pm and let V be a subspace.
⊆ Rd
i) If N ∈ S d−1 is a simple characteristic vector for P with {Re∇Pm (N ), Im∇Pm (N )} ⊆ V ⊥
then σP (V ) = 0. ii) If P is of principal type then σP (V ) = 0 if and only if σP0 (V ) = 0 if and only if {Re∇Pm (N ), Im∇Pm (N )} ⊆ V ⊥
for some N ∈ S d−1 ∩{ξ ∈ Rd ; Pm (ξ) = 0}.
In particular, if P is of real principal type then σP (V ) = 0 if and only if there is N ∈ {ξ ∈ Rd ; Pm (ξ) = 0}\{0} with ∇Pm (N ) ∈ V ⊥ which is the case e.g. for V = span{N } for N 6= 0, Pm (N ) = 0.
46
Surjectivity of augmented operators
Proof.
By lemma 3.12 for every simply characteristic
constant
Q ∈ LN (P )
N ∈ S d−1
there is a non-
of the form
β + Pm−1 (N ) + h∇Pm (N ), xi Q(x) = p |β + Pm−1 (N )|2 + |∇Pm (N )|2 β ∈ λ1 (N ). Hence, if {Re∇Pm (N ), Im∇Pm (N )} ⊆ V ⊥ d constant in V but not constant in R . Therefore
for some
Q
is
it follows that
˜ V (0, t) Q = 0, ˜ t) t≥1 Q(0, inf
σP (V ) = 0 by lemma 2.6. This proves i). Now if P is of principal type it follows from theorem 3.10 and the remark 0 preceding it that σP (V ) = 0 if and only if σP (V ) = 0. Moreover, if σP (V ) = 0 it follows from lemma 2.6 and lemma 3.12 i) that there are sequences (tn )n∈N d−1 in [1, ∞), (Nn )n∈N in S ∩ {ξ ∈ Rd ; Pm (ξ) = 0} and (βn )n∈N in C such that thus
0
supx∈V,|x|≤tn |βn + Pm−1 (Nn ) + h∇Pm (Nn ), xi| n→∞ sup|x|≤tn |βn + Pm−1 (Nn ) + h∇Pm (Nn ), xi|
=
lim
=
lim
(Nn ) supx∈V,|x|≤1 | βn +Pm−1 + h∇Pm (Nn ), xi| tn
n→∞
(Nn ) + h∇Pm (Nn ), xi| sup|x|≤1 | βn +Pm−1 tn
in particular, for suitable
n→∞
(Nn )n∈N
(10)
c>0
0 = lim as the
,
(Nn ) | βn +Pm−1 | tn (Nn ) sup|x|≤1 | βn +Pm−1 |+c tn
,
are unit vectors. We conclude
βn + Pm−1 (Nn ) . n→∞ tn
0 = lim
Choosing a converging subsequence of
S d−1 ∩ {ξ ∈ Rd ; Pm (ξ) = 0} 0=
(Nn )n∈N
with limit
N
supx∈V,|x|≤1 |h∇Pm (N ), xi| supx∈V,|x|≤1 |h∇Pm (N ), xi| = sup|x|≤1 |h∇Pm (N ), xi| |∇Pm (N )|
so that
∀ x ∈ V : 0 = hRe∇Pm (N ), xi + ihIm∇Pm (N ), xi showing
{Re∇Pm (N ), Im∇Pm (N )} ⊆ V ⊥ .
On the other hand, applying i) it follows that
σP (V ) = 0
{Re∇Pm (N ), Im∇Pm (N )} ⊆ V ⊥ for some
we have
and equation (10) yields
N ∈ S d−1 ∩ {ξ ∈ Rd ; Pm (ξ) = 0}.
This proves ii).
if
N ∈
Surjectivity of augmented operators
As shown in the proof of [18, Theorem 10.4.5] there is a constant
C
d and the ξ ∈ Rd , t ≥ 1
such that for
only on all
47
maximal degree of the polynomials
P1 ,
and
P2
depending
1 ˜ ˜ ˜ P1 (ξ, t)P˜2 (ξ, t) ≤ P] 1 P2 (ξ, t) ≤ C P1 (ξ, t)P2 (ξ, t). C Herefrom follows immediately the next theorem which is [15, Theorem 6.7].
Theorem 3.15. For P1 , P2
∈ C[X1 , . . . , Xd ] and P (x) := P1 (x)P2 (x) the following hold for any subspace V ⊆ Rd . i) σP (V ) = 0 if and only if σP1 (V ) = 0 or σP2 (V ) = 0.
ii) σP0 (V ) = 0 if and only if σP0 1 (V ) = 0 or σP0 2 (V ) = 0. As a consequence of the above theorem together with theorems 3.4 and 3.10 as well as the results of chapter 2 we obtain the following theorem giving further su cient conditions for the augmented operator of a surjective dierential operator to be surjective again.
It will be seen in the next section that this
implication is not always true in general.
Theorem 3.16. Let X be an open subset of Rd and let the polynomials Q1 , . . . , Qn
be semi-elliptic or of generalized principal type. Set P := Q1 · · · Qn and denote its principal part by Pm . a) If for each x ∈ ∂X there is an open convex cone Γ 6= Rd such that (x + Γ◦ ) ∩ X = ∅ and Pm (y)σP (y) 6= 0 for all y ∈ Γ then P (D) : D 0 (X) → D 0 (X)
as well as P + (D) : D 0 (X × R) → D 0 (X × R)
are surjective. b) Let X0 ⊆ Rd be open and convex and let Γ1 , Γ2 , . . . be a sequence of open convex cones, all dierent from Rd . Moreover, let x1 , x2 . . . be a sequence T ◦ c in X0 . Denote by X the interior of X0 ∩ ∞ (x n=1 n + Γn ) and assume that for every n ∈ N we have εn > 0 such that Bεn (xn ) ∩ (xn + Γ◦n )c ⊆ X.
Then the following are equivalent. i) ii) iii) iv) Proof.
P (D) is surjective on D 0 (X). P + (D) is surjective on D 0 (X × R). Pm (y)σP (y) 6= 0 for all y ∈ ∪∞ n=1 Γn . 0 ∞ σP (y) 6= 0 for all y ∈ ∪n=1 Γn .
The surjectivity of
P (D) on D 0 (X) in a) follows from the hypothesis and
theorem 2.11. Moreover, we obtain from theorem 3.15 together with theorem 3.4 and theorem 3.10 that the hypothesis in a) together with theorem 2.11 imply the
P + -convexity for singular supports of X × R. P -convex for supports by theorem 1.5.
Thus a) follows since
X ×R
is also
In order to prove b) we observe that i) is equivalent to iii) by corollary 2.22. It follows from theorem 3.15, theorem 3.4, and theorem 3.10 that iii) and iv) are equivalent. Moreover, iii) and iv) together with corollary 2.22 imply ii). Finally, ii) implies i) by theorem 1.5.
48
Surjectivity of augmented operators
3.2 A surjective dierential operator on surjective augmented operator
D 0 (X)
with non-
P as well as X P (D) on D 0 (X) implies surjectivity of the augmented + 0 operator P (D) on D (X × R). Moreover, it will be shown in section 4.2 that 2 + this implication always holds in case of X ⊆ R . However, in general P (D) need not inherit surjectivity from P (D) as will be shown in this section. This In the previous section we gave some su cient conditions on such that surjectivity of
answers in the negative a problem posed by Bonet and Doma«ski in [5, Problem 9.1]. The example presented in this section will appear in [20]. As a special case of lemma 3.12 i) we have the following proposition.
Proposition 3.17. Let P be a homogeneous polynomial of degree m and ξ ∈ Rd P with P (ξ) = 0 and ∇P (ξ) 6= 0. Then x 7→ localization of P at in nity.
d j=1
∂j P (ξ)xj = h∇P (ξ), xi
In the example we now provide the polynomial hypoelliptic
P
P
is a
is hypoelliptic. Since for
the kernels of
P (D) : E (X) → E (X)
and
P (D) : D 0 (X) → D 0 (X)
coincide as locally convex spaces it is a Fréchet-Schwartz space. Hence it has property
(Ω)
if and only if it has property
(P Ω).
By the explanations given in
chapter 1 the following theorem therefore also gives an example of a surjective hypoelliptic dierential operator
P (D) : E (X) → E (X) such that its kernel does not have property
(Ω).
Therefore it also solves an
open problem from Varol [38, Section 3]. This should be compared with Vogt's classical result [39] that the kernel of an elliptic dierential operators always has
(Ω).
Theorem 3.18. For any d ≥ 3 there are an open subset X ⊆ Rd and a hypoel-
liptic polynomial P ∈ C[X1 , . . . , Xd ] such that P (D) is surjective on D 0 (X) but P + (D) is not surjective on D 0 (X × R). Therefore its kernel NP (X) does not have (Ω). Proof.
m Q(x) 6= 0 but σQ (x) = 0. It will be shown in section 4.1 that we have to choose d ≥ 3 for this. For 2 2 2 example, take Q(ξ) = ξ1 − ξ2 − . . . − ξd and x = ed = (0, . . . , 0, 1). Indeed, by proposition 3.17 applied to Q and ξ = (1, 1, 0, . . . , 0) it follows that Let
Q
be a homogeneous polynomial of real principal type of degree
which is not elliptic and let
x ∈ Rd , |x| = 1
such that
x 7→ h∇Q(ξ), xi = 2x1 − 2x2 is a localization of Q at in nity. Because h∇Q(ξ), ed i = 0 since d ≥ 3 we have σQ (ed ) = 0 by lemma 2.6 on the one hand and obviously Q(ed ) 6= 0 on the other hand. By [18, Theorem 11.1.12] there is a polynomial
R of degree 4m − 2 such that
P (ξ) := Q(ξ)4 + R(ξ)
(11)
Surjectivity of augmented operators
49
4m. Clearly, for its principal part P4m P4m (x) = Q4 (x) 6= 0. Moreover, we have
is a hypoelliptic polynomial of degree have
P4m = Q4
so
0 0 4 4 σP0 (x) ≤ σP0 4m (x) = σQ 4 (x) = (σQ (x)) ≤ (σQ (x)) = 0, where we have used lemma 3.1 iii), the obvious fact that
we
(12)
σP0 k (V ) = (σP0 (V ))k
k ∈ N, and lemma 3.1 i). P4m is homogeneous and P4m (x) 6= 0 there is an open proper convex d cone Γ 6= R with x ∈ Γ such that P4m (y) 6= 0 for every y ∈ Γ. If we set d ◦ X := R \Γ it follows from theorem 2.21 i) that X is P -convex for supports. Since P is hypoelliptic X is P -convex for singular supports as well. But because x ∈ Γ and σP0 (x) = 0 by inequality (12) X × R is not P + -convex for singular supports by theorem 2.21 iii). Thus, for the hypoelliptic polynomial P in (11) for any
Since
P (D) : D 0 (X) → D 0 (X)
is surjective
but
P + (D) : D 0 (X × R) → D 0 (X × R)
is not surjective.
Dierential operators in two variables
51
4 Linear partial dierential operators with constant coe cients in two independent variables A constant coe cient linear partial dierential operator
P (D) can be considered
as an endomorphism on various spaces of functions or distributions de ned on an open subset
X ⊆ Rd .
It is well-known and no surprise that the question of
surjectivity of this endomorphism depends on the space of functions or distributions under consideration. From results of Malgrange [25, Théorème 4] and
P (D)
Hörmander [13, Theorem 3.10] it follows that surjectivity of
on
D 0 (X)
E (X). On the other hand, it is well-known, that in genP (D) on E (X) is not su cient to guarantee surjectivity on
implies surjectivity on eral surjectivity of
D 0 (X),
as remarked in [13, Section 6]. (A concrete example for this is given in
example 2.24.) Moreover, convexity of on
E (X),
X
[25, Théorème 3], as well as on
is su cient for
D 0 (X),
P (D)
to be surjective
[26].
Proving a conjecture of De Giorgi and Cattabriga [8], it was shown by Pic-
∂ 2 /∂x21 + ∂ 2 /∂x22 is not surjective on the space A (Rd ) of real d analytic functions on R for every d ≥ 3. In particular, convexity of X is not su cient for P (D) to be surjective on A (X). Moreover, one can consider P (D) 0 as an endomorphism on the space of ultradistributions of Beurling type D(ω) (X) for a non-quasianalytic weight function ω . In this setting, it was shown by Lan0 genbruch [23, Example 3.13] that in general surjectivity of P (D) on D(ω) (X) for xed X depends explicitly on the weight function ω under consideration.
cinini [33] that
For all the above mentioned spaces of functions and distributions, characterizations of surjectivity of the endomorphism
P (D)
are available, although
evaluating these conditions in concrete examples is not an easy task, in general.
While surjectivity of
P (D)
on
E (X)
and
D 0 (X)
was characterized by
Malgrange in [25] and Hörmander in [13], respectively, a characterization of the surjectivity of
P (D)
in the setting of ultradistributions of Beurling type has
P (D) A (X) by means of an application of the Projk -functors of Palamodov [31, 32]
been given by Björck [1]. Moreover, a characterization of surjectivity of on
(see also Vogt [42]) is due to Langenbruch [24]. In case of a convex open set
X
a dierent characterization of surjectivity of
P (D)
on
A (X) by means of a P was given by
Phragmén-Lindelö condition valid on the complex variety of Hörmander [16]. However, despite of the dierences in the surjectivity on it was shown by Zampieri in [44, 45] that in case of on
E (X)
d=2
E (X)
and
surjectivity of
A (X) P (D)
A (X). Moreover, Trèves conjectured X ⊆ R2 surjectivity of P (D) on E (X)
is equivalent to surjectivity on
in [35, Problem 2, page 389] that for implies surjectivity on
D 0 (X).
The content of the present chapter is to prove the following result stating that all the above mentioned dierences in the surjectivity of
P (D)
on the vari-
ous spaces of functions and distributions vanish in case of open subsets
Theorem A. Let X ⊆ R2 be open and P
equivalent. i) P (D) : E (X) → E (X) is surjective.
∈ C[X1 , X2 ].
ii) P (D) : A (X) → A (X) is surjective. iii) P (D) : D 0 (X) → D 0 (X) is surjective.
X
of
R2 .
Then the following are
52
Dierential operators in two variables
iv) P + (D) : D 0 (X × R) → D 0 (X × R) is surjective. 0 0 v) P (D) : D(ω) (X) → D(ω) (X) is surjective for every non-quasianalytic weight function ω. 0 0 vi) P (D) : D(ω) (X) → D(ω) (X) is surjective for some non-quasianalytic weight function ω.
vii) The intersection of every connected component of X with any characteristic line for P is convex. Herein, the equivalence of i) and vii) is due to Hörmander (see e.g. [18, Theorem 10.8.3]). dimension
d.
As mentioned above, iii) always implies i) no matter the
Moreover, as already stated above, the equivalence of i) and ii) is
due to Zampieri [44, Theorem 2]. The equivalence of i) and iii) proves in the a rmative Trèves' conjecture, while the su ciency of iii) for iv) shows that, again contrary to arbitrary dimension, the problem of Bonet and Doma«ski introduced in chapter 1 (see [5, Problem 9.1]) has a positive solution in the two dimensional case (compare with theorem 3.18). Note that iv) always implies iii) by theorem 1.5, again in arbitrary dimension
d.
That vi) always implies i) was
shown by Björck in [1, Theorem 3.4.12] (see also the remark preceding theorem 4.11). The aim of this chapter is to give proofs of the implications which are not yet proved. To be more precise, in section 4.1 we will prove Trèves' conjecture, thus showing that i) implies iii). Section 4.2 will be devoted to show that iii) implies iv), while in section 4.3 we will deal with the remaining non-trivial implication that i) su ces for v) to hold. The content of this chapter has been published in [21], [22], and [20].
4.1 On a conjecture of Trèves X ⊆ R2
Let
be open and
P ∈ C[X1 , X2 ]\{0} be of degree m. This section is P (D) on E (X) implies surjectivity of P (D)
devoted to prove that surjectivity of on
D 0 (X),
thus proving a conjecture of Trèves [35, Problem 2, page 389] and
showing that i) implies iii) in theorem A. As usual, we denote the principal part of
P
by
Pm .
polynomial
Applying the Fundamental Theorem of Algebra to the one-variable
z 7→ Pm (z, 1)
Pm (ξ1 , ξ2 ) = ξ2m Pm ( for some
of degree
k≤m
yields for
ξ2 6= 0
k k Y Y ξ1 ξ1 , 1) = ξ2m c ( − αj ) = cξ2m−k (ξ1 − αj ξ2 ) ξ2 ξ j=1 j=1 2
c ∈ C\{0}, αj ∈ C.
In particular it follows that
{ξ ∈ S 1 ; Pm (ξ) = 0} is a nite set. Therefore, the only characteristic surfaces for
P
are hyperplanes
and there are only a nite number of them, up to translations. We call them characteristic lines for obvious reasons. In
R2 P -convexity
for supports of
X
is completely characterized by the
following theorem due to Hörmander (cf. [18, Theorem 10.8.3]). for an elliptic polynomial Corollary 10.8.2]).
P
every open set is
P -convex
Recall that
for supports (cf. [18,
Dierential operators in two variables
53
Theorem 4.1. If
P is non-elliptic then the following conditions on an open connected set X ⊆ R2 are equivalent: i) X is P -convex for supports.
ii) The intersection of X with any characteristic line is convex. iii) Every x ∈ ∂X is the vertex of a closed proper convex cone C ⊆ R2 \X such that no characteristic line intersects C only at x. By proposition 2.10 it follows that the above condition iii) is equivalent to iii')
For every x ∈ ∂X there is an open convex cone Γ such that (x+Γ◦ )∩X = ∅ and Pm (y) 6= 0 for all y ∈ Γ.
Next, we want to prove a similar characterization for supports of an open, connected set
X ⊆ R2 .
not only in this task but also in proving that of
X ⊆ R2
follows from
P -convexity
P -convexity for singular
The following lemma will be useful
P -convexity
for singular supports
for supports.
Lemma 4.2. For a non-constant polynomial P principal part Pm we have
∈ C[X1 , X2 ]
of degree m with
{y ∈ R2 \{0}; σP (y) = 0} ⊆ {y ∈ R2 \{0}; Pm (y) = 0}.
In particular, {y ∈ S 1 ; σP (y) = 0} is nite. Proof. As for a hypoelliptic polynomial P the
σP
function
to 1 we can assume without loss of generality that
P
is constantly equal
is not hypoelliptic, hence
not elliptic.
{N ∈ S 1 ; Pm (N ) = 0} is nite. 1 Let us denote its elements by N1 , . . . , Nl . For each 1 ≤ j ≤ l choose xj ∈ S orthogonal to Nj . Take an arbitrary, non-constant Q ∈ L(P ) which exists because P is not hypoelliptic. By lemma 2.6 ii) there is 1 ≤ j ≤ l such that Q ∈ LNj (P ). By [18, Theorem 10.2.8] we have Q(ξ + sNj ) = Q(ξ) for any ξ ∈ R2 , s ∈ R. Hence Q(ξ) = Q(hξ, xj ixj ) for all ξ ∈ R2 . De ning As observed at the beginning of this section
q : R → C, s 7→ Q(sxj ) y ∈ S1
it follows that for xed
˜ span{y} (0, t) Q
and because
˜ t) Q(0,
Since
=
sup{|Q(λy)|; |λ| ≤ t} = sup{|Q(λhy, xj ixj )|; |λ| ≤ t}
=
sup{|q(λthy, xj i)|; |λ| ≤ 1},
|xj | = 1
we also have
=
sup{|Q(ξ)|; ξ ∈ R2 , |ξ| ≤ t} = sup{|Q(hξ, xj ixj )|; ξ ∈ R2 , |ξ| ≤ t}
=
sup{|Q(λxj )|; |λ| ≤ t} = sup{|q(λt)|; |λ| ≤ 1}.
Q ∈ L(P )
it follows that
q
is a polynomial of degree at most
m.
Because
of the fact that on the nite dimensional space of all polynomials in one variable of degree at most there is
C>0
m the norms sup|s|≤1 |p(s)| and
Pm
k=0
|p(k) (0)| are equivalent,
such that
C sup |p(s)| ≥ |s|≤1
m X k=0
|p(k) (0)| ≥ 1/C sup |p(s)| |s|≤1
54
Dierential operators in two variables
p ∈ C[X] with degree s 7→ q(st) and s 7→ q(sthy, xj i) for all
˜ span{y} (0, t) Q ˜ t) Q(0,
at most
|hy, xj i| ≤ 1
Applying this to the polynomials
Pm
≥ ≥
where we used
m.
gives
|q (k) (0)|tk |hy, xj i|k k=0 P m C 2 k=0 |q (k) (0)|tk |hy, xj i|m /C 2 ,
in the last inequality. We conclude for every
1≤
j≤l ˜ span{y} (0, t) Q |hy, xj i|m ≥ , ˜ t) C2 Q∈LNj (P ) Q(0, inf
C only depends on the degree m of P . It follows from {N ∈ S 1 ; Pm (N ) = 0} = {N1 , . . . , Nl } that for all t ≥ 1 where
lim inf ξ→∞
lemma 2.6 iii) and
˜ span{y} (0, t) Q P˜span{y} (ξ, t) |hy, xj i|m = min inf ≥ min . ˜ t) 1≤j≤l Q∈LNj (P ) 1≤j≤l C2 P˜ (ξ, t) Q(0,
Therefore, if for
y ∈ Rd \{0} 0 = σP (y) = inf lim inf t≥1 ξ→∞
y is orthogonal Pm (y) = 0.
it follows that which shows
to some
xj ,
P˜span{y} (ξ, t) P˜ (ξ, t)
hence
y
is a non-zero multiple of
Nj
P -convexity
for
As a rst application of the above lemma we characterize singular supports of open, connected
X ⊆ R2
P -convexity for supports in theorem 4.1.
similar to the characterization of
Its proof is mutatis mutandis identical
to that of theorem 4.1 but we include it for completeness' sake.
Theorem 4.3. For
P ∈ C[X1 , X2 ]
following are equivalent.
and an open connected set X ⊆ R2 the
i) X is P -convex for singular supports. ii) The intersection of X with every hyperplane H satisfying σP (H ⊥ ) = 0 is convex. iii) For every x ∈ ∂X there is an open convex cone Γ 6= R2 with σP (y) 6= 0 for all y ∈ Γ and (x + Γ◦ ) ∩ X = ∅. Proof.
We rst show that i) implies ii). It is enough to show that if (±1, 0) ∈ X σP ((0, 1)) = 0 (i.e. parallels to the x-axis are hyperplanes H with σP (H ⊥ ) = 0), then I = [−1, 1] × {0} ⊆ X . We join (−1, 0) and (1, 0) by a polygon γ in X without self-intersection, where we can assume that γ intersects the x1 -axis only at its end points. For if this is not the case we can decompose γ into several polygons meeting the x1 -axis only at the end points and treat them separately. Then I and γ are the boundary of a connected and compact set C . We de ne
and
Y = {y ∈ R; (x, y) ∈ C
for some
x ∈ R}
Y0 = {y ∈ Y ; (x, y) ∈ C ⇒ (x, y) ∈ X}.
Dierential operators in two variables
Y
55
Y0 is not empty since the Y0 . Since X is P -convex for singular supports it follows from [18, Corollary 11.3.2] that dX satis es the minimum principle in the hyperplane R × {y} for arbitrary y ∈ R. Therefore, if y ∈ Y0 then from the de nition of Y0 (x, y) ∈ C implies (x, y) ∈ X so that ∅= 6 C ∩ (R × {y}) ⊆ X ∩ (R × {y}) is compact. Hence for y ∈ Y0 and x with (x, y) ∈ C we have due to the minimum principle is a closed interval with non-empty interior and
end point of
Y
which is dierent from
0
belongs to
dX (x, y) ≥ dX (C ∩ (R × {y})) = dX (∂C ∩ (R × {y})) ≥ dX (γ ∩ (R × {y})) ≥
dist(γ, X
c
).
c ) > 0, i.e. if y ∈ Y0 then (x, y) ∈ C implies X c is bounded below by the positive constant c dist(γ, X ). From this it follows that Y0 is closed in Y . Since X is open, Y0 is also open in the interval Y . Because Y0 is not empty this implies Y = Y0 , hence 0 ∈ Y = Y0 , so that I = [−1, 1] × {0} ⊆ X . Next, we prove that ii) implies iii). If x ∈ ∂X and H is a hyperplane through x with σP (H ⊥ ) = 0 then one half ray H1 of H bounded by x is contained in X c by ii). If there is another hyperplane I through x with σP (I ⊥ ) = 0 such that H1 ∩ I = {x} then one of its half rays I1 bounded by x is contained in X c by ii) and since X is connected it can be chosen so that the convex hull Γ◦ of H1 and I1 is contained in X c (and obviously is a proper convex cone by H1 ∩ I = {x}). If there is a hyperplane K through x with σP (K ⊥ ) = 0 and ◦ ◦ with K ∩ Γ = {x} we continue extending Γ until there is no hyperplane L ⊥ ◦ with σP (L ) = 0 intersecting Γ only in x. Observe that by lemma 4.2 this
Because
γ ⊆X
we have dist(γ, X
that the distance form
(x, y)
to
procedure stops after a nite number of extensions so that the resulting closed convex cone is indeed proper! From proposition 2.10 it follows that for no we have
y∈Γ
σP (y) = 0.
To nish the proof, we show that iii) implies i). But this follows from theorem 2.11 ii) which itself was inspired by the proof of the corresponding implication of [18, Theorem 10.8.3]. With the aid of theorem 4.1, lemma 4.2, and theorem 4.3 we give now a proof of Trèves' conjecture.
Theorem 4.4. Let
be open and P ∈ C[X1 , X2 ] be a non-constant polynomial with principal part Pm . Then the following are equivalent. X ⊆ R2
i) P (D) : E (X) → E (X) is surjective. ii) P (D) : D 0 (X) → D 0 (X) is surjective. iii) The intersection of every characteristic line for P and any connected component of X is convex. iv) For every connected component X0 of X and every x ∈ ∂X0 there is an open convex cone Γ such that (x + Γ◦ ) ∩ X0 = ∅ and Pm (y) 6= 0 for all y ∈ Γ\{0}. Proof.
Without loss of generality, let
X
be connected.
If
P
is elliptic i) and
ii) are always satis ed. Moreover, in this case there is no characteristic line for
P
so that iii) is also satis ed. Choosing
always satis ed, too.
Γ = R2
in iv) we see that iv) is then
56
Dierential operators in two variables
We therefore assume that
P
is not elliptic. The equivalence of i), iii) and iv)
is theorem 4.1 and we only have to show that i) implies ii). If
P
is hypoelliptic,
are equivalent. If
P
X
is
P -convex
for singular support so that i) and ii)
is not hypoelliptic it follows from the equivalence of i) and
iii) together with lemma 4.2 and theorem 4.3 that supports. So i) implies ii) and the proof is nished.
X
is
P -convex
for singular
Dierential operators in two variables
57
4.2 The augmented dierential operator of a surjective dierential operator in two variables is again surjective The purpose of this section is to prove that iii) implies iv) in theorem A, thus showing that, contrary to dimension
d ≥ 3, for open sets X ⊆ R2
the problem of
Bonet and Doma«ski on surjectivity of augmented dierential operators posed in [5, Problem 9.1] has a positive solution (compare with section 3.2).
X ⊆ R2
So let
open and
P ∈ C[X1 , X2 ].
By theorem 1.5 we have to show
X implies P + -convexity for singular supports of X × R. Recall that even for d = 2 it has been shown in example 3.5 that P -convexity for singular supports of X is not enough to + ensure P -convexity for singular supports of X × R in general. Without loss of generality we can assume that X is connected so that the same is true for X × R. [1, Theorem 3.4.12] that
P -convexity for
supports and singular supports of
As in the proof of Trèves' conjecture, it will follow from theorem 4.1 that the su cient condition given in theorem 2.11 iii) for
X × R to be P + -convex for
X is P -convex for supports Pm (y) = 0, where as usual Pm is
singular supports is always satis ed if proved that part of
σP0 (y) = 0
implies
once we have the principal
P.
Lemma 4.5. Let P ∈ C[X1 , X2 ] be a non-constant polynomial of degree m with principal part Pm . Then we have {y ∈ R2 \{0}; σP0 (y) = 0} ⊆ {y ∈ R2 \{0}; Pm (y) = 0}.
In particular, {y ∈ S 1 ; σP0 (y) = 0} is nite. Proof.
R2 such that Pm (y) 6= 0. Then 0 6= σP (y) by 0 lemma 4.2. We assume that σP (y) = 0. Thus, denoting the span of y by [y], 2 there are sequences (ξn )n∈N in R and (tn )n∈N in [1, ∞) such that Let
y
be a unit vector in
P˜[y] (ξn , tn ) . ˜ (ξn , tn ) n→∞ P lim
If
(ξn )n∈N
is bounded, we can assume without restriction, that
Moreover, we can assume that
(tn )n∈N
in nity without loss of generality. For if
limn→∞ tn = t
for some
t ≥ 1,
limn→∞ ξn = ξ .
is unbounded - and therefore tends to
(tn )n∈N
is bounded, without restriction
so that
P˜[y] (ξn , tn ) P˜[y] (ξ, t) = . ˜ n→∞ P (ξ , t ) P˜ (ξ, t) n n
0 = lim But this means that
0 = sup |P (ξ + θy)|, |θ|≤t
P (ξ + sy) in s ∈ R is identically zero. Replacing ξn by ξ n for each n ∈ N we have a bounded sequence (ξn )n∈N in R2 and a (tn )n∈N in [1, ∞) tending to in nity such that
i.e. the polynomial and
tn
by
sequence
P˜[y] (ξn , tn ) . ˜ (ξn , tn ) n→∞ P
0 = lim
58
Dierential operators in two variables
Since
C>0
(ξn )n∈N
such that
(tn )n∈N tends to in nity, there Pm is a constant ξn + x)| ≤ C , where P = |P ( j tn j=0 Pj with Pj j=0 degree j . Using that tn ≥ 1 we obtain
is bounded and
sup|x|≤1
being homogeneous of
Pm
sup |P (ξn + x)| ≤ sup |x|≤tn
m X
|x|≤1 j=0
tjn |Pj (
ξn + x)| ≤ tm n C. tn
This gives
P˜[y] (ξn , tn ) P˜ (ξn , tn )
≥ ≥ =
Since
(ξn /tn )n∈N (tn )n∈N
fact that
|P (ξn + tn y)| sup|x|≤tn |P (ξn + x)| t−m n |P (ξn + tn y)| C Pm | j=0 tj−m Pj ( ξtnn + y)| n C
.
(ξn )n∈N and the 0 ≤ j ≤ m we have
converges to zero due to the boundedness of tends to in nity, it follows that for every
lim Pj (y + ξn /tn ) = Pj (y).
n→∞
Therefore, the right hand side of the above inequality converges to while the left hand side converges to zero contradicting So
(ξn )n∈N
|Pm (y)|/C
Pm (y) 6= 0.
has to be unbounded. It follows from [18, Proposition 10.2.10]
that for su ciently large
η ∈ R2
and
t≥1
we have
X Pη(α) (0) inf{ | − Q(α) (0)|t|α| ; Q ∈ L(P )} ≤ C|η|−2b , ˜η (0) P α C and b are positive constants. Thus, from of (ξ, t) 7→ P the continuity j j P˜[y] (ξ, t) and the equivalences of P˜[y] (ξ, t) and |hD, yi P (ξ)|t as well as j P (α) |α| ˜ P (ξ, t) and α |P (ξ)|t , for n su ciently large there are Qn ∈ L(P ) with
where
|
˜ n [y] (0, tn ) P˜[y] (ξn , tn ) Q − | < σP (y)/2 ˜ ˜ n (0, tn ) P (ξn , tn ) Q
which implies
˜ [y] (0, t) Q P˜[y] (ξn , tn ) Q˜n [y] (0, tn ) ≤ lim sup | − | ≤ σP (y)/2. ˜ t) t≥1 Q∈L(P ) Q(0, n→∞ P˜ (ξn , tn ) Q˜n (0, tn ) inf
inf
But by lemma 2.6 we have
˜ [y] (0, t) Q = σP (y), ˜ t) t≥1 Q∈L(P ) Q(0, inf
contradicting
inf
σP (y) > 0.
As an immediate consequence we obtain analogously to theorem 4.3 in section 4.1 a characterization of when
X ×R
is
P + -convex
for singular supports.
The proof is mutatis mutandis the same so that we omit it.
Dierential operators in two variables
Theorem 4.6. For
P ∈ C[X1 , X2 ]
following are equivalent.
59
and an open connected set X ⊆ R2 the
i) X × R is P + -convex for singular supports. ii) The intersection of X with every hyperplane H satisfying σP0 (H ⊥ ) = 0 is convex. iii) For every x ∈ ∂X there is an open convex cone Γ 6= R2 with σP0 (y) 6= 0 for all y ∈ Γ and (x + Γ◦ ) ∩ X = ∅. Moreover, combining the above theorem with theorem 4.1 and theorem 4.4 we obtain as in section 4.1 the following, proving that iii) implies iv) in theorem A.
Theorem 4.7. Let X ⊆ R2 be open and P
∈ C[X1 , X2 ].
on D 0 (X) then P + (D) is surjective on D 0 (X × R).
If P (D) is surjective
60
Dierential operators in two variables
4.3 Partial dierential operators on non-quasianalytic ultradistributions of Beurling type in two variables The main purpose of the present section is to prove an adaption of Trèves' conjecture to the setting of ultradistributions of Beurling type associated with a non-quasianalytic weight function condition i) implies v).
P -convexity
for
ω.
Thus, we will show that in theorem A
Furthermore, we will give some results dealing with
ω -singular
supports which are valid in arbitrary dimension.
Spaces of ultradistributions of Beurling type generalize classical distributions by allowing more exible growth conditions for the Fourier transforms of the corresponding test functions than the Paley-Wiener weights.
We choose
the ultradistributional framework in the sense of Braun, Meise, and Taylor, as introduced in [7]. We begin by recalling some well-known facts about ultradistributions.
De nition 4.8.
A continuous increasing function
ω : [0, ∞) → [0, ∞)
K≥1
for all
is called
(non-quasianalytic) weight function if it satis es the following properties
a
(α) there exists (β )
ω
R∞ 0
ω(t) 1+t2
with
ω(2t) ≤ K(1 + ω(t))
t ≥ 0,
dt < ∞, log t ω(t)
(γ )
limt→∞
(δ )
ϕ = ω ◦ exp
is extended to
= 0, is convex.
Cd
by setting
ω(z) := ω(|z|).
Since we are not dealing with
quasianalytic weight functions we simply speak of weight functions for brevity. For
K ⊆ Rd
compact let
D(ω) (K) = {f ∈ E (Rd ); supp f ⊆ K and Z |fˆ(x)| exp(λω(x)) dx < ∞
for all
λ ≥ 1}
Rd be equipped with its natural Fréchet space topology, and
D(ω) (X) =
S
the union being taken over all compact subsets of the open subset
D(ω) (K), X of Rd ,
equipped with its natural (LF)-space topology. The elements of its dual space
0 D(ω) (X)
are the
ultradistributions of Beurling type.
The associated local space in the sense of Hörmander [18, Theorem 10.1.19]
0 E(ω) (X) = D(ω) (X)loc = {u ∈ D(ω) (X); ϕu ∈ D(ω) (X)
for all
is the space of
ultradierentiable functions of Beurling type.
Remark 4.9.
i) For each weight function
ω
we have
ϕ ∈ D(ω) (X)}
limt→∞ ω(t)/t = 0
by the
remark following 1.3 of Meise, Taylor, and Vogt in [27]. ii) It is shown in [7] that condition
(β)
guarantees that
that there are partitions of unity consisting of elements of
D(ω) (X) 6= {0} D(ω) (X).
and
iii) By [7] we have
E(ω) (X) = {f ∈ E (X); for
all
k∈N
|f |k,K :=
and
sup α∈Nd 0 ,x∈K
K ⊆ X, K |f
(α)
compact
(x)| exp −kϕ
∗
|α| k
< ∞},
Dierential operators in two variables
where
61
ϕ∗ (s) = sup{st − ϕ(t); t ≥ 0} is the Young conjugate of ϕ. δ > 1 the function ω(t) = t1/δ is a weight function for which
iv) For
the cor-
responding class of ultradierentiable functions coincides with the small Gevrey class
γ δ (X) = {f ∈ E (X); ∀ K ⊆ X, K
compact
∀C≥1:
sup x∈K,α∈Nd 0
De nition 4.10.
E(ω) (X)
|f (α) (x)| < ∞}. α!δ C |α|
equipped with the seminorms
{| · |k,K : k ∈ N, K ⊆ X, K is a nuclear Fréchet space. Its dual
compact}
0 0 E(ω) (X) is equal to the space of u ∈ D(ω) (X)
for which supp u
= Rd \
is a compact subset of
[ {B ⊆ Rd
open;
u(ϕ) = 0
for all
ϕ ∈ D(ω) (B)}
X.
The next theorem characterizes surjectivity of a dierential operator on spaces of ultradistributions of Beurling type which is due to Björck [1, Theorem 3.4.12]. It should be noted that, although the weight functions considered here are slightly more general than the ones used in [1] the theorem is valid. More generally, complementing a result of Bonet, Galbis, and Meise [3], surjectivity of convolution operators between spaces of ultradistributions of Beurling type has been characterized by Frerick and Wengenroth in [11] and contains the following theorem as a special case. The formulation in [11] uses the notion of
P -convexity
for
(ω)-supports
instead of
P -convexity
for supports but these
coincide, as is easily seen (see e.g. [19, Remark 2.5 i)]).
Theorem 4.11. For X ω
⊆ Rd
the following are equivalent.
open, P ∈ C[X1 , . . . , Xd ] and a weight function
0 0 i) P (D) : D(ω) (X) → D(ω) (X) is surjective.
ii) X is P -convex for supports as well as P -convex for (ω)-singular supports. Recall, that an open subset
X
of
supports if for every compact subset of
X
such that for every
0 u ∈ E(ω) (X)
Rd K
(ω)-singular L sing supp (ω) u ⊆ L whenever
is called of
X
we have
P -convex
sing supp (ω) P (−D)u
⊆ K.
Remark 4.12.
is elliptic the same is obviously true for
If
P
Pˇ .
Hence P (−D) E which is analytic in R \{0}. Since analytic funcE(ω) (X) for each weight function ω (cf. [7, Proposition
d
has a fundamental solution tions are contained in
for
there is a compact subset
4.10]) we have in particular
ch(sing supp (ω) E) = ch(sing supp (ω) P (−D)δ0 ), where
ch(A)
A ⊆ Rd . By [2, Theorem 0 X ⊆ R and every u ∈ D(ω) (X) that
denotes the convex hull of a set
therefore follows for each open set
sing supp (ω) P (−D)u
d
= sing supp (ω) u.
2.1] it
62
Dierential operators in two variables
In particular,
X
is
P -convex for (ω)-singular supports. This and the well-known X of Rd is P -convex for supports for elliptic P imply
fact that every open subset
by theorem 4.11 the surjectivity of
0 0 (X) → D(ω) (X) P (D) : D(ω) whenever
P
is elliptic.
As in the classical case, lated to the continuation of
P -convexity for (ω)-singular supports is closely re(ω)-ultradierentiability of P (−D)u to u. Analo-
gously to the tools introduced by Hörmander in order to deal with the classical case (see e.g. [18, Section 11.3] and section 2.1) Langenbruch introduced the following notions in [23]. For a polynomial
P , a subspace V
let
σP,(ω) (V ) := inf lim inf t≥1 ξ→∞
of
Rd , and t > 0, ξ ∈ Rd
P˜V (ξ, tω(ξ)) . P˜ (ξ, tω(ξ))
ω ≡ 1, we obtain Hörmander's classical de nition of σP (V ) as σP,(ω) (y) instead σP,(ω) (span{y}) for y ∈ Rd .
If we formally set
studied in section 2.1. In order to simplify notation we write of
The next proposition is an immediate consequence of [23, Theorem 2.5] and the ultradistributional analogue of the part of proposition 2.5 used in section 2.2.
Proposition 4.13. Let X1 ⊆ X2 be open convex subsets of Rd . Assume that ev-
ery hyperplane H = {x ∈ Rd ; hx, N i = α}, N ∈ S d−1 , α ∈ R with σP,(ω) (N ) = 0 which intersects X2 already intersects X1 . 0 Then for every u ∈ D(ω) (X2 ) with P (D)u ∈ E(ω) (X2 ) as well as u|X1 ∈ E(ω) (X1 ) we already have u ∈ E(ω) (X2 ). satisfy P (D)u ∈ E(ω) (X2 ) and u|X1 ∈ E(ω) (X1 ). X2 is convex it follows from the Theorem of supports (see e.g. [17, Theorem 4.3.3]) and [3, Theorem A] that there is v ∈ E(ω) (X2 ) such that P (D)v = P (D)u 0 so that w := u − v ∈ D(ω) (X2 ) satis es P (D)w = 0 as well as w|X1 ∈ E(ω) (X1 ). Hence, by [23, Theorem 2.5] it follows that w ∈ E(ω) (X2 ) which proves the theorem. Proof. Let
0 u ∈ D(ω) (X2 )
Since
As in the classical case, when investigating
P -convexity for (ω)-singular sup-
ports by means of the above proposition it is necessary to study the zeros of
σP,(ω)
in
S d−1 .
In order to do so, recall the de nition of
ω -localizations of P at P and ξ ∈ Rd
in nity, as introduced by Langenbruch in [23]. For a polynomial we set
Pξ,ω (x) := P (ξ + ω(ξ)x) which is again a polynomial of the same degree as
C[X1 , . . . , Xd ]
P.
The set of all limits in
of the normalized polynomials
x 7→
Pξ,ω (x) P˜ξ,ω (0)
Lω (P ). More precisely, if N ∈ S d−1 then the set of limits where ξ/|ξ| → N (with ξ tending to in nity) is denoted by Lω,N (P ). as
ξ
tends to in nity is denoted by
Dierential operators in two variables
63
as well as Lω,N (P ) are closed subsets of the unit sphere of d variables of degree not exceeding the degree of P , equipped ˜ . The non-zero multiples of elements of Lω (P ) (resp. with the norm Q 7→ Q(0) of Lω,N (P )) are called ω -localizations of P at in nity (resp. ω -localizations of P at in nity in direction N ). Since ω(ξ) = ω(|ξ|), Q ∈ Lω,N (Pˇ ) if and only if ˇ ∈ Lω,−N (P ). Again, if we formally set ω ≡ 1 we obtain the well-known set Q L(P ) of localizations of P at in nity (see Hörmander [18, De nition 10.2.6] or Obviously,
Lω (P )
all polynomials in
section 2.1). For the classical case, i.e. if formally
ω ≡ 1,
the next result is lemma 2.6.
The proof is exactly the same as the one of 2.6 so that we omit it.
Lemma 4.14. Let P be of degree m with principal part Pm . i) For every subspace V of Rd and t ≥ 1 we have lim inf ξ→∞
˜ V (0, t) Q P˜V (ξ, tω(ξ)) = inf . ˜ t) Q∈Lω (P ) Q(0, P˜ (ξ, tω(ξ))
ii) Let N ∈ S d−1 and Q ∈ Lω,N (P ). If Pm (N ) 6= 0 then Q is constant. iii) If P is non-elliptic then for every subspace V of Rd and t ≥ 1 we have lim inf ξ→∞
P˜V (ξ, tω(ξ)) = inf N ∈S d−1 ,Pm (N )=0 P˜ (ξ, tω(ξ))
˜ V (0, t) Q . ˜ t) Q∈Lω,N (P ) Q(0, inf
iv) With the convention that the in mum taken over an empty subset of [0, 1] equals 1 we have σP,(ω) (V ) = inf
inf
t≥1 N ∈S d−1 ,Pm (N )=0
In case of
ω ≡ 1
˜ V (0, t) Q . ˜ t) Q∈Lω,N (P ) Q(0, inf
the corresponding result of the next proposition is due
to Hörmander [18, Theorem 10.2.8] and its proof uses the Tarski-Seidenberg theorem. In our case, the proof is rather elementary.
Lemma 4.15. If Q ∈ Lω,N (P ) then N ∈ Λ(Q). Proof.
Since
ω(ξ) = ω(|ξ|),
by a linear change of coordinates we can assume
N = e1 = (1, 0, . . . , 0). We denote the degree of P (e1 ) ≡ 0 we clearly have by Taylor's theorem that e1 ∈ Λ(P ) implies e1 ∈ Λ(Q) by the de nition of Lω (P ).
without loss of generality that
P
by
m.
In case of
which clearly
P (e1 )
(e )
Pξ,ω1 does not vanish d (α) identically either, for every ξ ∈ R . Since P → 7 (0)| is a norm on the α |P d space of all polynomials in d variables, it follows that for every ξ ∈ R Now, if
does not vanish identically it follows that
P
0 6=
X α
(e )
|Pξ,ω1 (0)| =
X α
|P (α+e1 ) (ξ)|ω(ξ)|α| =
X 0≤|α|≤m−1
|P (α+e1 ) (ξ)|ω(ξ)|α| ,
64
Dierential operators in two variables
because
P
has degree
m.
Hence, for every
ξ ∈ Rd , t ∈ R
we have by Taylor's
theorem
0
≤ = ≤ ≤
Since
Q ∈ Lω (P )
|P (e1 ) (ξ + ω(ξ)(x + se1 ))| P (α) (ξ)|ω(ξ)|α| α |P P 1 (x + se1 )α | | 0≤|α|≤m−1 P (α+e1 ) (ξ)ω(ξ)|α| α! P (α) (ξ)|ω(ξ)|α| α |P P 1 (α+e1 ) (ξ)|ω(ξ)|α| α! |(x + se1 )α | 0≤|α|≤m−1 |P P (α+e1 ) (ξ)|ω(ξ)1+|α| 0≤|α|≤m−1 |P 1 max0≤|α|≤m−1 α! |(x + se1 )α | . ω(ξ)
there is
(ξn )n∈N
tending to in nity such that
Q(x) = lim
n→∞
P (ξn + ω(ξn )x) . P˜ξn ,ω (0)
In particular, we also have
P (e1 ) (ξn + ω(ξn )x) . n→∞ P˜ξn ,ω (0)
Q(e1 ) (x) = lim The space of all polynomials in
d
variables of degree not exceeding
nite dimensional, all norms on it are equivalent. subsequence of and
(ξn )n∈N
if necessary, there is
c>0
m
being
Therefore, by passing to a such that for every
x ∈ Rd
s∈R |Q(e1 ) (x + se1 )|
=
|P (e1 ) (ξn + ω(ξn )(x + se1 ))| n→∞ P˜ξn ,ω (0) lim
|P (e1 ) (ξn + ω(ξn )(x + se1 ))| P (α+e1 ) (ξ )|ω(ξ )|α| n→∞ n n α |P
≤ c lim
1 max0≤|α|≤m−1 α! |(x + se1 )α | n→∞ ω(ξn ) 0.
≤ c lim =
x ∈ Rd the polynomial qx : R → C, s 7→ Q(x + se1 ) satis es (x+se1 ) = 0. Thus qx is constant which shows that e1 ∈ Λ(Q).
Hence, for each
qx0 (s)
(e1 )
=Q
With the aid of the previous lemma we can prove the next result exactly as lemma 4.2. Again we omit the proof.
Lemma 4.16. Let P part Pm . Then
∈ C[X1 , X2 ]
be a non-constant polynomial with principal
{y ∈ R2 \{0}; σP,(ω) (y) = 0} ⊆ {y ∈ R2 \{0}; Pm (y) = 0}.
In particular, the set {y ∈ S 1 ; σP,(ω) (y) = 0} is nite. Before we continue to discuss the two-dimensional case we provide a general su cient condition for
P -convexity
for
(ω)-singular
supports in arbitrary di-
mension similar to theorems 2.9 and 2.11. In order to do this, we rst recall an
Dierential operators in two variables
65
analogue to the classical characterization of
P -convexity for singular supports in
the context of ultradistributions of Beurling type which is due to Björck [1, Theorem 3.4.2 and Theorem 3.4.4]. Again, although the weight functions considered here are slightly more general than the ones in [1], the theorem is still valid in our context (see also [19, Theorem 4.2]).
Theorem 4.17.
0 i) If u ∈ E(ω) (Rd ) then
ch(sing supp (ω) u) = ch(sing supp (ω) P (−D)u).
ii) For an open subset X of Rd the following are equivalent. a) X is P -convex for (ω)-singular supports. 0 b) For each u ∈ E(ω) (X) one has dist(sing supp (ω) u, X c ) = dist(sing supp (ω) P (−D)u, X c ). From the above theorem it follows immediately that convex open sets are
P -convex
for
ω -singular
supports (see [1, Corollary 3.4.3]). Moreover, as in the
classical case (cf. [18, Theorem 10.6.4 and/or Theorem 10.7.4]) it follows that the interior of the intersection of any family of
P -convex sets for ω -singular supports
(see [1, Theorem 3.4.5]) as well as the set of points having a neighborhood
P -convex P -convex for ω -singular supports. of P -convexity for (ω)-singular supports given in
contained in all but nitely many members of a family of sets being for
ω -singular
supports is again
Using the characterization
4.17 ii) and theorem 4.13 we can prove part i) of the next theorem by exactly the same kind of arguments as theorem 2.9 ii). Part ii) follows from i) as theorem 2.11 ii) follows from theorem 2.9 ii). Because the proofs are verbatim the same we omit them.
Theorem 4.18. Let X ⊆ Rd be open and connected, P
∈ C[X1 , . . . , Xd ].
i) X is P -convex for (ω)-singular supports if for every x ∈ ∂X and any r > 0 there are convex sets C1 ⊆ C2 ⊆ Rd \X such that x ∈ C2 , C1 ⊆ Rd \B(0, r) and every hyperplane H with σP,(ω) (H ⊥ ) = 0 intersecting C2 already intersects C1 . ii) X is P -convex for (ω)-singular supports if for every x ∈ ∂X there is an open convex cone Γ 6= Rd such that (x + Γ◦ ) ∩ X = ∅ and σP,(ω) (y) 6= 0 for all y ∈ Γ. Now, changing the obvious in the proof of theorem 4.3 we obtain a characterization of
P -convexity
for
(ω)-singular
supports of open subsets
X ⊆ R2 .
Theorem 4.19. For a non-constant polynomial P ∈ C[X1 , X2 ] and an open connected set X ⊆ R2 the following are equivalent. i) X is P -convex for (ω)-singular supports. ii) The intersection of X with every hyperplane H satisfying σP,(ω) (H ⊥ ) = 0 is convex. iii) For every x ∈ ∂X there is an open convex cone Γ 6= R2 with σP,(ω) (y) 6= 0 for all y ∈ Γ and (x + Γ◦ ) ∩ X = ∅.
66
Dierential operators in two variables
As in section 4.1 we derive our next result from the above theorem, theorem 4.1 and the fact that for supports as well
0 P (D) is surjective on D(ω) (X) if and only if X as P -convex for (ω)-singular supports.
Theorem 4.20. Let X ⊆ R2 be open and let P
polynomial. The following are equivalent.
∈ C[X1 , X2 ]
is
P -convex
be a non-constant
i) P (D) : E (X) → E (X) is surjective. 0 0 ii) P (D) : D(ω) (X) → D(ω) (X) is surjective for some non-quasianalytic weight function ω. 0 0 iii) P (D) : D(ω) (X) → D(ω) (X) is surjective for each non-quasianalytic weight function ω.
iv) The intersection of every characteristic line with any connected component of X is convex. The next example shows that for
d ≥ 3
an analogous result to the above
theorem is not true in general. See also Langenbruch [23, Example 3.13], where it is even shown that surjectivity of on the weight function
Example 4.21.
ω
0 P (D) on D(ω) (X) for d ≥ 3 depends explicitly
in general.
d > 2 and P (x1 , . . . , xd ) = x21 − x22 − . . . − x2d . Moreover, 2 2 1/2 let Γ := {x ∈ R ; xd > (x1 + . . . + xd−1 ) }. Then Γ is an open convex cone d ◦ with Γ = Γ. Set X := R \Γ. As seen in 2.24, X is P -convex for supports but not P -convex for singular supports. Hence, P (D) is surjective on E (X) but P (D) is not surjective on D 0 (X). √ Moreover, as in 2.24 one checks that Q(ξ) = (ξ1 − ξ2 )/ 2 is a ω -localization √ at in nity in direction 1/ 2(1, 1, 0, . . . , 0). It therefore follows from lemma 4.14 Let
d
that
lim inf ξ→∞
where
P˜span{ed } (ξ, ω(ξ)) =0 P˜ (ξ, ω(ξ))
ed = (0, . . . , 0, 1). H = {x ∈ Rd ; hx, ed i = −1}
Setting
and
K := H ∩ {x ∈ Rd ; |x| ≤ 2} ∂X = ∂Γ to K is 1 while the distance of K relative H , strictly increases 1. Hence, it 0 that P (D) cannot be surjective on D(ω) (X).
it is easily seen that the distance of
∂Γ
to
∂H K ,
i.e. to the boundary of
follows from [23, Corollary 2.7]
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