P. Boggiatto - E. Schrohe
CHARACTERIZATION, SPECTRAL INVARIANCE AND THE
FREDHOLM PROPERTY OF MULTI-QUASI-ELLIPTIC
OPERATORS
Abstract. The class L0ρ,P(Rn) of pseudodifferential operators of zero order, modelled on a multi-quasi-elliptic weight, is shown to be a 9∗−algebra in the algebra B(L2(Rn)) of all bounded operators on
L2(Rn). Moreover, the Fredholm property is proven to characterize the elliptic elements in this algebra. This is achieved through a characteri-zation of these operators in terms of the mapping properties between the Sobolev spaces HPs(Rn)of their iterated commutators with multiplication operators and vector fields. We also prove and make use of the fact that order reduction holds in the scale of the HPs(Rn)-Sobolev spaces, that is every HPs(Rn)is homeomorphic to L2(Rn)through a suitable multi-quasi-elliptic operator of order s.
1. Introduction. Statement of the results.
Multi-quasi-elliptic polynomials were introduced in the seventies as a natural general-ization of elliptic and quasi-elliptic polynomials. They are an important subclass of the hypoelliptic polynomials of H¨ormander [15] and were studied by many authors, among them Friberg [9], Cattabriga [8], Zanghirati [24], Pini [17] and Volevic-Gindikin [23].
In [5] this theory is used to develop a pseudodifferential calculus for a class of operators on weighted Sobolev spaces inR2n based on the concept of a “Newton poly-hedron”.
DEFINITION 1. A complete Newton polyhedronP is a polyhedron of dimension d inRd+ = {r ∈ Rd : rj ≥ 0,j = 1, . . . ,d}with integer vertices and the following
properties:
(i) Ifv(k),k = 1, . . . ,N are the vertices of P, then {r ∈ Rd+ : rj ≤ v(k)j ,j =
1, . . . ,d} ⊆P.
(ii) There are finitely many elements a(l),l = 1, . . . ,M, with a(l)j > 0 for j = 1, . . . ,d, such that
P= {r∈Rd+:ha(l),ri ≤1,l=1, . . . ,M}.
In other words,Pmay contain only points with non-negative coordinates and must have one face on each coordinate hyperplane, while the other faces must have nor-mal vectors with all components strictly positive; in particular these faces must not be parallel to the coordinate hyperplanes.
Given a polynomial p =Pcγzγ in d complex variables, we associate with it the
polyhedronP consisting of the convex hull of allγ ∈ Nd
0 with cγ 6= 0. It is called
multi-quasi-elliptic if the associated polyhedron is a complete Newton polyhedron and
there exist constants C,R, such that:
X
γ∈P
|zγ| ≤C|p(z)| for|z| ≥R.
Here and in the following the notationPγ∈P indicates that we sum over all multi-indicesγ ∈Nn
0∩P, so that the sum is clearly finite.
The corresponding operators P(Dz) = Pγ∈PcγDγz are also said to be
multi-quasi-elliptic.
We now consider the case of even dimension d =2n,n≥1, and split z∈R2ninto
z=(x, ξ ), where x ∈Rn, ξ ∈Rn. Giving toξ the role of a covariable, we recover the
case of operators with polynomial cefficients:
p(x,D)= X
(α,β)∈P
xαDβx.
Easy but significant examples are the operators onRof the form:
P=x2h1+x2h0D2k0+D2k1.
If h0,h1,k0,k1∈Nsatisfy the conditions:
0<h0<h1, 0<k0<k1,
h0
h1
+k0
k1
>1,
then to P is associated the complete polyhedronP with vertices {(0,0), (2h1,0),
(2h0,2k0), (0,k1)}, and P is multi-quasi-elliptic with respect toP.
The symbolsσ (P)=x2h1+x2h0ξ2k0+ξ2k1 associated with these operators were originally considered by Gorcakov [10] and Pini [17] and later studied by Friberg in connection with differential operators with constants coefficients.
As an example of multi-quasi-elliptic operators in dimension n let us consider the operators of the form
1kx+ X |α+β|<k
x2αD2βx + |x|2k.
They are multi-quasi-elliptic. The associated polyhedron isP = {z ∈ Rn+×Rn+ : P2n
j=1zj ≤2k}.Unlike the class of quasi-elliptic operators, the space of
multi-quasi-elliptic operators is closed under composition: If A1and A2are multi-quasi-elliptic
with respect towP1 andwP2, then the operator A1◦A2is multi-quasi-elliptic with
In particular, any product of quasi-elliptic operators is multi-quasi-elliptic although it is, in general, no more quasi-elliptic. On the other hand there are multi-quasi-elliptic operators that are not product of quasi-elliptic operators (see Friberg [9]).
More generally, for a fixed complete polyhedronP, one can introduce pseudodif-ferential operators in classes suitably modelled on the polyhedronP. Before stating our results we summarize here some facts about this theory following [5], [3].
DEFINITION2. LetwP(z)=(Pγ∈Pz2γ)1/2, z=(x, ξ )∈ R2n, be the standard
weight function associated toP. For m∈R, ρ∈]0,1], we define the symbol classes: 3mρ,P(R2n)= {a∈C∞(R2n): |∂zγa(z)| ≤CγwPm−ρ|γ|(z)}.
The choice of the best constants defines a Fr´echet topology for3mρ,P(R2n).
For technical reasons we will also assume thatρdoes not exceed a certain value 1/µ < 1, whereµis the “formal order” of p(z), depending on the polyhedronP. This is of no importance for our purposes, see [3] for more details. The corresponding classes of operators are:
Lmρ,P = {A=O p(a): a∈3mρ,P(R2n)}. Here [O p(a)u](x)=(2π )−nR
Rnei xξa(x, ξ )uˆ(ξ )dξ, whileuˆ(ξ )=
R
e−i xξu(x)d x for u∈ S(Rn), the Schwartz space of all rapidly decreasing functions.
DEFINITION3. E3mρ,P is the space of all a∈3mρ,P(R2n)such that, with suitable constants C,R >0,
wmP(z)≤C|a(z)|, |z| ≥R.
The elements of E3mρ,P are called multi-quasi-elliptic symbols, the corresponding classes of operators are denoted with
ELmρ,P = {A=O p(a): a∈ E3mρ,P}.
Being a natural generalization of the classes considered by Friberg and Cattabriga, we call these operators multi-quasi-elliptic of order m.
For m=0, multi-quasi-ellipticity of a symbol a requires that|a(x, ξ )| ≥c>0, so multi-quasi-ellipticity then coincides with uniform ellipticity.
In caseP is the simplex with vertices the origin and the points{e(k)}2n
k=1 of the
canonical basis ofR2n, we are reduced to Shubin’s classesŴmρ(R2n), see Shubin [21], Chapter IV; Helffer [14], Chapter I.
DEFINITION4. Let3s =O p(wsP). For s∈Rwe define the Sobolev spaces
The pseudodifferential operators with symbols in3mρ,P(R2n)act continuously on these
spaces: For a∈3mρ,P(R2n)and s∈R,
O p(a): HPs(Rn)→ HPs−m(Rn)
is bounded and the operator norm can be estimated in terms of the symbol semi-norms of a.
The first result we present is that the pseudodifferential operators in L0ρ,P can be characterized by commutators, in fact we have
THEOREM1. A linear operator A : S(Rn)→ S′(Rn)belongs to L0ρ,Pif and only if, for all multi-indices,α, β ∈ Nn
0, the iterated commutators adα(x)adβ(Dx)A have
bounded extensions
adα(x)adβ(Dx)A : L2(Rn)→HPρ|α+β|(R n).
Theorem 1 is the key tool in the proof of the spectral invariance result for L0ρ,P: THEOREM2. Let A∈L0ρ,P, and suppose that A is invertible inB(L2(Rn)). Then A−1∈L0ρ,P.
The idea of characterizing pseudodifferential operators by commutators and using this for showing the spectral invariance goes back to R. Beals. In [1], Beals had in-troduced a calculus with pseudodifferential operators having symbols in very general classes S8,ϕλ . In [1] he stated the analogs of Theorems 1 and 2 under the restriction ϕ =8. Since the proof was not generally accepted, Ueberberg in 1988 published an article where he showed corresponding versions of Theorems 1 and 2 for the operators with symbols in H¨ormander’s classes S0ρ,δ,0 ≤ δ ≤ ρ ≤ 1, δ < 1. Let us point out, however, that the symbols we are considering are not contained in the class introduced by Beals, since our case corresponds to the choice
ϕ(x, ξ )=8(x, ξ )=ωρP(x, ξ ),
which does not satisfy Beals’ conditionϕ ≤const. In 1994, Bony and Chemin [6] proved analogous of the above theorems for a large class of symbols using the H¨orman-der-Weyl quantization:
[O pwa]u(x)=(2π )−n Z Z
ei(x−y)ξa(x+y
2 , ξ )u(y)d ydξ.
Multi-quasi-elliptic operators can be naturally re-considered in the frame of the Weyl-calculus, see [4], so that if one can show that the metric
gx,ξ =ωρP(x, ξ )(|d x|
satisfies the conditions ”Lenteur”, ”Principe d’Incertitude” (which are actually obvi-ous), and ”Temp´erence (Forte)” (not evident in our case) of [6], then Theorems 1 and 2 could be deduced from [6], Th´eor`emes 5.5 and 7.6. It seems however that the present direct proof, which relies on the spectral invariance result for S0,00 (Rn×Rn), shows an easier, alternative technique.
From Theorems 1 and 2 we will obtain
THEOREM3. L0ρ,P is a submultiplicative9∗-subalgebra ofL(L2(Rn)).
9∗algebras were introduced by Gramsch [11], Definition 5.1: A subalgebraAof
L(H), H a Hilbert, space is called a9∗-subalgebra, if
(i) it carries a Fr´echet topology which is stronger than the norm topology ofL(H), (ii) it is symmetric, i.e.,A∗=A, and
(iii) it is spectrally invariant, i.e.,A∩L(H)−1=A−1.
Here,A−1andL(H)−1denote the groups of invertible elements in the respective algebras. 9∗-algebras have many important features: in9∗-algebras there is a holo-morphic functional calculus in several complex variables, there are results in Fredholm and perturtation theory [11] as well as for periodic geodesics. Concerning decom-positions of inverses to analytic Fredholm functionals and the division problem for operator-valued distributions, see [12]. For further results on the9∗-property see also [13], [14], [18], [19], [20].
The Fr´echet topology on L0ρ,Pis the one induced from30ρ,P(R2n)via the injective mapping O p. A Fr´echet algebraAis called submultiplicative if there is a system of semi-norms{qj : j ∈N}which defines the topology and satisfies
qj(ab)≤qj(a)qj(b), a,b∈A.
We next show the existence of order reduction within this calculus, see Lemma 2. This allows us to extend the results on the characterization by commutators and spectral invariance to the case of arbitrary order.
Finally we characterize the Fredholm property by multi-quasi-ellipticity.
THEOREM 4. Let A ∈ Lmρ,P, s ∈ R. Then A : HPs(Rn) → HPs−m(Rn)is a Fredholm operator if and only if it is multi-quasi-elliptic of order m.
Let A∈Lmρ,P, s∈R. Then A : HPs(Rn)→ HPs−m(Rn)is a Fredholm operator if and only if it is multi-quasi-elliptic of order m.
Let p(x)be a positive multi-quasi-elliptic polynomial in the variables x ∈ Rn. Then the Schr¨odinger operator P = −1x + p(x), a generalization of the harmonic
oscillator of quantum mechanics, is multi-quasi-elliptic in the sense of our previous definition.
In both cases one would like to have spectral asymptotic estimates for these operators, in particular for the function N(λ)=Pλ
j≤λ1 counting the number of the eigenvalues
λj not exceedingλ([3] can be considered a first step in this direction). The idea is
to follow the approach taken by Helffer in [14] which is based on a thorough analysis of Shubin’s classes, and this makes the results of spectral invariance and their conse-quences particularly interesting.
2. Pseudodifferential operators in Lmρ,P
We now review the main properties of the multi-quasi-elliptic calculus assuming famil-iarity with the standard pseudodifferential calculus, cf. [16].
PROPOSITION4. (a) If a ∈ E3mρ,P then a−1 ∈ E3−ρ,mP and a−1∂αa ∈
E3−ρ,ρP|α|for allα(possibly after a modification of a(ζ )on a compact set).
(b) If a1∈ E3mρ,1Pand a2∈E3 m2
ρ,P then a1a2∈ E3 m1+m2
ρ,P .
(c) If A1∈ELmρ,1Pand A2∈EL m2
ρ,Pthen A1A2∈ EL m1+m2
ρ,P .
DEFINITION 6. An operator R ∈ Tm∈RLmρ,P is called regularizing or
smooth-ing. Regularizing operators define continuous maps R : S′(Rn)→ S(Rn).They have integral kernels R=R(x,y)∈ S(Rnx×Rny).
PROPOSITION5 (EXISTENCE OF THEPARAMETRIX). Let A∈ELmρ,P; then there exists an operator B∈EL−ρ,mPsuch that the operators R1= A B−I and R2=B A−I
both are regularizing. B is said to be a parametrix to A. If B′is another parametrix to the same operator A, then B−B′is a regularizing operator.
PROPOSITION6 (ELLIPTICREGULARITY). Let A∈ ELρ,mP. If Au∈ S(Rn)for some u∈S′(Rn)then necessarily u∈ S(Rn).
Definition 4 of the Sobolev spaces HPm(Rn)can be rephrased.
PROPOSITION 7. LetP be a fixed complete Newton polyhedron and let Am ∈
E Lmρ,P be a multi-quasi-elliptic operator; then
HPm(Rn)= Am−1(L2(Rn)).
Note that HPm(Rn)depends neither onρnor on the particular operator Am, but only
on m andP.
The main features of these spaces are the following.
PROPOSITION8. HPm(Rn)has a Hilbert space structure given by the inner prod-uct(u, v)P =(Amu,Amv)L2 +(Ru,Rv)L2. Here Amis an elliptic operator defining
the space HPm(Rn)according to Proposition 7, and R=I−AemAm, with a parametrix
e
Amof Am. We denote by||u||mthe norm of an element u in the space HPm(Rn).
Equiv-alently, we could define
HP1(Rn)= {u∈ S′(Rn): xαDβu ∈L2(Rn)for(α, β)∈P}.
with the inner product(u, v)3P =
P
(α,β)∈P(xαDβu,xαDβv)L2.
PROPOSITION9. (a) The topological dual HPm′(Rn)of HPm(Rn)is HP−m(Rn). (b) We have continuous imbeddings S(Rn) ֒→HPm(Rn) ֒→S′(Rn).
(c) We have compact imbeddings HPt (Rn) ֒→ HPs(Rn)if t>s.
(d) proj−limm∈RHPm(Rn)=S(Rn), ind−limm∈RHPm(R
3. Abstract characterization, order reduction, spectral invariance and the Fred-hold property
We now adress the central questions of this paper. We begin by proving that the opera-tors of order zero can be characterized via iterated commutaopera-tors.
DEFINITION7. Let A : S(Rn)→ S′(Rn)be a linear operator. For j =1, . . . ,n,
THEOREM 5 (ABSTRACT CHARACTERIZATION). A linear operator A :
S(Rn) → S′(Rn)belongs to L0ρ,P if and only if, for all multi-indicesα, β, the
iter-ated commutators BβαA have continuous extensions to linear maps: BβαA : L2(Rn)→
HPρ|α+β|(Rn).
Proof. If A ∈ L0ρ,P then the symbol of BβαA is ∂ξα∂xβa ∈ 3−ρ,ρP|α+β|(R2n), so that clearly the commutators extend to continuous maps: BβαA : L2(Rn)→ HPρ|α+β|(Rn).
Conversely assume that A admits the required continuous extensions: BβαA : L2(Rn)→HPρ|α+β|(Rn).
Letα0, β0 be arbitrary multi-indices and let 3ρ|α0+β0| = O p(w
ρ(|α0+β0|)
P )be as in Definition 4. For all multi-indicesα, β, we then have continuous maps:
(1) Bβα[3ρ|α0+β0|◦Bβα00A] : L
The continuity of the first operator is due to the hypothesis and the equality Bα1
β1B
α0
β0A =Bα1+α0
β1+β0A which is easily checked.
the symbol
In particular, choosingα0=β0=0, we see that A is a pseudodifferential operator and
b0,0=σ (A)=a∈ S0,00 (R2n).
Since w2P(x, ξ ) is a polynomial, there exist m ∈ R and ρ′ > 0 such that 3ρ|α0+β0|∈L
m
ρ′,0. For the symbol b we therefore have the asympotic expansion:
bα0,β0 ∼
In particular, the difference is a bounded function.
For 1 ≤ |α| ≤k the terms under the summation in (3) are products of derivatives
of a∈S00,0(R2n)and ofwρ
The operator with the symbol∂zkαalso satisfies the assymption of the theorem. Just as
before, we see that∂z2
Notice that we still have the subscript “0” instead of the desired “ρ”. Let us now suppose|α0+β0| = 2. The terms of (3) with 1 ≤ |α| ≤ k are now products of
derivatives of a(z)and ofwP2ρ(z) ∈ 32ρρ,P(R2n), so that, thanks to (4), they are still bounded. Proceeding as before, we conclude that the second derivatives of a(z)belong to3−0,2ρP(R2n). Iteration of the argument shows that∂zγa(z)∈ 3−|0,Pγ|ρ(R2n)for allγ, which implies a∈30ρ,P(R2n).
COROLLARY1. A linear operator A : S(Rn)→ S′(Rn)belongs to L0ρ,P if and only if, for all multi-indicesα, β and for all s ∈ R, the iterated commutators BβαA
have continuous extensions to linear maps: BβαA : HPs(Rn)→ HPs+ρ|α+β|(Rn).
As a preparation for the proof of the spectral invariance we need the following lemma. The proof is just as in the standard case. The crucial identity one has to verify is that
[ A−1, 3ε]= −A−1[ A, 3ε] A−1.
for all A ∈ L0ρ,P and a suitableε > 0. As before,3ε = O p(ωεP). Equality holds, because A ∈ S0,00 (R2n)andωP ∈ Sρm′,0(R2n)for suitable m, ρ′ > 0. For details see
[22] or [21], Section I.6.
LEMMA1. Let A∈L0ρ,P be invertible in the classB(L2(Rn))of bounded opera-tors on L2(Rn). Then A is invertible inB(HPs(Rn))for all s ∈R.
THEOREM6 (SPECTRALINVARIANCE ANDSUBMULTIPLICATIVITY)). Let A∈ L0ρ,P, and suppose that A is invertible inB(L2(Rn)). Then A−1 ∈ L0ρ,P. Moreover,
L0ρ,Pis a submultiplicative9∗-subalgebra ofB(L2(Rn)).
Proof. L0ρ,Pis a symmetric Fr´echet subalgebra ofB(L2(Rn))with a stronger topology. In order to show it is a9∗-subalgebra, we only have to check spectral invariance. Since
S0,00 (R2n)is a9∗−algebra, A−1necessarily belongs to S0,00 (R2n), hence [xj,A−1]= −A−1[xj,A] A−1,
[Dj,A−1]= −A−1[Dj,A] A−1,
(5)
cf. [20], Appendix. Using Leibniz’ rule and Lemma 1, these identities show that
BβαA−1: L2(Rn)→ HPρ|α+β|(Rn) is bounded. Hence A−1∈L0ρ,P by Theorem 5.
Finally let us check submultiplicativity. Corollary 1 suggests the following system of semi-norms{pα,β,s :α, β ∈Nn0,s∈N}for the topology of L0ρ,P:
pα,β,s(A)= kBβαAkL(Hs P(Rn),H
s+ρ|α+β|
P (Rn))
.
A priori, this topology is weaker than the topology induced from30ρ,P(R2n), since the
operator norm can be estimated in terms of the symbol semi-norms. The open mapping theorem yields that both are equivalent. The construction in [13], 3.4 ff, eventually shows how to derive submultiplicative semi-norms from the system{pα,β,s}.
LEMMA 2 (ORDER REDUCTION). For all s ∈ Rthere exists an operator T ∈
ELsρ,P such that T : HPs(Rn)→ L2(Rn)is a bicontinuous bijection.
Proof. Let us set A = O p(ws/2P ) ∈ ELs/2ρ,P. If A∗ is its L2-formal adjoint, then
A A∗ ∈ ELsρ,P and A A∗ : HPs(Rn)→ L2(Rn)is Fredholm. It can be shown easily that KerA A∗ = A A∗(HPs(Rn))⊥, where⊥means orthocomplementation in L2(Rn). In fact, KerA A∗ ⊆ S(Rn), is independent of s, and f ∈ KerA A∗ is equivalent to
f⊥A A∗(S(Rn))which in turn is equivalent to f⊥A A∗(HPs(Rn))as A A∗(S(Rn))is dense in A A∗(HPs(Rn)). Suppose now that {f1, ..,fk}is an orthonormal basis of the
finite dimensional vector space Ker A A∗=A A∗(HPs(Rn))⊥, viewed now as imbedded in both HPs(Rn)and L2(Rn). We consider the operator B = A A∗+ P where P
is the continuous extension to HPs(Rn)of P f = Pkj=1(f, fj)L2 fj, f ∈ S(Rn). P
is compact as it has finite rank and is smoothing since it has an integral kernel in
S(Rn×Rn). Then B is a Fredholm operator in ELsρ,P and index(B)=0. It can be easily checked that B : HPs(Rn)→ L2(Rn)is injective so that it is a bijection. The continuity of B−1follows from the open mapping theorem and the continuity of B.
COROLLARY2. Let A ∈L0ρ,P be invertible inB(HPs(Rn))for some s ∈ R, then A−1∈L0ρ,P.
Proof. If T : HPs(Rn)→ L2(Rn)is the order reduction, we know that B=T AT−1∈ L0ρ,P is invertible on L2(Rn). By Theorem 6 B−1 ∈ L0ρ,P, so A−1 = T B−1T−1 ∈ L0ρ,P.
REMARK 1. A consequence of Corollary 2 is that the spectrum of an operator
A∈L0ρ,P is independent of the space HPs(Rn). This is particularly relevant in view of the developement of a spectral theory for multi-quasi-elliptic operators.
The fact that multi-quasi-elliptic operators have the Fredholm property was proven in [5]; we show here that the converse holds.
THEOREM7. Let A∈Lm
ρ,P,m∈R. Then the following are equivalent:
(a) A∈ ELmρ,P.
(b) A : HPs(Rn)→HPs−m(Rn)is a Fredholm operator for all s∈R.
(c) A : Hs0
P(Rn)→ H s0−m
P (Rn)is a Fredholm operator for some s0∈R.
Proof. By [5], (a) implies (b), (b) trivially implies (c). In order to show that (c) implies
for suitableε >0,Cε>0,
ωρP(x, ξ )≥Cεhxiεhξiε.
This implies that30ρ,P(R2n)is embedded in the classS˜ε,00 (Rn×Rn)of slowly varying symbols, cf. Kumano-go [16], Chapter III, Definition 5.11. For these symbols it has been shown in [18], Theorem 1.8 that the Fredholm property on L2(Rn)implies uni-form ellipticity. This concludes the proof, for HP0(Rn)= L2(Rn), and the notion of uniform ellipticity coincides with that of multi-quasi-ellipticity of order zero.
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AMS Subject Classification: 47G30, 35S30. Paolo BOGGIATTO
Dipartimento di Matematica Universit`a di Torino Via C. Alberto 10 10123 Torino, ITALY
e-mail:paolo.boggiatto@unito.it
Elmar SCHROHE Institut f¨ur Mathematik Universit¨at Potsdam
14415-Potsdam, GERMANY
e-mail:schrohe@math.uni-potsdam.de
