5.8 Parametrices, ellipticity and hypoellipticity
5.8.2 Parametrix
with
N1 = π(I +R)ρ[αo]+mν Δαo{ψ(π(R))}E(Λ1,∞)σ−1(x, π)L(Hπ),
N2 = π(I +R)ρ[αo]+mν σo(x, π){Δαoσ(x, π)} E(Λ1,∞)σ−1(x, π)L(Hπ). For the first norm, we see that
N1 ≤ π(I +R)ρ[αo]+mν Δαo{ψ(π(R))}π(I +R)−mνL(Hπ)
π(I +R)mνE(Λ1,∞)σ−1(x, π)L(Hπ)
≤ CψC0−1,
since Δαo{ψ(π(R))} ∈S−∞ by Lemma 5.8.6. For the second norm, we see that N2 ≤ π(I +R)ρ[αo]+mν σo(x, π)π(I +R)−ρ[αo]ν L(Hπ)
π(I +R)ρ[αo]ν Δαoσ(x, π)π(I +R)−mνL(Hπ)
π(I +R)mν E(Λ1,∞)σ−1(x, π)L(Hπ)
≤ ψ∞Cρ−[1αo]C0−1σSmρ,δ,[αo],0,|m|.
Recursively on [αo], we can show similar properties for Δαo
ψ(π(R))σ(x, π)−1 , and obtain
σo(x, π)S−m
ρ,δ,ao,0,0
≤Cao,ψ
a1,a2≤ao
|γ|≤ρamaxo
Cγ−(a1+1)σ(x, π)aS2m
ρ,δ,ao,0,|m|. More generally, we have
XxβoΔαo{ψ(π(R))} =
[α1]+[α2]=[αo] [β1]+[β2]=[βo]
cβ1,β2cα1,α2 Xxβ1Δα1σo(x, π)
Xxβ2Δα2σ(x, π).
Because of the very first remark of this proof, we obtain XβoΔαoσo in terms of XβΔασo with [β]<[βo] and [α]<[αo] and of some derivatives ofψ(π(R)) and σ. If we assume that we can control all the seminormsσoS−m
ρ,δ,a,b,cwitha <[αo], b <[βo] and anyc∈R, then we can proceed as above introducing powers of I +R to obtain the estimate for the seminorms of ψ(π(R))σ(x, π)−1. Recursively this
shows Proposition 5.8.5.
Theorem 5.8.7. Let σ ∈ Sρ,δm be elliptic of elliptic order m with 1 ≥ ρ > δ ≥ 0.
We can construct a left parametrixB∈Ψ−mρ,δ for the operatorA= Op(σ), that is, there existsB∈Ψ−mρ,δ such that
BA−I∈Ψ−∞.
Comparing with two-sided parametrices in the case of compact Lie groups (Theorem 2.2.17), this parametrix is one-sided. It was also the case in [CGGP92].
Proof. We can adapt the proof in [Tay81,§0.4] to our setting. Letψ∈C∞(R) be such thatψ|(−∞,Λ1]= 0 and ψ|[Λ2,∞)= 1 for some Λ1,Λ2∈Rwith Λ<Λ1<Λ2. By Proposition 5.8.5,
ψ(π(R))σ−1(x, π)∈Sρ,δ−m.
Sinceψ(π(R)) =ψ(π(R))σ−1(x, π)σ(x, π), by Corollary 5.5.8, Op
ψ(π(R))σ−1(x, π)
A=ψ(R) modΨ−ρ,δ(ρ−δ);
nowψ(R) = I−(1−ψ)(R) and (1−ψ)∈ D([0,∞)) so (1−ψ)(R)∈Ψ−∞. This shows
Op
ψ(π(R))σ−1(x, π)
A = I modΨ−ρ,δ(ρ−δ). So we have
Op
ψ(π(R))σ−1(x, π)
A = I−U with U ∈Ψ−ρ,δ(ρ−δ). By Theorem 5.5.1, there existsT ∈Ψ0ρ,δ such that
T ∼I +U+U2+. . .+Uj+. . . By Theorem 5.5.3,
B:=T Op
ψ(π(R))σ−1
∈Ψ−mρ,δ . Therefore, we obtain
BA=T(I−U) = I modΨ−∞,
completing the proof.
It is not difficult to construct the following examples of elliptic operators satisfying Theorem 5.8.7 out of any Rockland operator. Indeed, combining Propo- sition 5.3.4 or Corollary 5.3.8 together with Proposition 5.8.2 yield
Example 5.8.8. LetRbe a positive Rockland operator of homogeneous degreeν. 1. For anym∈R, the operator (I +R)mν ∈Ψm is elliptic with respect toRof
elliptic orderm.
2. Iff1 andf2 are complex-valued smooth functions onGsuch that
x∈G,λ≥inf Λ
|f1(x) +f2(x)λ|
1 +λ >0 for some Λ≥0,
and such that Xα1f1, Xα2f2 are bounded for each α1, α2 ∈ Nn0, then the differential operator
f1(x) +f2(x)R ∈Ψν is (R,Λ, ν)-elliptic.
3. Letψ∈C∞(R) be such that
ψ|(−∞,Λ1] = 0 and ψ|[Λ2,∞)= 1,
for some real numbers Λ1,Λ2 satisfying 0 < Λ1 < Λ2, Then the operator ψ(R)R ∈Ψν is (R,Λ2, ν)-elliptic.
More generally, iffis a smooth complex-valued function onGsuch that infG|f|>0 and thatXαf is bounded onGfor every α∈Nn0, then
f(x)ψ(R)R ∈Ψν is elliptic with respect toRof elliptic orderν.
Hence all the operators in Example 5.8.8 admit a left parametrix.
We will see other concrete examples of elliptic differential operators on the Heisenberg group in Section 6.6.1, see Example 6.6.2.
In fact we can prove the existence of left parametrices for symbols which are elliptic with an elliptic order lower than their order. Indeed, we can modify the hypothesis of the ellipticity in Section 5.8.1 to obtain the analogue of H¨ormander’s theorem about hypoellipticity involving lower order terms, similar to Theorem 2.2.18 in the compact case.
Theorem 5.8.9. Letσ∈Sρ,δm with1≥ρ > δ≥0. We assume thatσis elliptic with respect to a positive Rockland operatorRin the sense of Definition 5.8.1, and that its elliptic order is mo≤m.
We also assume that the following hypothesis on the lower order terms holds:
there is Λ ∈ R such that for any γ ∈ R, x ∈ G, μ-almost all π ∈ G, and any u∈ H∞π,Λ, we have
π(I +R)ρ[α]−δ[β]+γ
ν
ΔαXβσ(x, π)
π(I +R)−γνuHπ
≤Cα,β,γ σ(x, π)uHπ, (5.84) with Cα,β,γ =Cα,β,γ,σ,R,m o,Λ,γ independent of (x, π)∈G×G andu∈ Hπ,∞Λ.
Then we can construct a left parametrix B ∈ Ψ−mρ,δo for the operator A = Op(σ), that is, there existsB∈Ψ−mρ,δo such that
BA−I∈Ψ−∞.
Proceeding as in Corollary 5.8.4, we can show easily that it suffices to assume (5.79) and (5.84) for a countable sequenceγ which goes to +∞and−∞.
Proof. Let ψ ∈ C∞(R) be such that ψ|(−∞,Λ1] = 0 and ψ|[Λ2,∞) = 1 for some Λ1,Λ2∈Rwith Λ<Λ1<Λ2. Proceeding as in the proof of Proposition 5.8.5, we see that
σo(x, π) :=ψ(π(R))σ−1(x, π)∈Sρ,δ−mo, with similar estimates for the seminorms ofσo andσ.
With similar ideas, using (5.84), we claim that, for any multi-indexβo∈Nn0, we have
Xβoσ(x, π)σo(x, π)∈Sδρ,δ[βo]. Indeed, from the proof of Proposition 5.8.5, we know that
Xσo=−σoXσ E(Λ,∞)σ−1, hence
X
Xβoσ(x, π)σo(x, π)
=XXβoσ(x, π)σo(x, π) +Xβoσ(x, π)Xσo(x, π)
=XXβoσ(x, π)σo(x, π)−Xβoσ(x, π)σoXσ E(Λ,∞)σ−1,
and we can use the hypothesis (5.84) on each term to control theSρ,δm-seminorms of the expression on the right-hand side. For the difference operators, from the proof of Proposition 5.8.5, we know with |αo|= 1, that
Δαoσo= Δαoψ(π(R))E(Λ,∞)σ−1−σo Δαoσ E(Λ,∞)σ−1. Hence
Δαo
Xβoσ(x, π)σo(x, π)
=XβoΔαoσ(x, π)σo(x, π) +Xβoσ(x, π) Δαoσo(x, π)
=XβoΔαoσ(x, π)σo(x, π)−Xβoσ(x, π)σo Δαoσ E(Λ,∞)σ−1 +Xβoσ(x, π) Δαoψ(π(R))ψo(π(R))σ−1,
where ψo ∈ C∞(R) is a fixed smooth function such that ψo|[Λ1,∞) = 1 and ψo|(−∞,Λ1/2)= 0. While we can use the hypothesis (5.84) on the first two terms, we use Lemma 5.8.6 for the last term which is then smoothing. Proceeding recur- sively as in the proof of Proposition 5.8.5, we obtain the estimates for the sum on the right-hand side.
We now define recursively σn(x, π) :=
⎛
⎝
0<[α]≤n
Δασn−[α]Xασ
⎞
⎠σo, n= 1,2, . . .
It is easy to check that each symbolσn(x, π) is inSρ,δ−mo−n(ρ−δ)and that as in the compact case,
Op(σo)Op(σ)−I−Op(σ1)Op(σ)−. . .−Op(σn)Op(σ)∈Ψm−mρ,δ 0−n. Therefore, the operatorB∈Ψ−mρ,δo whose symbol is given by the asymptotic sum σo−∞
j=1σj is a left parametrix forA= Op(σ).
We will see a concrete example of hypoelliptic differential operators on the Heisenberg group in Section 6.6.2, see Example 6.6.4.
We now note the following generalisation of Proposition 5.8.5 that we have already used in the proof of Theorem 5.8.9.
Proposition 5.8.10. Assume 1 ≥ρ ≥ δ ≥ 0. Let σ ∈ Sρ,δm be a symbol which is (R,Λ, mo)-elliptic with respect to a positive Rockland operatorR. If ψ∈C∞(R) is such that
ψ|(−∞,Λ1] = 0 and ψ|[Λ2,∞)= 1,
for some real numbersΛ1,Λ2 satisfying Λ<Λ1<Λ2, then the symbol {ψ(π(R))σ−1(x, π), (x, π)∈G×G},
given by
ψ(π(R))σ(x, π)−1:=ψ(π(R))Eπ(Λ1,∞)σ−1(x, π), is inS−mρ,δo. Moreover, for any ao, bo∈N0, we have
ψ(π(R))σ−1(x, π)S−mo
ρ,δ ,ao,bo,0
≤C
a1,a2≤ao
b1,b2≤bo
|γ|≤ρamaxo+δbo
Cγ,σ,a1+Λb11+1σ(x, π)aS2m+b2 ρ,δ,ao,bo,|m|,
whereC >0 is a positive constant depending onao, bo, ψ, and where the constant Cγ,σ,Λ1 was given in (5.79).
Here the elliptic ordermoand the symbol ordermare different but the same results holds: one can construct a symbolψ(π(R))σ−1(x, π)∈Sρ,δ−mo. The proof is easily obtained by generalising the proof of Proposition 5.8.5.
We now show that Theorem 5.8.7 has a partial inverse.
Proposition 5.8.11. Suppose that the operator A = Op(σ)∈ Ψmρ,δ, with 1 ≥ ρ >
δ≥0, admits a left parametrix B ∈Ψ−mρ,δ, i.e.BA−I∈Ψ−∞. Thenσis elliptic of order m, that is, there exist a positive Rockland operator R of homogeneous degreeν, andΛ∈Rsuch that for anyγ∈R,x∈G,μ-almost allπ∈G, and any u∈ H∞π,Λ we have
π(I +R)γνσ(x, π)uHπ ≥Cγπ(I +R)γνπ(I +R)mνuHπ.
Moreover, if this property holds for one positive Rockland operator then it holds for any Rockland operator.
Proof. Let A and B be as in the statement. Let σ and τ be their respective symbols. Then the symbol
ε := τσ−I
= (τσ−Op−1(BA))−(I−Op−1(BA)), is in Sρ,δ−(ρ−δ), and we can write
π(I +R)m+γν τσ=π(I +R)m+γν +0π(I +R)−ρ−δν π(I +R)m+γν , where
ε0:=π(I +R)m+γν επ(I +R)ρ−δν −m+γν ∈Sρ,δ0 . For anyu∈ H∞π, (x, π)∈G×G, we thus have
π(I +R)m+γν τ(x, π)σ(x, π)uHπ
=
π(I +R)m+γν +0(x, π)π(I +R)−ρ−δν π(I +R)m+γν
uHπ.
We can bound the left hand side by π(I +R)m+γν τ(x, π)σ(x, π)uHπ
≤ π(I +R)m+γν τ(x, π)π(I +R)−γνL(Hπ)π(I +R)γνσ(x, π)uHπ
≤ τS−m
0,0,|γ|π(I +R)γνσ(x, π)uHπ, and the right hand side below by
π(I +R)m+γν +0(x, π)π(I +R)−ρ−δν π(I +R)m+γν uHπ
≥ π(I +R)m+γν uHπ− 0(x, π)π(I +R)−ρ−δν π(I +R)m+γν uHπ
≥ π(I +R)m+γν uHπ
−0(x, π)L(Hπ)π(I +R)−ρ−δν π(I +R)m+γν uHπ. Hence ifu∈E(Λ,∞)H∞π where Λ≥0 then
τS−m
0,0,|γ|π(I +R)γνσ(x, π)uHπ
≥ π(I +R)m+γν uHπ
−0(x, π)L(Hπ)(1 + Λ)−ρ−δν π(I +R)m+γν uHπ. Clearlyτ≡0 andτS−m
0,0,|γ| = 0. Furthermore
0(x, π)L(Hπ)≤ 0Sρ,δ0 ,0,0,0<∞,
hence we can choose Λ≥0 such that
0(x, π)L(Hπ)(1 + Λ)−ρ−δν ≤ 0Sρ,δ0 ,0,0,0(1 + Λ)−ρ−δν ≤ 1 2,
in view ofρ > δ. We have therefore obtained foru∈E(Λ,∞)Hπ∞with the chosen Λ, that
π(I +R)γνσ(x, π)uHπ ≥ 1 2τS−m
0,0,|γ|
π(I +R)m+γν uHπ,
which is the required statement.