Square roots of perturbed sub-elliptic operators on Lie groups

Square roots of perturbed sub-elliptic operators on Lie
groups
Lashi Bandara
(Joint work with Tom ter Elst, Auckland and Alan McIntosh, ANU)
Centre for Mathematics and its Applications
Australian National University
August 13, 2012
POSTI/Mprime Seminar
University of Calgary
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Square roots of operators on Lie groups
August 13, 2012
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Lie groups
Let G be a Lie group of dimension n and g is Lie algebra.
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Square roots of operators on Lie groups
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Lie groups
Let G be a Lie group of dimension n and g is Lie algebra.
We let dµ denote the left invariant Haar measure.
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Square roots of operators on Lie groups
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Algebraic basis and vectorfields
A set {a1 , . . . , ak } ⊂ g is an algebraic basis if we can recover a basis for g
by multi-commutation.
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Square roots of operators on Lie groups
August 13, 2012
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Algebraic basis and vectorfields
A set {a1 , . . . , ak } ⊂ g is an algebraic basis if we can recover a basis for g
by multi-commutation.
We assume that the {ai } are linearly independent.
Lashi Bandara (ANU)
Square roots of operators on Lie groups
August 13, 2012
3 / 28
Algebraic basis and vectorfields
A set {a1 , . . . , ak } ⊂ g is an algebraic basis if we can recover a basis for g
by multi-commutation.
We assume that the {ai } are linearly independent.
Let Ai denote the left translation of ai .
Lashi Bandara (ANU)
Square roots of operators on Lie groups
August 13, 2012
3 / 28
Algebraic basis and vectorfields
A set {a1 , . . . , ak } ⊂ g is an algebraic basis if we can recover a basis for g
by multi-commutation.
We assume that the {ai } are linearly independent.
Let Ai denote the left translation of ai .
The vectorfields {Ai } are linearly independent and global.
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Square roots of operators on Lie groups
August 13, 2012
3 / 28
Distance
Theorem of Carathéodory-Chow tells us that for any two points x, y ∈ G,
we can find a curve γ : [0, 1] → G such that
X
γ̇(t) =
γ̇ i (t)Ai (γ(t)) ∈ span {Ai (γ(t))} .
i
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Square roots of operators on Lie groups
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Distance
Theorem of Carathéodory-Chow tells us that for any two points x, y ∈ G,
we can find a curve γ : [0, 1] → G such that
X
γ̇(t) =
γ̇ i (t)Ai (γ(t)) ∈ span {Ai (γ(t))} .
i
The length of such a curve then is given by
ˆ 1 X
i 2 1
γ̇ (t) ) 2 dt
`(γ) =
(
0
Lashi Bandara (ANU)
i
Square roots of operators on Lie groups
August 13, 2012
4 / 28
Distance
Theorem of Carathéodory-Chow tells us that for any two points x, y ∈ G,
we can find a curve γ : [0, 1] → G such that
X
γ̇(t) =
γ̇ i (t)Ai (γ(t)) ∈ span {Ai (γ(t))} .
i
The length of such a curve then is given by
ˆ 1 X
i 2 1
γ̇ (t) ) 2 dt
`(γ) =
(
0
i
Define distance d(x, y) as the infimum over the length of all such curves.
Lashi Bandara (ANU)
Square roots of operators on Lie groups
August 13, 2012
4 / 28
Distance
Theorem of Carathéodory-Chow tells us that for any two points x, y ∈ G,
we can find a curve γ : [0, 1] → G such that
X
γ̇(t) =
γ̇ i (t)Ai (γ(t)) ∈ span {Ai (γ(t))} .
i
The length of such a curve then is given by
ˆ 1 X
i 2 1
γ̇ (t) ) 2 dt
`(γ) =
(
0
i
Define distance d(x, y) as the infimum over the length of all such curves.
The measure dµ is Borel-regular with respect to d and we consider
(G, d, dµ) as a measure metric space.
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Square roots of operators on Lie groups
August 13, 2012
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Sub-Laplacian
Define an associated sub-Laplacian by:
X
∆=−
A2i .
i
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Square roots of operators on Lie groups
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Sub-Laplacian
Define an associated sub-Laplacian by:
X
∆=−
A2i .
i
This is a densely-defined, self-adjoint operator on L2 (G).
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Square roots of operators on Lie groups
August 13, 2012
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We say that a Lie group is nilpotent if
g1 = [g, g], g2 = [g, g1 ], g3 = [g1 , g2 ], . . .
is eventually 0. That is, there is a k such that gk = 0.
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We say that a Lie group is nilpotent if
g1 = [g, g], g2 = [g, g1 ], g3 = [g1 , g2 ], . . .
is eventually 0. That is, there is a k such that gk = 0.
On such spaces, we consider the uniformly elliptic second order operator
X
Ai bij Aj
DH = −b
i,j
where b, bij ∈ L∞ (G).
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Square roots of operators on Lie groups
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The main theorem for nilpotent Lie groups
Theorem (B.-E.-Mc)
Let G be a connected nilpotent and suppose there exist κ1 , κ2 > 0 such
that
ˆ X
X
Re b(x) ≥ κ1 and Re
bij Ai uAj u ≥ κ2
kAi uk2
G i,j
i
for almost all x ∈ G and u ∈ H1 (G). Then,
√
D(Ai ) = H1 (G), and
(i) D( DH ) = ∩m
√
Pi=1
m
(ii) k DH uk ' i=1 kAi uk for all u ∈ H1 (G).
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Stability
Theorem (B.-E.-Mc)
Let 0 < ηi < κi and suppose that b̃, b̃ij ∈ L∞ (G) such that kb̃k∞ ≤ η1 and
k(b̃ij )k∞ ≤ η2 . Then,
q
k
X
p
k DH u − D̃H uk . (kb̃k∞ + k(b̃ij )k∞ )
kAi uk,
i=1
for u ∈ H1 (G) and where
D̃H = (b + b̃)
k
X
Ai (bij + b̃ij )Aj .
i,j=1
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Operator theory
Procedure in [AKMc].
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Square roots of operators on Lie groups
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Operator theory
Procedure in [AKMc]. Let H be a Hilbert space.
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Square roots of operators on Lie groups
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Operator theory
Procedure in [AKMc]. Let H be a Hilbert space.
(H1) The operator Γ : D(Γ) ⊂ H → H is closed, densely-defined and
nilpotent (Γ2 = 0).
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Square roots of operators on Lie groups
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Operator theory
Procedure in [AKMc]. Let H be a Hilbert space.
(H1) The operator Γ : D(Γ) ⊂ H → H is closed, densely-defined and
nilpotent (Γ2 = 0).
(H2) The operators B1 , B2 ∈ L(H ) satisfy
Re hB1 u, ui ≥ κ1 kuk
whenever u ∈ R(Γ∗ )
Re hB2 u, ui ≥ κ2 kuk
whenever u ∈ R(Γ)
where κ1 , κ2 > 0 are constants.
Lashi Bandara (ANU)
Square roots of operators on Lie groups
August 13, 2012
9 / 28
Operator theory
Procedure in [AKMc]. Let H be a Hilbert space.
(H1) The operator Γ : D(Γ) ⊂ H → H is closed, densely-defined and
nilpotent (Γ2 = 0).
(H2) The operators B1 , B2 ∈ L(H ) satisfy
Re hB1 u, ui ≥ κ1 kuk
whenever u ∈ R(Γ∗ )
Re hB2 u, ui ≥ κ2 kuk
whenever u ∈ R(Γ)
where κ1 , κ2 > 0 are constants.
(H3) The operators B1 , B2 satisfy B1 B2 (R(Γ)) ⊂ N (Γ) and
B2 B1 (R(Γ∗ )) ⊂ N (Γ∗ ).
Lashi Bandara (ANU)
Square roots of operators on Lie groups
August 13, 2012
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Operator theory
Procedure in [AKMc]. Let H be a Hilbert space.
(H1) The operator Γ : D(Γ) ⊂ H → H is closed, densely-defined and
nilpotent (Γ2 = 0).
(H2) The operators B1 , B2 ∈ L(H ) satisfy
Re hB1 u, ui ≥ κ1 kuk
whenever u ∈ R(Γ∗ )
Re hB2 u, ui ≥ κ2 kuk
whenever u ∈ R(Γ)
where κ1 , κ2 > 0 are constants.
(H3) The operators B1 , B2 satisfy B1 B2 (R(Γ)) ⊂ N (Γ) and
B2 B1 (R(Γ∗ )) ⊂ N (Γ∗ ).
Let Γ∗B = B1 Γ∗ B2 , ΠB = Γ + Γ∗B , and Π = Γ + Γ∗ .
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Square roots of operators on Lie groups
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Harmonic analysis and Kato square root type estimates
Theorem (Kato square root type estimate)
Suppose that (Γ, B1 , B2 ) satisfy (H1)-(H3) and
ˆ ∞
dt
' kuk2
ktΠB (1 + t2 Π2B )−1 uk2
t
0
for all u ∈ R(ΠB ) ⊂ H . Then,
+
−
(i) There is a spectral decomposition H = N (ΠB ) ⊕ EB
⊕ EB
, where
±
EB are spectral subspaces and the sum is in general non-orthogonal,
and
q
(ii) D(Γ) ∩ D(Γ∗B ) = D(ΠB ) = D( Π2B ) with
q
kΓuk + kΓB uk ' kΠB uk ' k Π2B uk for all u ∈ D(ΠB ).
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Homogeneous conditions
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Homogeneous conditions
(H4) Let X be a complete, connected metric space and µ a Borel-regular
measure on X that is doubling. Then set H = L2 (X , CN ; dµ).
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Homogeneous conditions
(H4) Let X be a complete, connected metric space and µ a Borel-regular
measure on X that is doubling. Then set H = L2 (X , CN ; dµ).
(H5) The operators Bi are matrix-valued pointwise multiplication operators
such that the function x 7→ Bi (x) are L∞ (X , L(CN )).
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Square roots of operators on Lie groups
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Homogeneous conditions
(H4) Let X be a complete, connected metric space and µ a Borel-regular
measure on X that is doubling. Then set H = L2 (X , CN ; dµ).
(H5) The operators Bi are matrix-valued pointwise multiplication operators
such that the function x 7→ Bi (x) are L∞ (X , L(CN )).
(H6) For every bounded Lipschitz function ξ : X → C, multiplication by ξ
preserves D(Γ) and Mξ = [Γ, ξI] is a multiplication operator.
Furthermore, there exists a constant m > 0 such that
|Mξ (x)| ≤ m |Lip ξ(x)| for almost all x ∈ X .
Lashi Bandara (ANU)
Square roots of operators on Lie groups
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Homogeneous conditions
(H4) Let X be a complete, connected metric space and µ a Borel-regular
measure on X that is doubling. Then set H = L2 (X , CN ; dµ).
(H5) The operators Bi are matrix-valued pointwise multiplication operators
such that the function x 7→ Bi (x) are L∞ (X , L(CN )).
(H6) For every bounded Lipschitz function ξ : X → C, multiplication by ξ
preserves D(Γ) and Mξ = [Γ, ξI] is a multiplication operator.
Furthermore, there exists a constant m > 0 such that
|Mξ (x)| ≤ m |Lip ξ(x)| for almost all x ∈ X .
(H7) For each open ball B, we have
ˆ
Γu dµ = 0 and
B
ˆ
Γ∗ v dµ = 0
B
for all u ∈ D(Γ) with spt u ⊂ B and for all v ∈ D(Γ∗ ) with
spt v ⊂ B.
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(H8) -1 (Poincaré hypothesis)
There exists C 0 > 0, c > 0 and an operator
Ξ : D(Ξ) ⊂ L2 (X , CN ) → L2 (X , CM ) such that D(Π) ∩ R(Π) ⊂ D(Ξ)
and
ˆ
ˆ
2
2
|u − uB | dµ ≤ C 0 r2
|Ξu| dµ
B
B
for all balls B = B(x, r) and u ∈ D(Π) ∩ R(Π).
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(H8) -1 (Poincaré hypothesis)
There exists C 0 > 0, c > 0 and an operator
Ξ : D(Ξ) ⊂ L2 (X , CN ) → L2 (X , CM ) such that D(Π) ∩ R(Π) ⊂ D(Ξ)
and
ˆ
ˆ
2
2
|u − uB | dµ ≤ C 0 r2
|Ξu| dµ
B
B
for all balls B = B(x, r) and u ∈ D(Π) ∩ R(Π).
-2 (Coercivity hypothesis)
There exists C̃ > 0 such that for all u ∈ D(Π) ∩ R(Π),
kΞuk ≤ C̃kΠuk.
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(H8) -1 (Poincaré hypothesis)
There exists C 0 > 0, c > 0 and an operator
Ξ : D(Ξ) ⊂ L2 (X , CN ) → L2 (X , CM ) such that D(Π) ∩ R(Π) ⊂ D(Ξ)
and
ˆ
ˆ
2
2
|u − uB | dµ ≤ C 0 r2
|Ξu| dµ
B
B
for all balls B = B(x, r) and u ∈ D(Π) ∩ R(Π).
-2 (Coercivity hypothesis)
There exists C̃ > 0 such that for all u ∈ D(Π) ∩ R(Π),
kΞuk ≤ C̃kΠuk.
This is slightly different from (H8) in [Bandara].
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Theorem (B.)
Let X , (Γ, B1 , B2 ) satisfy (H1)-(H8). Then, ΠB satisfies the quadratic
estimate
ˆ ∞
dt
ktΠB (1 + t2 Π2B )−1 uk2
' kuk2
t
0
for all u ∈ R(ΠB ) ⊂ L2 (X , CN ).
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Geometric setup
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Geometric setup
Define the bundle W = span {Ai } ⊂ TG and complexify it.
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Geometric setup
Define the bundle W = span {Ai } ⊂ TG and complexify it.
Equip W with the inner product h(Ai , Aj ) = δij .
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Geometric setup
Define the bundle W = span {Ai } ⊂ TG and complexify it.
Equip W with the inner product h(Ai , Aj ) = δij .
Equip G with the sub-connection
∇f = Ak f Ak
where Ak = Ak ∗ ∈ W ∗ .
Lashi Bandara (ANU)
Square roots of operators on Lie groups
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Geometric setup
Define the bundle W = span {Ai } ⊂ TG and complexify it.
Equip W with the inner product h(Ai , Aj ) = δij .
Equip G with the sub-connection
∇f = Ak f Ak
where Ak = Ak ∗ ∈ W ∗ .
Equip W with the sub-connection
˜ i Ai ) = (∇ui ) ⊗ Ai
∇(u
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Square roots of operators on Lie groups
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Geometric setup
Define the bundle W = span {Ai } ⊂ TG and complexify it.
Equip W with the inner product h(Ai , Aj ) = δij .
Equip G with the sub-connection
∇f = Ak f Ak
where Ak = Ak ∗ ∈ W ∗ .
Equip W with the sub-connection
˜ i Ai ) = (∇ui ) ⊗ Ai
∇(u
We have that W ∼
= Ck and L2 (G) ⊕ L2 (W) ∼
= L2 (Ck+1 ).
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Operator setup
Define: Γ : D(Γ) ⊂ L2 (G) ⊕ L2 (W ∗ ) → L2 (G) ⊕ L2 (W ∗ ) by
0 0
Γ=
.
∇ 0
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Operator setup
Define: Γ : D(Γ) ⊂ L2 (G) ⊕ L2 (W ∗ ) → L2 (G) ⊕ L2 (W ∗ ) by
0 0
Γ=
.
∇ 0
Then,
0 − div
Γ =
0
0
∗
and Π =
0 − div
,
∇
0
where we define div = −∇ ∗ .
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Operator setup
Define: Γ : D(Γ) ⊂ L2 (G) ⊕ L2 (W ∗ ) → L2 (G) ⊕ L2 (W ∗ ) by
0 0
Γ=
.
∇ 0
Then,
0 − div
Γ =
0
0
∗
and Π =
0 − div
,
∇
0
where we define div = −∇ ∗ .
Let B = (bij ). Then, define
b 0
B1 =
0 0
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and B2 =
0 0
.
0 B
Square roots of operators on Lie groups
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Proof of the homogeneous problem
Set X = G and H = L2 (G) ⊕ L2 (W).
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Proof of the homogeneous problem
Set X = G and H = L2 (G) ⊕ L2 (W).
(H1) The sub-connection ∇ is densely-defined and closed and so is Γ.
Nilpotency is by construction.
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Proof of the homogeneous problem
Set X = G and H = L2 (G) ⊕ L2 (W).
(H1) The sub-connection ∇ is densely-defined and closed and so is Γ.
Nilpotency is by construction.
(H2) By accretivity assumptions.
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Proof of the homogeneous problem
Set X = G and H = L2 (G) ⊕ L2 (W).
(H1) The sub-connection ∇ is densely-defined and closed and so is Γ.
Nilpotency is by construction.
(H2) By accretivity assumptions.
(H3) By construction.
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Proof (cont.)
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Proof (cont.)
(H4) The measure dµ is Borel-regular and the nilpotency of G implies that
it is doubling.
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Proof (cont.)
(H4) The measure dµ is Borel-regular and the nilpotency of G implies that
it is doubling.
(H5) By assumption.
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Proof (cont.)
(H4) The measure dµ is Borel-regular and the nilpotency of G implies that
it is doubling.
(H5) By assumption.
(H6) It is an easy fact that for all bounded Lipschitz ξ : G → C,
|Mξ (x)| = |[Γ, ξ(x)I]| = |∇ξ(x)| ≤ k Lip ξ(x)
for for almost all x ∈ G.
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Proof (cont.)
(H4) The measure dµ is Borel-regular and the nilpotency of G implies that
it is doubling.
(H5) By assumption.
(H6) It is an easy fact that for all bounded Lipschitz ξ : G → C,
|Mξ (x)| = |[Γ, ξ(x)I]| = |∇ξ(x)| ≤ k Lip ξ(x)
for for almost all x ∈ G.
(H7) By the left invariance of the measure dµ.
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Proof (cont.)
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Proof (cont.)
(H8)
-1 The nilpotency of G implies the following Poincaré inequality
ˆ
ˆ
2
2
|f − fB | dµ . r2
|∇f | dµ
B
B
∞
for all balls B, and f ∈ C (B). See [SC, (P.1), p118].
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Proof (cont.)
(H8)
-1 The nilpotency of G implies the following Poincaré inequality
ˆ
ˆ
2
2
|f − fB | dµ . r2
|∇f | dµ
B
B
∞
for all balls B, and f ∈ C (B). See [SC, (P.1), p118].
˜ 2 ).
Define Ξu = (∇u1 , ∇u
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Proof (cont.)
(H8)
-1 The nilpotency of G implies the following Poincaré inequality
ˆ
ˆ
2
2
|f − fB | dµ . r2
|∇f | dµ
B
B
∞
for all balls B, and f ∈ C (B). See [SC, (P.1), p118].
˜ 2 ).
Define Ξu = (∇u1 , ∇u
-2 The crucial fact needed here is the regularity result [ERS, Lemma 4.2]
which gives
kAi Aj f k . k∆f k
for f ∈ H2 (G) = D(∆).
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Inhomogeneous problem
For general Lie groups, we need to consider operators with lower order
terms.
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Inhomogeneous problem
For general Lie groups, we need to consider operators with lower order
terms.
Let b, bij , ck , dk , e ∈ L∞ (G). Define the following uniformly elliptic second
order operator
DI = −b
m
X
ij=1
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Ai bij Aj u − b
m
X
Ai ci u − b
i=1
Square roots of operators on Lie groups
m
X
di Ai u − beu.
i=1
August 13, 2012
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Theorem (B.-E.-Mc)
Let G be a connected Lie group and suppose there exists κ1 , κ2 > 0 such
that
Re b(x) ≥ κ1 ,


!
ˆ
m
m
m
X
X
X
ci u +
bij Aj u Ai u dµ
Re
eu +
di Ai u u +
G
i=1
i=1
j=1
≥ κ2
2
kuk +
m
X
!
2
kAi uk
i=1
for almost all x ∈ G and u ∈ H1 (G). Then,
√
1
(i) D( DI ) = ∩m
i=1 D(Ai ) = H (G), and
√
Pm
(ii) k DI uk ' kuk + i=1 kAi uk for all u ∈ H1 (G).
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Spaces of exponential growth
(X , d, µ) an exponentially locally doubling measure metric space. That is:
there exist κ, λ ≥ 0 and constant C ≥ 1 such that
0 < µ(B(x, tr)) ≤ Ctκ eλtr µ(B(x, r))
for all x ∈ X , r > 0 and t ≥ 1.
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Changes to (H7) and (H8)
The following (H7) from [Morris]:
(H7) There exist c > 0 such that for all open balls B ⊂ X with r ≤ 1,
ˆ
ˆ
Γu dµ ≤ cµ(B) 21 kuk and Γ∗ v dµ ≤ cµ(B) 21 kvk
B
B
for all u ∈ D(Γ), v ∈ D(Γ∗ ) with spt u, spt v ⊂ B.
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We introduce the following local (H8):
(H8) -1 (Local Poincaré hypothesis)
There exists C 0 > 0, c > 0 and an operator
Ξ : D(Ξ) ⊂ L2 (X , CN ) → L2 (X , CM ) such that D(Π) ∩ R(Π) ⊂ D(Ξ)
and
ˆ
ˆ
2
2
2
|u − uB | dµ ≤ C 0 r2 (|Ξu| + |u| ) dµ
B
B
for all balls B = B(x, r) and for u ∈ D(Π) ∩ R(Π).
-2 (Coercivity hypothesis)
There exists C̃ > 0 such that for all u ∈ D(Π) ∩ R(Π),
kΞuk + kuk ≤ C̃kΠuk.
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Square roots of operators on Lie groups
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Theorem (Morris)
Let X , (Γ, B1 , B2 ) satisfy (H1)-(H8). Then, ΠB satisfies the quadratic
estimate
ˆ ∞
dt
ktΠB (1 + t2 Π2B )−1 uk2
' kuk2
t
0
for all u ∈ R(ΠB ) ⊂ L2 (X , CN ).
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Square roots of operators on Lie groups
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Setup
Set X = G and H = L2 (G) ⊕ L2 (G) ⊕ L2 (W) ∼
= L2 (Ck+2 ).
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Square roots of operators on Lie groups
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Setup
Set X = G and H = L2 (G) ⊕ L2 (G) ⊕ L2 (W) ∼
= L2 (Ck+2 ).
Let S = (I, ∇), S ∗ = [I − div].
Lashi Bandara (ANU)
Square roots of operators on Lie groups
August 13, 2012
25 / 28
Setup
Set X = G and H = L2 (G) ⊕ L2 (G) ⊕ L2 (W) ∼
= L2 (Ck+2 ).
Let S = (I, ∇), S ∗ = [I − div].
Let
Γ=
0 0
0 S∗
0 S∗
∗
∗
, Γ =
, and Π =
.
S 0
0 0
S 0
Lashi Bandara (ANU)
Square roots of operators on Lie groups
August 13, 2012
25 / 28
Setup
Set X = G and H = L2 (G) ⊕ L2 (G) ⊕ L2 (W) ∼
= L2 (Ck+2 ).
Let S = (I, ∇), S ∗ = [I − div].
Let
Γ=
0 0
0 S∗
0 S∗
∗
∗
, Γ =
, and Π =
.
S 0
0 0
S 0
Let B̃00 = e, B̃10 = (c1 , . . . , cm ), B̃01 = (d1 , . . . , dm )tr , B̃11 = (bij ), and
B = (B̃ij ).
Then, we can write
B1 =
Lashi Bandara (ANU)
b 0
0 0
0 0
and B2 =
.
0 B
Square roots of operators on Lie groups
August 13, 2012
25 / 28
Proof
The proofs of (H1)-(H6) are similar to the homogeneous situation.
(H7) The proof is the same as the homogeneous situation except the lower
1
order term introduces the term µ(B) 2 kuk on the right.
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Square roots of operators on Lie groups
August 13, 2012
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Proof
The proofs of (H1)-(H6) are similar to the homogeneous situation.
(H7) The proof is the same as the homogeneous situation except the lower
1
order term introduces the term µ(B) 2 kuk on the right.
(H8) -1 The existence of a local Poincaré inequality is guaranteed by [ER2,
Proposition 2.4]:
ˆ
ˆ
2
|f − fB | dµ . r
B
2
2
2
(|∇f | + |f | ) dµ
B
for all balls B = B(x, r) and where f ∈ C∞ (B).
Lashi Bandara (ANU)
Square roots of operators on Lie groups
August 13, 2012
26 / 28
Proof
The proofs of (H1)-(H6) are similar to the homogeneous situation.
(H7) The proof is the same as the homogeneous situation except the lower
1
order term introduces the term µ(B) 2 kuk on the right.
(H8) -1 The existence of a local Poincaré inequality is guaranteed by [ER2,
Proposition 2.4]:
ˆ
ˆ
2
|f − fB | dµ . r
B
2
2
2
(|∇f | + |f | ) dµ
B
for all balls B = B(x, r) and where f ∈ C∞ (B).
˜ 3 ).
Define Ξu = (∇u1 , ∇u2 , ∇u
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Square roots of operators on Lie groups
August 13, 2012
26 / 28
Proof
The proofs of (H1)-(H6) are similar to the homogeneous situation.
(H7) The proof is the same as the homogeneous situation except the lower
1
order term introduces the term µ(B) 2 kuk on the right.
(H8) -1 The existence of a local Poincaré inequality is guaranteed by [ER2,
Proposition 2.4]:
ˆ
ˆ
2
|f − fB | dµ . r
B
2
2
2
(|∇f | + |f | ) dµ
B
for all balls B = B(x, r) and where f ∈ C∞ (B).
˜ 3 ).
Define Ξu = (∇u1 , ∇u2 , ∇u
-2 The crucial fact needed here is the regularity result in [ER, Theorem 7.2],
kAi Aj uk2 . k∆uk2 + kuk2
for u ∈ H2 (G) = D(∆).
Lashi Bandara (ANU)
Square roots of operators on Lie groups
August 13, 2012
26 / 28
References I
[AKMc] Andreas Axelsson, Stephen Keith, and Alan McIntosh, Quadratic
estimates and functional calculi of perturbed Dirac operators, Invent.
Math. 163 (2006), no. 3, 455–497.
[Bandara] L. Bandara, Quadratic estimates for perturbed Dirac type
operators on doubling measure metric spaces, ArXiv e-prints (2011).
[Morris] Andrew Morris, Local hardy spaces and quadratic estimates for
Dirac type operators on Riemannian manifolds, Ph.D. thesis,
Australian National University, 2010.
[SC] L. Saloff-Coste and D. W. Stroock, Opérateurs uniformément
sous-elliptiques sur les groupes de Lie, J. Funct. Anal. 98 (1991),
no. 1, 97–121. MR 1111195 (92k:58264)
[ER] A. F. M. ter Elst and Derek W. Robinson, Subelliptic operators on
Lie groups: regularity, J. Austral. Math. Soc. Ser. A 57 (1994), no. 2,
179–229. MR 1288673 (95e:22019)
Lashi Bandara (ANU)
Square roots of operators on Lie groups
August 13, 2012
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References II
[ER2] A. F. M. Ter Elst and Derek W. Robinson, Second-order subelliptic
operators on Lie groups. I. Complex uniformly continuous principal
coefficients, Acta Appl. Math. 59 (1999), no. 3, 299–331. MR
1744755 (2001a:35043)
[ERS] A. F. M. ter Elst, Derek W. Robinson, and Adam Sikora, Heat
kernels and Riesz transforms on nilpotent Lie groups, Colloq. Math. 74
(1997), no. 2, 191–218. MR 1477562 (99a:35029)
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Square roots of operators on Lie groups
August 13, 2012
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