Geometry and the Kato square root problem
Lashi Bandara
Centre for Mathematics and its Applications
Australian National University
7 June 2013
Geometric Analysis Seminar
University of Wollongong
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Geometry and the Kato problem
7 June 2013
1 / 28
Outline
Brief overview of the Kato square root problem on Rn .
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Geometry and the Kato problem
7 June 2013
2 / 28
Outline
Brief overview of the Kato square root problem on Rn .
A motivating application to hyperbolic PDE.
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
2 / 28
Outline
Brief overview of the Kato square root problem on Rn .
A motivating application to hyperbolic PDE.
Recent progress on the Kato square root problem on smooth
manifolds by McIntosh and B.
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
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Outline
Brief overview of the Kato square root problem on Rn .
A motivating application to hyperbolic PDE.
Recent progress on the Kato square root problem on smooth
manifolds by McIntosh and B.
Recent progress on subelliptic Kato square root problems on Lie
groups by ter Elst, McIntosh, and B.
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Geometry and the Kato problem
7 June 2013
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Outline
Brief overview of the Kato square root problem on Rn .
A motivating application to hyperbolic PDE.
Recent progress on the Kato square root problem on smooth
manifolds by McIntosh and B.
Recent progress on subelliptic Kato square root problems on Lie
groups by ter Elst, McIntosh, and B.
Kato square root problem on smooth manifolds with non-smooth
metrics, connection to geometric flows and PDEs.
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Geometry and the Kato problem
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The Kato square root problem
Let A ∈ L∞ (Rn , L(CN )) and a ∈ L∞ (Rn ).
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Geometry and the Kato problem
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The Kato square root problem
Let A ∈ L∞ (Rn , L(CN )) and a ∈ L∞ (Rn ). Suppose that there exists
κ1 , κ2 > 0 such that for all u ∈ W1,2 (Rn ),
Re a(x) ≥ κ1
Lashi Bandara (ANU)
and
Re hA∇u, ∇ui ≥ κ2 kuk2 .
Geometry and the Kato problem
7 June 2013
3 / 28
The Kato square root problem
Let A ∈ L∞ (Rn , L(CN )) and a ∈ L∞ (Rn ). Suppose that there exists
κ1 , κ2 > 0 such that for all u ∈ W1,2 (Rn ),
Re a(x) ≥ κ1
and
Re hA∇u, ∇ui ≥ κ2 kuk2 .
The Kato square root problem on Rn is the statement that
p
D( −a div A∇) = W1,2 (Rn )
p
k −a div A∇uk ' k∇uk.
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Geometry and the Kato problem
(K1)
7 June 2013
3 / 28
The Kato square root problem
Let A ∈ L∞ (Rn , L(CN )) and a ∈ L∞ (Rn ). Suppose that there exists
κ1 , κ2 > 0 such that for all u ∈ W1,2 (Rn ),
Re a(x) ≥ κ1
and
Re hA∇u, ∇ui ≥ κ2 kuk2 .
The Kato square root problem on Rn is the statement that
p
D( −a div A∇) = W1,2 (Rn )
p
k −a div A∇uk ' k∇uk.
(K1)
This was answered in the positive in 2002 by Pascal Auscher, Steve
Hofmann, Michael Lacey, Alan McIntosh and Phillipe Tchamitchian in
[AHLMcT].
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If further A = A∗ , (K1) is a trivial consequence of the Lax-Milgram
Theorem.
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If further A = A∗ , (K1) is a trivial consequence of the Lax-Milgram
Theorem.
√
√
Solution to (K1) implies that D( − div A∇) = D( − div A∗ ∇).
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Geometry and the Kato problem
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If further A = A∗ , (K1) is a trivial consequence of the Lax-Milgram
Theorem.
√
√
Solution to (K1) implies that D( − div A∇) = D( − div A∗ ∇).
We can ask a more abstract question for accretive operators L on a
Hilbert space H .
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Geometry and the Kato problem
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If further A = A∗ , (K1) is a trivial consequence of the Lax-Milgram
Theorem.
√
√
Solution to (K1) implies that D( − div A∇) = D( − div A∗ ∇).
We can ask a more abstract question for accretive operators
L on
√
√a
Hilbert space H . There, the question is whether D( L∗ ) = D( L).
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
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If further A = A∗ , (K1) is a trivial consequence of the Lax-Milgram
Theorem.
√
√
Solution to (K1) implies that D( − div A∇) = D( − div A∗ ∇).
We can ask a more abstract question for accretive operators
L on
√
√a
Hilbert space H . There, the question is whether D( L∗ ) = D( L).
In general, this is not true by a counterexample of McIntosh in 1972
in [Mc72].
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
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If further A = A∗ , (K1) is a trivial consequence of the Lax-Milgram
Theorem.
√
√
Solution to (K1) implies that D( − div A∇) = D( − div A∗ ∇).
We can ask a more abstract question for accretive operators
L on
√
√a
Hilbert space H . There, the question is whether D( L∗ ) = D( L).
In general, this is not true by a counterexample of McIntosh in 1972
in [Mc72].
A second related question is the following.
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
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If further A = A∗ , (K1) is a trivial consequence of the Lax-Milgram
Theorem.
√
√
Solution to (K1) implies that D( − div A∇) = D( − div A∗ ∇).
We can ask a more abstract question for accretive operators
L on
√
√a
Hilbert space H . There, the question is whether D( L∗ ) = D( L).
In general, this is not true by a counterexample of McIntosh in 1972
in [Mc72].
A second related question is the following. Suppose that Jt is a
family of closed, densely-defined, Hermitian forms on H with domain
W and L(t) the associated self-adjoint operators to Jt with domain
W.
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
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If further A = A∗ , (K1) is a trivial consequence of the Lax-Milgram
Theorem.
√
√
Solution to (K1) implies that D( − div A∇) = D( − div A∗ ∇).
We can ask a more abstract question for accretive operators
L on
√
√a
Hilbert space H . There, the question is whether D( L∗ ) = D( L).
In general, this is not true by a counterexample of McIntosh in 1972
in [Mc72].
A second related question is the following. Suppose that Jt is a
family of closed, densely-defined, Hermitian forms on H with domain
W and L(t) the associated self-adjoint operators to Jt with domain
W.pIf t 7→ Jt extends to holomorphic family (for small z), then is
∂t L(t) : V → H a bounded operator?
Lashi Bandara (ANU)
Geometry and the Kato problem
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If further A = A∗ , (K1) is a trivial consequence of the Lax-Milgram
Theorem.
√
√
Solution to (K1) implies that D( − div A∇) = D( − div A∗ ∇).
We can ask a more abstract question for accretive operators
L on
√
√a
Hilbert space H . There, the question is whether D( L∗ ) = D( L).
In general, this is not true by a counterexample of McIntosh in 1972
in [Mc72].
A second related question is the following. Suppose that Jt is a
family of closed, densely-defined, Hermitian forms on H with domain
W and L(t) the associated self-adjoint operators to Jt with domain
W.pIf t 7→ Jt extends to holomorphic family (for small z), then is
∂t L(t) : V → H a bounded operator?
Counterexample to this second question by McIntosh in 1982 in
[Mc82].
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Geometry and the Kato problem
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Motivations from PDE
For k = 1, 2, let Lk = − div Ak ∇ where Ak ∈ L∞ (Rn , L(Cn ))
non-negative self-adjoint and Lk uniformly elliptic.
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Motivations from PDE
For k = 1, 2, let Lk = − div Ak ∇ where Ak ∈ L∞ (Rn , L(Cn ))
non-negative self-adjoint and Lk uniformly elliptic.
√
√
As aforementioned, D( Lk ) = W1,2 (Rn ) and k Lk uk ' k∇uk for
u ∈ W1,2 (Rn ).
Lashi Bandara (ANU)
Geometry and the Kato problem
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Motivations from PDE
For k = 1, 2, let Lk = − div Ak ∇ where Ak ∈ L∞ (Rn , L(Cn ))
non-negative self-adjoint and Lk uniformly elliptic.
√
√
As aforementioned, D( Lk ) = W1,2 (Rn ) and k Lk uk ' k∇uk for
u ∈ W1,2 (Rn ).
Let uk be solutions to the wave equation with respect to Lk with the same
initial data.
Lashi Bandara (ANU)
Geometry and the Kato problem
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Motivations from PDE
For k = 1, 2, let Lk = − div Ak ∇ where Ak ∈ L∞ (Rn , L(Cn ))
non-negative self-adjoint and Lk uniformly elliptic.
√
√
As aforementioned, D( Lk ) = W1,2 (Rn ) and k Lk uk ' k∇uk for
u ∈ W1,2 (Rn ).
Let uk be solutions to the wave equation with respect to Lk with the same
initial data. That is,
∂t2 uk + Lk uk = 0
∂t uk | t=0 = g ∈ L2 (Rn )
uk (0) = f ∈ W1,2 (Rn ).
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Suppose there exists a C > 0
√
√
k L1 u − L2 uk ≤ CkA1 − A2 k∞ k∇uk.
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Geometry and the Kato problem
(P)
7 June 2013
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Suppose there exists a C > 0
√
√
k L1 u − L2 uk ≤ CkA1 − A2 k∞ k∇uk.
(P)
Then, whenever t > 0, the following estimate holds:
ˆ
ku1 (t) − u2 (t)k + k
t
∇(u1 (s) − u2 (s)) dsk
0
≤ CtkA1 − A2 k∞ (k∇f k + kgk).
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Suppose there exists a C > 0
√
√
k L1 u − L2 uk ≤ CkA1 − A2 k∞ k∇uk.
(P)
Then, whenever t > 0, the following estimate holds:
ˆ
ku1 (t) − u2 (t)k + k
t
∇(u1 (s) − u2 (s)) dsk
0
≤ CtkA1 − A2 k∞ (k∇f k + kgk).
See [Aus].
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The estimate (P) is related to the second question of Kato.
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The estimate (P) is related to the second question of Kato.
By solving the Kato square root problem (K1) for complex coefficients A,
we are able to automatically obtain (P) from (K1).
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Kato square root problem on manifolds
Let M be a smooth, complete Riemannian manifold with metric g,
Levi-Civita connection ∇, and volume measure µ.
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Kato square root problem on manifolds
Let M be a smooth, complete Riemannian manifold with metric g,
Levi-Civita connection ∇, and volume measure µ.
Write div = −∇ ∗ in L2 and let S = (I, ∇).
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Kato square root problem on manifolds
Let M be a smooth, complete Riemannian manifold with metric g,
Levi-Civita connection ∇, and volume measure µ.
Write div = −∇ ∗ in L2 and let S = (I, ∇).
Assume a ∈ L∞ (M) and A = (Aij ) ∈ L∞ (M, L(L2 (M) ⊕ L2 (T∗ M)).
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Geometry and the Kato problem
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Kato square root problem on manifolds
Let M be a smooth, complete Riemannian manifold with metric g,
Levi-Civita connection ∇, and volume measure µ.
Write div = −∇ ∗ in L2 and let S = (I, ∇).
Assume a ∈ L∞ (M) and A = (Aij ) ∈ L∞ (M, L(L2 (M) ⊕ L2 (T∗ M)).
Consider the following second order differential operator
LA : D(LA ) ⊂ L2 (M) → L2 (M) defined by
LA u = aS ∗ ASu = −a div(A11 ∇u) − a div(A10 u) + aA01 ∇u + aA00 u.
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The main theorem on manifolds
Theorem (B.-Mc, 2012)
Let M be a smooth, complete Riemannian manifold with |Ric| ≤ C and
inj(M ) ≥ κ > 0. Suppose the following ellipticity condition holds: there
exists κ1 , κ2 > 0 such that
Re hav, vi ≥ κ1 kvk2
Re hASu, Sui ≥ κ2 kuk2W1,2
√
2
1,2
D(∇) = W1,2 (M)
for v ∈
√L (M) and u ∈ W (M). Then, D( LA ) = 1,2
and k LA uk ' k∇uk + kuk = kukW1,2 for all u ∈ W (M).
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Lipschitz estimates
Since we allow the coefficients a and A to be complex, we obtain the
following stability result as a consequence:
Theorem (B.-Mc, 2012)
Let M be a smooth, complete Riemannian manifold with |Ric| ≤ C and
inj(M) ≥ κ > 0. Suppose that there exist κ1 , κ2 > 0 such that
Re hav, vi ≥ κ1 kvk2 and Re hASu, Sui ≥ κ2 kuk2W1,2 for v ∈ L2 (M) and
u ∈ W1,2 (M). Then for every ηi < κi , whenever kãk∞ ≤ η1 , kÃk∞ ≤ η2 ,
the estimate
q
p
k LA u − LA+Ã uk . (kãk∞ + kÃk∞ )kukW1,2
holds for all u ∈ W1,2 (M). The implicit constant depends in particular on
A, a and ηi .
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The Hodge-Dirac operator
Let Ω(M) denote the algebra of differential forms over M under the
exterior product ∧.
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The Hodge-Dirac operator
Let Ω(M) denote the algebra of differential forms over M under the
exterior product ∧.
Let d be the exterior derivative as an operator on L2 (Ω(M)) and d∗ its
adjoint, both of which are nilpotent operators.
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The Hodge-Dirac operator
Let Ω(M) denote the algebra of differential forms over M under the
exterior product ∧.
Let d be the exterior derivative as an operator on L2 (Ω(M)) and d∗ its
adjoint, both of which are nilpotent operators.
The Hodge-Dirac operator is then the self-adjoint operator D = d +d∗ .
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Geometry and the Kato problem
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The Hodge-Dirac operator
Let Ω(M) denote the algebra of differential forms over M under the
exterior product ∧.
Let d be the exterior derivative as an operator on L2 (Ω(M)) and d∗ its
adjoint, both of which are nilpotent operators.
The Hodge-Dirac operator is then the self-adjoint operator D = d +d∗ .
The Hodge-Laplacian is then D2 = d d∗ + d∗ d.
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The Hodge-Dirac operator
Let Ω(M) denote the algebra of differential forms over M under the
exterior product ∧.
Let d be the exterior derivative as an operator on L2 (Ω(M)) and d∗ its
adjoint, both of which are nilpotent operators.
The Hodge-Dirac operator is then the self-adjoint operator D = d +d∗ .
The Hodge-Laplacian is then D2 = d d∗ + d∗ d.
For an invertible A ∈ L∞ (L(Ω(M))), we consider perturbing D to obtain
the operator DA = d +A−1 d∗ A.
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Curvature endomorphism for forms
Let θi be an orthonormal frame at x for Ω1 (M) = T∗ M.
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Curvature endomorphism for forms
Let θi be an orthonormal frame at x for Ω1 (M) = T∗ M.
Denote the components of the curvature tensor in this frame by Rmijkl .
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Curvature endomorphism for forms
Let θi be an orthonormal frame at x for Ω1 (M) = T∗ M.
Denote the components of the curvature tensor in this frame by Rmijkl .
The curvature endomorphism is then the operator
R ω = Rmijkl θi ∧ (θj x (θk ∧ (θl x ω)))
for ω ∈ Ωx (M).
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Curvature endomorphism for forms
Let θi be an orthonormal frame at x for Ω1 (M) = T∗ M.
Denote the components of the curvature tensor in this frame by Rmijkl .
The curvature endomorphism is then the operator
R ω = Rmijkl θi ∧ (θj x (θk ∧ (θl x ω)))
for ω ∈ Ωx (M).
This can be seen as an extension of Ricci curvature for forms, since
g(R ω, η) = Ric(ω [ , η [ ) whenever ω, η ∈ Ω1x (M) and where
[ : T∗ M → TM is the flat isomorphism through the metric g.
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Geometry and the Kato problem
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Curvature endomorphism for forms
Let θi be an orthonormal frame at x for Ω1 (M) = T∗ M.
Denote the components of the curvature tensor in this frame by Rmijkl .
The curvature endomorphism is then the operator
R ω = Rmijkl θi ∧ (θj x (θk ∧ (θl x ω)))
for ω ∈ Ωx (M).
This can be seen as an extension of Ricci curvature for forms, since
g(R ω, η) = Ric(ω [ , η [ ) whenever ω, η ∈ Ω1x (M) and where
[ : T∗ M → TM is the flat isomorphism through the metric g.
The Weitzenböck formula then asserts that D2 = tr12 ∇ 2 + R .
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Theorem (B., 2012)
Let M be a smooth, complete Riemannian manifold and let β ∈ C \ {0}.
Suppose there exist η, κ > 0 such that |Ric| ≤ η and inj(M) ≥ κ.
Furthermore, suppose there is a ζ ∈ R satisfying g(R u, u) ≥ ζ |u|2 , for
u ∈ Ωx (M) and A ∈ L∞ (L(Ω(M))) and κ1 > 0 satisfying
Re hAu, ui ≥ κ1 kuk2 .
q
Then, D( D2A + |β|2 ) = D(DA ) = D(d) ∩ D(d∗ A) and
q
k D2A + |β|2 uk ' k DA uk + kuk.
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Lie groups
Let G be a Lie group of dimension n with Lie algebra g and equipped with
the left-invariant Haar measure µ.
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Lie groups
Let G be a Lie group of dimension n with Lie algebra g and equipped with
the left-invariant Haar measure µ.
We say that a linearly independent a = {a1 , . . . , ak } ⊂ g is an algebraic
basis if we can recover a basis for g through multi-commutation.
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Lie groups
Let G be a Lie group of dimension n with Lie algebra g and equipped with
the left-invariant Haar measure µ.
We say that a linearly independent a = {a1 , . . . , ak } ⊂ g is an algebraic
basis if we can recover a basis for g through multi-commutation.
Let Ai denote the right-translation of ai and Ai = A∗i . Let
span {A1 , . . . , Ak } = A ⊂ TG be
bundleobtained through the
the
∗
1
right-translation of a and A = A , . . . , Ak the dual of A.
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Subelliptic distance
Theorem of Carathéodory-Chow tells us that for any two points x, y ∈ G,
we can find an absolutely continuous curve γ : [0, 1] → G such that
X
γ̇(t) =
γ̇ i (t)Ai (γ(t)) ∈ A.
i
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Subelliptic distance
Theorem of Carathéodory-Chow tells us that for any two points x, y ∈ G,
we can find an absolutely continuous curve γ : [0, 1] → G such that
X
γ̇(t) =
γ̇ i (t)Ai (γ(t)) ∈ A.
i
The length of such a curve then is given by
ˆ 1 X
i 2 1
γ̇ (t) ) 2 dt
`(γ) =
(
0
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i
Geometry and the Kato problem
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Subelliptic distance
Theorem of Carathéodory-Chow tells us that for any two points x, y ∈ G,
we can find an absolutely continuous curve γ : [0, 1] → G such that
X
γ̇(t) =
γ̇ i (t)Ai (γ(t)) ∈ A.
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.
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Subelliptic distance
Theorem of Carathéodory-Chow tells us that for any two points x, y ∈ G,
we can find an absolutely continuous curve γ : [0, 1] → G such that
X
γ̇(t) =
γ̇ i (t)Ai (γ(t)) ∈ A.
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 µ is Borel-regular with respect to d.
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Subelliptic operators
For f ∈ C∞ (G), define
∇f = Ai f Ai .
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Subelliptic operators
For f ∈ C∞ (G), define
∇f = Ai f Ai .
This defines an sub-connection on C∞ (M).
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Subelliptic operators
For f ∈ C∞ (G), define
∇f = Ai f Ai .
This defines an sub-connection on C∞ (M).
Each vector field Ai is a skew-adjoint differential operator.
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Subelliptic operators
For f ∈ C∞ (G), define
∇f = Ai f Ai .
This defines an sub-connection on C∞ (M).
Each vector field Ai is a skew-adjoint differential operator. We consider it
as a unbounded operator on L2 (G) with domain D(Ai ).
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Subelliptic operators
For f ∈ C∞ (G), define
∇f = Ai f Ai .
This defines an sub-connection on C∞ (M).
Each vector field Ai is a skew-adjoint differential operator. We consider it
as a unbounded operator on L2 (G) with domain D(Ai ).
By also considering ∇ as a closed, densely-defined operator on L2 (M), we
obtain the first-order Sobolev space W1,2 (G)0 = D(∇) = ∩ki=1 D(Ai ).
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Geometry and the Kato problem
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Subelliptic operators
For f ∈ C∞ (G), define
∇f = Ai f Ai .
This defines an sub-connection on C∞ (M).
Each vector field Ai is a skew-adjoint differential operator. We consider it
as a unbounded operator on L2 (G) with domain D(Ai ).
By also considering ∇ as a closed, densely-defined operator on L2 (M), we
obtain the first-order Sobolev space W1,2 (G)0 = D(∇) = ∩ki=1 D(Ai ).
We write the divergence as div = −∇ ∗ .
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Geometry and the Kato problem
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Subelliptic operators
For f ∈ C∞ (G), define
∇f = Ai f Ai .
This defines an sub-connection on C∞ (M).
Each vector field Ai is a skew-adjoint differential operator. We consider it
as a unbounded operator on L2 (G) with domain D(Ai ).
By also considering ∇ as a closed, densely-defined operator on L2 (M), we
obtain the first-order Sobolev space W1,2 (G)0 = D(∇) = ∩ki=1 D(Ai ).
We write the divergence as div = −∇ ∗ . Then, the subelliptic Laplacian
associated to A is
k
X
∆ = − div ∇ = −
A2i .
i=1
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Nilpotent Lie groups
The Lie group G is nilpotent if the inductively defined sequence
g1 = [g, g], g2 = [g1 , g], . . . is eventually zero.
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Geometry and the Kato problem
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Nilpotent Lie groups
The Lie group G is nilpotent if the inductively defined sequence
g1 = [g, g], g2 = [g1 , g], . . . is eventually zero.
Theorem (B.-E.-Mc., 2012)
Let (G, d, µ) be a connected, nilpotent Lie group with a an algebraic basis,
d the associated sub-elliptic distance, and µ the left Haar measure.
Suppose that a, A ∈ L∞ and that there exist κ1 , κ2 > 0 satisfying
Re hav, vi ≥ κ1 kvk2 ,
and
Re hA∇u, ∇ui ≥ κ2 k∇uk2 .
0 . Then,
for √
every v ∈ L2 (G) and u ∈ W1,2 (G)
√
D( −a div A∇) = W1,2 (G)0 and k −a div A∇uk ' k∇uk for
u ∈ W1,2 (G)0 .
Lashi Bandara (ANU)
Geometry and the Kato problem
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General Lie groups
Let S = (I, ∇) as in the manifold case.
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Geometry and the Kato problem
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General Lie groups
Let S = (I, ∇) as in the manifold case.
Theorem (B.-E.-Mc., 2012)
Let (G, d, µ) be a connected Lie group, a an algebraic basis, d the
associated sub-elliptic distance, and µ the left Haar measure. Let
a, A ∈ L∞ such that
Re hav, vi ≥ κ1 kvk2 ,
and
Re hASu, Sui ≥ κ2 kukW1,2 0
√
1,2
0
2
1,2
0
∗
for every
√ v ∈ L (G) and u ∈ W (G) . Then, D( aS AS) = W (G)
with k aS ∗ ASuk ' kukW1,2 0 = kuk + k∇uk.
Lashi Bandara (ANU)
Geometry and the Kato problem
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Operator theory
We adapt the framework due to Axelsson (Rosén), Keith, McIntosh in
[AKMc].
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
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Operator theory
We adapt the framework due to Axelsson (Rosén), Keith, McIntosh in
[AKMc].
Let H be a Hilbert space and Γ : H → H a closed, densely-defined,
nilpotent operator.
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
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Operator theory
We adapt the framework due to Axelsson (Rosén), Keith, McIntosh in
[AKMc].
Let H be a Hilbert space and Γ : H → H a closed, densely-defined,
nilpotent operator.
Suppose that B1 , B2 ∈ L(H ) such that here exist κ1 , κ2 > 0 satisfying
Re hB1 u, ui ≥ κ1 kuk2
and
Re hB2 v, vi ≥ κ2 kvk2
for u ∈ R(Γ∗ ) and v ∈ R(Γ).
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
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Operator theory
We adapt the framework due to Axelsson (Rosén), Keith, McIntosh in
[AKMc].
Let H be a Hilbert space and Γ : H → H a closed, densely-defined,
nilpotent operator.
Suppose that B1 , B2 ∈ L(H ) such that here exist κ1 , κ2 > 0 satisfying
Re hB1 u, ui ≥ κ1 kuk2
and
Re hB2 v, vi ≥ κ2 kvk2
for u ∈ R(Γ∗ ) and v ∈ R(Γ).
Furthermore, suppose the operators B1 , B2 satisfy B1 B2 R(Γ) ⊂ N (Γ)
and B2 B1 R(Γ∗ ) ⊂ N (Γ∗ ).
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
19 / 28
Operator theory
We adapt the framework due to Axelsson (Rosén), Keith, McIntosh in
[AKMc].
Let H be a Hilbert space and Γ : H → H a closed, densely-defined,
nilpotent operator.
Suppose that B1 , B2 ∈ L(H ) such that here exist κ1 , κ2 > 0 satisfying
Re hB1 u, ui ≥ κ1 kuk2
and
Re hB2 v, vi ≥ κ2 kvk2
for u ∈ R(Γ∗ ) and v ∈ R(Γ).
Furthermore, suppose the operators B1 , B2 satisfy B1 B2 R(Γ) ⊂ N (Γ)
and B2 B1 R(Γ∗ ) ⊂ N (Γ∗ ).
The primary operator we consider is ΠB = Γ + B1 Γ∗ B2 .
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Geometry and the Kato problem
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If the quadratic estimates
ˆ ∞
ktΠB (1 + t2 Π2B )−1 uk2 ' kuk
(Q)
0
hold for every u ∈ R(ΠB ), then, H decomposes into the spectral
subspaces of ΠB as H = N (ΠB ) ⊕ E+ ⊕ E− and
q
D( Π2B ) = D(ΠB ) = D(Γ) ∩ D(Γ∗ B2 )
q
k Π2B uk ' kΠB uk ' kΓuk + kΓ∗ B2 uk.
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If the quadratic estimates
ˆ ∞
ktΠB (1 + t2 Π2B )−1 uk2 ' kuk
(Q)
0
hold for every u ∈ R(ΠB ), then, H decomposes into the spectral
subspaces of ΠB as H = N (ΠB ) ⊕ E+ ⊕ E− and
q
D( Π2B ) = D(ΠB ) = D(Γ) ∩ D(Γ∗ B2 )
q
k Π2B uk ' kΠB uk ' kΓuk + kΓ∗ B2 uk.
The Kato problems are then obtained by letting
H = L2 (M) ⊕ (L2 (M) ⊕ L2 (T∗ M)) and letting
0 0
0 S∗
a 0
0 0
∗
Γ=
, Γ =
, B1 =
, B2 =
.
0 0
S 0
0 0
0 A
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
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Geometry and harmonic analysis
Harmonic analytic methods are used to prove quadratic estimates (Q).
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Geometry and the Kato problem
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Geometry and harmonic analysis
Harmonic analytic methods are used to prove quadratic estimates (Q).
The idea is to reduce the quadratic estimate (Q) to a Carleson measure
estimate.
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Geometry and the Kato problem
7 June 2013
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Geometry and harmonic analysis
Harmonic analytic methods are used to prove quadratic estimates (Q).
The idea is to reduce the quadratic estimate (Q) to a Carleson measure
estimate. This is achieved via a local T (b) argument.
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
21 / 28
Geometry and harmonic analysis
Harmonic analytic methods are used to prove quadratic estimates (Q).
The idea is to reduce the quadratic estimate (Q) to a Carleson measure
estimate. This is achieved via a local T (b) argument.
Geometry enters the picture precisely in the harmonic analysis.
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
21 / 28
Geometry and harmonic analysis
Harmonic analytic methods are used to prove quadratic estimates (Q).
The idea is to reduce the quadratic estimate (Q) to a Carleson measure
estimate. This is achieved via a local T (b) argument.
Geometry enters the picture precisely in the harmonic analysis. We need to
perform harmonic analysis on vector fields, not just functions.
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
21 / 28
Geometry and harmonic analysis
Harmonic analytic methods are used to prove quadratic estimates (Q).
The idea is to reduce the quadratic estimate (Q) to a Carleson measure
estimate. This is achieved via a local T (b) argument.
Geometry enters the picture precisely in the harmonic analysis. We need to
perform harmonic analysis on vector fields, not just functions.
One can show that this is not artificial - the Kato problem on functions
immediately provides a solution to the dual problem on vector fields.
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Geometry and the Kato problem
7 June 2013
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Elements of the proofs
Similar in structure to the proof of [AKMc] which is inspired from the
proof in [AHLMcT].
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Geometry and the Kato problem
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Elements of the proofs
Similar in structure to the proof of [AKMc] which is inspired from the
proof in [AHLMcT].
A dyadic decomposition of the space
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
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Elements of the proofs
Similar in structure to the proof of [AKMc] which is inspired from the
proof in [AHLMcT].
A dyadic decomposition of the space
A notion of averaging (in an integral sense)
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
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Elements of the proofs
Similar in structure to the proof of [AKMc] which is inspired from the
proof in [AHLMcT].
A dyadic decomposition of the space
A notion of averaging (in an integral sense)
Poincaré inequality - on both functions and vector fields
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
22 / 28
Elements of the proofs
Similar in structure to the proof of [AKMc] which is inspired from the
proof in [AHLMcT].
A dyadic decomposition of the space
A notion of averaging (in an integral sense)
Poincaré inequality - on both functions and vector fields
Control of ∇2 in terms of ∆.
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
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The case of non-smooth metrics on manifolds
We let M be a smooth, complete manifold as before but now let g be a
C0 metric. Let µg denote the volume measure with respect to g.
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
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The case of non-smooth metrics on manifolds
We let M be a smooth, complete manifold as before but now let g be a
C0 metric. Let µg denote the volume measure with respect to g.
Let h ∈ C0 (T (2,0) M). Then, define
khkop,g = sup
sup
|hx (u, v)| .
x∈M |u|g =|v|g =1
If g̃ is another C0 metric satisfying kg − g̃kop,g ≤ δ < 1, then
L2 (M, g) = L2 (M, g̃) and W1,2 (M, g) = W1,2 (M, g̃) with comparable
norms.
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Geometry and the Kato problem
7 June 2013
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ΠB under a change of metric
The operator Γg does not change under the change of metric.
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ΠB under a change of metric
The operator Γg does not change under the change of metric. However,
Γ∗g = C −1 Γg̃∗ C
where C is the bounded, invertible, multiplication operator on
L2 (M) ⊕ L2 (M) ⊕ L2 (T∗ M).
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
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ΠB under a change of metric
The operator Γg does not change under the change of metric. However,
Γ∗g = C −1 Γg̃∗ C
where C is the bounded, invertible, multiplication operator on
L2 (M) ⊕ L2 (M) ⊕ L2 (T∗ M).
Thus,
ΠB,g = Γg + B1 Γ∗g B2 = Γg̃ + B1 C −1 Γg̃∗ CB2 .
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
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ΠB under a change of metric
The operator Γg does not change under the change of metric. However,
Γ∗g = C −1 Γg̃∗ C
where C is the bounded, invertible, multiplication operator on
L2 (M) ⊕ L2 (M) ⊕ L2 (T∗ M).
Thus,
ΠB,g = Γg + B1 Γ∗g B2 = Γg̃ + B1 C −1 Γg̃∗ CB2 .
This allows us to reduce the study of ΠB,g for a C0 metric g to the study
of ΠB̃,g̃ = Γg̃ + B̃1 Γg̃∗ B̃2 where B̃1 = B1 C −1 and B̃2 = CB2 , but now
with a smooth metric g̃.
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
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Connection to geometric flows
Given a C0 metric g on a smooth compact manifold, we are able to always
find C∞ metric g̃.
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Geometry and the Kato problem
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Connection to geometric flows
Given a C0 metric g on a smooth compact manifold, we are able to always
find C∞ metric g̃.
The metric g̃ has inj(M, g̃) > κ and |Ric(g̃)|g̃ ≤ η so we obtain a
corresponding Kato square root estimate in this setting.
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
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Connection to geometric flows
Given a C0 metric g on a smooth compact manifold, we are able to always
find C∞ metric g̃.
The metric g̃ has inj(M, g̃) > κ and |Ric(g̃)|g̃ ≤ η so we obtain a
corresponding Kato square root estimate in this setting.
The non-compact situation poses issues.
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
25 / 28
Connection to geometric flows
Given a C0 metric g on a smooth compact manifold, we are able to always
find C∞ metric g̃.
The metric g̃ has inj(M, g̃) > κ and |Ric(g̃)|g̃ ≤ η so we obtain a
corresponding Kato square root estimate in this setting.
The non-compact situation poses issues.
Smooth the metric via mean curvature flow for, say, a C2 imbedding?
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
25 / 28
Connection to geometric flows
Given a C0 metric g on a smooth compact manifold, we are able to always
find C∞ metric g̃.
The metric g̃ has inj(M, g̃) > κ and |Ric(g̃)|g̃ ≤ η so we obtain a
corresponding Kato square root estimate in this setting.
The non-compact situation poses issues.
Smooth the metric via mean curvature flow for, say, a C2 imbedding?
Smooth the metric via Ricci flow in the general case? Regularity of the
initial metric?
Lashi Bandara (ANU)
Geometry and the Kato problem
7 June 2013
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Application to PDE
In the case we are able to find a suitable C∞ metric near the C0 one, then
we have Lipschitz estimates.
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Geometry and the Kato problem
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Application to PDE
In the case we are able to find a suitable C∞ metric near the C0 one, then
we have Lipschitz estimates.
Possible application to hyperbolic PDE?
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Geometry and the Kato problem
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Application to PDE
In the case we are able to find a suitable C∞ metric near the C0 one, then
we have Lipschitz estimates.
Possible application to hyperbolic PDE?
“Stability” of geometries with Ricci bounds and injectivity radius bounds?
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Geometry and the Kato problem
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References I
[AHLMcT] Pascal Auscher, Steve Hofmann, Michael Lacey, Alan
McIntosh, and Ph. Tchamitchian, The solution of the Kato square
root problem for second order elliptic operators on Rn , Ann. of Math.
(2) 156 (2002), no. 2, 633–654.
[AKMc-2] Andreas Axelsson, Stephen Keith, and Alan McIntosh, The
Kato square root problem for mixed boundary value problems, J.
London Math. Soc. (2) 74 (2006), no. 1, 113–130.
, Quadratic estimates and functional calculi of perturbed
[AKMc]
Dirac operators, Invent. Math. 163 (2006), no. 3, 455–497.
[Aus] Pascal Auscher, Lectures on the Kato square root problem, Surveys
in analysis and operator theory (Canberra, 2001), Proc. Centre Math.
Appl. Austral. Nat. Univ., vol. 40, Austral. Nat. Univ., Canberra, 2002,
pp. 1–18.
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References II
[Christ] Michael Christ, A T (b) theorem with remarks on analytic capacity
and the Cauchy integral, Colloq. Math. 60/61 (1990), no. 2, 601–628.
[Mc72] Alan McIntosh, On the comparability of A1/2 and A∗1/2 , Proc.
Amer. Math. Soc. 32 (1972), 430–434.
[Mc82]
, On representing closed accretive sesquilinear forms as
1/2
∗1/2
(A u, A
v), Nonlinear partial differential equations and their
applications. Collège de France Seminar, Vol. III (Paris, 1980/1981),
Res. Notes in Math., vol. 70, Pitman, Boston, Mass., 1982,
pp. 252–267.
[Morris] Andrew J. Morris, The Kato square root problem on
submanifolds, J. Lond. Math. Soc. (2) 86 (2012), no. 3, 879–910.
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