Download Geometry and the Kato square root problem

Geometry and the Kato square root problem
Lashi Bandara
Centre for Mathematics and its Applications
Australian National University
29 July 2014
Geometric Analysis Seminar
Beijing International Center for Mathematical Research
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Geometry and the Kato problem
1/34
History of the problem
In the 1960’s, Kato considered the following abstract evolution
equation
∂t u(t) + A(t)u(t) = f (t), t ∈ [0, T ].
on a Hilbert space H .
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Geometry and the Kato problem
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History of the problem
In the 1960’s, Kato considered the following abstract evolution
equation
∂t u(t) + A(t)u(t) = f (t), t ∈ [0, T ].
on a Hilbert space H .
This has a unique strict solution u = u(t) if
D(A(t)α ) = const
for some 0 < α ≤ 1 and A(t) and f (t) satisfy certain smoothness
conditions.
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Typically A(t) is defined by an associated sesquilinear form
Jt : W × W → C
where W ⊂ H .
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Geometry and the Kato problem
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Typically A(t) is defined by an associated sesquilinear form
Jt : W × W → C
where W ⊂ H .
Suppose 0 ≤ ω ≤ π/2. Then Jt is ω-sectorial means that
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Geometry and the Kato problem
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Typically A(t) is defined by an associated sesquilinear form
Jt : W × W → C
where W ⊂ H .
Suppose 0 ≤ ω ≤ π/2. Then Jt is ω-sectorial means that
(i) W ⊂ H is dense,
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Geometry and the Kato problem
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Typically A(t) is defined by an associated sesquilinear form
Jt : W × W → C
where W ⊂ H .
Suppose 0 ≤ ω ≤ π/2. Then Jt is ω-sectorial means that
(i) W ⊂ H is dense,
(ii) Jt [u, u] ∈ Sω+ = {ζ ∈ C : |arg ζ| ≤ ω} ∪ {0},
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Geometry and the Kato problem
3/34
Typically A(t) is defined by an associated sesquilinear form
Jt : W × W → C
where W ⊂ H .
Suppose 0 ≤ ω ≤ π/2. Then Jt is ω-sectorial means that
(i) W ⊂ H is dense,
(ii) Jt [u, u] ∈ Sω+ = {ζ ∈ C : |arg ζ| ≤ ω} ∪ {0}, and
(iii) W is complete under the norm
kuk2W = kuk2 + Re Jt [u, u].
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Geometry and the Kato problem
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T : D(T ) → H is called ω-accretive if
(i) T is densely-defined and closed,
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Geometry and the Kato problem
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T : D(T ) → H is called ω-accretive if
(i) T is densely-defined and closed,
(ii) hT u, ui ∈ Sω+ for u ∈ D(T ),
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Geometry and the Kato problem
4/34
T : D(T ) → H is called ω-accretive if
(i) T is densely-defined and closed,
(ii) hT u, ui ∈ Sω+ for u ∈ D(T ), and
(iii) σ(T ) ⊂ Sω+ .
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Geometry and the Kato problem
4/34
T : D(T ) → H is called ω-accretive if
(i) T is densely-defined and closed,
(ii) hT u, ui ∈ Sω+ for u ∈ D(T ), and
(iii) σ(T ) ⊂ Sω+ .
A 0-accretive operator is non-negative and self-adjoint.
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Geometry and the Kato problem
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Let A(t) : D(A(t)) → H be defined as the operator with largest
domain such that
Jt [u, v] = hA(t)u, vi
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u ∈ D(A(t)), v ∈ W.
Geometry and the Kato problem
5/34
Let A(t) : D(A(t)) → H be defined as the operator with largest
domain such that
Jt [u, v] = hA(t)u, vi
u ∈ D(A(t)), v ∈ W.
The theorem of Lax-Milgram guarantees that
Jt is ω-sectorial =⇒ A(t) is ω-accretive.
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Geometry and the Kato problem
5/34
Let A(t) : D(A(t)) → H be defined as the operator with largest
domain such that
Jt [u, v] = hA(t)u, vi
u ∈ D(A(t)), v ∈ W.
The theorem of Lax-Milgram guarantees that
Jt is ω-sectorial =⇒ A(t) is ω-accretive.
In 1962, Kato showed in [Kato] that for 0 ≤ α < 1/2 and
0 ≤ ω ≤ π/2,
D(A(t)α ) = D(A(t)∗α ) = D = const, and
kA(t)α uk ' kA(t)∗α uk,
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u ∈ D.
Geometry and the Kato problem
(Kα )
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Let A(t) : D(A(t)) → H be defined as the operator with largest
domain such that
Jt [u, v] = hA(t)u, vi
u ∈ D(A(t)), v ∈ W.
The theorem of Lax-Milgram guarantees that
Jt is ω-sectorial =⇒ A(t) is ω-accretive.
In 1962, Kato showed in [Kato] that for 0 ≤ α < 1/2 and
0 ≤ ω ≤ π/2,
D(A(t)α ) = D(A(t)∗α ) = D = const, and
kA(t)α uk ' kA(t)∗α uk,
u ∈ D.
(Kα )
Counter examples were known for α > 1/2 and for α = 1/2 when
ω = π/2.
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Geometry and the Kato problem
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Kato asked two questions. For ω < π/2,
(K1) Does (Kα ) hold for α = 1/2?
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Geometry and the Kato problem
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Kato asked two questions. For ω < π/2,
(K1) Does (Kα ) hold for α = 1/2?
p
(K2) For the case ω = 0, we know D( A(t)) = W and (K1) is true,
but is
p
k∂t A(t)uk . kuk
for u ∈ W?
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Geometry and the Kato problem
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Kato asked two questions. For ω < π/2,
(K1) Does (Kα ) hold for α = 1/2?
p
(K2) For the case ω = 0, we know D( A(t)) = W and (K1) is true,
but is
p
k∂t A(t)uk . kuk
for u ∈ W?
In 1972, McIntosh provided a counter example in [Mc72]
demonstrating that (K1) is false in such generality.
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Geometry and the Kato problem
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Kato asked two questions. For ω < π/2,
(K1) Does (Kα ) hold for α = 1/2?
p
(K2) For the case ω = 0, we know D( A(t)) = W and (K1) is true,
but is
p
k∂t A(t)uk . kuk
for u ∈ W?
In 1972, McIntosh provided a counter example in [Mc72]
demonstrating that (K1) is false in such generality.
In 1982, McIntosh showed that (K2) also did not hold in general in
[Mc82].
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Geometry and the Kato problem
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The Kato square root problem then became the following.
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Geometry and the Kato problem
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The Kato square root problem then became the following. Set
J[u, v] = hA∇u, ∇vi
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u, v ∈ W1,2 (Rn ),
Geometry and the Kato problem
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The Kato square root problem then became the following. Set
J[u, v] = hA∇u, ∇vi
u, v ∈ W1,2 (Rn ),
where A ∈ L∞ is a pointwise matrix multiplication operator
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Geometry and the Kato problem
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The Kato square root problem then became the following. Set
J[u, v] = hA∇u, ∇vi
u, v ∈ W1,2 (Rn ),
where A ∈ L∞ is a pointwise matrix multiplication operator satisfying
the following ellipticity condition:
Re J[u, u] ≥ κk∇uk,
Lashi Bandara
for some κ > 0.
Geometry and the Kato problem
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The Kato square root problem then became the following. Set
J[u, v] = hA∇u, ∇vi
u, v ∈ W1,2 (Rn ),
where A ∈ L∞ is a pointwise matrix multiplication operator satisfying
the following ellipticity condition:
Re J[u, u] ≥ κk∇uk,
for some κ > 0.
Under these conditions, is it true that
p
D( div A∇) = W1,2 (Rn )
p
k div A∇uk ' k∇uk.
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Geometry and the Kato problem
(K1)
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The Kato square root problem then became the following. Set
J[u, v] = hA∇u, ∇vi
u, v ∈ W1,2 (Rn ),
where A ∈ L∞ is a pointwise matrix multiplication operator satisfying
the following ellipticity condition:
Re J[u, u] ≥ κk∇uk,
for some κ > 0.
Under these conditions, is it true that
p
D( div A∇) = W1,2 (Rn )
p
k 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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Geometry and the Kato problem
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Kato square root problem for functions and forms
Let M be a smooth, complete Riemannian manifold with metric g,
Levi-Civita connection ∇, and volume measure µg .
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Geometry and the Kato problem
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Kato square root problem for functions and forms
Let M be a smooth, complete Riemannian manifold with metric g,
Levi-Civita connection ∇, and volume measure µg .
Write divg = −∇ ∗ in L2 .
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Geometry and the Kato problem
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Kato square root problem for functions and forms
Let M be a smooth, complete Riemannian manifold with metric g,
Levi-Civita connection ∇, and volume measure µg .
Write divg = −∇ ∗ in L2 .
Let Ω(M) denote the algebra of differential forms over M.
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Geometry and the Kato problem
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Kato square root problem for functions and forms
Let M be a smooth, complete Riemannian manifold with metric g,
Levi-Civita connection ∇, and volume measure µg .
Write divg = −∇ ∗ in L2 .
Let Ω(M) denote the algebra of differential forms over M.
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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Geometry and the Kato problem
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Kato square root problem for functions and forms
Let M be a smooth, complete Riemannian manifold with metric g,
Levi-Civita connection ∇, and volume measure µg .
Write divg = −∇ ∗ in L2 .
Let Ω(M) denote the algebra of differential forms over M.
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
8/34
Kato square root problem for functions and forms
Let M be a smooth, complete Riemannian manifold with metric g,
Levi-Civita connection ∇, and volume measure µg .
Write divg = −∇ ∗ in L2 .
Let Ω(M) denote the algebra of differential forms over M.
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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Geometry and the Kato problem
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Let S = (I, ∇).
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Geometry and the Kato problem
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Let S = (I, ∇).
Assume a ∈ L∞ (M) and
A = (Aij ) ∈ L∞ (M, L(L2 (M) ⊕ L2 (T∗ M)).
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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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Geometry and the Kato problem
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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.
The Kato square root problem for functions is then to determine:
√
√
D( LA ) = W1,2 (M) and k LA uk ' k∇uk + kuk = kukW1,2 for all
u ∈ W1,2 (M).
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Geometry and the Kato problem
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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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Geometry and the Kato problem
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For an invertible A ∈ L∞ (L(Ω(M))), we consider perturbing D to
obtain the operator DA = d +A−1 d∗ A.
The Kato square root problem for forms is then to determine the
following whenever 0 6= β ∈ C:
q
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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Geometry and the Kato problem
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Axelsson (Rosén)-Keith-McIntosh framework
(H1) The operator Γ : D(Γ) ⊂ H → H is a closed, densely-defined
and nilpotent operator, by which we mean R(Γ) ⊂ N (Γ),
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Axelsson (Rosén)-Keith-McIntosh framework
(H1) The operator Γ : D(Γ) ⊂ H → H is a closed, densely-defined
and nilpotent operator, by which we mean R(Γ) ⊂ N (Γ),
(H2) B1 , B2 ∈ L(H ) and there exist κ1 , κ2 > 0 satisfying the
accretivity conditions
Re hB1 u, ui ≥ κ1 kuk2 and Re hB2 v, vi ≥ κ2 kvk2 ,
for u ∈ R(Γ∗ ) and v ∈ R(Γ), and
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Geometry and the Kato problem
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Axelsson (Rosén)-Keith-McIntosh framework
(H1) The operator Γ : D(Γ) ⊂ H → H is a closed, densely-defined
and nilpotent operator, by which we mean R(Γ) ⊂ N (Γ),
(H2) B1 , B2 ∈ L(H ) and there exist κ1 , κ2 > 0 satisfying the
accretivity conditions
Re hB1 u, ui ≥ κ1 kuk2 and Re hB2 v, vi ≥ κ2 kvk2 ,
for u ∈ R(Γ∗ ) and v ∈ R(Γ), and
(H3) B1 B2 R(Γ) ⊂ N (Γ) and B2 B1 R(Γ∗ ) ⊂ N (Γ∗ ).
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Geometry and the Kato problem
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Axelsson (Rosén)-Keith-McIntosh framework
(H1) The operator Γ : D(Γ) ⊂ H → H is a closed, densely-defined
and nilpotent operator, by which we mean R(Γ) ⊂ N (Γ),
(H2) B1 , B2 ∈ L(H ) and there exist κ1 , κ2 > 0 satisfying the
accretivity conditions
Re hB1 u, ui ≥ κ1 kuk2 and Re hB2 v, vi ≥ κ2 kvk2 ,
for u ∈ R(Γ∗ ) and v ∈ R(Γ), and
(H3) B1 B2 R(Γ) ⊂ N (Γ) and B2 B1 R(Γ∗ ) ⊂ N (Γ∗ ).
Let us now define ΠB = Γ + B1 Γ∗ B2 with domain
D(ΠB ) = D(Γ) ∩ D(B1 Γ∗ B2 ).
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Geometry and the Kato problem
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Quadratic estimates
To say that ΠB satisfies quadratic estimates means that
ˆ ∞
dt
' kuk2 ,
ktΠB (I + t2 Π2B )−1 uk2
t
0
(Q)
for all u ∈ R(ΠB ).
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Geometry and the Kato problem
12/34
Quadratic estimates
To say that ΠB satisfies quadratic estimates means that
ˆ ∞
dt
' kuk2 ,
ktΠB (I + t2 Π2B )−1 uk2
t
0
(Q)
for all u ∈ R(ΠB ).
This implies that
q
D( Π2B ) = D(ΠB ) = D(Γ) ∩ D(Γ∗ B2 )
q
k Π2B uk ' kΠB uk ' kΓuk + kΓ∗ B2 uk
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Geometry and the Kato problem
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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
for √
v ∈ L2 (M) and u ∈ W1,2 (M). Then,
1,2
D(
√ LA ) = D(∇) = W (M) and
k LA uk ' k∇uk + kuk = kukW1,2 for all u ∈ W1,2 (M).
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Geometry and the Kato problem
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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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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.
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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 .
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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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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.
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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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Geometry and the Kato problem
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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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Geometry and the Kato problem
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The Kato problem for functions are captured in the AKM framework
on letting H = L2 (M) ⊕ (L2 (M) ⊕ L2 (T∗ M)) and letting
0 0
0 S∗
a 0
0 0
∗
Γ=
, Γ =
, B1 =
, B2 =
.
S 0
0 0
0 0
0 A
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Geometry and the Kato problem
17/34
The Kato problem for functions are captured in the AKM framework
on letting H = L2 (M) ⊕ (L2 (M) ⊕ L2 (T∗ M)) and letting
0 0
0 S∗
a 0
0 0
∗
Γ=
, Γ =
, B1 =
, B2 =
.
S 0
0 0
0 0
0 A
For the case of forms, the setup takes the form,
H = L2 (Ω(M)) ⊕ L2 (Ω(M)) and
A−1
0
A 0
d 0
δ β
∗
, B2 =
.
Γ=
,Γ =
, B1 =
0
A−1
0 A
β −d
0 −δ
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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).
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Geometry and the Kato problem
18/34
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.
Lashi Bandara
Geometry and the Kato problem
18/34
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
Geometry and the Kato problem
18/34
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
Geometry and the Kato problem
18/34
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
Geometry and the Kato problem
18/34
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
18/34
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
19/34
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
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Geometry and the Kato problem
19/34
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)
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Geometry and the Kato problem
19/34
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
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Geometry and the Kato problem
19/34
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 ∆.
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Geometry and the Kato problem
19/34
Rough metrics
Definition (Rough metric)
Let g be a (2, 0) symmetric tensor field with measurable coefficients
and that for each x ∈ M, there is some chart (U, ψ) near x and a
constant C ≥ 1 such that
C −1 |u|ψ∗ δ(y) ≤ |u|g(y) ≤ C |u|ψ∗ δ(y) ,
for almost-every y ∈ U and where δ is the Euclidean metric in ψ(U ).
Then we say that g is a rough metric, and such a chart (U, ψ) is said
to satisfy the local comparability condition.
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Geometry and the Kato problem
20/34
Metric perturbations
Definition
We say that two rough metrics g and g̃ are C-close if
C −1 |u|g̃(x) ≤ |u|g(x) ≤ C |u|g̃(x)
for almost-every x ∈ M where C ≥ 1. Two such metrics are said to
be C-close everywhere if this inequality holds for every x ∈ M.
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Geometry and the Kato problem
21/34
Metric perturbations
Definition
We say that two rough metrics g and g̃ are C-close if
C −1 |u|g̃(x) ≤ |u|g(x) ≤ C |u|g̃(x)
for almost-every x ∈ M where C ≥ 1. Two such metrics are said to
be C-close everywhere if this inequality holds for every x ∈ M.
We also say that g and g̃ are close if there exists some C ≥ 1 for
which they are C-close.
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Geometry and the Kato problem
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Metric perturbations
Definition
We say that two rough metrics g and g̃ are C-close if
C −1 |u|g̃(x) ≤ |u|g(x) ≤ C |u|g̃(x)
for almost-every x ∈ M where C ≥ 1. Two such metrics are said to
be C-close everywhere if this inequality holds for every x ∈ M.
We also say that g and g̃ are close if there exists some C ≥ 1 for
which they are C-close.
For two continuous metrics, C-close and C-close everywhere coincide.
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Proposition
Let g and g̃ be two rough metrics that are C-close. Then, there exists
B ∈ Γ(T∗ M ⊗ TM) such that it is symmetric, almost-everywhere
positive and invertible, and
g̃x (B(x)u, v) = gx (u, v)
for almost-every x ∈ M. Furthermore, for almost-every x ∈ M,
C −2 |u|g̃(x) ≤ |B(x)u|g̃(x) ≤ C 2 |u|g̃(x) ,
and the same inequality with g̃ and g interchanged. If g̃ ∈ Ck and
g ∈ Cl (with k, l ≥ 0), then the properties of B are valid for all
x ∈ M and B ∈ Cmin{k,l} (T∗ M ⊗ TM).
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The measure µg (x) = θ(x) dµg̃ (x), where θ(x) =
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p
det B(x).
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The measure µg (x) = θ(x) dµg̃ (x), where θ(x) =
Consequently,
p
det B(x).
(i) whenever p ∈ [1, ∞), Lp (T (r,s) M, g) = Lp (T (r,s) M, g̃) with
C
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n
− r+s+ 2p
kukp,g̃ ≤ kukp,g ≤ C
n
r+s+ 2p
Geometry and the Kato problem
kukp,g̃ ,
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The measure µg (x) = θ(x) dµg̃ (x), where θ(x) =
Consequently,
p
det B(x).
(i) whenever p ∈ [1, ∞), Lp (T (r,s) M, g) = Lp (T (r,s) M, g̃) with
C
n
− r+s+ 2p
kukp,g̃ ≤ kukp,g ≤ C
n
r+s+ 2p
kukp,g̃ ,
(ii) for p = ∞, L∞ (T (r,s) M, g) = L∞ (T (r,s) M, g̃) with
C −(r+s) kuk∞,g̃ ≤ kuk∞,g ≤ C r+s kuk∞,g̃ ,
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(iii) the Sobolev spaces W1,p (M, g) = W1,p (M, g̃) and
W01,p (M, g) = W01,p (M, g̃) with
C
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n
− 1+ 2p
kukW1,p ,g̃ ≤ kukW1,p ,g ≤ C
Geometry and the Kato problem
n
1+ 2p
kukW1,p ,g̃ ,
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(iii) the Sobolev spaces W1,p (M, g) = W1,p (M, g̃) and
W01,p (M, g) = W01,p (M, g̃) with
C
n
− 1+ 2p
kukW1,p ,g̃ ≤ kukW1,p ,g ≤ C
n
1+ 2p
kukW1,p ,g̃ ,
(iv) the Sobolev spaces Wd,p (M, g) = Wd,p (M, g̃) and
W0d,p (M, g) = W0d,p (M, g̃) with
C
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n
− n+ 2p
kukWd,p ,g̃ ≤ kukWd,p ,g ≤ C
Geometry and the Kato problem
n
n+ 2p
kukWd,p ,g̃ ,
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(iii) the Sobolev spaces W1,p (M, g) = W1,p (M, g̃) and
W01,p (M, g) = W01,p (M, g̃) with
C
n
− 1+ 2p
kukW1,p ,g̃ ≤ kukW1,p ,g ≤ C
n
1+ 2p
kukW1,p ,g̃ ,
(iv) the Sobolev spaces Wd,p (M, g) = Wd,p (M, g̃) and
W0d,p (M, g) = W0d,p (M, g̃) with
C
n
− n+ 2p
kukWd,p ,g̃ ≤ kukWd,p ,g ≤ C
n
n+ 2p
kukWd,p ,g̃ ,
(v) the divergence operators satisfy divD,g = θ−1 divD,g̃ θB and
divN,g = θ−1 divN,g̃ θB.
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Case of functions
Theorem (B, 2014)
Let g̃ be a smooth, complete metric and suppose that there exists
κ > 0 and η > 0 such that
(i) inj(M, g̃) ≥ κ and,
(ii) |Ric(g̃)| ≤ η.
Then, for any rough metric g that is close, the Kato square root
problem for functions has a solution on (M, g).
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Case of forms
Theorem (B, 2014)
Let g be a rough metric close to g̃, a smooth, complete metric, and
suppose that:
(i) there exists κ > 0 such that inj(M, g̃) ≥ κ,
(ii) there exists η > 0 such that |Ric(g̃)| ≤ η, and
(iii) there exists ζ ∈ R such that g̃(R ω, ω) ≥ ζ |ω|g̃2 .
Then, the Kato square root problem for forms has a solution on
(M, g).
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Compact manifolds with rough metrics
Theorem (B, 2014)
Let M be a smooth, compact manifold and g a rough metric. Then,
the Kato square root problem (on functions and forms, respectively)
has a solution.
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Cones and induced metrics
n be the n-cone of height h > 0 and radius r > 0.
Let Cr,h
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Cones and induced metrics
n be the n-cone of height h > 0 and radius r > 0. The cone
Let Cr,h
can be realised as the image of the graph function
|x|
.
Fr,h (x) = x, h 1 −
r
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Cones and induced metrics
n be the n-cone of height h > 0 and radius r > 0. The cone
Let Cr,h
can be realised as the image of the graph function
|x|
.
Fr,h (x) = x, h 1 −
r
Let U be an open set in Rn such that Br (0) ⊂ U . Then, define
Gr,h : U → Rn+1 as the map Fr,h whenever x ∈ Br (0) and (x, 0)
otherwise.
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Cones and induced metrics
n be the n-cone of height h > 0 and radius r > 0. The cone
Let Cr,h
can be realised as the image of the graph function
|x|
.
Fr,h (x) = x, h 1 −
r
Let U be an open set in Rn such that Br (0) ⊂ U . Then, define
Gr,h : U → Rn+1 as the map Fr,h whenever x ∈ Br (0) and (x, 0)
otherwise.
Then we obtain that the map Gr,h satisfies
p
|x − y| ≤ |Gr,h (x) − Gr,h (y)| ≤ 1 + (hr−1 )2 |x − y| .
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Proposition
Let γ : I → U be a smooth curve such that γ(0) 6∈ {0} ∪ ∂Br (0).
Then,
r
2 0 0
γ (0) ≤ (Gr,h ◦ γ) (0) ≤ 1 + h γ 0 (0) .
r2
Moreover, for u ∈ Tx U , x 6∈ {0} ∪ ∂Br (0) (and in particular for
almost-every x),
r
h2
|u|δ ≤ |u|g ≤ 1 + 2 |u|δ ,
r
where δ is the usual inner product on U induced by Rn .
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Proposition
Let γ : I → U be a smooth curve such that γ(0) 6∈ {0} ∪ ∂Br (0).
Then,
r
2 0 0
γ (0) ≤ (Gr,h ◦ γ) (0) ≤ 1 + h γ 0 (0) .
r2
Moreover, for u ∈ Tx U , x 6∈ {0} ∪ ∂Br (0) (and in particular for
almost-every x),
r
h2
|u|δ ≤ |u|g ≤ 1 + 2 |u|δ ,
r
where δ is the usual inner product on U induced by Rn .
A particular consequence is that the metrics g = G∗r,h δRn+1 and δRn
p
are 1 + (hr−1 )2 -close on U .
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Lemma
Given ε > 0, there exists two points x, x0 and distinct minimising
smooth geodesics γ1,ε and γ2,ε between x and x0 of length ε.
Furthermore, there are two constants C1,r,h,ε , C2,r,h,ε > 0 depending
on h, r and ε such that the geodesics γ1,ε and γ2,ε are contained in
Gr,h (Aε ) where Aε is the Euclidean annulus
{x ∈ Br (0) : C1,r,h,ε < |x| < C2,r,h,ε } .
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Theorem (B., 2014)
For any C > 1, there exists a smooth metric g which is C-close to the
Euclidean metric δ for which inj(R2 , g) = 0. Furthermore, the Kato
square root problem for functions can be solved for (R2 , g) under the.
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Geometry and the Kato problem
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In higher dimensions, we obtain a similar result since the
2-dimensional cone can be realised as a totally geodesic submanifold.
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Geometry and the Kato problem
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In higher dimensions, we obtain a similar result since the
2-dimensional cone can be realised as a totally geodesic submanifold.
Theorem (B., 2014)
Let M be a smooth manifold of dimension at least 2 and g a
continuous metric. Given C > 1, and a point x0 ∈ M, there exists a
rough metric h such that:
(i) it induces a length structure and the metric dg preserves the
topology of M,
(ii) it is smooth everywhere except x0 ,
(iii) the geodesics through x0 are Lipschitz,
(iv) it is C-close to g,
(v) inj(M \ {x0 } , h) = 0.
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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.
[AKMc]
, Quadratic estimates and functional calculi of
perturbed Dirac operators, Invent. Math. 163 (2006), no. 3,
455–497.
[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.
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References II
[Kato] Tosio Kato, Fractional powers of dissipative operators, J.
Math. Soc. Japan 13 (1961), 246–274. MR 0138005 (25 #1453)
[Mc72] Alan McIntosh, On the comparability of A1/2 and A∗1/2 ,
Proc. Amer. Math. Soc. 32 (1972), 430–434. MR 0290169 (44
#7354)
[Mc82]
, On representing closed accretive sesquilinear forms
as (A1/2 u, A∗1/2 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. MR 670278 (84k:47030)
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