The Kato square root problem on vector bundles with
generalised bounded geometry
Lashi Bandara
(Joint work with Alan McIntosh, ANU)
Centre for Mathematics and its Applications
Australian National University
7 August 2012
Recent trends in Geometric and Nonlinear Analysis
Banff International Research Station, Canada
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The Kato problem on GBG bundles
7 August 2012
1 / 30
History of the problem
In the 1960’s, Kato considered the following abstract evolution equation
du
+ A(t)u = f (t),
dt
t ∈ [0, T ].
on a Hilbert space H .
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The Kato problem on GBG bundles
7 August 2012
2 / 30
History of the problem
In the 1960’s, Kato considered the following abstract evolution equation
du
+ A(t)u = f (t),
dt
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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The Kato problem on GBG bundles
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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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The Kato problem on GBG bundles
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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
Lashi Bandara (ANU)
The Kato problem on GBG bundles
7 August 2012
3 / 30
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,
Lashi Bandara (ANU)
The Kato problem on GBG bundles
7 August 2012
3 / 30
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},
Lashi Bandara (ANU)
The Kato problem on GBG bundles
7 August 2012
3 / 30
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 = kuk + Re J[u, u].
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The Kato problem on GBG bundles
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T : D(T ) → H is called ω-accretive if
(i) T is densely-defined and closed,
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The Kato problem on GBG bundles
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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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The Kato problem on GBG bundles
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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 ), and
(iii) σ (T ) ⊂ Sω+ .
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The Kato problem on GBG bundles
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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 ), and
(iii) σ (T ) ⊂ Sω+ .
A 0-accretive operator is non-negative and self-adjoint.
Lashi Bandara (ANU)
The Kato problem on GBG bundles
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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.
The Kato problem on GBG bundles
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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.
Lashi Bandara (ANU)
The Kato problem on GBG bundles
7 August 2012
5 / 30
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.
The Kato problem on GBG bundles
(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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The Kato problem on GBG bundles
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Kato asked two questions. For ω < π/2,
(K1) Does (Kα ) hold for α = 1/2?
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The Kato problem on GBG bundles
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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
dp
. kuk
A(t)u
dt
for u ∈ W?
Lashi Bandara (ANU)
The Kato problem on GBG bundles
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6 / 30
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
dp
. kuk
A(t)u
dt
for u ∈ W?
In 1972, McIntosh provided a counter example in [Mc72] demonstrating
that (K1) is false in such generality.
Lashi Bandara (ANU)
The Kato problem on GBG bundles
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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
dp
. kuk
A(t)u
dt
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].
Lashi Bandara (ANU)
The Kato problem on GBG bundles
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The Kato square root problem then became the following.
Lashi Bandara (ANU)
The Kato problem on GBG bundles
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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 ∈ H1 (Rn ),
The Kato problem on GBG bundles
7 August 2012
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The Kato square root problem then became the following. Set
J[u, v] = hA∇u, ∇vi
u, v ∈ H1 (Rn ),
where A ∈ L∞ is a pointwise matrix multiplication operator
Lashi Bandara (ANU)
The Kato problem on GBG bundles
7 August 2012
7 / 30
The Kato square root problem then became the following. Set
J[u, v] = hA∇u, ∇vi
u, v ∈ H1 (Rn ),
where A ∈ L∞ is a pointwise matrix multiplication operator satisfying the
following ellipticity condition:
Re J[u, u] ≥ κ k∇uk ,
Lashi Bandara (ANU)
for some κ > 0.
The Kato problem on GBG bundles
7 August 2012
7 / 30
The Kato square root problem then became the following. Set
J[u, v] = hA∇u, ∇vi
u, v ∈ H1 (Rn ),
where A ∈ L∞ is a pointwise matrix multiplication operator satisfying the
following ellipticity condition:
Re J[u, u] ≥ κ k∇uk ,
for some κ > 0.
Under these conditions, is it true that
p
D( div A∇) = H1 (Rn )
p
div A∇u ' k∇uk
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The Kato problem on GBG bundles
(K1)
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The Kato square root problem then became the following. Set
J[u, v] = hA∇u, ∇vi
u, v ∈ H1 (Rn ),
where A ∈ L∞ is a pointwise matrix multiplication operator satisfying the
following ellipticity condition:
Re J[u, u] ≥ κ k∇uk ,
for some κ > 0.
Under these conditions, is it true that
p
D( div A∇) = H1 (Rn )
p
div A∇u ' k∇uk
(K1)
This was answered in the positive in 2002 by Auscher, Hofmann, Lacey,
McIntosh and Tchamitchian in [AHLMcT].
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The Kato problem on GBG bundles
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Setup
Let M be a smooth, complete Riemannian manifold with metric g,
Levi-Civita connection ∇, and volume measure dµ.
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The Kato problem on GBG bundles
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Setup
Let M be a smooth, complete Riemannian manifold with metric g,
Levi-Civita connection ∇, and volume measure dµ.
Write div = −∇ ∗ in L2 and let S = (I, ∇).
Lashi Bandara (ANU)
The Kato problem on GBG bundles
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Setup
Let M be a smooth, complete Riemannian manifold with metric g,
Levi-Civita connection ∇, and volume measure dµ.
Write div = −∇ ∗ in L2 and let S = (I, ∇).
Consider the following uniformly elliptic 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 Kato problem on GBG bundles
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Setup
Let M be a smooth, complete Riemannian manifold with metric g,
Levi-Civita connection ∇, and volume measure dµ.
Write div = −∇ ∗ in L2 and let S = (I, ∇).
Consider the following uniformly elliptic 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.
That is, we assume a and A = (Aij ) are L∞ multiplication operators and
that there exist κ1 , κ2 > 0 such that
Re hav, vi ≥ κ1 kvk2 , v ∈ L2
Re hASu, Sui ≥ κ2 (kuk2 + k∇uk2 ), u ∈ H1
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The Kato problem on GBG bundles
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The problem
The Kato square root problem on manifolds is to determine when the
following holds:
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The Kato problem on GBG bundles
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The problem
The Kato square root problem on manifolds is to determine when the
following holds:
(
√
D( LA ) = H1 (M)
√
LA u ' k∇uk + kuk = kuk
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H1
The Kato problem on GBG bundles
, u ∈ H1 (M)
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The main theorem
Theorem (B.-Mc)
Let M be a smooth, complete Riemannian manifold with |Ric| ≤ C and
inj(M ) ≥ κ > 0. Suppose there exist κ1 , κ2 > 0 such that
Re hav, vi ≥ κ1 kvk2
Re hASu, Sui ≥ κ2 kuk2H1
√
1
2
1
for
√ v ∈ L (M) and u ∈ H (M). Then, D( LA1) = D(∇) = H (M) and
k LA uk ' k∇uk + kuk = kukH1 for all u ∈ H (M).
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The Kato problem on GBG bundles
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Stability
Theorem (B.-Mc)
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
Re hASu, Sui ≥ κ2 kuk2H1
for v ∈ L2 (M) and u ∈ H1 (M). Then for every ηi < κi , whenever
kãk∞ ≤ η1 , kÃk∞ ≤ η2 , the estimate
p
q
LA u − LA+Ã u . (kãk∞ + kÃk∞ ) kukH1
holds for all u ∈ H1 (M). The implicit constant depends in particular on
A, a and ηi .
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The Kato problem on GBG bundles
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A more general problem
We can consider the Kato square root problem on vector bundles by
replacing M with V, a smooth, complex vector bundle of rank N over M
with metric h and connection ∇.
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The Kato problem on GBG bundles
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A more general problem
We can consider the Kato square root problem on vector bundles by
replacing M with V, a smooth, complex vector bundle of rank N over M
with metric h and connection ∇.
These theorems are obtained as special cases of corresponding theorems
on vector bundles.
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The Kato problem on GBG bundles
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A more general problem
We can consider the Kato square root problem on vector bundles by
replacing M with V, a smooth, complex vector bundle of rank N over M
with metric h and connection ∇.
These theorems are obtained as special cases of corresponding theorems
on vector bundles.
We use the adaptation of the first order systems approach in [AKMc],
which captures the Kato problem (and some other results of harmonic
analysis) in terms of perturbations of Dirac type operators.
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The Kato problem on GBG bundles
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Axelsson (Rosén)-Keith-McIntosh framework
(H1) Let Γ be a densely-defined, closed, nilpotent operator on a Hilbert
space H ,
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The Kato problem on GBG bundles
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Axelsson (Rosén)-Keith-McIntosh framework
(H1) Let Γ be a densely-defined, closed, nilpotent operator on a Hilbert
space H ,
(H2) 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),
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The Kato problem on GBG bundles
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Axelsson (Rosén)-Keith-McIntosh framework
(H1) Let Γ be a densely-defined, closed, nilpotent operator on a Hilbert
space H ,
(H2) 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),
(H3) The operators B1 , B2 satisfy B1 B2 R(Γ) ⊂ N (Γ) and
B2 B1 R(Γ∗ ) ⊂ N (Γ∗ ).
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The Kato problem on GBG bundles
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Axelsson (Rosén)-Keith-McIntosh framework
(H1) Let Γ be a densely-defined, closed, nilpotent operator on a Hilbert
space H ,
(H2) 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),
(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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The Kato problem on GBG bundles
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Growth restrictions
We say M has exponential volume growth if there exists c ≥ 1, κ, λ ≥ 0
such that
0 < µ(B(x, tr)) ≤ ctκ eλtr µ(B(x, r)) < ∞
(Eloc )
for all t ≥ 1, r > 0 and x ∈ M.
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The Kato problem on GBG bundles
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Growth restrictions
We say M has exponential volume growth if there exists c ≥ 1, κ, λ ≥ 0
such that
0 < µ(B(x, tr)) ≤ ctκ eλtr µ(B(x, r)) < ∞
(Eloc )
for all t ≥ 1, r > 0 and x ∈ M.
For instance, if Ric ≥ ηg, for η ∈ R, then (Eloc ) is satisfied.
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The Kato problem on GBG bundles
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Generalised bounded geometry
We want to set H = L2 (V), but we need to assume more structure in V.
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Generalised bounded geometry
We want to set H = L2 (V), but we need to assume more structure in V.
Definition (Generalised Bounded Geometry)
Suppose there exists ρ > 0, C ≥ 1 such that for each x ∈ M, there exists
a trivialisation ψ : B(x, ρ) × CN → π−1
V (B(x, ρ)) satisfying
C −1 I ≤ h ≤ CI
in the basis ei = ψ(x, êi ) , where êi is the standard basis for CN .
Then, we say that V has generalised bounded geometry or GBG. We call ρ
the GBG radius.
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The Kato problem on GBG bundles
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Further assumptions
(H4) The bundle V has GBG, H = L2 (V), and M grows at most
exponentially.
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The Kato problem on GBG bundles
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Further assumptions
(H4) The bundle V has GBG, H = L2 (V), and M grows at most
exponentially.
(H5) The operators B1 , B2 are matrix valued pointwise multiplication
operators. That is, Bi ∈ L∞ (M, L(V)) by which we mean that
Bi (x) ∈ L(π−1
V (x)) for every x ∈ M and there is a CBi > 0 so that
kBi (x)k∞ ≤ C for almost every x ∈ M.
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The Kato problem on GBG bundles
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Further assumptions
(H4) The bundle V has GBG, H = L2 (V), and M grows at most
exponentially.
(H5) The operators B1 , B2 are matrix valued pointwise multiplication
operators. That is, Bi ∈ L∞ (M, L(V)) by which we mean that
Bi (x) ∈ L(π−1
V (x)) for every x ∈ M and there is a CBi > 0 so that
kBi (x)k∞ ≤ C for almost every x ∈ M.
(H6) The operator Γ is a first order differential operator. That is, there
exists a CΓ > 0 such that whenever η ∈ C∞
c (M), we have that
ηD(Γ) ⊂ D(Γ) and Mη u(x) = [Γ, η(x)] u(x) is a multiplication
operator satisfying
|Mη u(x)| ≤ CΓ |∇η|T∗ M |u(x)|
for all u ∈ D(Γ) and almost all x ∈ M.
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The Kato problem on GBG bundles
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Dyadic cubes
Since we assume exponential growth of M, the work of Christ in [Christ]
and subsequently of Morris in [Morris] allows us to perform a dyadic
decomposition of the manifold below a fixed “scale.”
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The Kato problem on GBG bundles
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Dyadic cubes
Since we assume exponential growth of M, the work of Christ in [Christ]
and subsequently of Morris in [Morris] allows us to perform a dyadic
decomposition of the manifold below a fixed “scale.”
In particular, we are able to choose arbitrarily large J ∈ N so that for every
j ≥ J, Q j is an almost everywhere decomposition of M by open sets.
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The Kato problem on GBG bundles
7 August 2012
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Dyadic cubes
Since we assume exponential growth of M, the work of Christ in [Christ]
and subsequently of Morris in [Morris] allows us to perform a dyadic
decomposition of the manifold below a fixed “scale.”
In particular, we are able to choose arbitrarily large J ∈ N so that for every
j ≥ J, Q j is an almost everywhere decomposition of M by open sets.
b ∈ Q j such
If l > j then for every cube Q ∈ Q l , there is a unique cube Q
b
that Q ⊂ Q.
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The Kato problem on GBG bundles
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Dyadic cubes
Since we assume exponential growth of M, the work of Christ in [Christ]
and subsequently of Morris in [Morris] allows us to perform a dyadic
decomposition of the manifold below a fixed “scale.”
In particular, we are able to choose arbitrarily large J ∈ N so that for every
j ≥ J, Q j is an almost everywhere decomposition of M by open sets.
b ∈ Q j such
If l > j then for every cube Q ∈ Q l , there is a unique cube Q
b
that Q ⊂ Q.
Each such cube has a centre xQ .
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The Kato problem on GBG bundles
7 August 2012
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Dyadic cubes
Since we assume exponential growth of M, the work of Christ in [Christ]
and subsequently of Morris in [Morris] allows us to perform a dyadic
decomposition of the manifold below a fixed “scale.”
In particular, we are able to choose arbitrarily large J ∈ N so that for every
j ≥ J, Q j is an almost everywhere decomposition of M by open sets.
b ∈ Q j such
If l > j then for every cube Q ∈ Q l , there is a unique cube Q
b
that Q ⊂ Q.
Each such cube has a centre xQ .
Each cube Q ∈ Q j also has a diameter of at most C1 δ j , where C1 > 0
and δ ∈ (0, 1) are fixed, uniform quantities.
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The Kato problem on GBG bundles
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GBG coordinates
Since we assume the bundle satisfies GBG, we recall the uniform ρ > 0
from this criterion.
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The Kato problem on GBG bundles
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GBG coordinates
Since we assume the bundle satisfies GBG, we recall the uniform ρ > 0
from this criterion.
Choose J ∈ N such that C1 δ J ≤ ρ5 . We call tS = δ J the scale.
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The Kato problem on GBG bundles
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GBG coordinates
Since we assume the bundle satisfies GBG, we recall the uniform ρ > 0
from this criterion.
Choose J ∈ N such that C1 δ J ≤ ρ5 . We call tS = δ J the scale.
Call the
system of trivialisations
J the GBG
C = ψ : B(xQ , ρ) × CN → π−1
V (B(xQ , ρ)) s.t. Q ∈ Q
coordinates.
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The Kato problem on GBG bundles
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GBG coordinates (cont.)
Call the
n set of a.e. trivialisations
o
J the dyadic GBG
CJ = ϕ̃Q = ψ| Q : Q × CN → π−1
(Q)
s.t.
Q
∈
Q
V
coordinates.
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GBG coordinates (cont.)
Call the
n set of a.e. trivialisations
o
J the dyadic GBG
CJ = ϕ̃Q = ψ| Q : Q × CN → π−1
(Q)
s.t.
Q
∈
Q
V
coordinates.
b ∈ Q J satisfying Q ⊂ Q
b we call the
For any cube Q, the unique cube Q
GBG cube of Q.
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The Kato problem on GBG bundles
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GBG coordinates (cont.)
Call the
n set of a.e. trivialisations
o
J the dyadic GBG
CJ = ϕ̃Q = ψ| Q : Q × CN → π−1
(Q)
s.t.
Q
∈
Q
V
coordinates.
b ∈ Q J satisfying Q ⊂ Q
b we call the
For any cube Q, the unique cube Q
GBG cube of Q.
The GBG coordinate system of Q is then
ψ : B(xQb , ρ) × CN → π−1
b , ρ)).
V (B(xQ
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Machinery for the harmonic analysis
For t ≤ tS , we write Qt = Q j whenever δ j+1 < t ≤ δ j .
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The Kato problem on GBG bundles
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Machinery for the harmonic analysis
For t ≤ tS , we write Qt = Q j whenever δ j+1 < t ≤ δ j .
For j > J and Q ∈ Q j and u = ui ei ∈ L1loc (V) in the GBG
b Define, the cube integral
coordinates associated to Q.
ˆ
ˆ
u=
ui ei
Q
Q
inside B(xQb , ρ).
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The Kato problem on GBG bundles
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Machinery for the harmonic analysis
For t ≤ tS , we write Qt = Q j whenever δ j+1 < t ≤ δ j .
For j > J and Q ∈ Q j and u = ui ei ∈ L1loc (V) in the GBG
b Define, the cube integral
coordinates associated to Q.
ˆ
ˆ
u=
ui ei
Q
Q
inside B(xQb , ρ).
The cube average is then defined as uQ (y) =
and 0 otherwise.
Lashi Bandara (ANU)
The Kato problem on GBG bundles
ffl
Q
u, for y ∈ B(xQb , ρ)
7 August 2012
20 / 30
Machinery for the harmonic analysis
For t ≤ tS , we write Qt = Q j whenever δ j+1 < t ≤ δ j .
For j > J and Q ∈ Q j and u = ui ei ∈ L1loc (V) in the GBG
b Define, the cube integral
coordinates associated to Q.
ˆ
ˆ
u=
ui ei
Q
Q
inside B(xQb , ρ).
The cube average is then defined as uQ (y) =
and 0 otherwise.
ffl
Q
u, for y ∈ B(xQb , ρ)
Let At u(x) = uQ (x) whenever x ∈ Q ∈ Qt .
Lashi Bandara (ANU)
The Kato problem on GBG bundles
7 August 2012
20 / 30
Machinery for the harmonic analysis
For t ≤ tS , we write Qt = Q j whenever δ j+1 < t ≤ δ j .
For j > J and Q ∈ Q j and u = ui ei ∈ L1loc (V) in the GBG
b Define, the cube integral
coordinates associated to Q.
ˆ
ˆ
u=
ui ei
Q
Q
inside B(xQb , ρ).
The cube average is then defined as uQ (y) =
and 0 otherwise.
ffl
Q
u, for y ∈ B(xQb , ρ)
Let At u(x) = uQ (x) whenever x ∈ Q ∈ Qt .
For each w ∈ CN , let γt (x)w = (ΘB
t ω)(x) where ω(x) = w in the
GBG coordinates of each Q.
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The Kato problem on GBG bundles
7 August 2012
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Cancellation assumption
(H7) There exists c > 0 such that for all t ≤ tS and Q ∈ Qt ,
ˆ
ˆ
1
∗
Γu dµ ≤ cµ(Q) 2 kuk and Γ v dµ ≤ cµ(Q) 12 kvk
Q
Q
for all u ∈ D(Γ), v ∈ D(Γ∗ ) satisfying spt u, spt v ⊂ Q.
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The Kato problem on GBG bundles
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Dyadic Poincaré assumption
(H8) There exists CP , CC , c, c̃ > 0 and an operator
Ξ : D(Ξ) ⊂ L2 (V) → L2 (N ), where N is a normed bundle over M
with norm |· |N and D(Π) ∩ R(Π) ⊂ D(Ξ) satisfying for all
u ∈ D(Π) ∩ R(Π),
-1 (Dyadic Poincaré )
ˆ
ˆ
2
κ λcrt
2
|u − uQ | dµ ≤ CP (1 + r e
)(rt)
B
c̃B
2
2
|Ξu|N + |u|
dµ
for all balls B = B(xQ , rt) with r ≥ C1 /δ where Q ∈ Qt with t ≤ tS ,
and
-2 (Coercivity)
2
2
2
kΞukL2 (N ) + kukL2 (V) ≤ CC kΠukL2 (V) .
Lashi Bandara (ANU)
The Kato problem on GBG bundles
7 August 2012
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Kato square root type estimate
Proposition
Suppose M is a smooth, complete Riemannian manifold and V is a
smooth vector bundle over M. If (H1)-(H8) are satisfied, then
q
(i) D(Γ) ∩ D(Γ∗B ) = D(ΠB ) = D( Π2B ), and
q
(ii) kΓuk + kΓB uk ' kΠB uk ' k Π2B uk, for all u ∈ D(ΠB ).
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The Kato problem on GBG bundles
7 August 2012
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Kato square root problem on vector bundles
Theorem (B.-Mc.)
Suppose M grows at most exponentially and satisfies a local
Poincaré inequality on functions. Further, suppose that both V and T∗ M
have GBG, and
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The Kato problem on GBG bundles
7 August 2012
24 / 30
Kato square root problem on vector bundles
Theorem (B.-Mc.)
Suppose M grows at most exponentially and satisfies a local
Poincaré inequality on functions. Further, suppose that both V and T∗ M
have GBG, and
(i) the metric h and ∇ are compatible,
Lashi Bandara (ANU)
The Kato problem on GBG bundles
7 August 2012
24 / 30
Kato square root problem on vector bundles
Theorem (B.-Mc.)
Suppose M grows at most exponentially and satisfies a local
Poincaré inequality on functions. Further, suppose that both V and T∗ M
have GBG, and
(i) the metric h and ∇ are compatible,
(ii) thereexists
C > 0 such
that in each GBG chart we have that
∇ej , ∇dxi , ∂k hij , ∂k gij ≤ C a.e.,
Lashi Bandara (ANU)
The Kato problem on GBG bundles
7 August 2012
24 / 30
Kato square root problem on vector bundles
Theorem (B.-Mc.)
Suppose M grows at most exponentially and satisfies a local
Poincaré inequality on functions. Further, suppose that both V and T∗ M
have GBG, and
(i) the metric h and ∇ are compatible,
(ii) thereexists
C > 0 such
that in each GBG chart we have that
∇ej , ∇dxi , ∂k hij , ∂k gij ≤ C a.e.,
(iii) there exist κ1 , κ2 > 0 such that Re hau, ui ≥ κ1 kuk2 and
Re hASv, Svi ≥ κ2 kvk2H1 for all u ∈ L2 (V) and v ∈ H1 (V),
Lashi Bandara (ANU)
The Kato problem on GBG bundles
7 August 2012
24 / 30
Kato square root problem on vector bundles
Theorem (B.-Mc.)
Suppose M grows at most exponentially and satisfies a local
Poincaré inequality on functions. Further, suppose that both V and T∗ M
have GBG, and
(i) the metric h and ∇ are compatible,
(ii) thereexists
C > 0 such
that in each GBG chart we have that
∇ej , ∇dxi , ∂k hij , ∂k gij ≤ C a.e.,
(iii) there exist κ1 , κ2 > 0 such that Re hau, ui ≥ κ1 kuk2 and
Re hASv, Svi ≥ κ2 kvk2H1 for all u ∈ L2 (V) and v ∈ H1 (V), and
(iv) we have
that D(∆) ⊂ H2 (V), and there exist C 0 > 0 such that
∇ 2 u ≤ C 0 k(I + ∆)uk whenever u ∈ D(∆).
Lashi Bandara (ANU)
The Kato problem on GBG bundles
7 August 2012
24 / 30
Kato square root problem on vector bundles
Theorem (B.-Mc.)
Suppose M grows at most exponentially and satisfies a local
Poincaré inequality on functions. Further, suppose that both V and T∗ M
have GBG, and
(i) the metric h and ∇ are compatible,
(ii) thereexists
C > 0 such
that in each GBG chart we have that
∇ej , ∇dxi , ∂k hij , ∂k gij ≤ C a.e.,
(iii) there exist κ1 , κ2 > 0 such that Re hau, ui ≥ κ1 kuk2 and
Re hASv, Svi ≥ κ2 kvk2H1 for all u ∈ L2 (V) and v ∈ H1 (V), and
(iv) we have
that D(∆) ⊂ H2 (V), and there exist C 0 > 0 such that
∇ 2 u ≤ C 0 k(I + ∆)uk whenever u ∈ D(∆).
√
√
Then, D( LA ) = D(∇) = H1 (V) with k LA uk ' k∇uk + kuk = kukH1
for all u ∈ H1 (V).
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The Kato problem on GBG bundles
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Set H = L2 (V) ⊕ (L2 (V) ⊕ L2 (T∗ M ⊗ V)).
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The Kato problem on GBG bundles
7 August 2012
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Set H = L2 (V) ⊕ (L2 (V) ⊕ L2 (T∗ M ⊗ V)).
Define
0 0
a 0
0 0
Γ=
, B1 =
, and B2 =
.
S 0
0 0
0 A
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The Kato problem on GBG bundles
7 August 2012
25 / 30
Set H = L2 (V) ⊕ (L2 (V) ⊕ L2 (T∗ M ⊗ V)).
Define
0 0
a 0
0 0
Γ=
, B1 =
, and B2 =
.
S 0
0 0
0 A
Then,
0 S∗
Γ =
0 0
∗
Lashi Bandara (ANU)
and
Π2B
LA 0
=
0 ∗
The Kato problem on GBG bundles
7 August 2012
25 / 30
Kato square root problem for tensors
Theorem (B.-Mc.)
Let M be a smooth, complete Riemannian manifold with |Ric| ≤ C and
0
inj(M
that
2 ) ≥ κ 0> 0. Suppose that there exist C > 0 such
∇ u ≤ C k(I + ∆)uk whenever u ∈ D(∆) ⊂ H2 (T (p,q) M). Then,
√
√
D( LA ) = D(∇) = H1 (T (p,q) M) and k LA uk ' k∇uk + kuk = kukH1
for all u ∈ H1 (T (p,q) M).
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The Kato problem on GBG bundles
7 August 2012
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Ricci, injectivity bounds and GBG
Proposition
Suppose there is a κ, η > 0 such that inj(M) ≥ κ and |Ric| ≤ η.
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The Kato problem on GBG bundles
7 August 2012
27 / 30
Ricci, injectivity bounds and GBG
Proposition
Suppose there is a κ, η > 0 such that inj(M) ≥ κ and |Ric| ≤ η. Then for
A > 1 and α ∈ (0, 1), there exists rH (n, A, α, κ, η) > 0 such that for each
x ∈ M, there is a coordinate system corresponding to B(x, rH ) satisfying:
Lashi Bandara (ANU)
The Kato problem on GBG bundles
7 August 2012
27 / 30
Ricci, injectivity bounds and GBG
Proposition
Suppose there is a κ, η > 0 such that inj(M) ≥ κ and |Ric| ≤ η. Then for
A > 1 and α ∈ (0, 1), there exists rH (n, A, α, κ, η) > 0 such that for each
x ∈ M, there is a coordinate system corresponding to B(x, rH ) satisfying:
(i) A−1 δij ≤ gij ≤ Aδij as bilinear forms and,
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The Kato problem on GBG bundles
7 August 2012
27 / 30
Ricci, injectivity bounds and GBG
Proposition
Suppose there is a κ, η > 0 such that inj(M) ≥ κ and |Ric| ≤ η. Then for
A > 1 and α ∈ (0, 1), there exists rH (n, A, α, κ, η) > 0 such that for each
x ∈ M, there is a coordinate system corresponding to B(x, rH ) satisfying:
(i) A−1 δij ≤ gij ≤ Aδij as bilinear forms and,
X
(ii)
rH sup |∂l gij (y)|
y∈B(x,rH )
l
+
X
1+α
rH
l
Lashi Bandara (ANU)
|∂l gij (z) − ∂l gij (y)|
≤ A − 1.
d(y, z)α
y6=z∈B(x,rH )
sup
The Kato problem on GBG bundles
7 August 2012
27 / 30
Ricci, injectivity bounds and GBG
Proposition
Suppose there is a κ, η > 0 such that inj(M) ≥ κ and |Ric| ≤ η. Then for
A > 1 and α ∈ (0, 1), there exists rH (n, A, α, κ, η) > 0 such that for each
x ∈ M, there is a coordinate system corresponding to B(x, rH ) satisfying:
(i) A−1 δij ≤ gij ≤ Aδij as bilinear forms and,
X
(ii)
rH sup |∂l gij (y)|
y∈B(x,rH )
l
+
X
1+α
rH
l
|∂l gij (z) − ∂l gij (y)|
≤ A − 1.
d(y, z)α
y6=z∈B(x,rH )
sup
See the observation following Theorem 1.2 in [Hebey].
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The Kato problem on GBG bundles
7 August 2012
27 / 30
The proposition guarantees GBG coordinates for V = T (p,q) M.
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The Kato problem on GBG bundles
7 August 2012
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The proposition guarantees GBG coordinates for V = T (p,q) M.
The proposition gives ∇ej , ∇dxi , ∂k hij , ∂k gij ≤ C in each
GBG coordinate chart ej for T (p,q) M.
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The Kato problem on GBG bundles
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The proposition guarantees GBG coordinates for V = T (p,q) M.
The proposition gives ∇ej , ∇dxi , ∂k hij , ∂k gij ≤ C in each
GBG coordinate chart ej for T (p,q) M.
The Ricci bounds guarantee exponential volume growth and the local
Poincaré inequality.
Lashi Bandara (ANU)
The Kato problem on GBG bundles
7 August 2012
28 / 30
The proposition guarantees GBG coordinates for V = T (p,q) M.
The proposition gives ∇ej , ∇dxi , ∂k hij , ∂k gij ≤ C in each
GBG coordinate chart ej for T (p,q) M.
The Ricci bounds guarantee exponential volume growth and the local
Poincaré inequality.
By invoking Theorem 5 (on vector bundles), we obtain Theorem 6 (finite
rank tensors).
Lashi Bandara (ANU)
The Kato problem on GBG bundles
7 August 2012
28 / 30
The proposition guarantees GBG coordinates for V = T (p,q) M.
The proposition gives ∇ej , ∇dxi , ∂k hij , ∂k gij ≤ C in each
GBG coordinate chart ej for T (p,q) M.
The Ricci bounds guarantee exponential volume growth and the local
Poincaré inequality.
By invoking Theorem 5 (on vector bundles), we obtain Theorem 6 (finite
rank tensors).
Theorem 1 follows from Theorem 6 since ∇ 2 u . k(I + ∆)uk is a
consequence of the Bochner-Lichnerowicz-Weitzenböck identity, Ricci
curvature bounds and uniform lower bounds on injectivity radius.
Lashi Bandara (ANU)
The Kato problem on GBG bundles
7 August 2012
28 / 30
The proposition guarantees GBG coordinates for V = T (p,q) M.
The proposition gives ∇ej , ∇dxi , ∂k hij , ∂k gij ≤ C in each
GBG coordinate chart ej for T (p,q) M.
The Ricci bounds guarantee exponential volume growth and the local
Poincaré inequality.
By invoking Theorem 5 (on vector bundles), we obtain Theorem 6 (finite
rank tensors).
Theorem 1 follows from Theorem 6 since ∇ 2 u . k(I + ∆)uk is a
consequence of the Bochner-Lichnerowicz-Weitzenböck identity, Ricci
curvature bounds and uniform lower bounds on injectivity radius.
See Proposition 3.3 in [Hebey].
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The Kato problem on GBG bundles
7 August 2012
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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.
[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.
Lashi Bandara (ANU)
The Kato problem on GBG bundles
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References II
[Hebey] Emmanuel Hebey, Nonlinear analysis on manifolds: Sobolev
spaces and inequalities, Courant Lecture Notes in Mathematics, vol. 5,
New York University Courant Institute of Mathematical Sciences, New
York, 1999.
[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)
, On representing closed accretive sesquilinear forms as
[Mc82]
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. MR 670278 (84k:47030)
[Morris] Andrew J. Morris, The Kato Square Root Problem on
Submanifolds, ArXiv e-prints:1103.5089v1 (2011).
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