Definition and basic properties of heat
kernels I, An introduction
Zhiqin Lu,Department of Mathematics,UC Irvine, Irvine CA 92697
April 23, 2010
In this lecture, we will answer the following questions:
1 What is the heat kernel?
2 Why it is so difficult to understand the heat kernel?
3 The basic properties of the heat kernel.
In this lecture, we will answer the following questions:
1 What is the heat kernel?
2 Why it is so difficult to understand the heat kernel?
3 The basic properties of the heat kernel.
In this lecture, we will answer the following questions:
1 What is the heat kernel?
2 Why it is so difficult to understand the heat kernel?
3 The basic properties of the heat kernel.
In this lecture, we will answer the following questions:
1 What is the heat kernel?
2 Why it is so difficult to understand the heat kernel?
3 The basic properties of the heat kernel.
In this lecture, we will answer the following questions:
1 What is the heat kernel?
2 Why it is so difficult to understand the heat kernel?
3 The basic properties of the heat kernel.
S-T. Yau, R. SchoenDifferential GeometryAcademic Press, 2006
N. Berline, E. Getzler, M. VergneHeat kernels and Dirac operatorsSpringer 1992
E.B. Davies,One-parameter semi-groupsAcademic Press 1980
E.B. Davies,Heat kernels and spectral theoryCambridge University Press, 1990
The basic settings:Let M be a Riemannian manifold with the Riemannian metric
ds2 = gijdxi dxj.
The Laplace operator is defined as
∆ =1√g
∑ ∂
∂xi
(gij√g∂
∂xj
),
where (gij) = (gij)−1, g = det(gij).
The basic settings:Let M be a Riemannian manifold with the Riemannian metric
ds2 = gijdxi dxj.
The Laplace operator is defined as
∆ =1√g
∑ ∂
∂xi
(gij√g∂
∂xj
),
where (gij) = (gij)−1, g = det(gij).
At least three questions have to be addressed:
1 The Laplace operator as a densely defined self-adjointoperator;
2 The semi-groupof operators;
3 The existence of heat kernel.
At least three questions have to be addressed:
1 The Laplace operator as a densely defined self-adjointoperator;
2 The semi-groupof operators;
3 The existence of heat kernel.
At least three questions have to be addressed:
1 The Laplace operator as a densely defined self-adjointoperator;
2 The semi-groupof operators;
3 The existence of heat kernel.
Finite dimensional case
Let A be a positive definite matrix. Then A can bediagonalized. That is, up to a similar transformation by anorthogonal matrix, A is similar to the matrixλ1
. . .
λn
Let Ei be the eigenspace with respect to the eigenvalue λi andlet Pi : Rn → Ei be the orthogonal projection. Then we canwrite
A =n∑i=1
λiPi
Infinite dimensional case, an example
Let B be the space of L2 periodic functions on [−π, π]. Letf ∈ B. Then we have the Fourier expansion
f(x) ∼ a0
2+∞∑k=1
(ak cos kx+ bk sin kx).
If f is smooth, then the above expansion is convergent to thefunction.
The Laplace operator on one dimensional is
∆ =∂2
∂x2
If f is smooth, then
∆f =∞∑k=1
(−k2ak cos kx− k2bk sin kx)
Define
Pkf = ak cos kx
Qkf = bk sin kx
Then we can write
∆ = 0 · P0 −∞∑k=1
(k2Pk + k2Qk)
The Laplace operator on one dimensional is
∆ =∂2
∂x2
If f is smooth, then
∆f =∞∑k=1
(−k2ak cos kx− k2bk sin kx)
Define
Pkf = ak cos kx
Qkf = bk sin kx
Then we can write
∆ = 0 · P0 −∞∑k=1
(k2Pk + k2Qk)
The Laplace operator on one dimensional is
∆ =∂2
∂x2
If f is smooth, then
∆f =∞∑k=1
(−k2ak cos kx− k2bk sin kx)
Define
Pkf = ak cos kx
Qkf = bk sin kx
Then we can write
∆ = 0 · P0 −∞∑k=1
(k2Pk + k2Qk)
The Laplace operator on one dimensional is
∆ =∂2
∂x2
If f is smooth, then
∆f =∞∑k=1
(−k2ak cos kx− k2bk sin kx)
Define
Pkf = ak cos kx
Qkf = bk sin kx
Then we can write
∆ = 0 · P0 −∞∑k=1
(k2Pk + k2Qk)
In general, we can write
∆ = −∫ ∞
0
λ dE,
where E is the so-called spectral measure.
Let L2(M) be the space of L2 functions. The space C∞0 (M)is the space of smooth functions on M with compact support.
In general, we can write
∆ = −∫ ∞
0
λ dE,
where E is the so-called spectral measure.Let L2(M) be the space of L2 functions. The space C∞0 (M)is the space of smooth functions on M with compact support.
The Laplacian ∆ is defined on the space of smooth functionswith compact support.
. It is symmetric.That is∫∆f · g =
∫f ·∆g
However, C∞0 (M) is not a Banach space. We would like topick L2(M), the space of L2 functions.
Question
Can we extend the Laplacian onto L2(M)?
The Laplacian ∆ is defined on the space of smooth functionswith compact support.. It is symmetric.
That is∫∆f · g =
∫f ·∆g
However, C∞0 (M) is not a Banach space. We would like topick L2(M), the space of L2 functions.
Question
Can we extend the Laplacian onto L2(M)?
The Laplacian ∆ is defined on the space of smooth functionswith compact support.. It is symmetric.That is∫
∆f · g =
∫f ·∆g
However, C∞0 (M) is not a Banach space. We would like topick L2(M), the space of L2 functions.
Question
Can we extend the Laplacian onto L2(M)?
The Laplacian ∆ is defined on the space of smooth functionswith compact support.. It is symmetric.That is∫
∆f · g =
∫f ·∆g
However, C∞0 (M) is not a Banach space.
We would like topick L2(M), the space of L2 functions.
Question
Can we extend the Laplacian onto L2(M)?
The Laplacian ∆ is defined on the space of smooth functionswith compact support.. It is symmetric.That is∫
∆f · g =
∫f ·∆g
However, C∞0 (M) is not a Banach space. We would like topick L2(M), the space of L2 functions.
Question
Can we extend the Laplacian onto L2(M)?
The Laplacian ∆ is defined on the space of smooth functionswith compact support.. It is symmetric.That is∫
∆f · g =
∫f ·∆g
However, C∞0 (M) is not a Banach space. We would like topick L2(M), the space of L2 functions.
Question
Can we extend the Laplacian onto L2(M)?
Answer: No!
1 The Laplacian is an unbounded operator;
2 Like most differential operator, it is a closed graphoperator;
3 By the Closed Graph Theorem, if ∆ can be extended,then it must be a bounded operator, a contradiction.
Answer: No!
1 The Laplacian is an unbounded operator;
2 Like most differential operator, it is a closed graphoperator;
3 By the Closed Graph Theorem, if ∆ can be extended,then it must be a bounded operator, a contradiction.
Answer: No!
1 The Laplacian is an unbounded operator;
2 Like most differential operator, it is a closed graphoperator;
3 By the Closed Graph Theorem, if ∆ can be extended,then it must be a bounded operator, a contradiction.
Answer: No!
1 The Laplacian is an unbounded operator;
2 Like most differential operator, it is a closed graphoperator;
3 By the Closed Graph Theorem, if ∆ can be extended,then it must be a bounded operator, a contradiction.
Self-adjoint densely defined operatorDefinitionLet H be a Hilbert space, Given a densely defined linearoperator A on H, its adjoint A∗ is defined as follows:
1 The domain of A∗ consists of vectors x in H such that
y 7→ 〈x,Ay〉
is a bounded linear functional, where y ∈ Dom(∆);
2 If x is in the domain of A∗, there is a unique vector z inH such that
〈x,Ay〉 = 〈z, y〉
for any y ∈ Dom(∆). This vector z is defined to be A∗x.It can be shown that the dependence of z on x is linear.
If A∗ = A (which implies that Dom (A∗) = Dom (A)), then Ais called self-adjoint.
Define H10 (M) be the Sobolev space of the completion of the
vector space Λp(M) under the norm
||η||1 =
√∫M
|η|2dVM +
√∫M
|∇η|2 dVM .
We define the quadratic form Q on H10 (M) by
Q(ω, η) =
∫M
(〈dω, dη〉) dVM
for any ω, η ∈ H10 (M). Then we define
Dom(∆) =φ ∈ H1
0 (M) |∀ψ ∈ C∞(M),∃f ∈ L2(M), s.t. Q(φ, ψ) = −(f, ψ)
.
Proof.We first observe that for any ψ ∈ C∞(M) and φ ∈ H1
0 (M),we have
Q(φ, ψ) = −(φ,∆ψ).
Using this result, the proof goes as follows: for anyφ ∈ Dom(∆), the functionalψ 7→ (∆ψ, φ) = −Q(ψ, φ) = (f, ψ) is a bounded functional.Thus φ ∈ Dom(∆∗). On the other hand, if φ ∈ Dom(∆∗),then the functional ψ 7→ (∆ψ, φ) is bounded. By the Rieszrepresentation theorem, there is a unique f ∈ L2(M) suchthat (∆ψ, φ) = (f, ψ). Thus we must haveQ(φ, ψ) = −(∆ψ, φ) = −(f, ψ).
From the above discussion, we have proved that
TheoremThe Laplace operator has a self-adjoint extension.
DefinitionA one-parameter semigroup of operators on a complex Banachspace B is a family Tt of bounded linear operators, whereTt : B → B parameterized by real numbers t ≥ 0 and satisfiesthe following relations:
1 T0 = 1;
2 If 0 ≤ s, t <∞, then
TsTt = Ts+t.
3 The mapt, f → Ttf
from [0,∞)× B to B is jointly continuous.
The family of bounded operators
e∆t
forms a semi-group.
Even though e−∆t are all bounded operator, the kernel doesn’texist in general.
Definition of operator kernel
Let A be an operator on L2(M). If there is a function A(x, y)such that
Af(x) =
∫A(x, y)f(y)dy
for all functions f , then we call A(x, y) is the kernel of theoperator.
By the above definition, the kernel of an operator doesn’t existin general. For example, let B be a Banach space, and let Ibe the identity map. Then the kernel of I doesn’t exist.
The family of bounded operators
e∆t
forms a semi-group.Even though e−∆t are all bounded operator, the kernel doesn’texist in general.
Definition of operator kernel
Let A be an operator on L2(M). If there is a function A(x, y)such that
Af(x) =
∫A(x, y)f(y)dy
for all functions f , then we call A(x, y) is the kernel of theoperator.
By the above definition, the kernel of an operator doesn’t existin general. For example, let B be a Banach space, and let Ibe the identity map. Then the kernel of I doesn’t exist.
The family of bounded operators
e∆t
forms a semi-group.Even though e−∆t are all bounded operator, the kernel doesn’texist in general.
Definition of operator kernel
Let A be an operator on L2(M). If there is a function A(x, y)such that
Af(x) =
∫A(x, y)f(y)dy
for all functions f , then we call A(x, y) is the kernel of theoperator.
By the above definition, the kernel of an operator doesn’t existin general. For example, let B be a Banach space, and let Ibe the identity map. Then the kernel of I doesn’t exist.
The family of bounded operators
e∆t
forms a semi-group.Even though e−∆t are all bounded operator, the kernel doesn’texist in general.
Definition of operator kernel
Let A be an operator on L2(M). If there is a function A(x, y)such that
Af(x) =
∫A(x, y)f(y)dy
for all functions f , then we call A(x, y) is the kernel of theoperator.
By the above definition, the kernel of an operator doesn’t existin general. For example, let B be a Banach space, and let Ibe the identity map. Then the kernel of I doesn’t exist.
Definition of heat kernel
Let M be a complete Riemannian manifold. Then there existsheat kernel H(x, y, t) ∈ C∞(M ×M × R+) such that
(e∆tf)(x) =
∫M
H(x, y, t)f(y)dy
for any L2 function f . The heat kernel satisfies
1 H(x, y, t) = H(y, x, t)
2 limt→0+
H(x, y, t) = δx(y)
3 (∆− ∂∂t
)H = 0
4 H(x, y, t) =∫MH(x, z, t− s)H(x, y, s)dz, t > s > 0
An example
Let M be a compact manifold. Let fj(x) be an orthonormalbasis of eigenfunctions. Let
∆fj = −λjfj
Then we an write the heat kernel as a series
H(x, y, t) =∞∑j=0
e−λjtfj(x)fj(y)
Let Ω be a bounded domain of Rn with smooth boundary.
Consider the equation∂f
∂t= ∆f
f(0, x) = φ(x),
where φ is the initial value function.Then
f(t, x) =
∫M
H(x, y, t)φ(y)dy =∞∑j=1
e−λjtajfj(x),
where aj =∫Mφ(x)fj(x)dx.
Conclusion: If a1 6= 0, then eλ1tf(t, x)→ a1f1(x) 6= 0
Let Ω be a bounded domain of Rn with smooth boundary.Consider the equation
∂f
∂t= ∆f
f(0, x) = φ(x),
where φ is the initial value function.
Then
f(t, x) =
∫M
H(x, y, t)φ(y)dy =∞∑j=1
e−λjtajfj(x),
where aj =∫Mφ(x)fj(x)dx.
Conclusion: If a1 6= 0, then eλ1tf(t, x)→ a1f1(x) 6= 0
Let Ω be a bounded domain of Rn with smooth boundary.Consider the equation
∂f
∂t= ∆f
f(0, x) = φ(x),
where φ is the initial value function.Then
f(t, x) =
∫M
H(x, y, t)φ(y)dy =∞∑j=1
e−λjtajfj(x),
where aj =∫Mφ(x)fj(x)dx.
Conclusion: If a1 6= 0, then eλ1tf(t, x)→ a1f1(x) 6= 0
Let Ω be a bounded domain of Rn with smooth boundary.Consider the equation
∂f
∂t= ∆f
f(0, x) = φ(x),
where φ is the initial value function.Then
f(t, x) =
∫M
H(x, y, t)φ(y)dy =∞∑j=1
e−λjtajfj(x),
where aj =∫Mφ(x)fj(x)dx.
Conclusion: If a1 6= 0, then eλ1tf(t, x)→ a1f1(x) 6= 0
A theorem of Brascamp-Lieb
Theorem [Brascamp-Lieb]
Let Ω be a bounded convex domain with smooth boundary inRn. Let f1(x) > 0 be the first eigenfunction. Then log f1(x) isa concave function.
Ideas of the proof
1 Construct a log-concave function φ(x) with a1 6= 0;
2 Prove that along the flow, the log-concavity is preserved.
A theorem of Brascamp-Lieb
Theorem [Brascamp-Lieb]
Let Ω be a bounded convex domain with smooth boundary inRn. Let f1(x) > 0 be the first eigenfunction. Then log f1(x) isa concave function.
Ideas of the proof
1 Construct a log-concave function φ(x) with a1 6= 0;
2 Prove that along the flow, the log-concavity is preserved.
A theorem of Brascamp-Lieb
Theorem [Brascamp-Lieb]
Let Ω be a bounded convex domain with smooth boundary inRn. Let f1(x) > 0 be the first eigenfunction. Then log f1(x) isa concave function.
Ideas of the proof
1 Construct a log-concave function φ(x) with a1 6= 0;
2 Prove that along the flow, the log-concavity is preserved.
Construction of the log-concavity
function
1 Since Ω is a convex domain, the local defining functionϕ(x) is log-concave neat the boundary.
2 Let φ = ϕe−C|x|2. Then for sufficient large C, φ is
log-concave.
Construction of the log-concavity
function
1 Since Ω is a convex domain, the local defining functionϕ(x) is log-concave neat the boundary.
2 Let φ = ϕe−C|x|2. Then for sufficient large C, φ is
log-concave.
Maximum Principle
Let g = log f . Then
∂g
∂t= ∆g + |∇g|2.
For any i, we have
∂(−gii)∂t
= ∆(−gii) + 2gkiigk + 2∑k
g2ki