Heat kernel estimates for general boundary problems

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Heat kernel estimates for general boundary problems

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Li, L and Strohmaier, A orcid.org/0000-0002-8446-3840 (2016) Heat kernel estimates for

general boundary problems. Journal of Spectral Theory, 6 (4). pp. 903-919. ISSN

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LIANGPAN LI AND ALEXANDER STROHMAIER

Dedicated to the memory of Yuri Safarov (1958–2015)

Abstract. We show that not feeling the boundary estimates for heat kernels hold for any non-negative self-adjoint extension of the Laplace operator acting on vector-valued compactly supported functions on a domain in Rd

. They are therefore valid for any choice of boundary condition and we show that the implied constants can be chosen independent of the self-adjoint extension. The method of proof is very general and is based on finite propagation speed estimates and explicit Fourier Tauberian theorems obtained by Y. Safarov.

1.

Introduction

Let

U

be an open set in

R

d

_(d

_≥

_{2) and consider the Dirichlet Laplace operator}

∆

D

on

L

2

(U

). Then the fundamental solution

K(t), t

≥

0 of the heat equation with

Dirichlet boundary conditions can be constructed via spectral calculus as

K

D

(t) = exp(

−

t∆

D

).

The integral kernel

K

D

(x, y

;

t) of

K

D

(t) defined by

(K

D

(t)f

)(x) =

Z

U

K

D

(x, y;

t)f

(y)dy

is a positive smooth function on

U

_×

U

_×

R

+

_{. It describes the propagation of heat}

from the point

x

to the point

y

in time

t. In case

U

=

R

d

_{the heat kernel is explicitly}

given by

K

0

(x, y;

t) = (4πt)

−d/2

exp(

−

|

x

₋

y

_|

2

4t

).

On physical grounds one expects that for small times the heat kernel is dominated

by local contributions that do not involve the boundary of

U

. This is essentially the

principle of not feeling the boundary by Kac ([19]). Both qualitative and explicit

quantitative versions of this principle have been obtained by some authors ([2,

4,

18])

by exploiting the probabilistic interpretation of the heat kernel ([27,

28,

31]). The

2010 Mathematics Subject Classification. 35K08.

Key words and phrases. Heat kernel, vector-valued Laplacian, finite propagation speed, spectral function, Neumann boundary problem.

best estimate for the Dirichlet Laplacian we are aware of was obtained in [4] and

reads

(1.1)

1

_≥

K

D

(x, y;

t)

K

0(x, y;

t)

≥

1

−

e

−δ2_/t

d

X

j=1

2

j

(j

₋

1)!

δ

2

t

j

−1

.

Here

δ

is the distance of the convex hull of

_{

x, y

_}

to the boundary

∂U

of

U

. It is

also known (see e.g. [4,

17]) that

lim

t→0+

t

log

1

₋

K

D

(x, x;

t)

K0(x, x;

t)

=

₋

ρ

2

(x),

where

ρ(x) is the distance of

x

to

∂U

. These estimates show that as

t

goes to 0

the error in approximating the heat kernel by

K0(x, y;

t) is exponentially small with

decay rate determined by the distance to the boundary.

Explicit estimates like these are important in spectral geometry and mathematical

physics. For example the meromorphic extension of the local spectral zeta function is

usually based on the expansion of the heat kernel ([16]). The above estimates directly

lead to bounds on the local spectral zeta functions or other spectral invariants ([15]).

A particular example of such a local spectral function is the Casimir energy density

that plays a distinguished role in physics. For these applications it is important to

allow for boundary conditions other than Dirichlet. For example Casimir interaction

between two conducting obstacles is described by the Casimir energy density of the

photon field. This is obtained from the Laplace operator acting on one forms with

relative boundary conditions. To be able to deal with Laplace operators of such type

one needs to consider self-adjoint extensions of the vector-valued Laplace operator

on domains that are not simply sums of Laplace operator on functions. In order to

illustrate this let us discuss briefly the example of the propagation of electromagnetic

waves, or in a quantum field theoretic description the propagation of a photon. To

keep things simple assume that

U

is either

R

3

_\

_{K, where}

_K

_⊂

_R

3

_{is a compact}

subset with smooth boundary, or a bounded domain with smooth boundary. If the

boundary is modelled as a perfect conductor then separation of variables in the

wave equation results in the Laplace operator acting on

C

3

_{-valued functions and the}

following boundary conditions for the electromagnetic vector potential

A

(x) of the

form

n

(x)

_×

A

(x) = 0,

_∇

A

(x) = 0

for all

x

_∈

∂U

. Here

n

(x) is the outward pointing unit-vector-field on the boundary

∂U

of

U

. These boundary conditions define a self-adjoint extension of the Laplace

operator acting on

C

3

_{-valued smooth compactly supported functions. This}

The aim of this note is to show that explicit not feeling the boundary estimates

can be obtained for any self-adjoint extension of the Laplace operator acting on

vector-valued functions on a domain. They can be derived from a combination of

finite propagation speed estimates and explicit Fourier Tauberian theorems that

were found by Safarov in [29]. The idea of using finite propagation speed estimates

in this context is not new and is already present in the classical paper [10]. It has

since been used by many authors to derive heat kernel bounds on manifolds (see e.g.

[11,

14,

23,

26,

30]). The combination with the estimates of the spectral function

obtained via Fourier Tauberian arguments gives bounds that in some regimes are

better than the known estimates for the Dirichlet heat kernel. The implied constants

are independent of the boundary conditions.

To describe the main result let us assume that, as before,

U

is an open set in

R

d

_{, d}

_≥

_{2 and denote by}

_{ρ(x) the distance from}

_x

_∈

_U

_{to the boundary of}

_U.

Let

N

be a positive integer. Consider in the Hilbert space

L

2

_(U

_;

_C

N

_{) an arbitrary}

non-negative self-adjoint extension ∆

U

of the Laplacian

−

∂

2

∂x

2 1

+

_{· · ·}

+

∂

2

∂x

2

d

:

C

_c∞

(U;

C

N

₎

_→

_C

∞

c

(U

;

C

N

)

acting component-wise. The heat kernel for ∆

U

, denoted by

K

_U

(x, y;

t) =







K

_U(11)

(x, y;

t)

_{· · ·}

K

_U(1N)

(x, y;

t)

...

. ..

...

K

_U(N1)

(x, y

;

t)

_{· · ·}

K

_U(N N)

(x, y;

t)







,

is the integral kernel of

e

−t∆U

_{, t >}

0 defined by the functional calculus of

self-adjoint operators. When

U

=

R

d

_{the counterpart for ∆}

_U

_and

_K

_U

_{(x, y;}

_{t) is denoted}

respectively by ∆0

and

K

0(x, y;

t). Of course,

K

0

(x, y;

t) = (4πt)

−d/2

exp(

−

|

x

₋

y

_|

2

4t

)

1

.

Theorem 1.1.

There exist two positive constants

C

1

, C

2

depending only

d

such that

if

t

_≤

(ρ(x)+₈ρ(y))2

then

k

K

U

(x, y;

t)

−

K

0(x, y;

t)

k ≤

C1ρ(x, y)

−d

+

C2

·

exp

−

(ρ(x) +

ρ(y))

2

4t

t

2⌈d+12 ⌉− 1 2

.

Here

ρ(x, y) = min(ρ(x), ρ(y))

.

The constants

C

1

, C

2

can be explicitly given, but we refer the reader to the relevant

section of this paper for the full description. As a corollary, we are able to answer

a question raised in [21] about an upper estimate for the Neumann heat kernel.

2.

Vector-valued Laplacians

Throughout we fix some notations: Let

m

_∈

N

_with

_{m >}

d

(1 + ∆

V

)

−m

. If

N

= 1 we also write

G

(Vm)

for

G

(m)

V

. By (local) elliptic regularity

G

(_Vm)

is continuous on the open set

U

_×

U

. For any

R >

0 define

(2.1)

J

m

(R;

t) = inf

ψ∈AR

J

m

(ψ;

t) (R >

0),

where

A

R

is the set of real-valued functions

ψ

in

C

2m

(

R

) such that Supp(1

−

ψ)

⊂

(

₋

R, R), and

(2.2)

J

m

(ψ;

t) =

Z

R

(1

₋

d

2

ds

2

)

m

_ψ(s)e

−s42t

ds.

Any matrix of size

N

_×

N

can be naturally regarded as a linear operator on the

Hilbert space

C

N

_{, so we let}

_{k · k}

_{denote its operator norm.}

2.1.

Finite propagation speed.

Before we start let us make some notational

remarks.

Let

x, y

_∈

U

,

v, w

_∈

C

N

_.

_{We denote}

_δ

_x(v)

₌

_δ

_x

_⊗

_v

_{, where}

_δ

_x

_is

the Dirac delta distribution at

x. Strictly speaking,

δ

x

⊗

v

is not in the domain

of the self-adjoint operators

e

−t∆V

_{and cos(s}

√

_∆

V

). We understand however

ex-pressions such as cos(s

√

∆

V

)(δ

x

⊗

v) as distributions (in

x) with values in the

Hilbert space

L

2

₍

_U

_;

_C

N

_{). Pairing with the test function}

_ϕ

_∈

_C

∞

c

(U) is defined

as cos(s

√

∆

V

)(ϕ

⊗

v). As usual, the expression cos(s

√

∆

V

)(δ

x

) is then understood

as a distribution with values in

L

2

(

_U

;

C

N

_⊗

₍

C

N

₎

∗

_{) =}

_L

2

₍

_U

_{; Mat(N,}

C

_{)) With this}

notation the distributional integral kernel

k

_{∈ D}

′

(U

_×

U

; Mat(N,

C

_{)) of an operator}

K

is

k

(x, y) =

_h

δ

x

, Kδ

y

i

. In particular, expressions of the form

h

δ

(xv)

, Kδ

y(w)

i

are

bi-distributions and the pairing with test functions

ϕ1

_⊗

ϕ2

_∈

C

_c∞

(U

_×

U

) is given

by

_h

ϕ

1

⊗

v, Kϕ

2

⊗

w

i

=

h

v,

k

w

i

CN

(ϕ

₁

⊗

ϕ

2).

Alternatively, one can also understand cos(s

√

∆

V

)(δ

x

⊗

v) as the distributional

limit of a sequence cos(s

√

∆

V

)(ϕ

n

⊗

v

), where

ϕ

n

is a

δ-family centered at

x. Note

that cos(s

√

∆

V

) is formally self-adjoint, and a continuous map from

C

c∞

(U

;

C

N

) to

C

∞

_(U;

C

N

_{). This follows from (local) elliptic regularity because ∆}

m V

cos(s

√

∆

V

) =

cos(s

√

∆

V

)∆

mV

is a continuous map from

C

0∞

(

U

)

→

L

2

(

U

) for any

m

∈

N

. It

therefore extends by duality to a continuous map from

_E

′

_(U

_;

C

N

_{) to}

_D

′

_(U

_;

C

N

_{). As}

usual, here

_D

′

_(U

_;

C

N

_{) denotes the space of distributions with values in}

C

N

_{, and}

E

′

_(U;

C

N

_{) denotes the subspace of distributions of compact support.}

Theorem 2.1.

The following pointwise estimate holds for the heat kernel:

k

K

U

(x, y;

t)

−

K

0(x, y;

t)

k

≤

k

G

(_Um)

(x, x)

_kk

G

_U(m)

(y, y)

_k

12

₊

Γ(m

−

d

2

)

(4π)

d2

(m

−

1)!

_J

_m

_{(ρ(x) +}

_ρ(y);

_t)

Proof.

Let

ψ

_∈

A

R

where

R

=

ρ(x) +

ρ(y). It is well-known (see e.g. [34]) that

e

−t∆V

₌

1

2

√

πt

Z

R

cos(s

p

∆

V

)e

− s2

4t

ds

=

1

2

√

πt

Z

R

cos(s

p

∆

V

)(1

−

ψ(s))e

− s2

4t

ds

+

1

2

√

πt

Z

R

(1 + ∆

V

)

−m

cos(s

p

∆

V

)(1

−

d

2

ds

2

)

m

_(ψ(s)e

−s2

4t

)ds.

(2.3)

Finite propagation speed for the wave equation implies that if

_|

s

1

|

< ρ(x) then

cos(s

1

√

∆0)δ

x(v)

has compact support in

U

and agrees with cos(s

1

√

∆

U

)δ

x(v)

. Note

that any

s

_∈

R

_with

_|

_s

_|

_{< ρ(x) +}

_{ρ(y) can be written as}

_s

₌

_s

₁

₊

_s

₂

_with

_|

_s

₁

_|

_{< ρ(x),}

|

s

2

|

< ρ(y),

s

1

s

2

≥

0. With this decomposition available and by considering

cos(s

p

∆

V

) = 2 cos(s

1

p

∆

V

) cos(s

2

p

∆

V

)

−

cos((s

1

−

s

2)

p

∆

V

) cos(0

p

∆

V

)

as well as

_|

s

1

−

s

2

|

<

max

{

ρ(x), ρ(y)

}

, 0

<

min

{

ρ(x), ρ(y)

}

, one obtains

h

δ

_x(v)

,

cos(s

p

∆

U

)δ

y(w)

i − h

δ

(xv)

,

cos(s

p

∆0)δ

_y(w)

_i

= 0

for any

s

_∈

R

_with

_|

_s

_|

_{< ρ(x) +}

_{ρ(y). As supp(1}

₋

_ψ)

_⊂

₍

₋

_ρ(x)

₋

_{ρ(y), ρ(x) +}

_ρ(y)),

we get

(2.4)

(1

₋

ψ(s))

_h

δ

_x(v)

,

cos(s

p

∆

U

)δ

y(w)

i −

(1

−

ψ

(s))

h

δ

x(v)

,

cos(s

p

∆

0

)δ

(yw)

i

= 0

for any

s

_∈

R

_{. On the other hand, note that}

h

δ

_x(v)

,

(1 + ∆

V

)

−m

cos(s

p

∆

V

)δ

y(w)

i

=

h

(1 + ∆

V

)

− m

2

δ

(v)

x

,

cos(s

p

∆

V

)(1 + ∆

V

)

− m

2

δ

(w)

y

i

.

Applying the Cauchy-Schwarz inequality several times this gives

(2.5)

_|h

δ

_x(v)

,

(1 + ∆

V

)

−m

cos(s

p

∆

V

)δ

y(w)

i| ≤ |

v

||

w

| k

G

(m)

V

(x, x)

kk

G

(m)

V

(y, y)

k

1/2

for any

s

_∈

R

_{. Combining (}

_2.4

_{), (}

_2.5

_{), (}

_2.10

_{) with (}

_2.3

_{) suffices to conclude the}

proof.

2.2.

Safarov’s estimate.

Since ∆

U

is a non-negative self-adjoint operator, by the

spectral theorem

∆

U

=

Z

_∞

0

λ dΠ(λ),

where Π(λ) (λ

_≥

0) denotes the spectral projection of ∆

U

onto the interval [0, λ].

The so-called spectral function

e

(x, y;

λ), defined to be the integral kernel of Π(λ),

is smooth in

U

_×

U

for each fixed

λ.

If

N

= 1 we also write

e(x, y;

λ) for

e

(x, y;

λ). Safarov ([29, Cor. 3.1]) proved for

every

x

_∈

U

and all

λ >

0 that

(2.6)

e(x, x;

λ)

_≤

C

_d(1)

λ

d/2

+

C

(2)

d

ρ(x)

λ

1/2

+

C

(3)

d

ρ(x)

d

−1

,

where

C

_d(1)

,

C

_d(2)

,

C

_d(3)

are universal constants given respectively by

C

_d(1)

=

ω

d

(2π)

−d

with

ω

d

denoting the volume of the unit ball in

R

d

,

C

d(2)

=

dC

(1)

d

(2π

−1

(C

(3)

d

)

2

+

C

(3)

C

_d(3)

_≤

2m

d

3

1

2md

_{, where}

_m

d

=

⌈

d+1₂

⌉

(see [29, Lemma 2.6]). We should mention that

Safarov originally established (

2.6

) by understanding

e(x, x;

λ) as the integral kernel

of

Π(λ−0)+Π(₂ λ+0)

. But the right hand side of (

2.6

) is a continuous function of

λ >

0,

(

2.6

) also holds for our choice of the spectral function.

The key points for proving (

2.6

) are the fact (see [29, Lemma 2.7, Cor. 3.1])

that

χ

+(λ)e(x, x;

λ

2

_{) is a non-decreasing function of}

_λ

_on

_R

_{, and the cosine Fourier}

transform of

(C

_d(1)

)

−1

_·

d

dλ

χ+(λ)e(x, x;

λ

2

₎

coincides on the interval (

₋

ρ(x), ρ(x)) with the cosine Fourier transform of

dλ

d₊−1

.

Here

χ+

is the characteristic function of the positive axis. The latter property can

be seen from the finite propagation speed for the wave equation.

In the vector-valued situation, we claim as (non-negative) self-adjoint matrices,

(2.7)

e

(x, x;

λ)

_≤

C

_d(1)

λ

d/2

+

C

(2)

d

ρ(x)

λ

1/2

+

C

(3)

d

ρ(x)

d

−1

1

for every

x

_∈

U

and all

λ >

0. To this end we see once again from the finite

propagation speed for the wave equation that for each fixed unit vector

v

_∈

C

N

_{, the}

cosine Fourier transform of

(C

_d(1)

)

−1

_·

d

dλ

χ+(λ)

h

δ

(v)

x

,

Π(λ

2

)δ

(xv)

i

coincides on the interval (

₋

ρ(x), ρ(x)) with the cosine Fourier transform of

dλ

d₊−1

.

Also,

χ+(λ)

_h

δ

x(v)

,

Π(λ

2

)δ

(xv)

i

is a non-decreasing function of

λ

on

R

. So similar to

(

2.6

) we have

h

δ

_x(v)

,

Π(λ)δ

_x(v)

_{i ≤}

C

_d(1)

λ

d/2

+

C

(2)

d

ρ(x)

λ

1/2

+

C

(3)

d

ρ(x)

d

−1

(x

_∈

U, λ >

0),

which proves (

2.7

). For simplicity, applying H¨older’s and Young’s inequalities to the

right hand side of (

2.7

) gives for every

x

_∈

U

and all

λ >

0 that

(2.8)

e

(x, x;

λ)

_≤

(C

_d(4)

λ

d/2

+

C

_d(5)

ρ(x)

−d

)

1

,

where

C

_d(4)

= (C

_d(1)

+

d−_d1

2

d−2

_C

(2)

d

),

C

(5)

d

= 2

d−2

C

(2)

d

((C

(3)

d

)

d−1

+

1d

).

According to the functional calculus of self-adjoint operators, we have

(1 + ∆

U

)

−m

=

Z

_∞

0

1

(1 +

λ)

m

dΠ(λ).

For

m >

0 this integral can be understood as an operator-valued integral that

converges in the strong operator topology. For the purposes of this paper it is

enough to understand it in the weak sense as a statement about quadratic forms.

For

m >

d₂

the integral kernel

G

(_Um)

(x, y) of (1 + ∆

U

)

−m

is continuous on

U

×

U

and

we have

G

(_Um)

(x, x) =

Z

∞

0

1

Pointwise convergence of the integral can easily seen as follows. Choose

s

_∈

R

_such

that

m > s >

d₂

. The operator (1 + ∆

U

)

s/2

commutes with the spectral measure

and (1 + ∆

U

)

s/2

(1 + ∆

U

)

−m

(1 + ∆

U

)

s/2

is bounded. Therefore, its integral spectral

representation converges in the sense of quadratic forms. Since (1 + ∆

U

)

−s/2

maps

L

2

(U

;

C

N

_{) to}

_H

s

loc

(U

;

C

N

) it extends by duality to a continuous map

H

comp−s

(U

;

C

N

)

to

L

2

(U

;

C

N

_{). We conclude that}

(1 + ∆

U

)

−m

=

Z

_∞

0

1

(1 +

λ)

m

dΠ(λ)

converges in the sense of quadratic forms on

H

_comp−s

(U;

C

N

_{). By the Sobolev}

embed-ding theorem

δ

(xv)

is in

H

comp−s

(U

;

C

N

). Considering

m >

d2

, (

2.8

),

e

(x, x; 0)

≥

0, and

the following equivalent representation of the classical Beta function

B(α, β) =

Z

_∞

0

λ

α−1

(1 +

λ)

α+β

dλ

(Re(α)

>

0,

Re(β)

>

0),

one can use integration by parts to get

G

(_Um)

(x, x) =

Z

∞

0

1

(1 +

λ)

m

d

e

(x, x;

λ)

=

m

Z

∞

0

e

(x, x;

λ)

(1 +

λ)

m+1

dλ

−

e

(x, x; 0)

≤

m

Z

_∞

0

C

_d(4)

λ

d/2

+

C

_d(5)

ρ(x)

−d

(1 +

λ)

m+1

dλ

1

=

mC

_d(4)

B(1 +

d

2

, m

−

d

2

) +

C

(5)

d

ρ(x)

−d

1

.

(2.9)

On the other hand we have

e

₀

(x, x;

λ) =

C

_d(1)

λ

d/2

1

(see [33, Example 3.1]), where

e

₀

(x, y;

λ) denotes the spectral function of ∆

0

. Therefore, one obtains

G

(₀m)

(x, x) =

Γ(m

−

d

2

)

(4π)

d2

(m

−

1)!

1

.

(2.10)

Finally, by considering (

2.9

) and (

2.10

) and by introducing

C

_d(6)

=

mC

_d(4)

B

(1 +

d

2

, m

−

d

2

) +

Γ(m

₋

d₂

)

(4π)

d2

(m

−

1)!

,

we can deduce from Theorem

2.1

that

Theorem 2.2.

The following pointwise estimate holds for the heat kernel:

k

K

U

(x, y;

t)

−

K

0(x, y;

t)

k ≤

C

d(5)

ρ(x, y)

−d

+

C

(6)

d

·

J

m

(ρ(x) +

ρ(y);

t)

2.3.

Optimizing cut-off functions.

This section is devoted to proving Theorem

1.1

. To this end it suffices to bound

J

m

(R;

t) for

R

=

ρ(x) +

ρ(y). In general we

suppose

R >

0. The Hermite polynomials

H

n

(s) = (

−

1)

n

e

s

2

d

n

ds

n

e

−

s2

(n

= 0,

1,

2, . . .)

can be written as

H

n

(s) =

⌊n

2⌋

X

k=0

(

₋

1)

k

_n!

k!(n

₋

2k)!

(2s)

n−2k

_,

from which it is easy to deduce that

d

n

ds

n

(e

−s2

4t

) =

⌊n

2⌋

X

k=0

(

₋

1

2

)

n

₍

₋

₁₎

k

_n!

k!(n

₋

2k)!

t

k−n

_s

n−2k

_e

−s2

4t

(n

= 0,

1,

2, . . .).

Consequently, by Leibniz’s rule one gets for any non-negative integer

n

_≤

2m

that

d

n

ds

n

ψ(s)e

−s 2 4t

=

n

X

j=0 ⌊j2⌋

X

k=0

n

j

ψ

(n−j)

(

−

1

2

)

j

(

−

1)

k

j

!

k!(j

₋

2k)!

t

k−j

_s

j−2k

_e

−s2

4t

.

To optimize the choice of cutoff functions we first let

ψ

0

denote a fixed real-valued

function in

C

2m

₍

_R

_{) such that}

_ψ

0

(s) = 0 for

s

≤

0 and

ψ

0(s) = 1 for

s

≥

1. Later on

we will give concrete examples of

ψ

0

and thus

M

j

(ψ

0

) = max

0≤s≤1

d

j

_ψ0

ds

j

(s)

(j

= 0,

1, . . . ,

2m)

can be explicitly determined. Then for any 0

< ǫ

1

< ǫ

2

< R

define

ψ

ǫ1,ǫ2

(s) =

ψ

0

_|

_s

_{| −}

_ǫ

1

ǫ

2

−

ǫ

1

,

which is an even function in

C

2m

₍

_R

_{) with Supp(1}

₋

_ψ

ǫ1,ǫ2

)

⊂

(

−

R, R). We let the

parameters

ǫ

1

, ǫ

2

(depending on both

R

and

t) behave in the following way:

•

ǫ

2

→

R,

•

ǫ2

₋

ǫ1

_≡

2_Rt

.

With the help of Lemma

2.4

, it is not hard to show that if 0

< t

_≤

R₈2

, then

lim

ǫ2→R

Z

R

d

n

ds

n

(ψ

ǫ1,ǫ2

(s)e

−

s2

4t

)

ds

_≤

Z(n, ψ

0

, R;

t)e

−

R2

4t

(n

≤

2m),

(2.11)

where

Z

(n, ψ0, R;

t) is short for the rational function

⌊n

2⌋

X

k=0

n!

_⌈

n−2₂k−1

_⌉

!M0(ψ0)e

2

_R

n−2k−1

2

2n−2k−1

_k!(n

₋

_2k)!

t

1+k−n

₊

n−1

X

j=0 ⌊j2⌋

X

k=0

n!M

n−j

(ψ0

)eR

n−2k−1

2

n−2

_k!(j

₋

_2k)!(n

₋

_j)!

t

1+k−n

_.

Theorem 2.3.

Suppose

0

< t

_≤

R₈2

. Then

J

m

(R;

t)

≤

m

X

n=0

m

n

Z

(2n, ψ

0

, R;

t)e

−

R2

4t

.

Theorem

1.1

is an immediate consequence of Theorems

2.2

,

2.3

with

m

=

_⌈

d+1₂

_⌉

.

Lemma 2.4.

If

β

is a non-negative integer and if

ρ

_≥

2

√

t

, then

Z

∞

ρ

s

β

e

−s 2

4t

ds

≤

2e

l

_β

₋

₁

2

m

!

ρ

β−1

te

−ρ 2 4t

.

Proof.

Note first

Z

_∞

ρ

s

β

e

−s 2

4t

ds

= 2

β

t

β+1

2

Γ(

β

+ 1

2

,

ρ

2

4t

),

where Γ(

_·

,

_·

) is the upper incomplete Gamma function. If

β+1₂

is a positive integer

then it is known that Γ(

β+1₂

, r) = (

β−₂1

)!

e

−r

P

β−1

2

k=0 r

k

k!

for all

r >

0. This partially

proves the lemma simply by considering

ρ₄2_t

_≥

1. If

β+1₂

is a positive half-integer then

we can use Γ(

β+1₂

, r)

_≤

_√1

r

Γ(

β+2

2

, r) and the previous explicit formula for Γ(

β+2 2

, r)

to prove the remaining part of the lemma. This finishes the proof.

Although there are many test functions for

ψ0, we use an interpolating polynomial

because

M

j

(ψ

0) can be determined rather easily. For any

n

∈

N

, there exists a unique

polynomial

P

n

of degree

≤

2n

+ 1 such that

P

n

(0) = 0,

P

n

(1) = 1, and

d

i

ds

i

P

n

s=0

=

d

i

ds

i

P

n

s=1

= 0 (1

≤

i

≤

n).

We then define a function

P

e

n

on

R

such that it agrees with

P

n

on [0,

1], equals 0 on

(

_−∞

,

0), and equals 1 on (1,

_∞

). It is easy to check that

P

e

n

∈

C

n

(

R

). This means

that one can set

ψ0

=

P2

e

m

. A few examples of

P

n

are listed below:

P

1(s) = 3s2

₋

2s

3

,

P

2

(s) = 10s

3

−

15s

4

+ 6s

5

,

P3(s) = 35s

4

₋

84s

5

+ 70s

6

₋

20s

7

,

P4(s) = 126s

5

₋

420s

6

+ 540s

7

₋

315s

8

+ 70s

9

.

3.

Dirichlet boundary conditions

We denote by

K

_U(D)

(x, y;

t) the Dirichlet heat kernel for an open set

U

_⊂

R

d

_.

Michiel van den Berg’s (

1.1

) gives

(3.1)

_|

K

_U(D)

(x, y;

t)

₋

K

0

(x, y;

t)

| ≤

(4πt)

−d/2

exp(

−

|

x

₋

y

_|

2

+ 4δ

2

4t

)

d

X

j=1

To compare, Theorem

1.1

is a slight improvement of (

3.1

) for the short-time diagonal

elements of the Dirichlet heat kernel if

d

_≥

5.

It is known that

G

(_Um)

(x, x)

_≤

G

₀(m)

(x, x) for any

x

_∈

U

, where

G

(_Um)

is interpreted

in accordance with the choice that ∆

U

denotes the Dirichlet Laplacian on

U

. Thus

it follows from Theorem

2.1

and (

2.10

) that

(3.2)

_|

K

_U(D)

(x, y;

t)

₋

K0(x, y;

t)

_{| ≤}

Γ(m

−

d

2

)

(4π)

d2

(m

−

1)!

√

π

·

J

m

(ρ(x) +

ρ(y);

t)

√

t

.

We remark that two other estimates by Michiel van den Berg ([2]) reading

|

K

_U(D)

(x, x;

t)

₋

K

0

(x, x;

t)

| ≤

2d

(4πt)

d2

exp(

₋

ρ(x)

2

dt

),

(3.3)

|

K

_U(D)

(x, y;

t)

₋

K0(x, y;

t)

_{| ≤}

2d

(4πt)

d/2

exp(

−

(3

−

2

√

2)

(max

{

ρ(x), ρ(y)

}

)

2

dt

),

(3.4)

have been widely used in the study of short-time asymptotics of the heat trace (see

e.g. [3,

6,

32]) and some other related problems (see e.g. [5]).

4.

Neumann boundary conditions

Let

K

_U(N)

(x, y;

t) denote the Neumann heat kernel for a smooth bounded open set

U

_⊂

R

d

_{. As an application of Theorem}

_1.1

_{(or Theorem}

_4.1

_with

_m

₌

_⌈

d+1

2

⌉

), there

exists a positive function

g

on

U

such that if 0

< t

_≤

ρ(x₂)2

then

(4.1)

K

_U(N)

(x, x;

t)

_≤

(4πt)

−d/2

₊

_g(x)

_·

_t

−α

_·

_exp(

₋

ρ(x)

2

t

),

where

α

= 2

_⌈

d+1₂

_{⌉ −}

1₂

. This answers a question raised by Lacey ([21]) who

conjec-tured that for the class of smooth bounded strictly star-shaped domains (

4.1

) holds

for some

α >

d₂

as long as time

t

is sufficiently small. Lacey also asked to extend

the main result in [21] to unbounded domains, domains with non-smooth boundary,

or more general boundary conditions. Because of Theorem

1.1

this is indeed doable

for the diagonal element of the corresponding Neumann heat kernel.

In the rest of the section we also give a replacement of (

2.9

) for

G

(_Um)

(x, x) without

using Safarov’s estimate (

2.6

). Here

G

(_Um)

is interpreted in accordance with the choice

that ∆

U

denotes the Neumann Laplacian on

U

. This can be done by appealing to

partial domain monotonicity of the Neumann heat kernel.

For simplicity we assume that

U

_⊂

R

d

_{is a smooth bounded open set. Note first}

(see e.g. [8, (3.33)], [12,

_§

3.4])

(4.2)

G

(_Um)

(x, x) =

1

(m

₋

1)!

Z

_∞

0

t

m−1

e

−t

K

_U(N)

(x, x;

t)dt,

monotonicity principle ([1]) claiming for any

U2

_⊂

U1

that

K

_U(N₁)

(x, y;

t)

_≤

K

_U(N₂)

(x, y;

t) ((x, y, t)

_∈

U

2

×

U

2

×

R

+

).

Even though, Kac’s original idea ([19]) of comparing

K

_U(D)

(x, x;

t) with

K

_Bx(D)

(x, x;

t)

where

B

x

is chosen here

1

to be the ball in

R

d

with center

x

and radius

ρ(x), still

works for the Neumann boundary problems. This is exactly a result by Kendall

([20], see also [24]; if

U

is convex then see [9]) stating

(4.3)

K

_U(N)

(x, x;

t)

_≤

K

_Bx(N)

(x, x;

t) ((x, t)

_∈

U

_×

R

+

_),

which combined with (

4.2

) yields

(4.4)

G

(_Um)

(x, x)

_≤

G

(_Bxm)

(x, x) (x

_∈

U

).

Let

U

_d

_{(x, y;}

_{t) denote the Neumann heat kernel for the}

_{d-dimensional unit ball. The}

Pascu-Gageonea resolution ([25]) of the Laugesen-Morpurgo conjecture ([22]) says

that

(4.5)

U

_d

_{(x, x;}

_t)

_<

U

_d

_{(y, y;}

_t)

holds for all

t >

0 and all

x, y

in the

d-dimensional unit ball with

_|

x

_|

<

_|

y

_|

. This

result implies that

(4.6)

U

_d

_(0,

_0;

_t)

_<

Tr(e

−t∆(_dN)

₎

ω

d

,

where ∆

(_dN)

is short for the Neumann Laplacian on the

d-dimensional unit ball.

Now let

x

_∈

U

be fixed. It is straightforward to verify that

(4.7)

K

_Bx(N)

(x, x;

t) =

U

_d

_0,

_0;

t

ρ(x)

2

ρ(x)

d

.

Hence by considering (

4.4

), (

4.2

) with

U

replaced by

B

x

, (

4.7

) and (

4.6

), we get

G

(_Um)

(x, x)

_≤

1

(m

₋

1)!

·

Z

_∞

0

t

m−1

e

−t

Tr(e

− t

ρ(x)2∆ (N)

d

₎

ω

d

ρ(x)

d

dt

=

ρ(x)

2m−d

(m

₋

1)!ω

d

·

Z

∞

0

t

m−1

e

−tρ(x)2

Tr(e

−t∆(dN)

)dt

≤

ρ(x)

2m−d

(m

₋

1)!ω

d

·

Z

1

0

t

m−1

Tr(e

−t∆(dN)

)dt

+ Tr(e

−∆

(N)

d

)

Z

∞

0

t

m−1

e

−tρ(x)2

dt

=

ρ(x)

2m−d

(m

₋

1)!ω

d

·

Z

1

0

t

m−1

Tr(e

−t∆(dN)

)dt

+

Tr(e

−∆(_dN)

₎

ω

d

ρ(x)

d

,

where in the last inequality we have used the fact Tr(e

−t∆(_dN)

₎

_≤

_Tr(e

−∆(_dN)

_{) for all}

t

_≥

1. This estimate together with (

2.10

) gives from Theorem

2.1

the following

1_{To be precise, Kac ([19]) setB}

Theorem 4.1.

Let

U

_⊂

R

d

_{a smooth bounded open set and let}

_m

_∈

N

_with

_{m >}

d

2

.

For any

t >

0

and any

x, y

in

U

one has

K

_U(N)

(x, y;

t)

₋

(4πt)

−d/2

exp(

₋

|

x

−

y

|

2

4t

)

_≤

N

d

(x, y)

·

J

m

(ρ(x) +

ρ(y);

t)

2

√

πt

,

where

N

d

(x, y) =

Z

1

0

t

m−1

Tr(e

−t∆(dN)

)dt

(m

₋

1)!ω

d

·

(max

_{

ρ(x), ρ(y)

_}

)

2m−d

+

Γ(m

−

d

2

)

(4π)

d2

(m

−

1)!

+

Tr(e

−∆(_dN)

₎

ω

d

·

(min

_{

ρ(x), ρ(y)

_}

)

−d

.

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Department of Mathematical Sciences, Loughborough University, LE11 3TU, UK