Spectral properties of differential operators

Adam Sikora

Macquarie University in Sydney, Australia

(Wrocław, October 8-10, 2012)

SPECTRAL PROPERTIES OF DIFFERENTIAL OPERATORS -METHOD OF WAVE EQUATION IN HARMONIC ANALYSIS AND

PDE’S

ADAM SIKORA

1. Overview

We will discuss here the notion of the finite speed propagation of solutions of waveequations, which is also called sometimes the weak Huygens principle. Finite speedpropagation property is very common phenomenon in our every day experience.One can mention such apparent examples as the speed of sound and light or waterwaves. Here we describe efficient and elegant way of obtaining rigorous proof offinite propagation speed in a very wide variety of self-adjoint second order differentialoperators.

We start with a brief discussion of the standard Laplace operator and the cor-responding wave and heat equation. Then we recall notion of spectral theory anddiscuss equivalence of so called Davies-Gaffney property and finite speed propaga-tion. This equivalence provides very efficient tool to obtain rigorous proofs of thefinite speed property in the wide range of situation. Next we discuss application offinite speed property in harmonic analysis and PDEs including heat kernels theory,Riesz transform, spectral multipliers and Schrodinger propagator.

The main point which I would like to make in these lecture is that thanks to itssimple geometric interpretation the notion of finite speed propagation can be use invery efficient and intuitive wave to study various aspects of harmonic analysis andPDEs. In many cases this simple geometric picture behind finite speed propagationallow us to design the efficient framework for calculation before working our allnecessary and sometimes tedious details. In other words it is very useful tool to”understand” sometimes quite complicated calculations.

2. Heat Equation

We consider an infinite homogeneous wire. Suppose that at time t ≥ 0 and pointx ∈ R the temperature is equal to T (t, x). Then the amount of heat energy containedin the interval [a, b] is equal to ∫ b

a

T (t, x)dx.

and the ratio of change of this quantity is equal to

∂t

∫ b

a

T (t, x)dx =

∫ b

a

∂tT (t, x)dx.

Date: October 6, 2012.1

2 ADAM SIKORA

Now the heat energy flow through point a and b should be proportional to ∂xT (t, a)and ∂xT (t, b) respectively so

∂t

∫ b

a

T (t, x)dx =

∫ b

a

∂tT (t, x)dx = ∂xT (t, b)− ∂xT (t, a) =

∫ b

a

∂2xT (t, x)dx.

However, the above equality holds for all intervals [a, b] so it follows that

∂tT (t, x) = ∂2xT (t, x).

This equation is called the heat equation and our argument above shows that thisequation governs the heat propagation in a infinite homogeneous wire. Note thatin a more physical discussion we should consider a couple of constants such as heatcapacity and heat conductivity. To simplify the discussion we will just assume thatall constants are equal to 1. Now we are going to use the Divergence Theorem toargue that similar equations should govern the heat propagation in two, three (ormore) dimensional objects. For example we consider the heat propagation in thehomogeneous space R3. Let V be any domain (subset of R3) and ∂V boundarysurface of V . The amount of heat energy enclosed by V is equal to∫

V

T (t, x, y, z)dV,

where T (t, x, y, z) is the temperature at point (x, y, z) at time t. Next one can expectthat the heat energy flow through the surface ∂V is proportional to∫

∂V

nT (t, x, y, z)dS =

∫∂V

∇T (t, x, y, z) · ndS

Hence by the Divergence Theorem∫∂V

∇T (t, x, y, z) · ndS =

∫V

div(∇T (t, x, y, z))dV.

A simple calculation shows that

div(∇f)(x, y, z) = ∂2xf(x, y, z) + ∂2

yf(x, y, z) + ∂2zf(x, y, z)

Often the following notation is used

∆f(x, y, z) = ∂2xf(x, y, z) + ∂2

yf(x, y, z) + ∂2zf(x, y, z)

and ∆ is called the Laplace operator. In other words∫∂V

∇T (t, x, y, z) · ndS =

∫V

(∂2xT (t, x, y, z) + ∂2

yT (t, x, y, z) + ∂2zT (t, x, y, z)

)dV

=

∫V

∆T (t, x, y, z)dV

The same argument as in the one dimensional case shows that∫V

∆T (t, x, y, z)dV =

∫∂V

∇T (t, x, y, z) · ndS = ∂t

∫V

T (t, x, y, z)dV

=

∫V

∂tT (t, x, y, z)dV.

METHOD OF WAVE EQUATION IN HARMONIC ANALYSIS AND PDE’S 3

However, the above equality holds for all subsets V ⊂ R3 so the temperature distri-bution T (t, x, y, z) satisfies the following equation

∂tT (t, x, y, z) = ∆T (t, x, y, z).

This equation governs heat propagation in a homogeneous three dimensional medium.It is also an important and significant example of a Partial Differential Equation(PDE). PDEs are any equations which involve partial derivatives.

2.1. Initial conditions. An natural physical problem related to our discussion ofheat propagation is as follows: Suppose that one knows that initial temperature attime 0 is described by function T (0, x, y, z) = f(x, y, z). Find the temperature attime t > 0 at (x, y, z) that is T (t, x, y, z).

Mathematically we can formulate the problem in the following way: Find thefunction T (t, x, y, z) such that

∂tT (t, x, y, z) = ∆T (t, x, y, z)

and T (0, x, y, z) = f(x, y, z). The function f is called an initial condition for theconsidered heat equation.

2.2. Boundary conditions. It would be more natural and more practical to con-sider finite interval [a, b] or finite subset Ω ⊂ R3 as our medium. That is we wouldlike to consider the temperature function T : R+ × [a, b] → R or T : R+ × V → R.How should we describe the heat propagation at points a and b or at ∂V ? There aresome different choices, which correspond to different physical possibilities. VectorCalculus can help us to understand the situation. Suppose that you wrap your in-terval [a, b] or your solid V in a perfect thermal isolator. Then for an interval thisshould mean that

(1) ∂xT (t, a) = 0 and ∂xT (t, b) = 0

that is the heat energy does not flow through points a and b.For V it should mean that heat energy does not flow through ∂V so

∇T (t, x, y, z) · ndS = 0

for all (x, y, z) ∈ ∂V . That is

(2) ∇T (t, x, y, z) · n = nT (t, x, y, z) = 0

for all (x, y, z) ∈ ∂V . Conditions (1) and (2) are called Neumann boundary condi-tions.

One can also think about different boundary conditions like for example

T (t, a) = 0 and T (t, b) = 0

or for V

T (t, x, y, z) = 0

for all (x, y, z) ∈ ∂V . These conditions are called Dirichlet boundary conditions.

4 ADAM SIKORA

3. Energy and Finite speed propagation for the wave equation

Now we are going to discuss the wave equation. This equation has the followingform

∂2t F = ∆F.

For example the expression

∂2t F (t, x) = ∂2

xF (t, x)

is the one dimensional wave equation;

(3) ∂2t F (t, x, y) = ∆F (t, x, y) = ∂2

xF (t, x, y) + ∂2xF (t, x, y)

is the two dimensional version. We discuss the wave equation in R3 below. Waveequation can be used to model such physical phenomena as vibration of a string orthread, propagation of acoustic sound, or electromagnetic waves.

3.1. Initial conditions for wave equation. When one considers the heat equa-tions it is natural to expect that it is enough to know the initial temperature T (0, x)to determine the heat distribution at any future time. To determine propagation ofthe solution of the wave equation one has to know the initial position F (0, x) as wellas the values of Ft(0, x) = ∂tF (0, x). An illuminating comparison can be made withthe behaviour of system in Newton mechanic, where we have to know not only theinitial positions but also the velocity of all elements of the considered system.

3.2. Finite speed of propagation of solutions of the wave equation. Next wewill discuss a fundamental property of the wave equation - the finite speed propaga-tion of solutions. In our discussion we will use the Divergence Theorem again.

Theorem 3.1. Suppose that function F : Rn → R satisfies the wave equation. Thenfinite speed propagation property holds for F . That is if at point t = to and y ∈ Rn

one has

F (to, x) = 0 and Ft(to, x) = 0 ∀x∈∈B(r,y)⊂Rn

then for all t ≥ to

F (t, x) = 0 and Ft(t, x) = 0 ∀x∈B(r+t−to,y),

where B(r, y) is the ball or radius r defined by B(r, y) = x ∈ Rn :∑

i(xi−yi)2 ≤ r2.

The above theorem is a special case of more general results which we discusslater. Here we would like to explain main point of the classical proof based onenergy estimates and for simplicity we consider here only the straight line and theEuclidean space, n = 1 or n = 3. Suppose that the function F : R×Rn → R (n = 1or n = 3) is the solution of the wave equation

∂2t F (t, x) = ∆xF (t, x) = div∇F (t, x).

Consider the following quantity

E(t) =

∫Rn

(|Ft(t, x)|2 + |∇F (t, x)|2)dx.

METHOD OF WAVE EQUATION IN HARMONIC ANALYSIS AND PDE’S 5

We will show that E(t) does not depend on t. To do this it is enough to show that∂tE(t) = 0 for all t ∈ R.

∂tE(t) = ∂t

∫Rn

(|Ft(t, x)|2 + |∇F (t, x)|2)dx

=

∫Rn

∂t(|Ft(t, x)|2 + |∇F (t, x)|2)dx

=

∫Rn

2(Ft(t, x)Ftt(t, x) +∇Ft(t, x) · ∇F (t, x))dx

=

∫Rn

2Ft(t, x)(Ftt(t, x)−∆F (t, x)) = 0.

To verify the above calculation note that by product rule

∂t|Ft(t, x)|2 = 2Ft(t, x)Ftt(t, x) and ∂t|∇F (t, x)|2 = 2∇Ft(t, x) · ∇F (t, x)

and that integration by parts gives∫Rn

∇F (x) · ∇G(x)dx = −∫

Rn

G(x)∆F (x)dx.

Thus E does not depend on time and this quantity has a natural interpretation asthe energy of the system. One can think about the function

|Ft(t, x)|2 + |∇F (t, x)|2

as a density (distribution in the space) of the energy at time t.

Proof of the finite propagation property in the one and three dimensional cases.We will use the idea of the energy corresponding to the wave equation to prove finitespeed propagation property of the wave equation. First we consider the case n = 1and for every r ∈ R and s > 0 we set

Er,s(t) =

∫ r+s

r−s|Ft(t, x)|2 + |∇F (t, x)|2

Next we calculate ∂tEr,t(t). We set u = r + t and v = r − t. In a similar way asbefore we conclude that

∂tEr,t(t) = ∂t

∫ u

v

(|Ft(t, x)|2 + |∇F (t, x)|2)dx(4)

=

∫ u

v

∂t(|Ft(t, x)|2 + |∇F (t, x)|2)dx

+|Ft(t, v)|2 + |∇F (t, v)|2 + |Ft(t, u)|2 + |∇F (t, u)|2.

The last four terms come from the change in the domain of integration. Indeed recallthat by Fundamental Theorem of Analysis

d

dx

∫ x

a

f(s)ds = f(x).

6 ADAM SIKORA

We discuss the one-dimensional case so we write F ′ instead of ∇F . Integrating byparts we get (recall that u = r + t and v = r − t)∫ u

v

∂t(|Ft(t, x)|2 + |F ′(t, x)|2)dx(5)

=

∫ u

v

2(Ft(t, x)Ftt(t, x) + F ′t(t, x)F ′(t, x))

=

∫ u

v

2(Ft(t, x)Ftt(t, x)− Ft(t, x)F ′′(t, x))

+2Ft(t, u)F ′(t, u)− 2Ft(t, v)F ′(t, v)

= 2Ft(t, u)F ′(t, u)− 2Ft(t, v)F ′(t, v) ≥−(|Ft(t, v)|2 + |F ′(t, v)|2 + |Ft(t, u)|2 + |F ′(t, u)|2

)To verify the last estimates we use the inequality between arithmetic and geometricmeans (a simple version of the Schwarz inequality)

p2 + q2 ≥ 2pq.

By (4) and (5) we get ∂tEr,t(t) ≥ 0 that is the function e(t) = Er,t(t) is an increasingfunction of t. This property has a very useful consequence. Namely we note E(t) = Eis constant so E − Er,t(t) is a decreasing function. However

E − Er,t(t) ≥ 0 and E − Ea+b2, b−a

2(to) = 0.

because we assume that F (to, x) = Ft(to, x) = 0 for all x such that x ≤ a or x ≥ b.Hence

E − Ea+b2, b−a

2+|t−t−o|(t) = 0.

Thus if at time t = to, F (to, x) = Ft(to, x) = 0 for all x such that x ≤ a or x ≥ b forsome a < b then F (t, x) = Ft(t, x) = 0 for all t ≥ to and all x ≤ a−|t− t0| and for allx ≥ b+ |t− t0|. This means that the solution of the wave equation propagates onlywith unit speed. This observation is called the finite speed of propagation propertyof the solution of the wave equation.

Next, we are going to extend our discussion of the energy corresponding to the waveequation and finite speed propagation for the wave equation to three dimensionalcase. We start by defining

EΩ(t) =

∫Ω

(|Ft(t, x)|2 + |∇F (t, x)|2)dx,

where Ω is a connected domain in R3. We note that

div(F∇G) = ∇F · ∇G+ F∆G

and by Divergence theorem∫∂Ω

F∇G · ndσ =

∫Ω

div(F∇G)dx =

∫Ω

(∇F · ∇G+ F∆G)dx

METHOD OF WAVE EQUATION IN HARMONIC ANALYSIS AND PDE’S 7

where dσ is the surface area of the boundary ∂Ω of Ω. Hence

∂tEΩ(t) = ∂t

∫Ω

(|Ft(t, x)|2 + |∇F (t, x)|2)dx

=

∫Ω

∂t(|Ft(t, x)|2 + |∇F (t, x)|2)dx

=

∫Ω

2Ft(t, x)(Ftt(t, x) +∇Ft(t, x) · ∇F (t, x))dx

=

∫Ω

2Ft(t, x)(Ftt(t, x)−∆F (t, x)) +

∫∂Ω

2Ft∇F · ndσ =

∫∂Ω

2Ft∇F · ndσ.

It means that the vector field 2Ft∇F describes the flow of the energy in the system.Next we denote by B(x, t) the ball of radius t and center at x and we calculate

∂tEB(x,t)(t) = ∂t

∫B(x,t)

(|Ft(t, x)|2 + |∇F (t, x)|2)dx

=

∫B(x,t)

2(Ft(t, x)Ftt(t, x) +∇Ft(t, x) · ∇F (t, x))dx

+

∫∂B(x,t)

(|Ft(t, x)|2 + |∇F (t, x)|2)dσ.

The second term comes from the change in the domain of integration. Hence

∂tEB(x,t)(t) = ∂t

∫∂B(x,t)

2Ft(t, x)∇F (t, x) · ndσ(x).

+

∫∂B(x,t)

(|Ft(t, x)|2 + |∇F (t, x)|2)dσ(x)

≥ −∫∂B(x,t)

(|Ft(t, x)|2 + |∇F (t, x)|2)dσ(x)

+

∫∂B(x,t)

(|Ft(t, x)|2 + |∇F (t, x)|2)dσ(x) ≥ 0.

The above calculation shows that e(t) = EB(x,t)(t) is an increasing function and this,in turn, implies the finite speed of propagation of the solutions of the wave equation.

Remark It is not difficult to check that the above proof without any changes workfor all dimensions n = 2, 3, 4, . . ..

4. Spectral theory

There is a significant need to generalize notion of the standard Laplace operatorand the corresponding wave and heat equations. For example one would like to con-sider different geometries of underling ambient space, to introduce non-homogeneousheat capacity or to consider Neumann and Dirichlet boundary conditions mentionabove. We would also would like to be able to consider Schrodinger operators withsome potentials which significance comes from their interpretation in quantum me-chanics.

All mentioned above operators are examples of self-adjoint operators. We nextdiscuss such operators in more systematic way.

8 ADAM SIKORA

Recall that, if L is a non-negative self-adjoint operator on L2(X, dµ), one canconstruct the spectral decomposition EL(λ) of the operator L. For any boundedBorel function m : [0,∞) → C, one then defines the operator m(L) : L2(M,dµ) →L2(X, dµ) by the formula

m(L) =

∫ ∞0

m(λ) dEL(λ).

Now, for z ∈ C+ and mz(λ) = exp(−zλ), one sets mz(L) = exp(−zL), z ∈ C+.By spectral theory the family exp(−zL) = exp(−zL) : z ∈ C+, (which is calledsemigroup of operators generated by L) is a family of contraction on L2(X). Thatis ‖ exp(−zL)‖2→2 ≤ 1 for all z ∈ C+. Now the solution of the heat equation

∂tF (t, x) = −LxF (t, x)

with initial condition

F (x, 0) = f(x)

can be written down using the notion of semigroup exp(−zL) as

F (x, t) = exp(−tL)f(x).

Indeed at least formally

d

dtexp(−tL)f(x) = −L exp(−tL)f(x)

and exp(0L) = Id. Similarly the solution of wave equation

∂2t F (t, x) = −LxF (t, x)

with initial condition

F (x, 0) = f(x) and Ft(x, 0) = g(x)

can expressed in terms of spectral multipliers defined above as

F (x, t) = cos(−t√L)f(x) +

sin(t√L)√

Lg(x).

There is a simple relation between the wave and heat equation which we going touse extensively later and which can be in simple manner stated in the following way

exp(−sL) =

∫ ∞0

cos(t√L)

2√πse−

t2

4s dt.

Indeed in virtue of spectral theorem it is enough to show that the above relationholds if L in the the above equality is replace by λ2 for any λ ∈ R. Then it justbecomes a standard exercise in Fourier Transform.

5. Examples

The range of possible examples of operator L which we would like to discussincludes the following operators

(a) The standard Laplace operator ∆ =∑

i ∂2i .

(b) ∆Ω the Laplace operator with Dirichlet or Neumann boundary condition forsome subset Ω ⊂ Rn.

METHOD OF WAVE EQUATION IN HARMONIC ANALYSIS AND PDE’S 9

(c) Uniformly elliptic second order operator in divergence form. For any f ∈C∞c (Rn) we define

Lf(x) =n∑

i,j=1

∂i(aij(x)∂jf).

Uniform ellipticity assumption means that

C|ξ|2 ≤n∑

i,j=1

aij(x)ξjξi ≤ c|ξ|2

for all x ∈ Rn and ξ ∈ Rn.(d) Laplace-Beltrami operator on complete Riemannian manifolds.(e) Hodge Laplacian operator on complete Riemannian manifolds and other op-

erators acting on forms.(f) Schrodinger operators

∆ + V

where V : Rn → R is a real potential function.(g) The operators corresponding to local or strongly local Dirichlet quadratic

forms and in particularly MMD, measurable metric Dirichlet spaces.

6. Heat and integral kernels

Let (X, d, µ) be a metric measure space, that is µ is a Borel measure with respectto the topology defined by the metric d. Next let B(x, r) = y ∈ X, d(x, y) < r bethe open ball with center x ∈ X and radius r > 0. For 1 ≤ p ≤ +∞, we denote thenorm of a function f ∈ Lp(X, dµ) by ‖f‖p, by 〈., .〉 the scalar product in L2(M,dµ),and if T is a bounded linear operator from Lp(X, dµ) to Lq(X, dµ), 1 ≤ p, q ≤ +∞,we write ‖T‖p→q for the operator norm of T .

If there is a locally integrable function KT : X ×X → C such that

〈Tf1, f2〉 =

∫X

Tf1 f2 dµ =

∫X

KT (x, y) f1(y) f2(x) dµ(y) dµ(x)

for all f1 and f2 in Cc(X), then we say that T is a kernel operator with kernel KT .It is well known that if T is bounded from L1(X) to Lq(X), where q > 1, then T isa kernel operator, and

‖T‖L1→Lq = supy∈X‖KT (·, y)‖Lq ;

vice versa, if T is a kernel operator and the right hand side of the above inequalityis finite, then T is bounded from L1(X) to Lq(X), even if q = 1.

One of especially important instant of this notion is a heat kernel, that is thekernels corresponding to the semigroup operators exp(−zL) which are often denoteby pt. More precisely

e−tLf(x) =

∫X

pt(x, y)f(y)dµ(y), f ∈ L2(X,µ), µ− a.e. x ∈M.

In the Euclidean space Rn, pt is given by the classical Gauss-Weierstrass kernel:

pt(x, y) =1

(4πt)n/2exp

(−|x− y|

2

4t

), t > 0, x, y ∈ Rn.

10 ADAM SIKORA

On general non-compact manifolds, where of course no such formula is available,the subject of upper estimates of the heat kernel has led to an intense activity in thelast few decades (see for instance [6], [7], [13], [14] for references and background).

7. Theorems of Phragmen-Lindelof type

Our discussion of finite propagation speed for the solutions of the wave equationrequires some rather simple observation related to the Phragmen-Lindelof theorem.Let us start with stating the Phragmen-Lindelof theorem for sectors.

Theorem 7.1. Let S be the open region in C bounded by two rays meeting at anangle π/α, for some α > 1/2. Suppose that F is analytic on S, continuous on S,and satisfies |F (z)| ≤ C exp(c|z|β) for some β ∈ [0, α) and for all z ∈ S. Then thecondition |F (z)| ≤ B on the two bounding rays implies |F (z)| ≤ B for all z ∈ S.

For the proof see [10, Theorem 7.5, p.214, vol.II] or [17, Lemma 4.2, p.108]. Propo-sition 7.2 is a simple consequence of Theorem 7.1.

Proposition 7.2. Suppose that F is an analytic function on C+. Assume that, forgiven numbers A,B, γ > 0, a ≥ 0,

(6) |F (z)| ≤ B, ∀ z ∈ C+, and

(7) |F (t)| ≤ Aeate−γt , ∀ t ∈ R+.

Then

(8) |F (z)| ≤ B exp(−Re

γ

z

), ∀ z ∈ C+.

Proof. Consider the function

(9) u(ζ) = F

(γ

ζ

),

which is also defined on C+. By (6),

|u(ζ)eζ | ≤ B exp |ζ|, ∀ ζ ∈ C+.

Again by (6) we have, for any ε > 0,

(10) supReζ=ε

|u(ζ)eζ | ≤ Beε.

By (7),

(11) supζ∈[ε,∞)

|u(ζ)eζ | ≤ Aeaγ/ε.

Hence, by Phragmen-Lindelof theorem with angle π/2 and β = 1, applied to

S+ε = z ∈ C : Rez > ε and Imz > 0

and

S−ε = z ∈ C : Rez > ε and Imz < 0,one obtains

supReζ≥ε

|u(ζ)eζ | ≤ maxAeaγ/ε, Beε, ∀ ε > 0.

METHOD OF WAVE EQUATION IN HARMONIC ANALYSIS AND PDE’S 11

Now by the Phragmen-Lindelof theorem with angle π and β = 0,

(12) supReζ≥ε

|u(ζ)eζ | ≤ Beε, ∀ ε > 0.

Letting ε→ 0 we obtain

supReζ>0

|u(ζ)eζ | ≤ B.

This proves (8) by putting ζ = γz.

Note that the estimate (8) does not depend on constants A, a in (7).

8. Finite speed propagation of solutions of wave eqaution

Set

Dρ = (x, y) ∈ X ×X : d(x, y) ≤ ρ.Given an operator T from Lp(X) to Lq(X), we write

(13) suppKT ⊆ Dρif 〈Tf1, f2〉 = 0 for all open sets Ui ⊂ M , fi ∈ L2(Ui, dµ), i = 1, 2, where r =d(U1, U2). Note that if T is an integral operator with a kernel KT , then (13) coincideswith the standard meaning of suppKT ⊆ Dρ, that is KT (x, y) = 0 for all (x, y) /∈ Dρ.

Given a non-negative self-adjoint operator L on L2(X). We say that L satisfiesthe finite speed propagation property if

(14) suppKcos(t√L) ⊆ Dt ∀t > 0 .

For U1, U2 ⊂ M open subsets of M , let d(U1, U2) = infx∈U1,y∈U2 d(x, y). We saythat the family exp(−zL) : z ∈ C+ satisfies the Davies-Gaffney estimate if

(15) |〈exp(−tL)f1, f2〉| ≤ exp

(−r

2

4t

)‖f1‖2‖f2‖2

for all t > 0, Ui ⊂M , fi ∈ L2(Ui, dµ), i = 1, 2 and r = d(U1, U2). Note that we onlyassume that (15) holds for positive real t.

Theorem 8.1. Let L be a self-adjoint positive definite operator acting on L2(X).Then conditions (14) and (15) are equivalent.

Remark. The connection of the heat and the wave equation has a long history(see [1, 11], see also [16] and the third proof of [6, Theorem 3.2, p. 157]). For theorigin of the L2 Gaussian estimates (15) so-called the Davies or the Davies-Gaffneyestimates see [3].

Proof. Suppose that supp fn ⊆ Un for n = 1, 2, and that 0 ≤ r < d(U1, U2). Put

u(z) =< exp(−L/(4z))f1, f2 > .

L is a self-adjoint positive definite operator so u is an analytic function on the complexhalf-plane Re z > 0, continuous and bounded on the set z ∈ C : Re z ≥ 0, z 6= 0,and

sup Re z=0 |er2zu(z)| ≤ ‖f1‖L2(TX)‖f2‖L2(TX).

By (15)

sup z∈R+ |er2zu(z)| ≤ C‖f1‖L2(TX)‖f2‖L2(TX).

12 ADAM SIKORA

Hence, by Phragmen-Lindelof theorem for an angle (see [10, Theorem 7.5, p.214, vol.II] or [17, Lemma 4.2, p.107])

|er2zu(z)| ≤ ‖f1‖L2(TX)‖f2‖L2(TX)

and

(16) |u(z)| ≤ exp(−r2Re z)‖f1‖L2(TX)‖f2‖L2(TX)

for all z such that Re z > 0. Next we note that

(17) < exp(−sL)f1, f2 >=

∫ ∞0

< cos(t√L)f1, f2 >

2√πse−

t2

4s dt.

By change of variable t :=√t in integral (17) and putting s := 1/(4s) we get

(18) s−1/2 < exp(− L

4s

)f1, f2 >= 2

∫ ∞0

(πt)−1/2 < cos(√t√L)f1, f2 > e−st dt,

so the function v(z) = z−1/2u(z) is a Fourier-Laplace transform of the function

w(t) = (√πt)−1 < cos(

√t√L)f1, f2 >. Now by (16) and the Paley-Wiener Theorem

(Theorem 7.4.3 [8])

(19) supp w ⊆ [r2,∞).

This proves that (15) implies (14). Now if (14) holds, then by (17)

| < exp(−sL)f1, f2 > | ≤∫ ∞

0

| < cos(t√L)f1, f2 > |

2√πse−

t2

4s dt

=

∫ ∞r

| < cos(t√L)f1, f2 > |

2√πse−

t2

4s dt ≤ ‖f1‖L2(X,µ)‖f2‖L2(X,µ)

∫ ∞r

2√πse−

t2

4s dt

≤ e−r2

4s ‖f1‖L2(X,µ)‖f2‖L2(X,µ).

The following lemma is a very simple but useful consequence of (14).

Lemma 8.2. Assume that L satisfies (14) and that F is the Fourier transform of

an even bounded Borel function F with supp F ⊆ [−r, r]. Then supp KF (√L) ⊆ Dr.

Proof. Since F is even, by the Fourier inversion formula,

F (√L) =

1

2π

∫ +∞

−∞F (t) cos(t

√L) dt.

But supp F ⊆ [−r, r] and Lemma 8.2 follows from (14).

9. Davies-Gaffney estimates

A slightly different form of Davies-Gaffney estimate is mostly considered in theliterature (see for instance [3] or [6]): in our notation, it reads

(20) |〈exp(−tL)χU1 , χU2〉| ≤ exp

(−r

2

4t

)√µ(U1)µ(U2)

METHOD OF WAVE EQUATION IN HARMONIC ANALYSIS AND PDE’S 13

where χU denotes the characteristic function of the set U . Of course, (20) followsfrom (15) by taking f1 = χU1 and f2 = χU2 . Conversely, assume (20) and letfi =

∑j c

ijχAij , where Aij ⊂ Ui. Then

< exp(−tL)f1, f2 > ≤∑j

∑`

|c1jc

2` |(µ(A1

j)µ(A2`))

1/2 exp

(−d2(A1

j , A2`)

4t

)≤

∑j

∑`

|c1jc

2l |(µ(A1

j)µ(A2`))

1/2 exp

(−d

2(U1, U2)

4t

).

By spectral theorem < exp(−zL)f1, f2 >≤ ‖f1‖2‖f2‖2. Proposition 7.2 then yields(15) for such f1, f2, and one concludes by density.

One may wonder what is the justification of the constant 4 in (15); we shall seein Theorem 8.1 below that in the case where exp(−zL) is a semigroup e−zL, 4 is thegood normalisation between the operator L and the distance d, namely it translatesthe fact that the associated wave equation has propagation speed 1.

Examples. As we already said, condition (15) holds for all kinds of self-adjoint, el-liptic, second order like operators. Condition (15) is well-known to hold for Laplace-Beltrami operators on all complete Riemannian manifolds. More precisely, Condi-tion (20) is proved for such operators in [3] and [6]. See also the remark after Theorem9.3. In the more general setting of Laplace type operators acting on vector bundles,condition (15) is proved in [16]. Another important class of semigroups satisfyingcondition (15) are semigroups generated by Schrodinger operators with real potentialand magnetic field (see for example [15], as well as Theorem 9.3 below).

Estimates (15) also hold in the setting of local Dirichlet forms (see for example[5, Theorem 2.8], and also [18], [19]). In this case the metric measure spaces underconsideration are possibly not equipped with any differential structure. However, thesemigroups associated with these Dirichlet forms do satisfy in general Davies-Gaffneyestimates with respect to an intrinsic distance.

9.1. Self-improving properties of Davies-Gaffney estimates. It is enough totest (15) on balls only. Then we observe that any additional multiplicative constantor even additional exponential factor in (15) can be replaced by the 1.

Lemma 9.1. Suppose that (M,d, µ) is a separable metric space and that the analyticfamily exp(−zL) : z ∈ C+ of bounded operators on L2(M,dµ) satisfies condition(15) restricted to all balls Ui = B(xi, ri), i = 1, 2, for all x1, x2 ∈ M , r1, r2 > 0.Then it satisfies condition (15) for all open subsets U1, U2.

Proof. Let U1 and U2 be arbitrary open subsets of M ; set r = d(U1, U2). Let

f =∑k

i=1 fi, where for all 1 ≤ i ≤ k, fi ∈ L2(B(xi, ri), dµ), B(xi, ri) ⊂ U1, and

fi1(x)fi2(x) = 0 for all x ∈ M , 1 ≤ i1 < i2 ≤ k. Similarly let g =∑`

j=1 gj where

gj ∈ L2(B(yj, sj), dµ), B(yj, sj) ⊂ U2 for all 1 ≤ j ≤ `, and gj1(x)gj2(x) = 0 for allx ∈ M , 1 ≤ j1 < j2 ≤ `. Note that d(B(xi, ri), B(yj, sj)) ≥ r. Now if condition (15)

14 ADAM SIKORA

holds for balls then

|〈exp(−tL)f, g〉| = |〈exp(−tL)k∑i=1

fi,∑j=1

gj〉|

=k∑i=1

∑j=1

|〈exp(−tL)fi, gj〉|

≤k∑i=1

∑j=1

e−r2

4t ‖fi‖2‖gj‖2

≤ e−r2

4t

(k∑i=1

‖fi‖2

)(∑j=1

‖gj‖2

)

≤ e−r2

4t

√k`

(k∑i=1

‖fi‖22

)1/2(∑j=1

‖gj‖22

)1/2

= e−r2

4t

√k`‖f‖2‖g‖2.

Now if we put F (z) = 〈exp(−zL)f, g〉 then Proposition 7.2 shows that the termCkl in the above inequality can be replaced by 1. This means that (15) holds forf and g. Now to finish the proof of the lemma, it is enough to note that, sinceM is separable, the space of all possible finite linear combinations of functions fsuch that suppf ⊂ B(x, r) ⊂ U is dense in L2(U, dµ). Moreover, if f =

∑ki=1 fi and

fi ∈ L2(B(xi, ri), dµ) for all 1 ≤ i ≤ k then there exist functions fi ∈ L2(B(xi, ri), dµ)

such that f =∑k

i=1 fi and in addition, for all 1 ≤ i1 < i2 ≤ k, fi1(x)fi2(x) = 0 forall x ∈M .

Lemma 9.2. Suppose that, for some C ≥ 1 and some a > 0,

(21) |〈exp(−tL)f1, f2〉| ≤ Ceate−r2

4t ‖f1‖2‖f2‖2, ∀t > 0,

whenever fi ∈ L2(M,dµ), suppfi ⊆ B(xi, ri), i = 1, 2, and r = d(B(x1, r1), B(x2, r2)).Then the family exp(−zL) : z ∈ C+ satisfies condition (15).

Proof. Lemma 9.2 is a straightforward consequence of Proposition 7.2 and Lemma 9.1.

Let us give an application of Lemma 9.2 by giving yet another example whereDavies-Gaffney estimates hold, namely Schrodinger semigroups with real potential.Suppose that ∆ is the non-negative Laplace-Beltrami operator on a Riemannianmanifold M with Riemannian measure µ and geodesic distance d, and consider theoperator ∆ + V acting on C∞c (M), where V ∈ L1

loc(M,dµ). If we assume that∆ + V ≥ 0 then we can define the Friedrichs extension of ∆ + V , which with someabuse of notation we also denote by ∆ + V (see for example [2, Theorem 1.2.8].

Theorem 9.3. Suppose that ∆ is the Laplace-Beltrami operator on a Riemannianmanifold M , that V ∈ L1

loc(M,dµ) and that ∆ + V ≥ 0 as a quadratic form. Thenthe semigroup exp(−zL) = exp(−z(∆ + V)) : z ∈ C+ satisfies condition (15).

METHOD OF WAVE EQUATION IN HARMONIC ANALYSIS AND PDE’S 15

Proof. We start our proof with the additional assumption V ≥ 0. For f ∈ L2(M,dµ),t > 0, x ∈ M , we put ft(x) = f(t, x) = exp(−t(∆ + V))f(x). Let κ > 0, and afunction ξ ∈ C∞(M), both to be chosen later, such that |∇ξ| ≤ κ, where ∇ is theRiemannian gradient on M . Next, as in [4, 6, 16], we consider the integral

E(t) =

∫M

|f(t, x)|2eξ(x) dµ(x).

Then

E ′(t)

2= Re

∫M

∂tf(t, x)f(t, x)eξ(x) dµ(x) = −Re

∫M

((∆ + V)ft) fteξ dµ

= −Re

∫M

(∇ft · ∇(fte

ξ) + |ft|2Veξ)dµ

= −Re

∫M

(|∇ft|2 +∇ft · ft∇ξ + |ft|2V

)eξ dµ

≤∫M

(−|∇ft|2 + |∇ft||∇ξ||ft|

)eξ dµ

≤ 1

4

∫M

|ft|2|∇ξ|2eξ dµ ≤κ2E(t)

4

(note that the non-negativity of V is used in the first inequality). Hence E(t) ≤exp(κ2t/2)E(0).

Consider now two disjoints open sets U1 and U2 in M . Choose ξ = κd(., U1). Onehas |∇ξ| ≤ κ, ξ ≡ 0 on U1, and, for any g ∈ L2

loc(M,dµ),∫U2

|g|2eξ dµ ≥ eκr∫U2

|g|2 dµ,

where r = d(U1, U2). Hence if supp f ⊆ U1 then, taking g = ft,∫U2

|ft|2 dµ ≤ e−κrE(t) ≤ exp

(κ2t

2− κr

)E(0) = exp

(κ2t

2− κr

)∫U1

|f |2 dµ.

Choosing finally κ = r/t we obtain∫U2

| exp(−tL)f |2 dµ ≤ exp

(−r

2

2t

)∫U1

|f |2 dµ,

that is, for all f ∈ L2(U1, dµ),

supg∈L2(U2,dµ), ‖g‖2=1

|〈exp(−tL)f, g〉|2 =

∫U2

|Ψ(t)f |2 dµ ≤ exp

(−r

2

2t

)‖f‖2

2,

which yields (15). Next, we consider a potential V ∈ L1loc(M) such that ∆ + V ≥ 0.

We put Va(x) = maxV(x),−a and La = ∆ + Va. When a goes to ∞ then Laconverges to L = ∆ + V in the strong resolvent sense (see [9, Theorem VIII.3.3,p.454] or [12, Theorem S.16 p.373]). Hence by [12, Theorem VIII.20, p.286] or by [9,Theorem VIII.3.11, p.459 and Theorem IX.2.16, p.504 ], exp(−tLa)f converges toexp(−tL)f = exp(−t(∆ + V))f for any f ∈ L2(M). Therefore it is enough to prove(15) for exp(−tLa)f , with a given in R. Finally we note that Va + a ≥ 0, thus itfollows from the first part of the proof that

exp(−t(∆ + Va + a)) = e−at exp(−tLa)

16 ADAM SIKORA

satisfies condition (15). But this implies that the semigroup exp(−tLa) satisfiescondition (21) and Theorem 9.3 follows from Lemma 9.2.

Remark : Note that the case V = 0 is allowed in Theorem 9.3, in other wordsit yields a proof of (15) for the Laplace-Beltrami operator on complete Riemannianmanifolds.

10. Davies-Gaffney estimates for Hodge Laplacian

Suppose that M is a complete Riemannian manifold and µ is an absolutely con-tinuous measure with a smooth density not equal to zero at any point of M . ByΛkT ∗M we denote the bundle of k-forms on M . For fixed β, β∗ ∈ L2(Λ1T ∗M) andγ ∈ L2(ΛkT ∗M) We define the operator L (L = Lβ,β∗,γ) acting on L2(ΛkT ∗M) bythe formula

(22) 〈Lω, ω〉 =

∫M

|dkω + ω ∧ β|2 + |dn−k ∗ ω + ∗ω ∧ β∗|2 + | ∗ ω ∧ γ|2 dµ(x),

where ω is a smooth compactly supported k-form and ∗ is the Hodge star operator.With some abuse of notation we also denote by L its Friedrichs extension. Note thatfor example the Hodge Laplacian and Schrodinger operators with electromagneticfields can be defined by (22)).

Theorem 10.1. The operator L defined by (22) satisfies (14) and (15).

Proof. We put ωt(x) = ω(t, x) = exp(−tL)ω. Then we fix some function ξ ∈ C∞(M)such that |dξ| ≤ κ and we consider the integral

E(t) =

∫M

(ω(t, x), ω(t, x))eξ(x) dµ(x).

Next we note that for every k-form η and 1-form ζ we have |ζ∧η|2+|ζ∧∗η|2 = |η|2|ζ|2and

E ′(t)

2= Re

∫M

(∂tω(t, x), ω(t, x))eξ dµ(x) = −Re

∫M

(Lωt, ωteξ) dµ

= −Re

∫M

[(dkωt + ωt ∧ β, dk(ωteξ) + (ωte

ξ) ∧ β) + (∗ωt ∧ γ, ∗ωt ∧ γ)eξ]

dµ

−Re

∫M

(dn−k ∗ ωt + ∗ωt ∧ β∗, dn−k(∗ωteξ) + (∗ωteξ) ∧ β∗) dµ

= −∫M

[|dkωt + ωt ∧ β|2eξ + |dn−k ∗ ωt + ∗ωt ∧ β|2eξ + | ∗ ωt ∧ γ|2eξ

]dµ

−Re

∫M

(dkωt + ωt ∧ β, d0ξ ∧ ωt)eξ dµ−∫M

(dn−k ∗ ωt + ∗ωt ∧ β∗, d0ξ ∧ ∗ωt)eξ dµ

≤ −∫M

[|dkωt + ωt ∧ β|2eξ + |dn−k ∗ ωt + ∗ωt ∧ β∗|2eξ + | ∗ ωt ∧ γ|2eξ

]dµ

+

∫M

[|dkωt + ωt ∧ β|2eξ + |dn−k ∗ ωt + ∗ωt ∧ β∗|2eξ + | ∗ ωt ∧ γ|2eξ

]dµ

+1

4

∫M

[|d0ξ ∧ ωt|2eξ + |d0ξ ∧ ∗ωt|2eξ

]dµ =

1

4

∫M

|ωt|2|d0ξ|2eξ dµ ≤ κ2E(t)

4.

METHOD OF WAVE EQUATION IN HARMONIC ANALYSIS AND PDE’S 17

Hence E(t) ≤ exp(κ2t/2)E(0). Now we say that ξ ∈ Θκ ⊆ C∞(M) if ξ(x) = 0 forx ∈ B(x1, r1) and |dξ| ≤ κ. Next assume that 0 ≤ r < ρ(x1, x2)− (r1 + r2). Then

supξ∈Θκ

∫B(x2,r2)

|ω|2eξ dµ ≥ erκ∫B(x2,r2)

|ω|2 dµ.

Hence if supp ω0 ⊆ B(x1, r1) then∫B(x2,r2)

|ωt|2 dµ ≤ exp(κ2t

2− rκ

)∫M

|ω0|2 dµ

Putting κ = r/t we get

(23)

∫B(x2,r2)

|ωt|2 dµ ≤ exp(−r2

2t

)∫M

|ω0|2 dµ = exp(−r2

2t

)∫B(x1,r1)

|ω0|2 dµ

Now (15) is a straightforward consequence of (23).

11. Gaussian bounds

In this section we assume that X satisfies the doubling property, i.e. there existsa constant C such that

(24) µ(B(x, 2r)) ≤ Cµ(B(x, r))

uniformly for all x ∈ X and for all r > 0. Note also that (24) implies that there existpositive constants C and D such that

(25) µ(B(x, γr)) ≤ C(1 + γ)Dµ(B(x, r)) ∀γ > 0, x ∈ X, r > 0.

In the sequel the value D always refers to the constant in (25).Before we discuss our main result in this section we discuss the following very

simply but useful technical results.

Theorem 11.1. Let X be a measurable metric space with the doubling conditionand let L be a self-adjoint positive definite operator. The following conditions areequivalent:

(26) ‖Kexp(−tL)(x, · )‖2L2(X) ≤ Cµ(B(x, t1/2))−1 ∀t > 0, x ∈ X;

(27) ‖K(I+tL)−m/4(x, · )‖2L2(X) ≤ Cmµ(B(x, t1/2))−1 ∀t > 0, x ∈ X

for any m > D, where D is the constant from condition (25) .

Proof. Note that

(I + (tL))−m/4 =1

Γ(m/4)

∫ ∞0

e−s sm/4−1 exp(−s(tL)) ds.

Hence by (25)

‖K(I+tL)−m/4(x, · )‖L2(X) ≤1

Γ(m/4)

∫ ∞0

e−s sm/4−1‖Kexp(−tsL)(x, · )‖L2(X) ds

≤ 1

Γ(m/4)

∫ ∞0

e−s sm/4−1µ(B(x, (st)1/2))−1/2 ds

≤ 1

Γ(m/4)µ(B(x, t1/2))−1/2

∫ ∞0

e−s sm/4−1(1 + 1/s)D/4 ds

= Cµ(B(x, t1/2))−1/2.

18 ADAM SIKORA

To prove that (27) implies (26) we note that

‖Kexp(−tL)(x, · )‖L2(X) ≤ ‖ exp(−tL)(1 + tL)m‖L2→L2‖K(I+tL)−m(x, · )‖L2(X)

≤ supλ∈R+

e−tλ(1 + tλ)m‖K(I+tL)−m(x, · )‖L2(X) ≤ C‖K(I+tL)−m(x, · )‖L2(X).(28)

Theorem 11.2. Suppose that for some number N ∈ N and points x, y ∈ X thereexist functions Vx, Vy : R+ 7→ R such that

(29) ‖K(I+t2L)−N/4(z, · )‖L2(X) ≤ Vz(t) ∀t > 0, z = x, y.

Then, there exists a constant CN such that for all t < d(x, y)2

Kexp(−tL)(x, y) ≤ CNVx

(t

d(x, y)

)Vy

(t

d(x, y)

)exp(−d(x,y)2

4t

)d(x, y)t−1/2

.(30)

Thus if L satisfies (26) or (27), then

|Kexp(−tL)(x, y)| ≤ Cµ(B(x,

t

d(x, y)

))− 12µ(B(y,

t

d(x, y)

))− 12

exp(−d(x,y)2

4t

)d(x, y)t−1/2

(31)

for all t < d(x, y)2.

Proof. For s > 1, we define the family of functions φs by the formula

φs(x) = ψ(s(|x| − s)),

where ψ ∈ C∞(R) and

ψ(x) =

0 if x ≤ −11 if x ≥ −1/2 .

Finally we define functions Fs and Rs by the following formula

Fs(x) =1√4π

exp (−x2

4)−Rs(x) = φs(x)

1√4π

exp (−x2

4)

so that Fs(λ) + Rs(λ) = exp(−λ2) and

(32) Fs(√tL) + Rs(

√tL) = exp(−tL).

Integration by parts N times yields∫φs(x)e−

x2

4 e−ixλ =

∫ ( 1

x/2 + iλ

(. . .( 1

x/2 + iλφs(x)

)′. . .)′)′

︸ ︷︷ ︸N

e−x2

4−iλxdx.

Hence for any natural number N and s > 1

(33) |Fs(λ)| ≤ C ′N1

s(1 + λ2/s2)N/2e−

s2

4 ,

where C ′N is a constant depending only on N . Next we note that supp Rs ⊆ [−s +12s, s − 1

2s], so if we put sxy = d(x, y)t−1/2, then Lemma 8.2 KdRsxy (

√tL)(x, y) = 0.

METHOD OF WAVE EQUATION IN HARMONIC ANALYSIS AND PDE’S 19

Hence by (32) 1

(34) Kexp(−tL)(x, y) = KdFsxy (√tL)(x, y).

Now let Jsxy be a function such that Jsxy(λ)2 = Fsxy(t1/2λ). By (33)

supλ≥0

∣∣∣∣Jsxy(λ)

(1 +

λ2t2

d(x, y)2

)N/4∣∣∣∣ ≤ Cexp

(−d(x,y)2

8t

)√d(x, y)t−1/2

.

Note that

(35) ‖KF1F2(L)(x, · )‖L2(X) ≤ ‖F1‖L∞‖KF2(L)(x, · )‖L2(X).

Now by (35)

‖|KJsxy (√L)(x, · )|‖L2(X) ≤ C

exp(−d(x,y)2

8t

)√d(x, y)t−1/2

∥∥∥∣∣∣K“I+ t2L

d(x,y)2

”−N/4(x, · )∣∣∣∥∥∥L2(X)

≤ CVx

(t

d(x, y)

)exp(−d(x,y)2

8t

)√d(x, y)t−1/2

.(36)

Note that Note also that

KF1F2(L)(x, y)| ≤ ‖KF1(L)(x, · )‖L2(X)‖KF2(L)( · , y)‖L2(X).(37)

Finally by (37)(38)|Kexp(−tL)(x, y)| = |KdFsxy (

√tL)(x, y)| ≤ ‖KJsxy (

√L)(x, · )‖L2(X)‖KJsxy (

√L)(y, · )‖L2(X)

and (30) follows from (36) and (38).

12. Spectral multipliers

We make the following assumptions about L and (X, d, µ):

• The space X is separable and has dimension n in the sense of the volumegrowth of balls: that is, there exist constants 0 < c1 < c2 <∞ such that

(39) c1ρn ≤ µ(B(x, ρ)) ≤ c2ρ

n

for every x ∈ X and ρ > 0;• cos(t

√L) satisfies finite speed propagation in the sense that

(40) supp cos(t√

L) ⊂ Dt := (z1, z2) ⊂ X ×X | d(z1, z2) ≤ |t|.

The meaning of this statement is that 〈f1, cos(t√

L)f2〉 = 0 whenever supp f1 ∈B(z1, ρ1), supp f2 ∈ B(z2, ρ2) and |t|+ ρ1 + ρ2 ≤ d(z1, z2).• L satisfies restriction estimates, which come in a strong and a weak form.

We say that L satisfies Lp to Lp′

restriction estimates for all energies if thespectral measure dE√L(λ) maps Lp(X) to Lp

′(X) for some p satisfying 1 ≤

p < 2 and all λ > 0, with an operator norm estimate

(41)∥∥dE√L(λ)

∥∥Lp(X)→Lp′ (X)

≤ Cλn(1/p−1/p′)−1, for all λ > 0.

1(34) shows that the remainder Rsxy(√

tL) does not contribute to the value of the heat kernelKexp(−tL)(x, y). Subtracting the remainder from the heat propagator is the main idea of the proof.

20 ADAM SIKORA

Lemma 12.1. Suppose that (x, d, µ) satisfies (39) and S is an bounded linear oper-ator from Lp(X)→ Lq(X) such that

suppS ⊂ Dρfor some ρ > 0. Then for any any 1 ≤ p < q ≤ ∞ there exists a constant C = Cp,qsuch that

‖S‖p→p ≤ Cρn(1/p−1/q)‖S‖p→q.

Proof. We fix ρ > 0. Then first we choose a sequence xn ∈ M such that d(xi, xj) >ρ/10 for i 6= j and supx∈X infi d(x, xi) ≤ ρ/10. Such sequence exists because M is

separable. Second we define Bi by the formula

(42) Bi = B(xi,

ρ

10

)−(∪j<iB

(xj,

ρ

10

)),

where B (x, ρ) = y ∈ M : d(z, z′) ≤ ρ. Third we put χi = χ eBi , where χ eBi is the

characteristic function of set Bi. Fourth we define the operator Mχi by the formulaMχig = χig.

Note that for i 6= j B(xi,ρ20

) ∩B(xi,ρ20

) = ∅. Hence

K = supi

#j; d(xi, xj) ≤ 2ρ ≤ supx

|B(x, 2ρ)|∣∣B (x, ρ20

)∣∣ < 40nc2

c1

<∞.

It is not difficult to see that

Dρ ⊂ ∪i,j; d(xi,xj)<2ρBi × Bj ⊂ D4ρ

so

Sf =∑

d(xi,xj)<2ρ

MχiSMχjf.

Hence by Holder inequality

‖Sf‖pp =∥∥ ∑d(xi,xj)<2ρ

MχiSMχjf∥∥pLp

=∑i

∥∥ ∑j; d(xi,xj)<2ρ

MχiSMχjf∥∥pp

≤∑i

|Bi|p(1/p−1/q)∥∥ ∑j; d(xi,xj)<2ρ

MχiSMχjf∥∥pq

≤ Cρnp(1/p−/q)∑i

∥∥ ∑j; d(xi,xj)<2ρ

MχiSMχjf∥∥pq

≤ CKp−1ρnp(1/p−1/q)∑i

∑j; d(xi,xj)<2ρ

∥∥MχiSMχjf∥∥pq

≤ CKpρnp(1/p−1/q)∑j

∥∥SMχjf∥∥pq

≤ CKpρnp(1/p−1/q)‖S‖pp→q∑j

∥∥Mχjf∥∥pp

= CKpρnp(1/p−1/q)‖S‖pp→q‖f‖ppThis finishes the proof of Lemma 12.1.

METHOD OF WAVE EQUATION IN HARMONIC ANALYSIS AND PDE’S 21

Theorem 12.2. Suppose that (X, d, µ) and L satisfy (39) and (40), and that Lsatisfies Lp to Lp

′restriction estimates for all energies, (41), for some p with 1 ≤

p < 2. Let s > n(1/p − 1/2) be a Sobolev exponent. Then there exists C dependingonly on n, p, s, and the constant in (41) such that, for every even F ∈ Hs(R)

supported in [−1, 1], F (√

L) maps Lp(X)→ Lp(X), and

(43) supα>0

∥∥F (α√

L)∥∥p→p ≤ C‖F‖Hs .

Proof of Theorem 12.2. We first assume that L satisfies Lp to Lp′

restriction esti-mates for all energies. We take η ∈ C∞c (−4, 4) even and such that∑

l∈Z

η( t

2l)

= 1 for all t 6= 0.

Then we set φ(t) =∑

l≤0 η(2−lt),

F0(λ) =1

2π

∫ +∞

−∞φ(t)F (t) cos(tλ) dt

and

(44) Fl(λ) =1

2π

∫ +∞

−∞η( t

2l)F (t) cos(tλ) dt.

Note that by virtue of the Fourier inversion formula

F (λ) =∑l≥0

Fl(λ)

and by Lemma 8.2

suppFl(α√

L) ⊂ D2l+2α.

Now by Lemma 12.1,

(45)∥∥F (α

√L)∥∥p→p ≤

∑l≥0

∥∥Fl(α√L)∥∥p→p ≤ C

∑l≥0

(2lα)n(1/p−1/2)∥∥Fl(α√L)

∥∥p→2

.

Unfortunately, Fl is no longer compactly supported. To remedy this we choose afunction ψ ∈ C∞c (−4, 4) such that ψ(λ) = 1 for λ ∈ (−2, 2) and note that∥∥Fl(α√L)

∥∥p→2≤∥∥(ψFl)(α

√L)∥∥p→2

+∥∥((1− ψ)Fl)(α

√L)∥∥p→2

.

To estimate the norm ‖ψFl(α√

L)‖p→2 we use our restriction estimates (41). Usinga T ∗T argument and the fact that suppψ ⊂ [−4, 4], we note that∥∥ψFl(α√L)

∥∥2

p→2=∥∥|ψFl|2(α

√L)∥∥p→p′ ≤

∫ 4/α

0

|ψFl(αλ)|2∥∥dE√L(λ)

∥∥p→p′ dλ

≤ C

α

∫ 4

0

|ψFl(λ)|2∥∥dE√L(λ/α)

∥∥p→p′ dλ.(46)

It follows from the above calculation and (41) that

(47) αn(1/p−1/2)∥∥ψFl(α√L)

∥∥p→2≤ C‖ψFl‖2,

for all α > 0. As a consequence, we obtain∑l≥0

2ln(1/p−1/2)αn(1/p−1/2)∥∥ψFl(α√L)

∥∥p→2≤∑l≥0

2ln(1/p−1/2)‖ψFl‖2

22 ADAM SIKORA

for all α > 0. Now let us recall that by definition of Besov space∑l≥0

2ln(1/p−1/2)‖ψFl‖2 ≤∑l≥0

2ln(1/p−1/2)‖Fl‖2 = ‖F‖Bn(1/p−1/2)1,2

.

See, e.g., [20, Chap. I and II] for more details. We also recall that if s > s′ thenHs ⊂ Bs′

1,2 and ‖F‖Bn(1/p−1/2)1,2

≤ Cs‖F‖Hs for all s > n(1/p − 1/2), see again [20].

Therefore, we have shown that

(48)∑l≥0

2ln(1/p−1/2)αn(1/p−1/2)∥∥ψFl(α√L)

∥∥p→2≤ C‖F‖Hs .

Now it follows from standard Gaussian bounds that

(49)∥∥E√L[0, λ]

∥∥Lp(X)→Lp′ (X)

≤ Cλn(1/p−1/p′), λ ≥ λ0

with a uniform C. (Here E√L[0, λ] is the same as χ[0,λ](√L).)

Next we obtain bounds for the part of estimate (45) corresponding to the term

‖(1−ψ)Fl(α√

L)‖p→2. This only requires the spectral projection estimates (49). Wewrite

|(1− ψ)Fl|2(α√

L) =

∫ ∞0

∣∣(1− ψ)(αλ)Fl(αλ)∣∣2dE√L(λ)

= −∫ ∞

0

( ddλ

∣∣(1− ψ)(αλ)Fl(αλ)∣∣2)E√L(λ) dλ

= −∫ ∞

0

( ddλ

∣∣(1− ψ)(λ)Fl(λ)∣∣2)E√L(λ/α) dλ.

Hence, using (49),

(50)∥∥(1− ψ)Fl(α

√L)∥∥2

p→2≤ C

∫ ∞0

( ddλ|(1− ψ)(λ)Fl(λ)|2

)(λα

)n(1/p−1/p′)dλ.

We write

Fl(λ) =1

2π

∫eit(λ−λ

′)η( t

2l)F (λ′) dλ′ dt,

use the identityeit(λ−λ

′) = i−N(λ− λ′)−N(d/dt)Neit(λ−λ′),

and integrate by parts N times. Note that if λ ∈ supp 1 − ψ and λ′ ∈ suppF thenλ ≥ 2 and λ′ ≤ 1, and hence λ− λ′ ≥ λ/2. It follows that∣∣((1− ψ)Fl)(λ)

∣∣ ≤ Cλ−N2−N(l−1)‖F‖2

with C independent of N . Similarly,∣∣ ddλ

((1− ψ)Fl)(λ)∣∣ ≤ Cλ−N2−N(l−1)2l‖F‖2.

Using this in (50) with N sufficiently large and l ≥ 2, we obtain

(2lα)n(1/p−1/2)∥∥((1− ψ)Fl)(α

√L)∥∥p→2≤ C2−l‖F‖2.

Therefore, we have

(51)∑l

(2lα)n(1/p−1/2)∥∥((1− ψ)Fl)(α

√L)∥∥p→2≤ C‖F‖2 ≤ C‖F‖Hs .

Equations (45), (48) and (51) prove (43).

METHOD OF WAVE EQUATION IN HARMONIC ANALYSIS AND PDE’S 23

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