Schwartz kernel theorem - Local spectral asymptotics and heat kernel bounds for Dirac and Lapla

u²+ (∂_tu)²+ |∇_xu|²

dx (0 ≤ t < r₀) in the same manner as above, then apply Gronwall’s lemma appropriately.

2.5 Schwartz kernel theorem

Let U ⊂ R^d be an open set. The Schwartz kernel theorem (e.g. [79, Thm. 5.2.1]) says that every continuous linear map T from C_c^∞(U )³ to its dual spaceD⁰(U )⁴ is uniquely determined by an element K_T ∈D⁰(U × U )in the following way:

hT φ, ψi = hK_T, ψ ⊗ φi (φ, ψ ∈ C_c^∞(U )).

Here h·, ·i denotes the dual between C_c^∞(U )and D⁰(U )or C_c^∞(U × U ) and D⁰(U × U ). One calls K_T the integral or operator kernel of T . If S is a bounded linear operator on L²(U ), then it follows immediately from

|hSφ, ψi_L2(U )| ≤ kSkq

hφ, φi_L2(U )

hψ, ψi_L2(U )

that the restriction T of S onto C_c^∞(U )⁵ is a continuous linear map from C_c^∞(U )⁶ toD⁰(U )⁷. Since C_c^∞(U ) is a dense subset of L²(U ), we see that S is uniquely determined by T or its integral kernel K_T. For simplicity we do not distinguish S and T , and write K_Sfor K_T. In general, let T be a continuous linear map from C_c^∞(U ; C^N)to its dual spaceD⁰(U ; C^N), or be a bounded linear operator on L²(U ; C^N). For each fixed pair (i, j), 1 ≤ i, j ≤ N , one can introduce a

3Here C_c^∞(U )is endowed with the following topology structure: a sequence of functions {φn}^∞_n=1 in C_c^∞(U ) is said to converge to φ0 in C_c^∞(U )if there exists a compact subset of U containing all of the supports of φnas subsets, and {φn}^∞n=1converges to φ in C^k(U )for all k ∈ N.

4An element ofD⁰(U )is also called a distribution on U . The topology structure onD⁰(U )is endowed as follows:

a sequence of distributions {Zn}^∞n=1on U is said to converge to Z0inD⁰(U )if Zn(φ) → Z0(φ)for all φ ∈ Cc^∞(U ).

5Here C_c^∞(U )is only regarded as a subset of L²(U ).

6Here C_c^∞(U )is endowed with its own standard topology structure, not the one inherited from L²(U ).

7This is due to the fact that L²(U )is continuously embedded inD⁰(U ).

continuous linear map T_ij from C_c^∞(U )toD⁰(U )by defining

where φ appears at the j-th position in



, ψ appears at the i-th position in



then have a matrix of integral kernels

K_T=

which is called the integral or operator kernel of T. Obviously, T is uniquely determined by its integral kernel because of linearity:

We end this section with a few remarks.

First, let U₂ be another open set in a possibly different Euclidean space R^d². The original Schwartz kernel theorem actually says that every continuous linear map T from C_c^∞(U ) to D⁰(U2)is uniquely determined by an element K_T ∈D⁰(U2× U ) in the following way:

hT φ, ψi = hK_T, ψ ⊗ φi (φ ∈ C_c^∞(U ), ψ ∈ C_c^∞(U2)).

Following the previous argument, one can define integral kernel for continuous linear maps from C_c^∞(U ; C^N)toD⁰(U₂; C^N²)or bounded linear operators from L²(U ; C^N)to L²(U₂; C^N²), where N2 is an arbitrary positive integer.

Second, we should remind the reader that the Schwartz kernel theorem is a local statement.

For example, given an operator acting on smooth sections of a vector bundle, one can induce locally-defined operators from one coordinate system to another, and define the corresponding integral kernels. This has been done many times in §2.2.2 and §2.2.4.

Finally, note that in various situations the integral kernels can be realized partially or globally as continuous or smooth (scalar or matrix-valued) functions. For example, the integral kernel of any pseudo-differential operator on U is smooth off the diagonal, and the Dirichlet heat kernel for U is smooth on U × U for any fixed time t > 0.

Spectral counting functions

Let M be a closed smooth Riemannian manifold of dimension d and metric g. Let E be a smooth complex hermitian vector bundle over M . As usual we denote by C^∞(M ; E)the space of smooth sections of E, and by L²(M ; E) the Hilbert space of square integrable sections equipped with the natural inner product defined by the hermitian structure on the fibres and the metric measure µ_g on M .

We first recall some basic facts about operators of Laplace type. A second order partial differential operator P : C^∞(M ; E) → C^∞(M ; E)is said to be of Laplace type if its principal symbol σ_P is of the form σ_P(ξ) = g_x(ξ, ξ)id_E_x for all covectors ξ ∈ T_x^∗M. In local coordinates this means that P is of the form

P = −g^ij(x)∂i∂j+ a^k(x)∂k+ b(x), (3.1) where a^k, b are smooth matrix-valued functions, and we have used Einstein’s sum convention.

Given a Laplace type operator P , it is known that there exist a unique connection ∇ on E and a unique bundle endomorphism V ∈ C^∞(M ; End(E)) such that P = ∆^∇+ V. We assume that P is self-adjoint and non-negative. Thus there exists an orthonormal basis {φ_j}^∞_j=1 for L²(M ; E)consisting of smooth eigensections such that P φ_j = λ²_jφj,where λ_j are chosen to be non-negative and correspond to the eigenvalues of the operator√

Let A be a classical pseudo-differential operator of order m ∈ R. The (microlocalized) spectral counting function NA(λ)of P is defined as

NA(λ) = X

λj<λ

hAφ_j, φji. (3.2)

Let χ ∈ S (R) be a Schwartz function such that the Fourier transform F χ of χ is 1 near the origin and supp(F χ) ⊂ (−δ, δ), where δ is a positive constant smaller than half the radius of injectivity of M . It is well known (e.g. [40, 84, 85, 132, 133, 162] for various special cases) that

(χ ∗ N_A⁰)(λ) =

∞

j=1

hAφ_j, φjiχ(λ − λ_j) ∼

∞

k=0

Ak(A, P )λ^d+m−k−1 (λ → ∞), (3.3)

where the spectral counting coefficientsAk(A, P )do not depend on the choice of χ, and are lo-cally computable in terms of the local full symbols of A and P . This can be derived from studying the Fourier integral operator representation of Ae^−it

√

P via the stationary phase method.

Apart from the Fourier integral operator representation method, there exist several other ways to recover the mollified spectral counting coefficientsAk(A, P ).

First, the (microlocalized) spectral zeta function ζ(s, A, P ) is defined by ζ(s, A, P ) = X

λj>0

hAφ_j, φ_ji

λ^s_j (Re(s) > d + m). (3.4)

It is well known (e.g. [40, 162]) that ζ(s, A, P ) admits a meromorphic continuation to C whose only singularities are simple poles at s = d + m − k (k = 0, 1, 2, . . .) with residuesAk(A, P ).

Second, the Mellin transform of

tr(Ae^−tP) − X

λj=0

hAφ_j, φji (t ∈ (0, ∞))

admits a meromorphic continuation ζ(2s, A, P )Γ(s) to C whose singularities can be completely determined from those of ζ(s, A, P ) and Γ(s). Here Γ(s) denotes the classical Gamma function.

After establishing a suitable vertical decay estimate for ζ(2s, A, P )Γ(s), one can deduce from the inverse Mellin transform theorem the following widely used heat expansion (e.g. [64, 67, 68, 101, 137])

tr(Ae^−tP) ∼

∞

k=0

Bk(A, P )t^k−d−m² +Ck(A, P )t^klog(t) +Dk(A, P )t^k

(t → 0⁺). (3.5) The above notation system may bring confusion to the reader as it could happen that there are non-negative integers k such that ^k−d−m₂ are non-negative integers. In this case one can simply setBk(A, P ) = 0, thus (3.5) is well-defined. The relation between the mollified counting coefficients and some of the heat coefficients can be summarized as follows:

Case 1: If the order m of A is an integer, then

• Bk(A, P ) = ^Γ(

d+m−k 2 )

2 ·Ak(A, P ) (d + m − k is positive or negative but odd);

• Ck(A, P ) = 0 (d + m + 2k < 0);

• Ck(A, P ) = ⁽⁻¹⁾_2·k!^k+1 ·Ad+m+2k(A, P ) (d + m + 2k ≥ 0).

Case 2: If the order m of A is not an integer, then for all non-negative integers k:

• Bk(A, P ) = ^Γ(

d+m−k 2 )

2 ·Ak(A, P );

• Ck(A, P ) = 0.

Thus the heat expansion (3.5) contains all information about {Ak(A, P )}^∞_k=0.

In exactly the same way, the following resolvent trace expansion (e.g. [65, 67, 137]) tr(A(1+tP )⁻^N²) ∼

∞

k=0

B_k^{(N )}(A, P )t^k−d−m² +C_k^{(N )}(A, P )t^klog(t)+D_k^{(N )}(A, P )t^k(t → 0⁺) (3.6)

also contains all information about {Ak(A, P )}^∞_k=0, where N is any complex number such that Re(N ) > max{d + m, 0}. Similar to the unambiguousness of (3.5), one can setB^{(N )}_k (A, P ) = 0 whenever ^k−d−m₂ is a non-negative integer to guarantee (3.6) is well-defined.

To summarize, there exist at least four different ways, such as studying

• spectral counting functions,

• spectral zeta functions,

• heat expansions, and

• resolvent trace expansions,

to retrieve all the information about {Ak(A, P )}^∞_k=0. For example, using parametrix constructions in any of these methods results in the well-known leading term

A0(A, P ) = 1 (2π)^d

T₁^∗M

Tr(σ_A). (3.7)

In the second chapter we discussed the concepts of invariantly-defined principal and sub-principal symbols. In theory one can use parametrix constructions in any of these methods to expressA1(A, P )in terms of the principal and sub-principal symbols of both A and P .

The mollified spectral counting coefficientsAk(A, P )do not depend on the choice of χ, and are locally computable in terms of the local full symbols of A and P . But as we do not have invariantly-defined concepts of “sub-sub-principal symbol”, “sub-sub-sub-principal symbol” and so on, it is not so convenient to regardAk(A, P ) (k ≥ 2) from global viewpoint. In particular, the expressions of Ak(A, P ) normally involve many summands of derivatives of the local full symbols of A and P , thus their geometric meanings are not easy to be retrieved.

In this chapter we will see that the Wodzicki residue can provide a clear interpretation of Ak(A, P )for all k ≥ 0. Actually, there exist smooth functions f_k(A, P )on M such that

Ak(A, P ) = Z

fk(A, P )dµg (3.8)

for all k ≥ 0. In practice, one can extract microlocal information about P from f_k(A, P ) with A ranging all classical pseudo-differential operators or endomorphisms of the given bundle.

In the next chapter we will specialize in Dirac type operators. Let D be a self-adjoint Dirac type operator. There exists a discrete spectral resolution {φ_j, µj}^∞_j=1 of D, where {φ_j}^∞_j=1 is an orthonormal basis for L²(M ; E), and Dφ_j = µ_jφ_j for each j. Obviously, φ_jwill be eigensections of P = P with eigenvalues µ²_j. Therefore, using the notation from before λ_j = |µ_j|. By setting

B^±= Sign(D) ± Id_E

2 ,

we see that Ak(B^±, P )carry microlocal information about the positive (negative) spectrum of D. Later on we will extract this information from studying f_k(F B^±, P )with F ranging all smooth endomorphisms of E.

This chapter is arranged as follows. In Sections 3.1, 3.3 and 3.4 we provide proofs of (3.3), (3.5) and (3.6), respectively, and in Section 3.2 we determine the singularity structure of spectral zeta functions (3.4). The author should clearly state that he does not claim any originality over these four classical results. All of the other sections are devoted to providing explicit formulae forAk(A, P ). This will be performed by two methods: one is Wodzicki’s residue in Section 3.5, the other is complex powers of elliptic operators in Section 3.7. In Section 3.8 we specialize in A1(A, P ). By the way, we study a case where A is a partial differential operator in Section 3.6.

3.1 FIO method

Formula (3.3) is essentially Proposition 2.1 in [40], Corollary 2.2 in [84], Theorem 2.2 in [133]

and Proposition 1.1 in [162], except the authors either consider scalar operators or assume A is of order zero. Recall that χ ∈S (R) is chosen so that F χ = 1 near the origin and supp(F χ) ⊂ (−δ, δ), where δ is smaller than half the radius of injectivity of M . If t is sufficiently small, say

|t| < δ₁ < δ, then locally the integral kernel (Ae^−it

√P)(t, x, y)of the operator Ae^−itP^1/2 is well known to have the form

(Ae^−it

√

P)(t, x, y) = 1 (2π)^d

R^d

a(t, x, y, ξ)eiθ(t,x,y,ξ)dξ,

where a is a classical (matrix-valued) symbol of order m. This can be seen from (2.32) and the rule of product between a classical pseudo-differential operator and a Fourier integral operator.

The scalar-valued phase function θ is of the form

θ(t, x, y, ξ) = ψ(x, y, ξ) − tq0(y, ξ), where q₀ denotes the (scalar) principal symbol of√

ψ(x, y, ξ) = (x − y) · ξ + O(|x − y|²|ξ|).

For details see (2.25) and (2.31). It is also known that tr(Ae^−it

√

P)is smooth in (−δ, δ)\{0}, so we introduce a cut-off function % ∈S (R) satisfying %(t) = 1 if |t| < ^δ₂¹ and supp(%) ⊂ (−δ₁, δ₁).

Using integration by parts one gets (χ ∗ N_A⁰ )(λ) = 1

2π Z

(F χ)(t)(%(t) + 1 − %(t))tr(Ae^−it^√^P)e^iλtdt

= 1

(2π)^d+1 Z

R^d

(F χ)(t)%(t)Tr(a(t, y, y, ξ))e^−itq⁰^(y,ξ)e^iλtdydξdt + o(λ^−∞) (λ → ∞),

where o(λ^−∞)is short for o(λ^−h)for any positive integer h. Consider I(y, λ) =

R^d

(F χ)(t)%(t)Tr(a(t, y, y, ξ))e^−itq⁰^(y,ξ)e^iλtdξdt.

We here pass to polar coordinates by putting ξ = λrω, |ω| = 1, dξ = λ^dr^d−1drdω. Then I(y, λ) = λ^d+m

S^d−1

R⁺

(F χ)(t)%(t)Tr(a(t, y, y, ω))r^d+m−1e^iλ(t−trq⁰^(y,ω))drdti dω.

Here we apply the stationary phase method to the two-dimensional drdt integral. The phase function is Φ(r, t) = t − trq₀(y, ω), whose unique critical point close to t = 0 is given by

(r₀, t₀) = 1 q₀(y, ω), 0

At this critical point the Hessian matrix Φ⁰⁰of the phase function satisfies det(Φ⁰⁰) = −q0(y, ω)² <

0. Applying the stationary phase method (e.g. [62, Prop. 2.3], [149, p. 344]) yields a full asymp-totic expansion for I(y, λ) and proves (3.3) as a consequence. We should mention that to cor-rectly apply the stationary phase method one needs to introduce a suitable cut-off function of the variable r. The details of this more careful treatment can be seen in [62, p.136].

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