The Heat Kernel — Regularity of Operator

6.1 Regularity of Operator

We will fix a manifold NN, which will be thought of as M\partial M, and consider the manifold N×R0N \times \mathbb {R}_{\geq 0}. We will write yNy \in N and uR0u \in \mathbb {R}_{\geq 0}. We let EE be a Hermitian vector bundle on NN.

Let AA be a first-order self-adjoint elliptic operator on EE, and consider the differential operator

D=u+A. D = \frac{\partial }{\partial u} + A.

This is the differential operator of interest on N×R0N \times \mathbb {R}_{\geq 0}. While the definition resembles that of a heat operator, it may not be the best idea to think of it as a heat operator. We will later introduce a further time variable tt and consider etDDe^{-t D^*D}, and confusion might arise.

The basic problem we want to solve is the equation

Df=g. Df = g.

For now, we assume gΓc(E×R0)g \in \Gamma _ c(E \times \mathbb {R}_{\geq 0}), and later extend to more general functions after we establish the right bounds.

As before, since AA is self-adjoint and elliptic, we can find an eigenbasis {ψλ}\{ \psi _\lambda \} of AA. We write

f(y,u)=λfλ(u)ψλ(y),g(y,t)=λgλ(u)ψλ(y). f(y, u) = \sum _\lambda f_\lambda (u) \psi _\lambda (y),\quad g(y, t) = \sum _\lambda g_\lambda (u) \psi _\lambda (y).

The we see that for the equation to hold, we must have

(ddu+λ)fλ(u)=gλ(u). \left(\frac{\mathrm{d}}{\mathrm{d}u} + \lambda \right) f_\lambda (u) = g_\lambda (u).

We have previously decided that the boundary condition we want is fλ(0)=0f_\lambda (0) = 0 for λ0\lambda \geq 0. There is another way we can think about this boundary condition. Observe that the solution to this differential equation is well defined up to adding CeλuC e^{-\lambda u}. If λ<0\lambda < 0, then there is at most one solution with reasonable growth as uu \to \infty . However, if λ0\lambda \geq 0, then we can add any multiples of eλue^{-\lambda u} and have a sensible solution (λ=0\lambda = 0 is somewhat of an edge case). Thus, fixing fλ(0)=0f_\lambda (0) = 0 gives us a unique “sensible” solution.

For convenience, we write PP for the composition

H1(E×R0)rH0(E×{0})H0(E×{0}), H^1(E \times \mathbb {R}_{\geq 0}) \overset {r}{\to } H^0(E \times \{ 0\} ) \to H^0(E \times \{ 0\} ),

where rr is restriction and the last map is the spectral projection onto the subspace spanned by the eigenvectors with non-negative eigenvalues. Then our boundary condition is Pf=0Pf = 0. By an abuse of notation, we write 1P1 - P for rPr - P, and the boundary condition for DD^* will then by (1P)f=0(1 - P)f = 0. We shall write Γ(E×R0;P)\Gamma (E\times \mathbb {R}_{\geq 0}; P) for the space of smooth sections ff such that Pf=0Pf = 0.

Imposing this boundary condition, we can write down explicit solutions for fλf_\lambda :

fλ(u)={0ueλ(vu)gλ(v)  dvλ0ueλ(vu)gλ(v)  dvλ<0. f_\lambda (u) = \begin{cases} \displaystyle \int _0^ u e^{\lambda (v - u)} g_\lambda (v) \; \mathrm{d}v & \lambda \geq 0\\ \displaystyle -\int _ u^\infty e^{\lambda (v - u)} g_\lambda (v) \; \mathrm{d}v & \lambda < 0 \end{cases}.

We check that for large uu, the function fλ(u)f_\lambda (u) is either exponentially decreasing (if λ>0\lambda > 0), constant (if λ=0\lambda = 0) or identically zero (if λ<0\lambda < 0). The constant term wrecks our hope that this takes values in HsH^ s, but we can still hope it takes values in HlocsH^ s_{\mathrm{loc}}.

Note also that for λ<0\lambda < 0, the value of fλ(u)f_\lambda (u) depends on the future values of gλg_\lambda , which ties in with our previous point — at large uu, all future values of gλg_\lambda are zero, hence fλ(u)=0f_\lambda (u) = 0.

We let Q:Γc(E×R0)Γ(E×R0;P)Q\colon \Gamma _ c(E \times \mathbb {R}_{\geq 0}) \to \Gamma (E \times \mathbb {R}_{\geq 0}; P) be the function that sends gλg_\lambda to fλf_\lambda above, which is clearly linear. The next proposition will, in particular, show that this map is well-defined, i.e. QfQf is actually smooth:

Proposition 6.1

QQ extends to a continuous map HsHlocs+1H^ s \to H^{s + 1}_{\mathrm{loc}}.

Proof

Recall that since AA is elliptic, we have

fHs+1C(fHs+AfHs). \| f\| _{H^{s + 1}} \leq C (\| f\| _{H^ s} + \| Af\| _{H^ s}).

So we can equivalently define the H1H^1 norm by

fH12=fL22+ufL22+AfL22. \| f\| _{H^1}^2 = \| f\| ^2_{L^2} + \| \partial _ u f\| ^2_{L^2} + \| Af\| ^2_{L^2}.

Using that the eigenspace decomposition is orthogonal in L2L^2 and the defining equations

Aψλ=λψλ,fλu=gλλfλ, A \psi _\lambda = \lambda \psi _\lambda ,\quad \frac{\partial f_\lambda }{\partial u} = g_\lambda - \lambda f_\lambda ,

we get an inequality

fH12Cλ(1+λ2)fλL22+gλL22. \| f\| _{H^1}^2 \leq C \cdot \sum _\lambda (1 + \lambda ^2) \| f_\lambda \| ^2_{L^2} + \| g_\lambda \| ^2_{L^2}.

To bound fλL2\| f_\lambda \| _{L^2} in terms of gλL2\| g_\lambda \| _{L^2}, we use the (rotated) Laplace transform

f^λ(ξ)=0eiuξgλ(u)  du. \hat{f}_\lambda (\xi ) = \int _0^\infty e^{-iu\xi } g_\lambda (u) \; \mathrm{d}u.

One checks easily that up to a constant which we omit, we have Parseval's identity:

f^λL2=fλL2. \| \hat{f}_\lambda \| _{L^2} = \| f_\lambda \| _{L^2}.

We can explicitly compute

f^λ(ξ)=g^λ(ξ)+fλ(0)λ+iξ. \hat{f}_\lambda (\xi ) = \frac{\hat{g}_\lambda (\xi ) + f_\lambda (0)}{\lambda + i \xi }.

Using Parseval's identity, and the fact that

fλ(0)=0eλugλ(u)  du if λ<0, f_\lambda (0) = -\int _0^\infty e^{\lambda u} g_\lambda (u) \; \mathrm{d}u\text{ if } \lambda < 0,

we obtain bounds

λfλL22gλL2. |\lambda | \| f_\lambda \| _{L^2} \leq 2 \| g_\lambda \| _{L^2}.

So if we ignore the λ=0\lambda = 0 term, QQ would send H0H^0 into H1H^1. The f0f_0 term is eventually constant, given by the integral of the compactly supported function 0g0(v)  dv\int _0^\infty g_0(v)\; \mathrm{d}v. So QQ maps H0H^0 into Hloc1H^1_{\mathrm{loc}}.

Using the equation

dsdus(ddu+λ)fλ=dsgλdus \frac{\mathrm{d}^ s}{\mathrm{d}u^ s} \left(\frac{\mathrm{d}}{\mathrm{d}u} + \lambda \right) f_\lambda = \frac{\mathrm{d}^ s g_\lambda }{\mathrm{d}u^ s}

and calculating as above, we see that HsH^ s gets mapped into Hlocs+1H^{s + 1}_{\mathrm{loc}}.

The main takeaway from the section is then the following theorem:

Theorem 6.2

There is a linear operator

Q:Γc(E×R0)Γ(E×R0;P) Q\colon \Gamma _ c(E \times \mathbb {R}_{\geq 0}) \to \Gamma (E \times \mathbb {R}_{\geq 0}; P)

such that

DQg=g for all gΓc(E×R0),QDf=f for all fΓc(E×R0;P). \begin{aligned} DQg & = g\text{ for all }g \in \Gamma _ c(E \times \mathbb {R}_{\geq 0}),\\ QDf & = f\text{ for all }f \in \Gamma _ c(E \times \mathbb {R}_{\geq 0}; P). \end{aligned}

Moreover, QQ extends to a continuous map HsHlocs+1H^ s \to H^{s + 1}_{\mathrm{loc}} for all integral ss. It is given by convolution with a kernel Qu(y,z)Q_ u(y, z), where uu is now allowed to take negative values. The kernel is smooth away from u=0u = 0.