We define \( H_j:\ \mathcal{S}(X) \longrightarrow \mathcal{S}(X\times Y^+) \) by its action on the frequent space of \( X \), in particular, set \[ \tilde{H_j h}(\xi,\eta):= H_j(\xi,y) \tilde h(\xi) \] where \( \tilde f\) is the partial (in \( x \)) Fourier transform of \( f \) and \( H_j(\xi,y) \) is given by \[ H_j(\xi,y):= (1-\psi(\xi))\int_\Gamma \sum_{\alpha= 0}^{m-1}e^\alpha_j(\xi) \frac{A_\alpha^+(\xi,\eta)}{A^+(\xi,\eta)}e^{i\eta y}d\eta \] where \( \Gamma\subset \mathbb{C} \) is a curve enclosing all zeros of \( A(\xi,\eta) \) with \( \im \eta >0\), \( (e^\alpha_j(\xi))_{\alpha,j} \) is the inverse matrix of \( (c^j_\beta(\xi))_{j,\beta} \) and \( A_\alpha^+(\xi,\eta):= \sum_{\beta=0}^{m-\alpha-1} a^+_{\alpha+\beta+1}(\xi)\eta^\beta\).

In fact \( G \) is defined as follows: \[ G_0(\xi,\eta):= \frac{1-w(\xi,\eta)}{A(\xi,\eta)} \] which is smooth at \( (0,0) \), where \( 1- w \) has a zero of infinite order. Then \( G_0(D): \mathcal{S}(X\times Y) \longrightarrow \mathcal{S}(X\times Y) \) extends to \( W^{k-r,p}(X\times Y) \longrightarrow W^{k,p}(X\times Y) \). Finally, take \( G=C_+ G E_+ \), which maps \( \mathcal{S}(X\times Y^+) \longrightarrow \mathcal{S}(X\times Y^+) \) by first extending a function to the whole plan, applying \( G_0 \) and finally cutting-off.

We will use the change of coordinates \( \tilde x_i = \lambda^{-\sigma/\sigma_i}x_i, \) which gives a diffeomorphism \( h_\lambda \) from \( B_{\epsilon_0} \) to \( B_{\epsilon_0/\lambda} \), in which the derivative operators are \[ \left(\frac{\partial}{\partial \tilde{x_i}}\right)^\alpha_i = \lambda^{\alpha_i\sigma/\sigma_i}\left(\frac{\partial}{\partial x_i}\right)^\alpha_i,\quad \tilde D^\alpha = \lambda^{\|\alpha\|}D^\alpha \]

The operator \( A \), viewed in \( h_\lambda \), i.e. the operator \( f\mapsto A(f\circ h_\lambda) \), is \( \sum_{\|\alpha\|\leq r} a_\alpha(\lambda^{\sigma/\sigma_i}\tilde{x_i}) \lambda^{-\|\alpha\|} \tilde D^\alpha \). We pose

where \( \varphi \) is radial in \( \tilde x \), equals \( 1 \) for \( \|\tilde x\|\leq 1 \) and \( 0 \) for \( \|\tilde x\| \geq 2 \).

The coefficient before \( \tilde D^\alpha \) of \( \tilde A_\lambda^* \) is \( \lambda^{r-\|\alpha\|} \left[\varphi(\tilde x) a_\alpha(\lambda^{\sigma/\sigma_i} \tilde{x_i}) + (1-\varphi(\tilde x))a_\alpha(0)\delta_{\|\alpha\|=r}\right]\) is the same as that of \( \tilde A_0 \) for \( \tilde x \) outside of \( B_2 \) and \( C^0 \)-converges to that of \( \tilde A_0 \) inside \( B_1 \). Hence for \( \lambda \) sufficiently small the corresponding "\( k,l \)" diagram of \( \tilde A^*_\lambda \) is exact, hence so is the diagram of \( \lambda^{-r}\tilde A_\lambda^* \). Choose \(\epsilon = \lambda\) and \( A^\# \) to be \( \lambda^{-r}\tilde A_\lambda^* \) viewed in \( X \) through \( h\lambda \).

Let \( \psi \) be a cut-off function that equals \( 1 \) on \( B_\delta \) and \( 0 \) outside of \( B_\epsilon \) and \( A^\# \) be the differential operator on \( X \) with exact "\( k,l \)" diagram given by Proposition prop:interior-cube which equals \( A \) on \( B_\epsilon \). The idea of the remaining computation is to use the exactness of \( A^\# \) on \( \psi f \) and the reason for which the local-global passage is not trivial is that the operator \( A^\# \) and the multiplication by \( \psi \) do not commute. The commutator \( [A^\#,\psi] \), however, is of weight at least \( 1 \) less than \( A \) and with the choice \( l\geq k-1 \) the norm \( \|[\psi, A^\#]f\|_{W^{k-r,p}(X)} \) is dominated by \( \|f\|_{W^{l,p}} \).

1. If \( f\in W^{k,p}(B_\epsilon) \) then \( \psi f\in W^{k,p}(B_\epsilon, \partial) \) and

   \begin{align*} \|f\|_{W^{k,p}(B_\delta)} &\leq \| \psi f\|_{W^{k,p}(X)} \leq C \left( \|A^\#\psi f\|_{W^{k-r,p}(X)} + \|\psi f\|_{W^{l,p}(X)} \right)\\ &\leq C \left( \|\psi A^\# f\|_{W^{k-r,p}(X)} + \| [\psi, A^\#] f \|_{W^{k-r,p}(X)} + \|\psi f \|_{W^{l,p}(X)} \right)\\ &\leq C' \left( \| Af \|_{W^{k-r,p}(B_\epsilon)} + \| f \|_{W^{l,p}(B_\epsilon)} \right) \end{align*}
2. Given \( f\in W^{l,p}(B_\epsilon) \) and \( Af\in W^{k-r,p}(B_\epsilon) \), one has \( \psi f\in W^{l,p}(X) \). Also, \( [A^\#, \psi] f \in W^{l-r+1,p}(X)\subset W^{k-r,p}(X) \) and \( \psi A^\# f = \psi A f\in W^{k-r,p}(X) \), therefore \( A^\#(\psi f)\in W^{k-r,p}(X) \). By exactness of \( A^\# \), one has \( \psi f\in W^{k,r}(X) \), so \( f\in W^{k,r}(B_\delta) \).
3. If \( g\in W^{l-r,p}(B_\delta/ \partial) \subset W^{l-r,p}(X) \), by exactness of \( A^\# \) we can find \( \tilde f \in W^{l,p}(X) \) such that \(g-A^\# \tilde f \in W^{k-r,p}(X)\). Choose \( f = \psi \tilde f \in W^{l,p}(B_\epsilon/ \partial)\) then \[ g- Af = g - A^\#(\psi \tilde f) = \psi(g- A^\# \tilde f) + [\psi, A^\#]\tilde f \in W^{k-r,p}(B_\epsilon) \] since \( \psi(g- A^\# \tilde f)\in W^{k-r,p}(B_\epsilon) \) and \( [\psi, A^\#]\tilde f \in W^{l-r+1,p}(B_\epsilon) \subset W^{k-r,p}(B_\epsilon)\).

We can suppose \( l\geq k-1 \), the general case follows using

Now covering \( M \) by finitely many charts of type \( B_\delta\subset B_\epsilon \) and \( B^+_\delta\subset B^+_\epsilon \) such that the interior of \( B_{\delta} \) and of \( B^+_{\delta} \) cover \( M \). Also, choose a partition of unity \( \sum \psi = 1 \) subordinated to \( B_\delta \) and \( B^+_{\delta} \). The exactness will be established if we can prove the analogue of the 2 last statements of Theorem thm:elliptic-const

For the regularity statement: If \( f\in W^{l,p}(M) \), \( Af\in W^{k-r,p}(M) \) and \( B^j f\in W^{k-r_j,p}(\partial M) \) then the same holds for \( \psi f \) in \( B_\epsilon \) and \( B_\epsilon^+ \) since \[ [A,\psi]f\in W^{l-r+1,p}\subset W^{k-r,p},\quad [B^j,\psi] f \in \partial W^{l-r_j+1,p}\subset \partial W^{k-r_j,p} \] Therefore \( \psi f \in W^{k,p}(B_\delta) \) or \( W^{k,p}(B^+_\delta) \) hence \( f\in W^{k,p}(M) \).

For the approximation: If \( g\in W^{l-r,p}(M) \) and \( h_j\in \partial W^{l-r_j,p}(\partial M) \) then \( \psi g\in W^{l-r,p}(B_\delta/ \partial) \) or \( W^{l-r,p}(B_\delta^+/ \partial_e) \) and \( \psi h_j\in \partial W^{l-r_j,p}(\partial_0 B^+_\delta / \partial)\). Then by Lemma lem:ell-loc-bndry, we can find \( \tilde f\in W^{l,p}(B_\epsilon/\partial) \) with \( \psi g - A \tilde f \in W^{k-r,p}(B_\epsilon/ \partial) \) or in a boundary cube \(\tilde f\in W^{l,p}(B^+_\epsilon/ \partial_e) \) with \( \psi g - A \tilde f \in W^{k-r,p}(B^+_\epsilon/ \partial_e) \) with \( \psi h_j - B^j \tilde f\in \partial W^{k-r_j,p}(\partial_0 B^+_\epsilon/ \partial) \). Then \( f:=\sum \tilde f \) makes sense and satisfies

Let \( \beta \leq \gamma \) be real numbers in \( [\alpha,\omega] \) and let \( \ker (\beta,\gamma) \) and \( \coker (\beta,\gamma) \) be the kernel and cokernel of operator \( \mathcal{C} \) on \( N\times [\beta,\gamma] \) with vanishing initial condition at \( \beta \). Since \( \dim \ker (\beta,\gamma) \) and \( \dim\coker (\beta,\gamma) \) are integer-valued, using the fact that \( \dim \ker (\beta,\omega) \) is decreasing in \( \beta \) and \( \dim\coker (\alpha,\gamma) \) is increasing in \( \gamma \), one can easily check that it suffices to show that the two functions are continuous in \( (\beta,\gamma) \) to prove that they are identically \( 0 \).

The following statements can be verified mechanically:

1. Monotonicity: \( \dim\ker(\beta,\gamma) \) is decreasing in \( \beta \), \( \dim\coker (\beta,\gamma) \) is increasing in \( \gamma \).
2. One-sided continuity: \( \dim \ker (\beta,\gamma) \) is left-continuous in \( \beta \), \( \dim\coker (\beta,\gamma) \) is right-continuous in \( \gamma \).
3. One-sided semi-continuity: \( \dim\ker (\beta,\gamma) \) is left upper semi-continuous in \( \gamma \), i.e. \[ \lim_{\gamma_1 \to \gamma_2^-} \inf \dim \ker (\beta,\gamma_1) \geq \dim\ker (\beta,\gamma_2) \] This is due to the left-continuity in first variable of \( \dim\ker \) and the exact sequence \[ 0 \longrightarrow \ker (\gamma_1,\gamma_2) \longrightarrow \ker(\beta,\gamma_2) \longrightarrow \ker(\beta,\gamma_1) \] where the last arrow is the restriction. Similar statement for \( \coker \): \[ \lim_{\beta_2 \to \beta_1^+} \dim \inf \coker (\beta_2,\gamma) \geq \dim\coker (\beta_1,\gamma) \]

This 3 statements suffice to finish the proof in the case where boundary conditions \( B^j \) on \( \partial_S M \) are of constant coefficients since \( \ker, \coker \) only depend on the difference \( \gamma -\beta \), up to a translation in time of the solutions.

In case \( B^j \) are of variable coefficients, the idea of making translation in time can be formulated using Index theory for Fredholm maps:

We recall that Fredholm maps between Banach spaces \( E, F \) are those in \( L(E,F) \) with closed image and finite dimensional kernel and cokernel. It is a classical result that

1. The set \( \mathcal{F} \) of Fredholm maps are open in \( L(E,F) \).
2. The index \( i(l):= \dim \ker l - \dim \coker l \) is continuous in \( \mathcal{F} \).

The difference \( \dim \ker (\beta,\gamma) -\dim\coker (\beta,\gamma) \) can be regarded as the index of a continuous family \( \mathcal{C}_{(\beta,\gamma)} \) of operators on the same space \( N\times [0,1] \) using the diffeomorphism \[ N\times [0,1] \xrightarrow{\sim} N\times [\beta,\gamma]. \] Hence \( \dim\ker (\beta,\gamma) - \dim \coker (\beta,\gamma) \) is constant. It follows that \( \dim \ker (\beta,\gamma) \) is both increasing and one-sided semi-continuous in \( \gamma \) hence is right-continuous in \( \gamma \), hence \( \dim \coker (\beta,\gamma) \) is continuous in \( \gamma \). Other continuities follows similarly.

Let us explain why the theorem is true in the case of no spatial boundary \(\partial N =\emptyset \). In this case, there is no distinction between \( l \) and \( l' \). Consider \( A \) as an elliptic operator on \( N\times [\tilde\pi,\omega] \) with \( \tilde\pi = \frac{\alpha +\pi}{2} \) and with no boundary operator, one has the following exact diagram: \[ \xymatrix{ W^{k,p}([\tilde\pi,\omega]) \ar@{^{(}->}[d] \ar@{->}[r]^<<<<<{{A}} & W^{k-r,p}([\tilde\pi,\omega]) \ar@{^{(}->}[d] \\ W^{l,p}([\tilde\pi,\omega]) \ar@{->}[r]^<<<<<{{A}} & W^{l-r,p}([\tilde\pi,\omega]) } \] Therefore the if \( f\in W^{l,p}([\alpha,\omega]) \) and \( Af \in W^{k-r,p}([\alpha,\omega]) \)then \( f \in W^{k,p}([\tilde\pi,\omega])\subset W^{k,p}([\pi,\omega]) \) and

It remains to check that we can get rid of the \( \| f \|_{W^{l,p}([\tilde\pi,\omega])} \) term on the right hand side. Suppose not, then there exists a sequence \( \{f_i\} \subset W^{l,p}([\alpha,\omega]) \) such that \( Af_i \to 0 \) in \(W^{k-r,p}([\alpha,\omega])\) and \( f_i \to 0\) in \( W^{l,p}([\alpha,\pi]) \) but \( \| f_i\|_{W^{l,p}([\tilde\pi,\omega])} = 1\). Then by \eqref{eq:reg-para}, \( \{f_i\} \) is a bounded sequence in \( W^{k,p}([\tilde\pi,\omega]) \) and, by Kondrachov's theorem, can be supposed to converge in \( W^{l,p}([\tilde\pi,\omega]) \) to a function \( \tilde f \) which has \( \| \tilde f \|_{W^{l,p}([\tilde\pi,\omega])} = 1 \) and \( A\tilde f = 0 \) on \( [\tilde\pi,\omega] \) because \( A \) commutes with the restriction. Moreover, since \( \|f_i\|_{W^{l,p}([\alpha,\pi])} \to 0 \), one has \( \tilde f\in W^{l,p}([\tilde\pi,\omega]/\tilde\pi) \) and the fact that \( \tilde f\ne 0 \) contradicts Theorem thm:para-causality).

Note that

is a direct sum of parabolic operators in each component, and hence an isomorphism, and

is a compact operator because it factors through \( W^{k+1}(N\times [\alpha,\omega]/\alpha) \). Therefore \( H+K \) is a Fredholm map with the same index as \( H \), which is \( 0 \). It is sufficient to check that the kernel of \( H+K \) is trivial.

Suppose that \( F = (f^1,\dots, d^n)\in \ker (H+K) \) then \( f^\alpha\in W^{2,p}(N\times [\alpha,\omega]/\alpha) \), so \( f^\alpha \) and \( \frac{\partial f^\alpha}{\partial x^i} \) are continuous function on \( N\times[\alpha,\omega] \). Since \[ \frac{\partial f^\alpha}{\partial t} + \Delta f^\alpha = -a^{\alpha i}_\beta \frac{\partial f^\beta}{\partial x^i} - b^\alpha_\beta f^\beta, \] by repeated use of Theorem thm:reg-parabolic the \( f^\alpha \) are smooth for \( t>\alpha \).

Let \( e:= \frac{1}{2}|F|^2 := \frac{1}{2}\sum_\alpha |f_\alpha|^2 \), then \( e \) is continuous on \( N\times [\alpha,\omega] \), vanishes on \( N\times \{\alpha\} \) and one has

where we used the inequality \( -u^2 - 2uv \leq v^2 \) to bound the second and third terms. We conclude that \( F=0 \) since \( e= 0 \) by the following Maximum principle.

We can suppose \( b \leq -1 \) and prove that \( \|f\|_{L^\infty(\partial_\omega M)} \leq \|f\|_{L^\infty(\partial_\alpha M)} \). Intuitively, this means that since heat spreads out, the largest density must be attained at time \( t=\alpha \). In fact, choose \( B = \max_M b + 1 \) and define \( \tilde f = f e^{-B(t-\alpha)} \) then \( \|\restr{\tilde f}{\alpha}\|_{L^\infty} = \| \restr{f}{\alpha} \|_{L^\infty} \) and \( \| \restr{\tilde f}{\omega}\|_{L^\infty} = e^{-B(\omega-\alpha)}\|\restr{f}{\omega}\|_{l^\infty} \). The function \( \tilde f \) satisfies the same heat equation \eqref{eq:infty-comparison} as \( f \), with \( b \) replaced by \( b-B \leq -1 \).

Now let us prove that under this supposition, \( |f| \) attains it maximum at time \( t=\alpha \). Since we can replace the solution \( f \) of \eqref{eq:infty-comparison} by \( -f \), we can suppose, for sake of contradiction, that \( |f| \) attains it maximum on \( N\times[\alpha,\omega] \) at \( (x^*, t^*) \) with \( |f(x^*,t^*)| = f(x^*,t^*) > 0 \) and \( t^* >\alpha \). Then one has

Plugging these in \eqref{eq:infty-comparison}, one has a contradiction.

Since \( L^1 \) is the dual space of \( L^\infty \), it is sufficient to prove that for all \( h\in C^\infty(N) \), one has \[ \int_{N\times \{\omega\}} fh \leq e^{B(\omega-\alpha)} \left\|\restr{f}{\alpha}\right\|_{L^1}. \|h\|_{L^\infty}. \]

Consider the backwards heat equation

which is just a heat equation on \( N\times[\alpha,\omega] \) with initial condition at \( \alpha \) if we pose \( \tilde g (t) := g(\omega+\alpha - t)\). The solution \( g \) exists and is smooth on \( N\times[\alpha,\omega] \). One has, at any time \( t \)

Therefore \[ \int_N f \frac{dg}{dt} + g \frac{df}{dt} = \int_N (a\nabla f)g - (\tilde a \nabla g)f + (b-\tilde b) fg \] Choose \( b=\tilde b \) and \( \tilde a = -a \) then the term \( (b-\tilde b)fg \) vanishes and the two first terms become \( \int_N \nabla_a (fg) = - \int_N fg \text{ div } a = 0 \) where \( \text{ div } a := \frac{\partial}{\partial x^i} a^i\) is the divergence. Therefore one has \( \frac{d}{dt}\int_N fg = 0 \), meaning that \[ \int_{N} \restr{f}{\omega}. h = \int_{N\times{\omega}} fg = \int_{N\times{\alpha}} fg \leq \|\restr{f}{\alpha} \|_{L^1}. \| h \|_{L^\infty} \leq e^{B(\omega-\alpha)} \|\restr{f}{\alpha}\|_{L^1}. \|h\|_{L^\infty} \] where we applied Theorem thm:infty-comparison to \( g \) (strictly speaking, to \( \tilde g \)) and \( B \) only depends on \( \tilde a = -a \) and \( \tilde b = b \).