$\bar{H}^{\bullet}I(\mathcal{F})$. Hence there is some net $\varphi_{l}\in
J(\mathcal{F})$ such that $R\alpha=\lim_{l}d_{\mathcal{F}}\varphi_{l}$ in
$J(\mathcal{F})$. We can assume $\varphi_{l}\in C^{\infty}_{\text{\rm
c}}(M^{1};\Lambda\mathcal{F})$ by the density of $C^{\infty}_{\text{\rm
c}}(M^{1};\Lambda\mathcal{F})$ in $J(\mathcal{F})$ (Section 2.6.7).

Using that $J(\mathcal{F})$ is compactly retractive (Section 2.6.7) and
arguing like in Section 5.2.1, we get that
$\\{R\alpha,d_{\mathcal{F}}\beta_{l}\\}_{l}$ is contained in some step
$J^{(s)}(\mathcal{F})$, and $R\alpha=\lim_{l}d_{\mathcal{F}}\varphi_{l}$ in
$J^{(s)}(\mathcal{F})$. Moreover, we can assume $d_{\mathcal{F}}\phi_{l}$ is a
sequence because $J^{(s)}(\mathcal{F})$ is a Fréchet space.

Consider the notation of Section 5.5.2. We have
$R\alpha=\lim_{l}d_{\mathcal{F}}\varphi_{l}$ in $J^{m}(\mathcal{F})$ by the
version of (2.5.36) for $J(\mathcal{F})$ (Section 2.6.7). We have
$\beta:=F_{m}\alpha\in ZJ^{(s^{\prime})}(\mathcal{F})\subset ZJ(\mathcal{F})$
by (5.5.5), obtaining a class
$\overline{[\beta]}\in\bar{H}^{\bullet}J(\mathcal{F})$.

Since the sequence $\varphi_{l}$ is in $C^{\infty}_{\text{\rm
c}}(M^{1};\Lambda\mathcal{F})$, it is also in $J^{m}(\mathcal{F})$ and in
$I^{(s^{\prime})}I(\mathcal{F})$, and we have $E_{m}\varphi_{l}=\varphi_{l}$
by the version of Corollary 2.5.6 with $J^{m}(\mathcal{F})$. Hence, by
Corollary 5.5.2,

$\beta=\alpha-
E_{m}R\alpha=\alpha-\lim_{l}E_{m}d_{\mathcal{F}}\varphi_{l}=\alpha-\lim_{l}d_{\mathcal{F}}E_{m}\varphi_{l}=\alpha-\lim_{l}d_{\mathcal{F}}\varphi_{l}$

in $I^{(s^{\prime})}(\mathcal{F})$, and therefore also in $I(\mathcal{F})$.
This shows that $\bar{\iota}_{*}(\overline{[\beta]})=\overline{[\alpha]}$ in
$\bar{H}^{\bullet}I(\mathcal{F})$.

######  Remark 5.5.4.

Using cohomology instead of reduced cohomology and a single element $\varphi$
instead of a net $\varphi_{l}$, the analogous argument gives $\ker
R_{*}\subset\operatorname{im}\iota_{*}$, obtaining $\ker
R_{*}=\operatorname{im}\iota_{*}$.

####  5.5.4. Injectivity of $\bar{\iota}_{*}$

Take any $\overline{[\alpha]}\in H^{\bullet}K(\mathcal{F})$ with
$\bar{\iota}_{*}(\overline{[\alpha]})=0$ in $\bar{H}^{\bullet}I(\mathcal{F})$.
Since $I(\mathcal{F})$ is compactly retractive,
$C^{\infty}(M;\Lambda\mathcal{F})$ is dense in $I(\mathcal{F})$ and every
$I^{(s)}(\mathcal{F})$ is a Fréchet space (Section 2.2.2), we get as above
that there is some $s$ and a sequence $\varphi_{l}$ in
$C^{\infty}(M;\Lambda\mathcal{F})$ such that $\alpha\in K^{(s)}(\mathcal{F})$
and $\alpha=\lim_{l}d_{\mathcal{F}}\varphi_{l}$ in $I^{(s)}(\mathcal{F})$.

Consider again the notation of Section 5.5.2. By (5.5.4),

$\alpha=F_{m}\alpha=\lim_{l}F_{m}d_{\mathcal{F}}\varphi_{l}=\lim_{l}d_{\mathcal{F}}F_{m}\varphi_{l}$

in $K^{(s)}(\mathcal{F})$, and therefore in $K(\mathcal{F})$. So
$\alpha\in\bar{B}K(\mathcal{F})=BK(\mathcal{F})$ by (5.1.3), and therefore
$[\alpha]=0$ in $H^{\bullet}K(\mathcal{F})$.

######  Remark 5.5.5.

Like in Remark 5.5.6, we also get the injectivity of $\iota_{*}$.

####  5.5.5. Surjectivity of $\bar{R}_{*}$

For any class $\overline{[\alpha]}\in\bar{H}^{\bullet}J(\mathcal{F})$, the
representative $\alpha$ is in some step $J^{(s)}(\mathcal{F})$, and therefore
it is also in $ZJ^{(s)}(\mathcal{F})$. With the notation of Section 5.5.2,
$\beta:=E_{m}\alpha\in ZI^{(s^{\prime})}(\mathcal{F})\subset ZI(\mathcal{F})$
by Corollary 5.5.2, and we have $R\beta=\alpha$. This shows that
$\overline{[\alpha]}=\bar{R}(\overline{[\beta]})$.

######  Remark 5.5.6.

Using cohomology instead of reduced cohomology, the analogous argument gives
the surjectivity of $R_{*}$.

###  5.6. Computations in the case of a suspension foliation

Consider the notation of Sections 4.2.1 to 4.2.5, where the case of a
suspension foliation with a simple foliated flow was considered.

####  5.6.1. Description of $(K(\mathcal{F}),d_{\mathcal{F}})$

For every $m\in\mathbb{N}_{0}$ and $s<-1/2$, consider the injection defined by
(2.6.38) for the vector bundle $\Lambda\mathcal{F}$,

(5.6.1) $C^{\infty}(L;\Lambda\otimes\Omega^{-1}NL)\to
K^{(s-m)}(M,L;\Lambda\mathcal{F})\;,\quad\alpha\mapsto\partial_{\rho}^{m}\delta_{L}^{\alpha}\;.$

######  Proposition 5.6.1.

Via (5.6.1), the operator $d_{\mathcal{F}}$ on
$K^{(s-m)}(M,L;\Lambda\mathcal{F})$ corresponds to the operator
$d_{L}-m{\eta\wedge}$ on $C^{\infty}(L;\Lambda\otimes\Omega^{-1}NL)$.

###### Proof.

Consider first the case $m=0$. According to (3.2.20), for some degree $v$,
take $\alpha\in C^{\infty}(L;\Lambda^{v}\otimes\Omega^{-1}NL)$ and $\beta\in
C^{\infty}_{\text{\rm c}}(M;\Lambda^{1,n-1-v})$. We can write
$\alpha=\alpha_{0}\otimes|\omega|^{-1}$ and $\beta=\beta_{0}\wedge\omega$ for
some $\alpha_{0}\in C^{\infty}(L;\Lambda^{v})$ and $\beta_{0}\in
C^{\infty}_{\text{\rm c}}(M;\Lambda^{0,n-1-v})$. By (2.9.5), (2.8.4), (2.9.8)
(or (2.9.2) and the Stokes’ theorem), (3.2.20) and (3.2.21), and since
$d\omega=\eta\wedge\omega$,

$\displaystyle\langle d_{\mathcal{F}}\delta_{L}^{\alpha},\beta\rangle$
$\displaystyle=-(-1)^{v}\langle\delta_{L}^{\alpha},d\beta\rangle=-(-1)^{v}\langle\delta_{L}^{\alpha},(d\beta_{0}+(-1)^{n-1-v}\beta_{0}\wedge\eta)\wedge\omega\rangle$
$\displaystyle=-(-1)^{v}\int_{L}\alpha_{0}\wedge((d+\eta\wedge)\beta_{0})|_{L}=-(-1)^{v}\int_{L}\alpha_{0}\wedge(d+\eta\wedge)(\beta_{0}|_{L})$
$\displaystyle=\int_{L}(d_{L}-\eta\wedge)\alpha_{0}\wedge\beta_{0}|_{L}=\int_{L}d_{L}\alpha\wedge\beta|_{L}=\langle\delta_{L}^{d_{L}\alpha},\beta\rangle\;.$

The general case follows from the previous case because
$d_{\mathcal{F}}\partial_{\rho}=\partial_{\rho}(d_{\mathcal{F}}-{\eta\wedge})$
on $C^{-\infty}(M;\Lambda\mathcal{F})$ by (4.2.8) and (4.2.9), and
$\eta\wedge\delta_{L}^{\alpha}=\delta_{L}^{\eta\wedge\alpha}$ by (2.8.2). ∎

######  Corollary 5.6.2.

Proposition 5.3.1 is true in this case.

###### Proof.

Apply (2.6.39), (2.6.40) and Proposition 5.6.1. ∎

####  5.6.2. A partial extension map on $M_{\pm}$

The notation of Sections 2.5.8, 2.5.9, 2.5.10, 2.5.11, 2.5.12 and 2.5.13,
concerning conormal distributions at the boundary, is also used here. By
Proposition 2.5.1, there is a continuous linear partial extension map,

$E_{m,\pm}:\mathcal{A}^{m}(M_{\pm};\Lambda\mathcal{F}_{\pm})\to\dot{\mathcal{A}}^{(s)}(M_{\pm};\Lambda\mathcal{F}_{\pm})\;,$

where $s=0$ if $m\geq 0$, and $m>s\in\mathbb{Z}^{-}$ if $m<0$. According to
the proof of Proposition 2.5.1 and Remarks 2.5.2 and 2.5.3, in the case
$0>m>s\in\mathbb{Z}^{-}$, the homomorphism $E_{m,\pm}$ can be given by the
composition

$\mathcal{A}^{m}(M_{\pm};\Lambda\mathcal{F}_{\pm})\xrightarrow{J^{N}}\mathcal{A}^{0}(M_{\pm};\Lambda\mathcal{F}_{\pm})\xrightarrow{E_{0,\pm}}\dot{\mathcal{A}}^{(0)}(M_{\pm};\Lambda\mathcal{F}_{\pm})\xrightarrow{\partial_{\rho}^{N}}\dot{\mathcal{A}}^{(s)}(M_{\pm};\Lambda\mathcal{F}_{\pm})\;,$

where $N=-s\in\mathbb{Z}^{+}$, $E_{0,\pm}$ is a continuous inclusion map, and
$J$ is the endomorphism of $C^{\infty}(M_{\pm};\Lambda\mathcal{F}_{\pm})$
given by

$J\alpha(\rho,y)=\int_{1}^{\rho}\alpha(\rho_{1},y)\,d\rho_{1}\;,$

using (2.5.32), (4.2.13) and the identity
$\Lambda_{(\rho,y)}\mathcal{F}\equiv\Lambda_{y}L$.

Consider also the endomorphism $\widetilde{J}$ of
$C^{\infty}(\widetilde{M}_{\pm};\Lambda\widetilde{\mathcal{F}}_{\pm})$ defined
like $J$,

$\widetilde{J}\tilde{\alpha}(\tilde{\rho},\tilde{y})=\int_{1}^{\tilde{\rho}}\tilde{\alpha}(\tilde{\rho}_{1},y)\,d\tilde{\rho}_{1}\;,$

using (4.2.12) and the identity
$\Lambda_{(\rho,\tilde{y})}\mathcal{F}_{\pm}\equiv\Lambda_{\tilde{y}}\widetilde{L}$.
Clearly, $\widetilde{J}$ corresponds to $J$ via

$\pi_{M_{\pm}}^{*}:C^{\infty}(M_{\pm};\Lambda\mathcal{F}_{\pm})\to
C^{\infty}(\widetilde{M}_{\pm};\Lambda\widetilde{\mathcal{F}}_{\pm})\;.$

Using (4.2.11), we can also write

(5.6.2)
$\widetilde{J}\tilde{\alpha}(\tau,\tilde{y})=e^{F(\tilde{y})}\int_{e^{-F(\tilde{y})}}^{\tau}\tilde{\alpha}(\tau_{1},\tilde{y})\,d\tau_{1}\;,$

with the change of variable $\tilde{\rho}_{1}=e^{F(\tilde{y})}\tau_{1}$,
because $\tilde{\rho}=\pm e^{F(\tilde{y})}\tau$.

For fixed $0<\epsilon<1$, consider the collar neighborhoods
$T_{\pm}:=T_{\pm,\epsilon}$ and
$\widetilde{T}_{\pm}=\widetilde{T}_{\pm,\epsilon}=\pi_{M_{\pm}}^{-1}(T_{\pm})$
of the boundaries in $M_{\pm}$ and $\widetilde{M}_{\pm}$. Using (3.2.3),
consider $\eta\in C^{\infty}(M_{\pm};\Lambda^{1}\mathcal{F}_{\pm})$ and
$\tilde{\eta}\in
C^{\infty}(\widetilde{M}_{\pm};\Lambda^{1}\widetilde{\mathcal{F}}_{\pm})$.

######  Proposition 5.6.3.

$Jd_{\mathcal{F}_{\pm}}=(d_{\mathcal{F}_{\pm}}-\eta\wedge)J$ on
$C^{\infty}_{\text{\rm c}}(T_{\pm};\Lambda\mathcal{F}_{\pm})$.

###### Proof.

Take open subsets $\widetilde{B}\subset\widetilde{L}$ such that
$\pi_{L}:\widetilde{B}\to B:=\pi_{L}(\widetilde{B})$ is a diffeomorphism.
Since the open sets of the form
$\varpi_{\pm}^{-1}(B)\equiv[0,\infty)_{\rho}\times\widetilde{B}_{\widetilde{\varpi}}$
cover $M_{\pm}$ and $J$ preserves the spaces $C^{\infty}_{\text{\rm
c}}(T_{\pm}\cap\varpi_{\pm}^{-1}(B);\Lambda\mathcal{F}_{\pm})$, it is enough
to prove the stated equality on $C^{\infty}_{\text{\rm
c}}(T_{\pm}\cap\varpi_{\pm}^{-1}(B);\Lambda\mathcal{F}_{\pm})$. In turn, this
follows by checking that
$\widetilde{J}d_{\widetilde{\mathcal{F}}_{\pm}}=(d_{\widetilde{\mathcal{F}}_{\pm}}-\tilde{\eta}\wedge)\widetilde{J}$
on $C^{\infty}_{\text{\rm
c}}(\widetilde{T}_{\pm};\Lambda\widetilde{\mathcal{F}}_{\pm})$ because

$\pi_{M_{\pm}}\equiv\operatorname{id}\times\pi_{L}:\widetilde{\varpi}_{\pm}^{-1}(\widetilde{B})\equiv[0,\infty)_{\tilde{\rho}}\times\widetilde{B}_{\widetilde{\varpi}}\to\varpi_{\pm}^{-1}(B)\equiv[0,\infty)_{\rho}\times
B_{\varpi}$

is a diffeomorphism. Let $\tilde{\alpha}\in C^{\infty}_{\text{\rm
c}}(\widetilde{T}_{\pm};\Lambda\widetilde{\mathcal{F}}_{\pm})$ and
$(\tau,\tilde{y})\in[0,\infty)\times\widetilde{L}$ with
$\tau<e^{-F(\tilde{y})}\epsilon$, which means that $(\tau,\tilde{y})$
corresponds to an element of $T_{\pm}$ via (4.2.11); in particular,
$\tilde{\alpha}(e^{-F(\tilde{y})},\tilde{y})=0$. So, by (5.6.2) and since
$d_{\widetilde{\mathcal{F}}_{\pm}}\tau=0$,

$\displaystyle\widetilde{J}d_{\widetilde{\mathcal{F}}_{\pm}}\tilde{\alpha}(\tau,\tilde{y})$
$\displaystyle=e^{F(\tilde{y})}\int_{e^{-F(\tilde{y})}}^{\tau}d_{\widetilde{\mathcal{F}}_{\pm}}\tilde{\alpha}(\tau_{1},\tilde{y})\,d\tau_{1}=e^{F(\tilde{y})}d_{\widetilde{\mathcal{F}}_{\pm}}\int_{e^{-F(\tilde{y})}}^{\tau}\tilde{\alpha}(\tau_{1},\tilde{y})\,d\tau_{1}$
$\displaystyle=d_{\widetilde{\mathcal{F}}_{\pm}}\,e^{F(\tilde{y})}\int_{e^{-F(\tilde{y})}}^{\tau}\tilde{\alpha}(\tau_{1},\tilde{y})\,d\tau_{1}-e^{F(\tilde{y})}\tilde{\eta}(\tilde{y})\wedge\int_{e^{-F(\tilde{y})}}^{\tau}\tilde{\alpha}(\tau_{1},\tilde{y})\,d\tau_{1}$
$\displaystyle=(d_{\widetilde{\mathcal{F}}_{\pm}}-\tilde{\eta}\wedge)\widetilde{J}\tilde{\alpha}(x,\tilde{y})\;.\qed$

######  Proposition 5.6.4.

We have
$\partial_{\rho}d_{\mathcal{F}_{\pm}}=(d_{\mathcal{F}_{\pm}}+\eta\wedge)\partial_{\rho}$
on $C^{-\infty}(M_{\pm};\Lambda\mathcal{F}_{\pm})$.

###### Proof.

Apply (3.2.3), (3.2.8) and (4.2.15). ∎

######  Corollary 5.6.5.

For all $m\in\mathbb{R}$,
$E_{m,\pm}d_{\mathcal{F}_{\pm}}=d_{\mathcal{F}_{\pm}}E_{m,\pm}$ on
$\mathcal{A}^{m}_{\text{\rm c}}(T_{\pm};\Lambda\mathcal{F}_{\pm})$.

###### Proof.

It is enough to consider the case $m<0$. Then apply Propositions 5.6.3 and
5.6.4, using the given definition of $E_{m,\pm}$ and the density of
$C^{\infty}_{\text{\rm c}}(T_{\pm};\Lambda\mathcal{F}_{\pm})$ in
$\mathcal{A}^{m}_{\text{\rm c}}(T_{\pm};\Lambda\mathcal{F}_{\pm})$ (see
Section 2.5.10). ∎

####  5.6.3. A partial extension map on $M$

Let us apply the notation of Section 2.6 to the suspension foliation (that
notation is compatible with the notation of Sections 3.1.12, 4.1.2 and 4.2).
Recall that $\boldsymbol{M}=M_{-}\sqcup M_{+}$, $\boldsymbol{\mathcal{F}}$ is
the combination of $\mathcal{F}_{\pm}$, and
$\boldsymbol{\pi}:\boldsymbol{M}\to M$ is the combination of
$\pi_{\pm}:M_{\pm}\to M$. The version of the commutative diagram (2.6.41) for
$\Lambda\boldsymbol{\mathcal{F}}\equiv\boldsymbol{\pi}^{*}\Lambda\mathcal{F}$
is

(5.6.3)
$\begin{CD}\mathcal{K}(\boldsymbol{M};\Lambda\boldsymbol{\mathcal{F}})@>{\iota}>{}>\dot{\mathcal{A}}(\boldsymbol{M};\Lambda\boldsymbol{\mathcal{F}})@>{R}>{}>\mathcal{A}(\boldsymbol{M};\Lambda\boldsymbol{\mathcal{F}})\phantom{\;.}\\\
@V{\boldsymbol{\pi}_{*}}V{}V@V{\boldsymbol{\pi}_{*}}V{}V@V{\boldsymbol{\pi}_{*}}V{\cong}V\\\
K(\mathcal{F})@>{\iota}>{}>I(\mathcal{F})@>{R}>{}>J(\mathcal{F})\;.\end{CD}$

Moreover $d_{\boldsymbol{\mathcal{F}}}\in\operatorname{Diff}_{\text{\rm
b}}(\boldsymbol{M};\Lambda\boldsymbol{\mathcal{F}})$ is the lift of
$d_{\mathcal{F}}$. Hence the operators defined by
$d_{\boldsymbol{\mathcal{F}}}$ on the spaces of the top row of (5.6.3)
correspond to the operators defined by $d_{\mathcal{F}}$ on the spaces of its
bottom row via the homomorphisms $\boldsymbol{\pi}_{*}$ (Section 2.6).
According to Section 2.5.14, $d_{\boldsymbol{\mathcal{F}}}$ preserves the
subspaces
$\dot{\mathcal{A}}^{(s)}(\boldsymbol{M};\Lambda\boldsymbol{\mathcal{F}})$ and
$\mathcal{A}^{m}(\boldsymbol{M};\Lambda\boldsymbol{\mathcal{F}})$.

The partial extension maps of Section 5.6.2,

$E_{m,\pm}:\mathcal{A}^{m}(M_{\pm};\Lambda\mathcal{F}_{\pm})\to\dot{\mathcal{A}}^{(s)}(M_{\pm};\Lambda\mathcal{F}_{\pm})\;,$

can be combined to define a continuous linear partial extension map

$\boldsymbol{E}_{m}:\mathcal{A}^{m}(\boldsymbol{M};\Lambda\boldsymbol{\mathcal{F}})\to\dot{\mathcal{A}}^{(s)}(\boldsymbol{M};\Lambda\boldsymbol{\mathcal{F}})\;.$

Then, according to Corollary 2.6.4 and its proof, a continuous linear partial
extension map $E_{m}:J^{m}(\mathcal{F})\to I^{(s)}(\mathcal{F})$ is given by
the composition

$J^{m}(\mathcal{F})\xrightarrow{\boldsymbol{\pi}_{*}^{-1}}\mathcal{A}^{m}(\boldsymbol{M};\Lambda\boldsymbol{\mathcal{F}})\xrightarrow{\boldsymbol{E}_{m}}\dot{\mathcal{A}}^{(s)}(\boldsymbol{M};\Lambda\boldsymbol{\mathcal{F}})\xrightarrow{\boldsymbol{\pi}_{*}}I^{(s)}(\mathcal{F})\;,$

which is a continuous inclusion map if $m\geq 0$. Recall that
$T\equiv(-\epsilon,\epsilon)_{\rho}\times L_{\varpi}$ and

$\boldsymbol{T}=T_{-}\sqcup
T_{+}=\boldsymbol{\pi}^{-1}(T)\equiv[0,\epsilon)_{\boldsymbol{\rho}}\times\partial\boldsymbol{M}_{\boldsymbol{\varpi}}\;.$

Like in Section 5.5.4, consider the restriction $E_{m,T}:J^{m}_{\text{\rm
c}}(T;\Lambda\mathcal{F})\to I^{(s)}_{\text{\rm c}}(T;\Lambda\mathcal{F})$ of
$E_{m}$. Suppose $\epsilon<1$, like in Section 5.6.2.

######  Corollary 5.6.6.

For all $m\in\mathbb{R}$, $E_{m,T}$ satisfies
$E_{m,T}d_{\mathcal{F}}=d_{\mathcal{F}}E_{m,T}$.

###### Proof.

Apply Corollary 5.6.5. ∎

###  5.7. Functoriality and leafwise homotopy invariance

####  5.7.1. Pull-back of conormal leafwise currents

Let $M^{\prime}$ be another closed manifold, and let $\phi:M^{\prime}\to M$ be
a smooth map transverse to $\mathcal{F}$. Then
$\mathcal{F}^{\prime}:=\phi^{*}\mathcal{F}$ is another transversely oriented
foliation of codimension one satisfying the conditions C and D in Section
4.1.2 with $M^{\prime\,0}:=\phi^{-1}(M^{0})$.

######  Remark 5.7.1.

The results of Section 5.7 have direct extensions to the case where $M$ or
$M^{\prime}$ may not be compact, with the condition that $M^{0}$ and
$M^{\prime\,0}$ are compact.

According to Section 2.2.8, the map (3.2.23) has a continuous extension

(5.7.1) $\phi^{*}:I(\mathcal{F})\to I(\mathcal{F}^{\prime})\;.$

defined as the composition

(5.7.2)
$I(\mathcal{F})\xrightarrow{\phi^{*}}I(M^{\prime},M^{\prime\,0};\phi^{*}\Lambda\mathcal{F})\xrightarrow{\phi^{*}}I(\mathcal{F}^{\prime})\;,$

like (2.8.19), using (2.2.20) with $E=\Lambda\mathcal{F}$. We can also
describe (5.7.1) as the restriction of (2.8.18) to conormal currents of
bidegree $(0,\bullet)$, like in (3.2.34). The map (5.7.1) is also a
restriction of (3.2.34).

Similarly, the analogs of (2.2.20) with $E=\Lambda\mathcal{F}$ for (2.6.42)
and (2.6.43) induce continuous homomorphisms

(5.7.3) $\displaystyle\phi^{*}:K(\mathcal{F})\to K(\mathcal{F}^{\prime})\;,$
(5.7.4) $\displaystyle\phi^{*}:J(\mathcal{F})\to J(\mathcal{F}^{\prime})\;.$

By passing to cohomology and reduced cohomology, we get continuous
homomorphisms,

(5.7.5) $\left\\{\begin{gathered}\phi^{*}:H^{\bullet}K(\mathcal{F})\to
H^{\bullet}K(\mathcal{F}^{\prime})\;,\\\ \begin{aligned}
\phi^{*}:&H^{\bullet}I(\mathcal{F})\to
H^{\bullet}I(\mathcal{F}^{\prime})\;,&\quad\phi^{*}:&\bar{H}^{\bullet}I(\mathcal{F})\to\bar{H}^{\bullet}I(\mathcal{F}^{\prime})\;,\\\
\phi^{*}:&H^{\bullet}J(\mathcal{F})\to
H^{\bullet}J(\mathcal{F}^{\prime})\;,&\quad\phi^{*}:&\bar{H}^{\bullet}J(\mathcal{F})\to\bar{H}^{\bullet}J(\mathcal{F}^{\prime})\;.\end{aligned}\end{gathered}\right.$

The assignment of the homomorphisms (5.7.1)–(5.7.5) is functorial.

####  5.7.2. Description of $\phi^{*}:K(\mathcal{F})\to
K(\mathcal{F}^{\prime})$

For $\omega^{\prime}=\phi^{*}\omega$ and $\eta^{\prime}=\phi^{*}\eta$, we have
$T\mathcal{F}^{\prime}=\ker\omega^{\prime}$ and
$d\omega^{\prime}=\eta^{\prime}\wedge\omega^{\prime}$ (the Frobenius
integrability condition for $\mathcal{F}^{\prime}$). Thus

(5.7.6) $\phi^{*}:C^{\infty}(M^{0};\Lambda)\to
C^{\infty}(M^{\prime\,0};\Lambda)$

is a cochain map for $d_{s\eta}$ and $d_{s\eta^{\prime}}$ ($s\in\mathbb{R}$)
(Section 2.9.3). In other words, $\phi$ induces

(5.7.7) $\phi^{*}:C^{\infty}(M^{0};\Lambda\otimes\Omega^{s}NM^{0})\to
C^{\infty}(M^{\prime\,0};\Lambda\otimes\Omega^{s}NM^{\prime\,0})\;,$

given by

(5.7.8)
$\phi^{*}(\alpha\otimes|\omega|^{s})=\phi^{*}\alpha\otimes|\omega^{\prime}|^{s}\;,$

which is another cochain map for the de Rham differentials defined with the
flat bundle structures of $\Omega^{s}NM^{0}$ and $\Omega^{s}NM^{\prime\,0}$.

If $\rho$ is a defining function of $M^{0}$ in some open neighborhood $T$,
then $\rho^{\prime}:=\phi^{*}\rho$ is a defining function of $M^{\prime\,0}$
in $T^{\prime}=\phi^{-1}(T)$, and (5.7.7) satisfies

(5.7.9)
$\phi^{*}(\alpha\otimes|d\rho|^{s})=\phi^{*}\alpha\otimes|d\rho^{\prime}|^{s}\;.$

Note the compatibility of (5.7.8) and (5.7.9) with (4.2.5). Furthermore, the
inverse image of $T:=T_{\epsilon}\equiv(-\epsilon,\epsilon)_{\rho}\times
M^{0}_{\varpi}$, for $\epsilon>0$ small enough, is a tubular neighborhood
$T^{\prime}\equiv(-\epsilon,\epsilon)_{\rho^{\prime}}\times
M^{\prime\,0}_{\varpi^{\prime}}$ of $M^{\prime\,0}$ in $M^{\prime}$, where
$\varpi^{\prime}:T^{\prime}\to M^{\prime\,0}$ satisfies
$\phi\varpi^{\prime}=\varpi\phi$ as maps $T^{\prime}\to M^{0}$. Thus

$\phi\equiv\operatorname{id}\times\phi:T^{\prime}\equiv(-\epsilon,\epsilon)\times
M^{\prime\,0}\to T\equiv(-\epsilon,\epsilon)\times M^{0}\;,$

which is proper because $M^{\prime\,0}$ is compact. We can use these tubular
neighborhoods to define the operators $\partial_{\rho}$ and
$\partial_{\rho^{\prime}}$ on $C^{\infty}_{\text{\rm
c}}(T;\Lambda\mathcal{F})$ and $C^{\infty}_{\text{\rm
c}}(T^{\prime};\Lambda\mathcal{F}^{\prime})$ (Section 2.6), which are used in
the identities of Proposition 5.3.1 for $K(\mathcal{F})$ and
$K(\mathcal{F}^{\prime})$ (Section 5.6.1). Clearly,

(5.7.10) $\partial_{\rho^{\prime}}\phi^{*}=\phi^{*}\partial_{\rho}\;,$

as maps $C^{\infty}_{\text{\rm c}}(T;\Lambda\mathcal{F})\to
C^{\infty}_{\text{\rm c}}(T^{\prime};\Lambda\mathcal{F}^{\prime})$,

######  Proposition 5.7.2.

According to Proposition 5.3.1, the map (5.7.3) is given by

$\phi^{*}\equiv\bigoplus_{k}\phi^{*}\equiv\bigoplus_{k}\phi^{*}\;,$

where the terms of the first direct sum are given by (5.7.6), and the terms of
the second direct sum are given by (5.7.7), taking $s=-k-1$.

###### Proof.

The second identity follows from the first one and (5.7.8). To prove the first
identity, by (5.7.10), it is enough to consider the term with $k=0$.

For $\alpha\in C^{\infty}(M^{0};\Lambda)$, let $u=\alpha\otimes|d\rho|^{-1}\in
C^{\infty}(M^{0};\Lambda\otimes\Omega^{-1}NM^{0})$. Using the first identity
of Proposition 5.3.1 for $k=0$, we have
$u\equiv\delta_{M^{0}}^{u}=\varpi^{*}\alpha\cdot\rho^{*}\delta_{0}$ in
$K(\mathcal{F})$, using Dirac sections (Section 2.2.6). Here,
$\rho^{*}\delta_{0}\in K(T,M^{0})$ is defined because
$\rho:T\to(-\epsilon,\epsilon)$ is transverse to $0$. Moreover
$u^{\prime}:=\phi^{*}u=\phi^{*}\alpha\cdot|d\rho^{\prime}|^{-1}$ by (5.7.9).
As before,
$u^{\prime}\equiv\delta_{M^{\prime\,0}}^{u^{\prime}}=\varpi^{\prime*}\phi^{*}\alpha\cdot\rho^{\prime*}\delta_{0}$
in $K(\mathcal{F}^{\prime})$. Take a sequence $f_{i}\in C^{\infty}_{\text{\rm
c}}(-\epsilon,\epsilon)$ converging to $\delta_{0}$ in $C^{-\infty}_{\text{\rm
c}}(-\epsilon,\epsilon)$. Then

$\displaystyle\phi^{*}\delta_{M^{0}}^{u}$
$\displaystyle=\phi^{*}(\varpi^{*}\alpha\cdot\rho^{*}\delta_{0})=\lim_{i}\phi^{*}(\varpi^{*}\alpha\cdot\rho^{*}f_{i})=\lim_{i}\phi^{*}\varpi^{*}\alpha\cdot\phi^{*}\rho^{*}f_{i}$
$\displaystyle=\lim_{i}\varpi^{\prime*}\phi^{*}\alpha\cdot\rho^{\prime*}f_{i}=\varpi^{\prime*}\phi^{*}\alpha\cdot\rho^{\prime*}\delta_{0}=\delta_{M^{\prime\,0}}^{u^{\prime}}\;.\qed$

######  Remark 5.7.3.

The equality $\partial_{\rho^{\prime}}\phi^{*}=\phi^{*}\partial_{\rho}$ has a
continuous extension as maps $C^{-\infty}_{\text{\rm
c}}(T;\Lambda\mathcal{F})\to C^{-\infty}_{\text{\rm
c}}(T^{\prime};\Lambda\mathcal{F}^{\prime})$, and the computations of the
above proof also work also with $\alpha\in C^{-\infty}(M^{0};\Lambda)$. So we
get similar expressions of
$\phi^{*}:C^{-\infty}_{M^{0}}(M;\Lambda\mathcal{F})\to
C^{-\infty}_{M^{\prime\,0}}(M^{\prime};\Lambda\mathcal{F}^{\prime})$ according
to Remark 5.3.4.

####  5.7.3. Push-forward of conormal leafwise currents

With the notation and conditions of Section 5.7.1, suppose that moreover
$\phi$ is a submersion such that the vertical bundle $\mathcal{V}$ is oriented
(Section 3.2.15). Thus $\phi:M^{\prime\,0}\to M^{0}$ is also a submersion
whose vertical bundle is $\mathcal{V}|_{M^{\prime\,0}}\subset TM^{\prime\,0}$,
also oriented. Then the case of (3.2.33) on smooth leafwise forms has a
continuous extension

(5.7.11) $\phi_{*}:I_{{\text{\rm c}}/{\text{\rm cv}}}(\mathcal{F}^{\prime})\to
I_{{\text{\rm c}}/{\cdot}}(\mathcal{F})\;.$

This map can be described as the restriction of the map (2.8.24) to conormal
currents of bidegree $(0,\bullet)$, like (3.2.33) in Section 3.2.15. We can
also describe (5.7.11) as the composition

$I_{{\text{\rm c}}/{\text{\rm
cv}}}(M^{\prime},L^{\prime};\Lambda\mathcal{F})\xrightarrow{\pi_{\text{\rm
top}}}I_{{\text{\rm c}}/{\text{\rm
cv}}}(M^{\prime},L^{\prime};\phi^{*}\Lambda\mathcal{F}\otimes\Omega_{\text{\rm
fiber}})\xrightarrow{\phi_{*}}I_{{\text{\rm
c}}/{\cdot}}(M,L;\Lambda\mathcal{F})\;,$

like in (3.2.35), where $\phi_{*}$ is given by (2.2.23) for
$E=\Lambda\mathcal{F}$. The map (5.7.11) is also a restriction of the case of
(3.2.33) for leafwise currents.

According to Section 2.6.15, the map (5.7.11) induces homomorphisms

(5.7.12) $\displaystyle\phi_{*}:K(\mathcal{F}^{\prime})\to K(\mathcal{F})\;,$
(5.7.13) $\displaystyle\phi_{*}:J_{{\text{\rm c}}/{\text{\rm
cv}}}(\mathcal{F}^{\prime})\to J_{{\text{\rm c}}/{\cdot}}(\mathcal{F})\;.$

Like in Section 5.7.1, we get induced continuous homomorphisms,

(5.7.14)
$\left\\{\begin{gathered}\phi_{*}:H^{\bullet}K(\mathcal{F}^{\prime})\to
H^{\bullet}K(\mathcal{F})\;,\\\ \begin{aligned}
\phi_{*}:&H^{\bullet}I_{\text{\rm c}}(\mathcal{F}^{\prime})\to
H^{\bullet}I_{\text{\rm
c}}(\mathcal{F})\;,&\quad\phi_{*}:&\bar{H}^{\bullet}I_{\text{\rm
c}}(\mathcal{F}^{\prime})\to\bar{H}^{\bullet}I_{\text{\rm
c}}(\mathcal{F})\;,\\\ \phi_{*}:&H^{\bullet}J_{\text{\rm
c}}(\mathcal{F}^{\prime})\to H^{\bullet}J_{\text{\rm
c}}(\mathcal{F})\;,&\quad\phi_{*}:&\bar{H}^{\bullet}J_{\text{\rm
c}}(\mathcal{F}^{\prime})\to\bar{H}^{\bullet}J_{\text{\rm
c}}(\mathcal{F})\;.\end{aligned}\end{gathered}\right.$

The assignments of homomorphisms (5.7.11)–(5.7.14) are clearly functorial.

####  5.7.4. Description of $\phi_{*}:K(\mathcal{F}^{\prime})\to
K(\mathcal{F})$

For $\phi$ as above, consider the notation of Section 5.7.2. Then

(5.7.15) $\phi_{*}:C^{\infty}(M^{\prime\,0};\Lambda)\to
C^{\infty}(M^{0};\Lambda)$

is a cochain map for $d_{s\eta}$ and $d_{s\eta^{\prime}}$ ($s\in\mathbb{R}$)
(Section 2.9.3). That is, $\phi$ induces

(5.7.16)
$\phi_{*}:C^{\infty}(M^{\prime\,0};\Lambda\otimes\Omega^{s}NM^{\prime\,0})\to
C^{\infty}(M^{0};\Lambda\otimes\Omega^{s}NM^{0})\;,$

given by

(5.7.17)
$\phi_{*}(\alpha\otimes|\omega^{\prime}|^{s})=\phi_{*}\alpha\otimes|\omega|^{s}\;,$

which is another cochain map for the de Rham differentials defined with the
flat bundle structures of $\Omega^{s}NM^{0}$ and $\Omega^{s}NM^{\prime\,0}$
induced by the Bott flat $T\mathcal{F}$-partial connection (Section 3.1.3).
Like in (5.7.9) and (5.7.10), we have

(5.7.18)
$\displaystyle\phi_{*}(\alpha\otimes|d\rho^{\prime}|^{s})=\phi_{*}\alpha\otimes|d\rho|^{s}\;,$
(5.7.19)
$\displaystyle\partial_{\rho}\phi_{*}=\phi_{*}\partial_{\rho^{\prime}}\;,$

where (5.7.19) holds as maps $C^{\infty}_{\text{\rm
c}}(T^{\prime};\Lambda\mathcal{F}^{\prime})\to C^{\infty}_{\text{\rm
c}}(T;\Lambda\mathcal{F})$.

######  Proposition 5.7.4.

According to Proposition 5.3.1, the map (5.7.12) is given by

$\phi_{*}\equiv\bigoplus_{k}\phi_{*}\equiv\bigoplus_{k}\phi_{*}\;,$

where the terms of the first direct sum are given by (5.7.15), and the terms
of the second direct sum are given by (5.7.16), taking $s=-k-1$.

###### Proof.

The second identity follows from the first one and (5.7.17). To prove the
first identity, by (5.7.19), it is enough to consider the term with $k=0$.

For $\beta\in C^{\infty}(M^{\prime\,0};\Lambda)$, let
$v^{\prime}=\beta\otimes|d\rho^{\prime}|^{-1}\in
C^{\infty}(M^{\prime\,0};\Lambda\otimes\Omega^{-1}NM^{\prime\,0})$. Like in
the proof of Proposition 5.7.2, we have
$v^{\prime}\equiv\delta_{M^{\prime\,0}}^{v^{\prime}}=\varpi^{\prime*}\beta\cdot\rho^{\prime*}\delta_{0}$
in $K(\mathcal{F}^{\prime})$. Moreover
$v:=\phi_{*}v^{\prime}=\phi_{*}\beta\cdot|d\rho|^{-1}$ by (5.7.18), with
$v\equiv\delta_{M^{0}}^{v}=\varpi^{\prime*}\phi_{*}\beta^{\prime}\cdot\rho^{*}\delta_{0}$
in $K(\mathcal{F})$. Take a sequence $f_{i}\in C^{\infty}_{\text{\rm
c}}(-\epsilon,\epsilon)$ converging to $\delta_{0}$ in $C^{-\infty}_{\text{\rm
c}}(-\epsilon,\epsilon)$. We get

$\displaystyle\phi_{*}\delta_{M^{\prime\,0}}^{v^{\prime}}$
$\displaystyle=\phi_{*}(\varpi^{\prime*}\beta\cdot\rho^{\prime*}\delta_{0})=\lim_{i}\phi_{*}(\varpi^{\prime*}\beta\cdot\rho^{\prime*}f_{i})=\lim_{i}\phi_{*}\varpi^{*}\beta\cdot\rho^{*}f_{i}$
$\displaystyle=\lim_{i}\varpi^{*}\phi_{*}\beta\cdot\rho^{*}f_{i}=\varpi^{*}\phi^{*}\beta\cdot\rho^{*}\delta_{0}=\delta_{M^{0}}^{v}\;.\qed$

The analog of Remark 5.7.3 for $\phi_{*}:K(\mathcal{F}^{\prime})\to
K(\mathcal{F})$ is true.

####  5.7.5. Leafwise homotopy invariance

With the notation of Section 3.2.16, let $H:(M^{\prime}\times
I,\mathcal{F}^{\prime}\times I)\to(M,\mathcal{F})$ ($I=[0,1]$) be a smooth
leafwise homotopy such that $H_{0}$ is transverse to $M^{0}$ and
$H_{0}^{-1}(M^{0})=M^{\prime\,0}$. Then, for every $p^{\prime}\in M^{\prime}$,
the map $H_{t*}:N_{p^{\prime}}\mathcal{F}^{\prime}\to
N_{H_{t}(p^{\prime})}\mathcal{F}$ is the composition of
$H_{0*}:N_{p^{\prime}}\mathcal{F}^{\prime}\to
N_{H_{0}(p^{\prime})}\mathcal{F}$ with the parallel transport along the
leafwise path $s\in[0,t]\mapsto H_{s}(p^{\prime})$. It follows that every
$H_{t}$ is transverse to $M^{0}$ and $H_{t}^{-1}(M^{0})=M^{\prime\,0}$. Hence
$H$ is transverse to $M^{0}$ and $H^{-1}(M^{0})=M^{\prime\,0}\times I$. Then,
by (3.2.38) and according to Sections 5.7.1 and 5.7.3, the corresponding
leafwise homotopy operator $\mathsf{h}:C^{\infty}(M;\Lambda\mathcal{F})\to
C^{\infty}(M^{\prime};\Lambda\mathcal{F}^{\prime})$ has continuous linear
extensions,

$\mathsf{h}:K(\mathcal{F})\to
K(\mathcal{F}^{\prime})\;,\quad\mathsf{h}:I(\mathcal{F})\to
I(\mathcal{F}^{\prime})\;,\quad\mathsf{h}:J(\mathcal{F})\to
J(\mathcal{F}^{\prime})\;.$

By continuity and according to Section 3.2.16, we have
$H_{1}^{*}-H_{0}^{*}=\mathsf{h}d_{\mathcal{F}}+d_{\mathcal{F}^{\prime}}\mathsf{h}$
with $H_{0}^{*}$ and $H_{1}^{*}$ given by (5.7.1), (5.7.3) and (5.7.4). Hence
we get the following.

######  Proposition 5.7.5.

Let $\phi,\psi:(M^{\prime},\mathcal{F}^{\prime})\to(M,\mathcal{F})$ be smooth
foliated maps transverse to $M^{0}$ with
$\phi^{-1}(M^{0})=\psi^{-1}(M^{0})=M^{\prime\,0}$. If $\phi$ is leafwise
homotopic to $\psi$, then $\phi$ and $\psi$ induce the same homomorphisms
(5.7.5).

###  5.8. Action of foliated flows on the conormal sequence

Let $\phi=\\{\phi^{t}\\}$ be a foliated flow with transversely simple
preserved leaves on a compact foliated manifold $(M,\mathcal{F})$. The
homomorphisms (5.7.5) induced by the maps $\phi^{t}$ define actions of
$\mathbb{R}$ on $H^{\bullet}K(\mathcal{F})$, $H^{\bullet}I(\mathcal{F})$ and
$H^{\bullet}J(\mathcal{F})$, denoted by $\phi^{*}=\\{\phi^{t*}\\}$, and
actions on $\bar{H}^{\bullet}I(\mathcal{F})$ and
$\bar{H}^{\bullet}J(\mathcal{F})$, denoted by
$\bar{\phi}^{*}=\\{\bar{\phi}^{t*}\\}$. By Proposition 5.7.5, they only depend
on the flow-leafwise-homotopy class of $\phi$ (Section 3.1.7).

With the notation of Section 4.3.3, the foliated flow $\xi=\\{\xi^{t}\\}$ has
transversely simple preserved leaves and satisfies $\bar{\xi}=\bar{\phi}$ and
$\xi^{t}=\operatorname{id}$ on $M^{0}$. By Proposition 3.1.4, there is a flow-
leafwise homotopy between $\phi$ and $\xi$, and therefore $\phi^{*}=\xi^{*}$
on $H^{\bullet}K(\mathcal{F})$. Consider the tubular neighborhood with
defining function, $T_{\epsilon}\equiv(-\epsilon,\epsilon)_{\rho}\times
M^{0}_{\varpi}$, like in Section 5.7.2.

######  Proposition 5.8.1.

According to Corollaries 5.3.2 and 5.3.3,

$\phi^{t*}\equiv\bigoplus_{k,L}e^{-(k+1)\varkappa_{L}t}\equiv\bigoplus_{k,L}e^{-(k+1)\varkappa_{L}t}$

on $H^{\bullet}K(\mathcal{F})$, where $k$ runs in $\mathbb{N}_{0}$ and $L$ in
$\pi_{0}M^{0}$.

###### Proof.

Since $\xi^{t*}\rho=e^{\varkappa_{L}t}\rho$ on every
$T_{L,\epsilon}\cap\xi^{-t}(T_{L,\epsilon})$ by (4.2.7), it follows from
(5.7.9) and Proposition 5.7.2 that

$\xi^{t*}\equiv\bigoplus_{k,L}e^{-(k+1)\varkappa_{L}t}\equiv\bigoplus_{k,L}e^{-(k+1)\varkappa_{L}t}$

on $K(\mathcal{F})$, according to Proposition 5.3.1 and Remark 5.3.3. Hence
$\phi^{t*}=\xi^{t*}$ has the stated expression on $H^{\bullet}K(\mathcal{F})$.
∎

Propositions 5.3.1, 5.3.2, 5.3.3 and 5.8.1 show Theorem 1.3.1.

##  Chapter 6 Dual-conormal leafwise reduced cohomology

###  6.1. Dual-conormal sequence of leafwise differential forms

Assume the conditions of Section 5.1 on $(M,\mathcal{F})$. According to
Section 2.3.4, the LCHS

$I^{\prime}(\mathcal{F})=I^{\prime}\Lambda^{\bullet}(\mathcal{F}):=I^{\prime}(M,M^{0};\Lambda\mathcal{F})$

is a topological complex with $d_{\mathcal{F}}$. It induces the _dual-conormal
leafwise cohomology_ and _dual-conormal leafwise reduced cohomology_ of
$\mathcal{F}$ (or of $(\mathcal{F},M^{0})$). The notation
$BI^{\prime}(\mathcal{F})$, $ZI^{\prime}(\mathcal{F})$,
$\bar{B}I^{\prime}(\mathcal{F})$, $H^{\bullet}I^{\prime}(\mathcal{F})$ and
$\bar{H}^{\bullet}I^{\prime}(\mathcal{F})$ is used like in Section 5.1.

For a leafwise flat vector bundle $E$, we can also consider the topological
complex

$I^{\prime}(\mathcal{F};E)=I^{\prime}\Lambda^{\bullet}(\mathcal{F};E)=I^{\prime}(M,M^{0};\Lambda\mathcal{F}\otimes
E)$

with $d_{\mathcal{F}}$. The LCHS $I^{\prime}(\mathcal{F};E)$ is also defined
for an arbitrary vector bundle $E$, missing the leafwise differential map
$d_{\mathcal{F}}$.

Moreover, the LCHSs

$I^{\prime\,(s)}(\mathcal{F})=I^{\prime\,(s)}\Lambda^{\bullet}(\mathcal{F})=I^{\prime\,(s)}(M,M^{0};\Lambda\mathcal{F})\quad(s\in\mathbb{R})\;.$

also become topological complexes with $d_{\mathcal{F}}$. The notation
$BI^{\prime\,(s)}(\mathcal{F})$, $ZI^{\prime\,(s)}(\mathcal{F})$,
$\bar{B}I^{\prime\,(s)}(\mathcal{F})$,
$H^{\bullet}I^{\prime\,(s)}(\mathcal{F})$ and
$\bar{H}^{\bullet}I^{\prime\,(s)}(\mathcal{F})$ is used like in Section 5.1.

######  Remark 6.1.1.

Although $\Omega M$ has no leafwise flat structure in general, we can assume
$\mathcal{F}$ is oriented by working locally or passing to the double cover of
orientations of $\mathcal{F}$. Then we can apply (3.2.19)–(3.2.21) and the
leafwise flat structure of $\Omega N\mathcal{F}$ to define $d_{\mathcal{F}}$
and $d_{\mathcal{F}}^{\text{\rm t}}$ on every $I^{(s)}(\mathcal{F};\Omega
M)\equiv I^{(s)}(\mathcal{F};\Omega N\mathcal{F})$. Since the condition of
being in $I^{(s)}(\mathcal{F};\Omega M)$ is local for elements of
$C^{-\infty}(M;\Lambda\mathcal{F}\otimes\Omega)$, this procedure gives the
definition of $d_{\mathcal{F}}=d_{\mathcal{F}}^{\text{\rm tt}}$.

For $s^{\prime}\leq s$ in $\mathbb{R}$, we have the continuous linear
restriction maps (Section 2.3.1)

(6.1.1) $j^{\prime}_{s}:I^{\prime}(\mathcal{F})\to
I^{\prime\,(s)}(\mathcal{F})\;,\quad
j^{\prime}_{s,s^{\prime}}:I^{\prime\,(s)}(\mathcal{F})\to
I^{\prime\,(s^{\prime})}(\mathcal{F})\;,$

where $j^{\prime}_{s}=j_{-s}^{\text{\rm t}}$ and
$j^{\prime}_{s,s^{\prime}}=j_{-s^{\prime},-s}^{\text{\rm t}}$ for the version
of (5.1.1) with $\Omega M$. The induced homomorphisms in cohomology and
reduced cohomology are denoted by $j^{\prime}_{s*}$,
$j^{\prime}_{s,s^{\prime}*}$, $\bar{\jmath}^{\prime}_{s*}$ and
$\bar{\jmath}^{\prime}_{s,s^{\prime}*}$. The homomorphisms
$j^{\prime}_{s,s^{\prime}*}$ and $\bar{\jmath}^{\prime}_{s,s^{\prime}*}$ form
projective spectra, giving rise to projective limits as $s\uparrow+\infty$.
Like in (5.1.2), the maps $j^{\prime}_{s*}$ and $\bar{\jmath}^{\prime}_{s*}$
induce canonical continuous linear maps,

(6.1.2) $\left\\{\begin{gathered}\tilde{\jmath}^{\prime}_{*}:=\varprojlim
j^{\prime}_{s*}:H^{\bullet}I^{\prime}(\mathcal{F})\to\widetilde{H}^{\bullet}I^{\prime}(\mathcal{F}):=\varprojlim
H^{\bullet}I^{\prime\,(s)}(\mathcal{F})\;,\\\
\hat{\jmath}^{\prime}_{*}:=\varprojlim\bar{\jmath}^{\prime}_{s*}:\bar{H}^{\bullet}I^{\prime}(\mathcal{F})\xrightarrow{\cong}\widehat{H}^{\bullet}I^{\prime}(\mathcal{F}):=\varprojlim\bar{H}^{\bullet}I^{\prime\,(s)}(\mathcal{F})\;,\end{gathered}\right.$

where the second one is a linear isomorphism (Section 6.2). The canonical maps
of the inductive limits to the steps are denoted by

$\tilde{\jmath}_{s*}:\widetilde{H}^{\bullet}I^{\prime}(\mathcal{F})\to
H^{\bullet}I^{\prime\,(s)}(\mathcal{F})\;,\quad\hat{\jmath}_{s*}:\widehat{H}^{\bullet}I^{\prime}(\mathcal{F})\to\bar{H}^{\bullet}I^{\prime\,(s)}(\mathcal{F})\;.$

Using the above type of notation, the LCHSs $J^{\prime}(\mathcal{F})$ and
$K^{\prime}(\mathcal{F})$ are also topological complexes with
$d_{\mathcal{F}}$ (Section 2.7.5), with corresponding spaces
$BJ^{\prime}(\mathcal{F})$, $ZJ^{\prime}(\mathcal{F})$,
$\bar{B}J^{\prime}(\mathcal{F})$, $H^{\bullet}J^{\prime}(\mathcal{F})$ and
$\bar{H}^{\bullet}J^{\prime}(\mathcal{F})$, and the same for
$K^{\prime}(\mathcal{F})$.

Similarly, we have topological complexes $J^{\prime\,(s)}(\mathcal{F})$,
$J^{\prime\,m}(\mathcal{F})$ and $K^{\prime\,(s)}(\mathcal{F})$
($s,m\in\mathbb{R}$) (Sections 2.7.4 and 2.6.13), with corresponding spaces
$BJ^{\prime\,(s)}(\mathcal{F})$, $ZJ^{\prime\,(s)}(\mathcal{F})$,
$\bar{B}J^{\prime\,(s)}(\mathcal{F})$,
$H^{\bullet}J^{\prime\,(s)}(\mathcal{F})$ and
$\bar{H}^{\bullet}J^{\prime\,(s)}(\mathcal{F})$, and the same for
$J^{\prime\,m}(\mathcal{F})$ and $K^{\prime\,(s)}(\mathcal{F})$. There are
obvious versions for $J^{\prime\,(s)}(\mathcal{F})$ and
$K^{\prime\,(s)}(\mathcal{F})$ of the maps (6.1.1) (Section 2.7.1), also
denoted by $j^{\prime}_{s}$ and $j^{\prime}_{s,s^{\prime}}$, giving rise to
projective spectra in cohomology and reduced cohomology, and the corresponding
projective limits. In the case of $J^{\prime}(\mathcal{F})$, the maps
$j^{\prime}_{s,s^{\prime}}$ and $j^{\prime}_{s}$ are continuous inclusions
(Section 2.7.1).

There are also continuous inclusion maps (Section 2.7.1)

(6.1.3)
$\left\\{\begin{gathered}j^{\prime}_{m}:J^{\prime}(\mathcal{F})\hookrightarrow
J^{\prime\,m}(\mathcal{F})\;,\quad
j^{\prime}_{m,m^{\prime}}:J^{\prime\,m}(\mathcal{F})\hookrightarrow
J^{\prime\,m^{\prime}}(\mathcal{F})\quad(m^{\prime}\leq m)\;,\\\
j^{\prime}_{m,s}:J^{m}(\mathcal{F})\hookrightarrow
J^{(s)}(\mathcal{F})\quad(m>s+n/2+1)\;,\\\
j^{\prime}_{s,m}:J^{(s)}(\mathcal{F})\hookrightarrow
J^{m}(\mathcal{F})\quad(s\geq m,0)\;,\end{gathered}\right.$

denoted like in (5.1.1) with some abuse of notation. The homomorphisms induced
by the maps $j^{\prime}_{m,m^{\prime}}$ in cohomology and reduced cohomology
form projective spectra whose inductive limits as $m\uparrow+\infty$ agree
with the previous ones for $J(\mathcal{F})$, and the maps $j^{\prime}_{m}$
induce a continuous linear isomorphism analogous to (6.1.2).

It will be shown (Corollary 6.3.2) that the canonical projections are TVS-
identities,

(6.1.4)
$\left\\{\begin{gathered}H^{\bullet}K^{\prime}(\mathcal{F})\equiv\bar{H}^{\bullet}K^{\prime}(\mathcal{F})\;,\quad
H^{\bullet}K^{\prime\,(s)}(\mathcal{F})\equiv\bar{H}^{\bullet}K^{\prime\,(s)}(\mathcal{F})\;,\\\
H^{\bullet}K^{\prime}(\mathcal{F})\equiv\varprojlim
H^{\bullet}K^{\prime\,(s)}(\mathcal{F})\;,\quad\bar{H}^{\bullet}K^{\prime}(\mathcal{F})\equiv\varprojlim\bar{H}^{\bullet}K^{\prime\,(s)}(\mathcal{F})\;.\end{gathered}\right.$

The version of the bottom row of (2.7.8) with $\Lambda\mathcal{F}$ is a short
exact sequence of continuous homomorphisms of topological complexes,

(6.1.5) $0\leftarrow
K^{\prime}(\mathcal{F})\xleftarrow{R^{\prime}}I^{\prime}(\mathcal{F})\xleftarrow{\iota^{\prime}}J^{\prime}(\mathcal{F})\leftarrow
0\;,$

using the notation $R^{\prime}=\iota^{\text{\rm t}}$ and
$\iota^{\prime}=R^{\text{\rm t}}$. The exactness of the induced sequences,

(6.1.6) $\displaystyle 0\leftarrow
H^{\bullet}K^{\prime}(\mathcal{F})\xleftarrow{R^{\prime}_{*}}H^{\bullet}I^{\prime}(\mathcal{F})\xleftarrow{\iota^{\prime}_{*}}H^{\bullet}J^{\prime}(\mathcal{F})\leftarrow
0\;,$ (6.1.7) $\displaystyle 0\leftarrow
H^{\bullet}K^{\prime}(\mathcal{F})\xleftarrow{\bar{R}^{\prime}_{*}}\bar{H}^{\bullet}I^{\prime}(\mathcal{F})\xleftarrow{\bar{\iota}^{\prime}_{*}}\bar{H}^{\bullet}J^{\prime}(\mathcal{F})\leftarrow
0\;,$

will be proved in Section 6.5; in particular, this shows Theorem 1.3.6.

Taking the transpose of the analog of (5.1.8) with $\Omega M$, we get
continuous linear maps

(6.1.8) $R^{\prime}_{s}:K^{\prime\,(s)}(\mathcal{F})\to
I^{\prime\,(s)}(\mathcal{F})\;,\quad\iota^{\prime}_{s}:I^{\prime\,(s)}(\mathcal{F})\to
J^{\prime\,(s)}(\mathcal{F})\;.$

Like in Section 5.1, the subscript “$s$” may be also added to the elements of
the cohomologies or reduced cohomologies of $K^{\prime\,(s)}(\mathcal{F})$,
$I^{\prime\,(s)}(\mathcal{F})$ or $J^{\prime\,(s)}(\mathcal{F})$.

###  6.2. Projective limits in reduced cohomology

The goal of this section is to prove the linear isomorphism (6.1.2), and its
version for $J^{\prime}(\mathcal{F})$. The case of $K^{\prime}(\mathcal{F})$
is given by (6.1.4).

To simplify the notation, we write

$\widetilde{H}^{\bullet}I^{\prime}(\mathcal{F})=\varprojlim
H^{\bullet}I^{\prime\,(s)}(\mathcal{F})\;,\quad\widehat{H}^{\bullet}I^{\prime}(\mathcal{F})=\varprojlim\bar{H}^{\bullet}I^{\prime\,(s)}(\mathcal{F})\;,$

and the canonical maps of the projective limits to the steps are denoted by

$\tilde{\jmath}^{\prime}_{s}:\widetilde{H}^{\bullet}I^{\prime}(\mathcal{F})\to
H^{\bullet}I^{\prime\,(s)}(\mathcal{F})\;,\quad\hat{\jmath}^{\prime}_{s}:\widehat{H}^{\bullet}I^{\prime}(\mathcal{F})\to\bar{H}^{\bullet}I^{\prime\,(s)}(\mathcal{F})\;.$

The same type of notation is used in the cases of $J^{\prime}(\mathcal{F})$
and $K^{\prime}(\mathcal{F})$.

######  Lemma 6.2.1.

$BI^{\prime}(\mathcal{F})$ is dense in every $BI^{\prime\,(s)}(\mathcal{F})$
is dense.

###### Proof.

Use that the image of $J^{\prime}(\mathcal{F})$ is dense in
$J^{\prime\,(s)}(\mathcal{F})$ (Section 2.3.1) and $d_{\mathcal{F}}$ is
continuous on $J^{\prime}(\mathcal{F})$ and $J^{\prime\,(s)}(\mathcal{F})$
(Section 2.7.5). ∎

Recall that $\bar{B}J^{\prime}(\mathcal{F})$ (resp.,
$\bar{B}J^{\prime\,(s)}(\mathcal{F})$) denotes the closure of
$BJ^{\prime}(\mathcal{F})$ (resp., $BJ^{\prime\,(s)}(\mathcal{F})$) in
$J^{\prime}(\mathcal{F})$ (resp., $J^{\prime\,(s)}(\mathcal{F})$).

######  Corollary 6.2.2.

As vector spaces,

$\bar{B}J^{\prime}(\mathcal{F})=\bigcap_{s}\bar{B}J^{\prime\,(s)}(\mathcal{F})\;.$

###### Proof.

By the definition of the projective topology of
$\bigcap_{s}J^{\prime\,(s)}(\mathcal{F})$ [Sch71, Section II.5] and using
Lemma 6.2.1, we get that $BJ^{\prime}(\mathcal{F})$ is dense in
$\bigcap_{s}\bar{B}J^{\prime\,(s)}(\mathcal{F})$. Moreover, this intersection
is closed in $J^{\prime}(\mathcal{F})$. Then the stated equality is true. ∎

######  Lemma 6.2.3.

As vector spaces,

$ZJ^{\prime}(\mathcal{F})=\bigcap_{s}ZJ^{\prime\,(s)}(\mathcal{F})\;.$

###### Proof.

Consider the commutative diagram

$\begin{CD}0@>{}>{}>\bigcap_{s}ZJ^{\prime\,(s)}(\mathcal{F})@>{}>{}>\bigcap_{s}J^{\prime\,(s)}(\mathcal{F})@>{d_{\mathcal{F}}}>{}>\bigcap_{s}BJ^{\prime\,(s)}(\mathcal{F})\\\
&&@A{}A{}A\Big{\|}@A{}A{}A&\\\
0@>{}>{}>ZJ^{\prime}(\mathcal{F})@>{}>{}>J^{\prime}(\mathcal{F})@>{d_{\mathcal{F}}}>{}>BJ^{\prime}(\mathcal{F})@>{}>{}>0\;.\end{CD}$

Here, the central vertical equality is the analog (2.5.54), the arrows that
are not given by $d_{\mathcal{F}}$ and do not go to $0$ denote inclusion maps,
and the bottom row is exact. Since the surjective maps
$d_{\mathcal{F}}:J^{\prime\,(s)}(\mathcal{F})\to
BJ^{\prime\,(s)}(\mathcal{F})$ form a homomorphism between projective spectra
whose kernel is the projective spectrum consisting of the spaces
$ZJ^{\prime\,(s)}(\mathcal{F})$, the top row is also exact [Wen03, Proposition
3.1.8]. Thus the left-hand-side vertical arrow is an equality of vector
spaces. ∎

######  Proposition 6.2.4.

The canonical map
$\bar{H}^{\bullet}J^{\prime}(\mathcal{F})\to\widehat{H}^{\bullet}J^{\prime}(\mathcal{F})$
is a linear isomorphism.

###### Proof.

Consider the commutative diagram

$\begin{CD}0@>{}>{}>\bigcap_{s}\bar{B}J^{\prime\,(s)}(\mathcal{F})@>{}>{}>\bigcap_{s}ZJ^{\prime\,(s)}(\mathcal{F})@>{}>{}>\widehat{H}^{\bullet}J^{\prime}(\mathcal{F})@>{}>{}>0\phantom{\;.}\\\
&&\Big{\|}\Big{\|}@A{}A{}A&\\\
0@>{}>{}>\bar{B}J^{\prime}(\mathcal{F})@>{}>{}>ZJ^{\prime}(\mathcal{F})@>{}>{}>\bar{H}^{\bullet}J^{\prime}(\mathcal{F})@>{}>{}>0\;.\end{CD}$

Here, Corollaries 6.2.2 and 6.2.3 give the vertical equalities of vector
spaces, the vertical arrow is canonical, and the other maps are canonical; in
particular, the bottom row is exact. Lemma 6.2.1 also shows that every
$BJ^{(s)}(\mathcal{F})$ is dense in $BJ^{(s^{\prime})}(\mathcal{F})$ for
$s^{\prime}<s$. Hence the right derived functor $\varprojlim^{1}$ satisfies
$\varprojlim^{1}\bar{B}J^{(s)}(\mathcal{F})=0$ as $s\uparrow+\infty$ [Wen03,
Theorem 3.2.1], obtaining that the top row is also exact by [Wen03, Corollary
3.1.5]. Then the result follows. ∎

On the other hand, the kind of arguments that will be given in Section 6.5 can
be adapted to show the exactness of the sequence

(6.2.1) $0\leftarrow
H^{\bullet}K^{\prime}(\mathcal{F})\xleftarrow{\widehat{R}^{\prime}_{*}}\widehat{H}^{\bullet}I^{\prime}(\mathcal{F})\xleftarrow{\hat{\iota}^{\prime}_{*}}\widehat{H}^{\bullet}J^{\prime}(\mathcal{F})\leftarrow
0\;,$

where $\widehat{R}^{\prime}_{*}=\varprojlim\bar{R}^{\prime}_{s*}$ and
$\hat{\iota}^{\prime}_{*}=\varprojlim\bar{\iota}^{\prime}_{s*}$, using the
homomorphisms induced by (6.1.8). This fits into a commutative diagram

$\begin{CD}0@<{}<{}<H^{\bullet}K^{\prime}(\mathcal{F})@<{}<{}<\bar{H}^{\bullet}I^{\prime}(\mathcal{F})@<{}<{}<\bar{H}^{\bullet}J^{\prime}(\mathcal{F})@<{}<{}<0\phantom{\;,}\\\
&&\Big{\|}@V{}V{}V@V{}V{\cong}V\\\
0@<{}<{}<H^{\bullet}K^{\prime}(\mathcal{F})@<{}<{}<\widehat{H}^{\bullet}I^{\prime}(\mathcal{F})@<{}<{}<\widehat{H}^{\bullet}J^{\prime}(\mathcal{F})@<{}<{}<0\;,\end{CD}$

where the top row is the exact sequence (6.1.7), and the vertical arrows are
canonical. The last vertical arrow is a linear isomorphism by Proposition
6.2.4. Then the central vertical arrow is also a linear isomorphism by the
five lemma.

###  6.3. Description of $H^{\bullet}K^{\prime}(\mathcal{F})$

As explained in Section 6.1, there is no loss of generality in assuming
$\mathcal{F}$ is oriented, and then we can apply (3.2.19)–(3.2.21) to get
$K^{(s)}(\mathcal{F};\Omega M)\equiv K^{(s)}(\mathcal{F};\Omega
N\mathcal{F})$, where we can consider $d_{\mathcal{F}}$ or
$d_{\mathcal{F}}^{\text{\rm t}}$ using the leafwise flat structure of $\Omega
N\mathcal{F}$.

Consider the notation of Section 5.3. Since $d\omega=\eta\wedge\omega$ and
$d_{\mathcal{F}}$ satisfies the derivation rule on products of smooth leafwise
currents and smooth leafwise forms (Section 3.2.8), it follows that the
version of Proposition 5.3.1 with coefficients in $\Omega N\mathcal{F}$ states
that

(6.3.1) $\left\\{\begin{gathered}K(\mathcal{F};\Omega
N\mathcal{F})\equiv\bigoplus_{k}C^{\infty}(M^{0};\Lambda)\equiv\bigoplus_{k}C^{\infty}(M^{0};\Lambda\otimes\Omega^{-k}NM^{0})\;,\\\
d_{\mathcal{F}}\equiv\bigoplus_{k}d_{-k}\equiv\bigoplus_{k}d\;,\end{gathered}\right.$

where $k$ runs in $\mathbb{N}_{0}$. Moreover the subcomplex
$K^{(s)}(\mathcal{F};\Omega N\mathcal{F})\subset K(\mathcal{F};\Omega
N\mathcal{F})$ corresponds to the finite direct sum with $k<-s-1/2$. Taking
dual spaces and transposing maps, using (2.8.4) and (2.9.8), we get the
following consequence.

######  Corollary 6.3.1.

We have identities of topological complexes,

$\displaystyle
K^{\prime}(\mathcal{F})\equiv\prod_{k}C^{-\infty}(M^{0};\Lambda)\equiv\prod_{k}C^{-\infty}(M^{0};\Lambda\otimes\Omega^{k}NM^{0})\;,$
$\displaystyle d_{\mathcal{F}}\equiv\prod_{k}d_{k}\equiv\prod_{k}d\;,$

where $k$ runs in $\mathbb{N}_{0}$. Moreover the quotient complex
$K^{\prime\,(s)}(\mathcal{F})$ corresponds to the finite direct sum with
$k<s-1/2$.

######  Corollary 6.3.2.

We have TVS-identities,

$H^{\bullet}K^{\prime}(\mathcal{F})\equiv\prod_{k}H_{k}^{\bullet}(M^{0})\equiv\prod_{k}H^{\bullet}(M^{0},\Omega^{k}NM^{0})\;,$

where $k$ runs in $\mathbb{N}_{0}$. Moreover
$H^{\bullet}K^{\prime\,(s)}(\mathcal{F})$ is the quotient space of
$H^{\bullet}K^{\prime}(\mathcal{F})$ given by the finite product with
$k<s-1/2$. In particular, (6.1.4) is satisfied.

######  Remark 6.3.3.

The differential complexes on $M^{0}$ used in Corollary 6.3.1 split into
direct sums of the same complexes given by leaves $L\subset M^{0}$. The same
applies to their cohomologies, used in Corollary 6.3.2.

######  Corollary 6.3.4.

There is a canonical TVS-isomorphism,

$H^{\bullet}K^{\prime}(\mathcal{F})\equiv H^{\bullet}(M^{0})\oplus
H^{n^{\prime\prime}-\bullet}K(\mathcal{F})^{\prime}\;,$

###### Proof.

Apply Corollaries 5.3.2 and 6.3.2, and (2.9.6). ∎

###  6.4. Description of $\bar{H}^{\bullet}J^{\prime}(\mathcal{F})$

Like in Section 5.4, by (2.7.4) and (2.7.5), for $m\in\mathbb{R}$,

(6.4.1) $\displaystyle
J^{\prime\,m}(\mathcal{F})\cong\boldsymbol{\rho}^{m}H_{\text{\rm
b}}^{-\infty}(\boldsymbol{M};\Lambda\boldsymbol{\mathcal{F}})\equiv\boldsymbol{\rho}^{m-\frac{1}{2}}H^{-\infty}(\mathring{\boldsymbol{M}};\Lambda\mathring{\boldsymbol{\mathcal{F}}})\;,$
(6.4.2) $\displaystyle
J^{\prime}(\mathcal{F})\cong\bigcap_{m}\boldsymbol{\rho}^{m}H_{\text{\rm
b}}^{-\infty}(\boldsymbol{M};\Lambda\boldsymbol{\mathcal{F}})=\bigcap_{m}\boldsymbol{\rho}^{m}H^{-\infty}(\mathring{\boldsymbol{M}};\Lambda\mathring{\boldsymbol{\mathcal{F}}})\;,$

as topological complexes with $d_{\mathcal{F}}$,
$d_{\boldsymbol{\mathcal{F}}}$ and $d_{\mathring{\boldsymbol{\mathcal{F}}}}$,
using the b-metric $\boldsymbol{g}$ to define $H_{\text{\rm
b}}^{-\infty}(\boldsymbol{M};\Lambda\boldsymbol{\mathcal{F}})$, and using
$\boldsymbol{g}|_{\mathring{\boldsymbol{M}}}$ to define
$H^{-\infty}(\mathring{\boldsymbol{M}};\Lambda\mathring{\boldsymbol{\mathcal{F}}})$.
The leafwise version of (2.9.4) (Section 3.5) also gives isomorphisms of
topological complexes,

(6.4.3)
$\boldsymbol{\rho}^{-m+\frac{1}{2}}:\big{(}\boldsymbol{\rho}^{m-\frac{1}{2}}H^{\infty}(\mathring{\boldsymbol{M}};\Lambda\mathring{\boldsymbol{\mathcal{F}}}),d_{\mathring{\boldsymbol{\mathcal{F}}}}\big{)}\xrightarrow{\cong}\big{(}H^{\infty}(\mathring{\boldsymbol{M}};\Lambda\mathring{\boldsymbol{\mathcal{F}}}),d_{\mathring{\boldsymbol{\mathcal{F}}},m-\frac{1}{2}}\big{)}\;.$

By (6.4.1) and (6.4.3), and the analog of (3.4.16) for
$\Delta_{\mathring{\boldsymbol{\mathcal{F}}},m-\frac{1}{2}}$ in
$H^{-\infty}(\mathring{\boldsymbol{M}};\Lambda\mathring{\boldsymbol{\mathcal{F}}})$
(Section 3.5), we get induced TVS-isomorphisms

(6.4.4) $\displaystyle\bar{H}^{\bullet}J^{\prime\,m}(\mathcal{F})$
$\displaystyle\cong\bar{H}^{\bullet}\big{(}\boldsymbol{\rho}^{m-\frac{1}{2}}H^{-\infty}(\mathring{\boldsymbol{M}};\Lambda\mathring{\boldsymbol{\mathcal{F}}}),d_{\mathring{\boldsymbol{\mathcal{F}}}}\big{)}$
(6.4.5)
$\displaystyle\cong\bar{H}^{\bullet}\big{(}H^{-\infty}(\mathring{\boldsymbol{M}};\Lambda\mathring{\boldsymbol{\mathcal{F}}}),d_{\mathring{\boldsymbol{\mathcal{F}}},m-\frac{1}{2}}\big{)}$
(6.4.6)
$\displaystyle\cong\ker\Delta_{\mathring{\boldsymbol{\mathcal{F}}},m-\frac{1}{2}}\;.$

By the analog of (6.1.2) for $J^{\prime}(\mathcal{F})$ and (6.4.6), the LCHS
$\bar{H}^{\bullet}J(\mathcal{F})$ is a projective limit of a sequence of
Hilbertian spaces, and therefore a Fréchet space. The isomorphisms (6.4.4) and
(6.4.5) are also true in cohomology.

Theorem 1.3.5 follows from the analog of (6.1.2) for $J^{\prime}(\mathcal{F})$
and (6.4.1)–(6.4.3).

###  6.5. Short exact sequence of dual-conormal reduced cohomology

The goal of this section is to prove the exactness of (6.1.7). Some remarks
will indicate how to adapt the proof to show also the exactness of (6.1.6) and
(6.2.1).

####  6.5.1. The maps $F^{\prime}_{m}$

For every $m\in\mathbb{R}$, let

$F^{\prime}_{m}=E_{-m}^{\text{\rm t}}:I^{\prime\,(s)}(\mathcal{F})\to
J^{\prime\,m}(\mathcal{F})\;,$

where $s=0$ if $m\leq 0$, and $m<s\in\mathbb{Z}^{+}$ if $m>0$, where

$E_{-m}:J^{-m}(\mathcal{F};\Omega M)\to I^{(-s)}(\mathcal{F};\Omega M)$

is given by the version of Corollary 5.5.2 with $\Omega M$ (see Remark 6.1.1);
thus

(6.5.1) $F^{\prime}_{m}d_{\mathcal{F}}=d_{\mathcal{F}}F^{\prime}_{m}\;.$

Since $s\geq m,0$, the map $j^{\prime}_{s,m}$ is defined, and we have

(6.5.2) $F^{\prime}_{m}\iota^{\prime}_{s}=E_{-m}^{\text{\rm
t}}R_{-s}^{\text{\rm t}}=(R_{-s}E_{-m})^{\text{\rm t}}=j_{-m,-s}^{\text{\rm
t}}=j^{\prime}_{s,m}\;.$

####  6.5.2. The maps $E^{\prime}_{m}$

For $s\in\mathbb{R}$ and $m>s+n/2+1$, let

$E^{\prime}_{m}=F_{-m}^{\text{\rm t}}:K^{\prime\,(s^{\prime})}(\mathcal{F})\to
I^{\prime\,(s)}(\mathcal{F})\;,$

where $s^{\prime}=0$ if $m\leq 0$, and $m<s^{\prime}\in\mathbb{Z}^{+}$ if
$m>0$. Here, we use the map

$F_{-m}:I^{(-s)}(\mathcal{F};\Omega M)\to K^{(-s^{\prime})}(\mathcal{F};\Omega
M)$

given be the version of Section 5.5.2 with coefficients in $\Omega M$ (Remark
5.5.3). Consider

(6.5.3)
$F^{\prime}_{m}=j^{\prime}_{m,s}F^{\prime}_{m}:I^{\prime\,(s^{\prime})}(\mathcal{F})\to
J^{\prime\,(s)}(\mathcal{F})\;,$

which is the transpose of the version of (5.5.2) with coefficients in $\Omega
M$,

$E_{-m}:J^{(-s)}(\mathcal{F};\Omega M)\to I^{(-s^{\prime})}(\mathcal{F};\Omega
M)\;.$

Then (6.5.2) becomes

(6.5.4) $F^{\prime}_{m}\iota^{\prime}_{s}=j^{\prime}_{s,s^{\prime}}\;.$

Transposing the versions of (5.5.3)–(5.5.5) with coefficients in $\Omega M$,
we get

(6.5.5)
$\displaystyle\iota^{\prime}_{s}F^{\prime}_{m}+E^{\prime}_{m}R^{\prime}_{s^{\prime}}=j^{\prime}_{s^{\prime},s}:I^{\prime\,(s^{\prime})}(\mathcal{F})\to
I^{\prime\,(s)}(\mathcal{F})\;,$ (6.5.6) $\displaystyle
R^{\prime}_{s}E^{\prime}_{m}=j^{\prime}_{s^{\prime},s}:K^{\prime\,(s^{\prime})}(\mathcal{F})\to
K^{\prime\,(s)}(\mathcal{F})\;,$ (6.5.7) $\displaystyle
E^{\prime}_{m}d_{\mathcal{F}}=d_{\mathcal{F}}E^{\prime}_{m}\;.$

Take greater numbers, $s_{1}>s$, $m_{1}>m$ and $s^{\prime}_{1}>s^{\prime}$,
satisfying the same inequalities as $s$, $m$ and $s^{\prime}$. Using (6.5.3),
the transposition of the versions of (5.5.6) and (5.5.7) with coefficients in
$\Omega M$ give

(6.5.8) $\displaystyle F^{\prime}_{m}j^{\prime}_{s^{\prime}_{1},s^{\prime}}$
$\displaystyle=j^{\prime}_{s_{1},s}F^{\prime}_{m_{1}}\;,$ (6.5.9)
$\displaystyle E^{\prime}_{m}j^{\prime}_{s^{\prime}_{1},s^{\prime}}$
$\displaystyle=j^{\prime}_{s_{1},s}E^{\prime}_{m_{1}}\;.$

####  6.5.3. The equality
$\ker\bar{R}^{\prime}_{*}=\operatorname{im}\bar{\iota}^{\prime}_{*}$

We already know that
$\ker\bar{R}^{\prime}_{*}\supset\operatorname{im}\bar{\iota}^{\prime}_{*}$. To
prove
$\ker\bar{R}^{\prime}_{*}\subset\operatorname{im}\bar{\iota}^{\prime}_{*}$,
take any class $\overline{[u]}\in\ker\bar{R}^{\prime}_{*}$ in
$\bar{H}^{\bullet}I^{\prime}(\mathcal{F})$. Thus there is some net $v_{l}$ in
$K^{\prime}(\mathcal{F})$ such that $R^{\prime}u=\lim_{l}d_{\mathcal{F}}v_{l}$
in $K^{\prime}(\mathcal{F})$. Write $u_{s}=j^{\prime}_{s}u\in
I^{\prime\,(s)}(\mathcal{F})$ and $v_{l,s}=j^{\prime}_{s}v_{l}\in
K^{\prime\,(s)}(\mathcal{F})$. Take $s$, $m$ and $s^{\prime}$ satisfying the
the conditions of Section 6.5.2, obtaining
$E^{\prime}_{m}:K^{\prime\,(s^{\prime})}(\mathcal{F})\to
I^{\prime\,(s)}(\mathcal{F})$ and
$F^{\prime}_{m}:I^{\prime\,(s^{\prime})}(\mathcal{F})\to
J^{\prime\,(s)}(\mathcal{F})$. Let $a_{s}=F^{\prime}_{m}u_{s^{\prime}}\in
J^{\prime\,(s)}(\mathcal{F})$ and $b_{l,s}=E^{\prime}_{m}v_{l,s^{\prime}}\in
I^{\prime\,(s)}(\mathcal{F})$. By (6.5.1),

$d_{\mathcal{F}}a_{s}=F^{\prime}_{m}d_{\mathcal{F}}u_{s^{\prime}}=0\;.$

Moreover, by (6.5.5) and (6.5.7),

$\displaystyle u_{s}$
$\displaystyle=j^{\prime}_{s^{\prime},s}u_{s^{\prime}}=\iota^{\prime}_{s}F^{\prime}_{m}u_{s^{\prime}}+E^{\prime}_{m}R^{\prime}_{s^{\prime}}u_{s^{\prime}}$
$\displaystyle=\iota^{\prime}_{s}a_{s}+\lim_{l}E^{\prime}_{m}d_{\mathcal{F}}v_{l,s^{\prime}}=\iota^{\prime}_{s}a_{s}+\lim_{l}d_{\mathcal{F}}b_{l,s}\;.$

Now consider the above notation for greater real numbers $s_{1}$, $m_{1}$ and
$s^{\prime}_{1}$, satisfying the same properties as $s$, $m$ and $s^{\prime}$.
By (6.5.8) and (6.5.9),

$\displaystyle
j^{\prime}_{s_{1},s}a_{s_{1}}=j^{\prime}_{s_{1},s}F^{\prime}_{m_{1}}u_{s^{\prime}_{1}}=F^{\prime}_{m}j^{\prime}_{s^{\prime}_{1},s^{\prime}}u_{s^{\prime}_{1}}=F^{\prime}_{m}u_{s^{\prime}}=a_{s}\;.$
$\displaystyle
j^{\prime}_{s_{1},s}b_{l,s_{1}}=j^{\prime}_{s_{1},s}E^{\prime}_{m_{1}}v_{l,s^{\prime}_{1}}=E^{\prime}_{m}j^{\prime}_{s^{\prime}_{1},s^{\prime}}v_{l,s^{\prime}_{1}}=E^{\prime}_{m}v_{l,s^{\prime}}=b_{l,s}\;.$

Therefore, taking $s\uparrow+\infty$, $m\uparrow+\infty$ and
$s^{\prime}\uparrow+\infty$, satisfying the above relations, the elements
$a_{s}\in J^{\prime\,(s)}(\mathcal{F})$ and $b_{l,s}\in
I^{\prime\,(s)}(\mathcal{F})$ define elements $a:=(a_{s})_{s}\in
ZJ^{\prime}(\mathcal{F})$ and $b_{l}:=(b_{l,s})_{s}\in
I^{\prime}(\mathcal{F})$, and we have
$u=\iota^{\prime}a+\lim_{l}d_{\mathcal{F}}b_{l}$. Hence
$\overline{[u]}=\bar{\iota}_{*}(\overline{[a]})$.

######  Remark 6.5.1.

A similar argument, taking an element $v\in K^{\prime}(\mathcal{F})$ instead
of a net $v_{l}$, shows the inclusion $\ker
R^{\prime}_{*}=\operatorname{im}\iota^{\prime}_{*}$ in
$H^{\bullet}I^{\prime}(\mathcal{F})$.

######  Remark 6.5.2.

As before, to prove
$\ker\widehat{R}^{\prime}_{*}=\operatorname{im}\hat{\iota}^{\prime}_{*}$ in
(6.2.1), we only have to prove “$\subset$”. For any
$\hat{u}:=(\overline{[u_{s}]}_{s})_{s}\in\ker\widehat{R}^{\prime}_{*}$, there
is some $v\in K^{\prime}(\mathcal{F})$ such that
$R^{\prime}_{s}u_{s}=d_{\mathcal{F}}v_{s}$, where $v_{s}=j^{\prime}_{s}v$.
Moreover,
$j^{\prime}_{s^{\prime},s}u_{s^{\prime}}=u_{s}+\lim_{l}d_{\mathcal{F}}g_{l,s,s^{\prime}}$
for some net $g_{l,s,s^{\prime}}$ in $I^{\prime\,(s)}(\mathcal{F})$. Take
$a_{s}$ and $b_{s,l}$ as above. The given argument shows that

$\displaystyle d_{\mathcal{F}}a_{s}=0\;,\quad
u_{s}+\lim_{l}d_{\mathcal{F}}g_{l,s^{\prime},s}=\iota^{\prime}_{s}a_{s}+\lim_{l}d_{\mathcal{F}}b_{l,s}\;,$
$\displaystyle
j^{\prime}_{s_{1},s}a_{s_{1}}=a_{s}+\lim_{l}d_{\mathcal{F}}F^{\prime}_{m}g_{l,s^{\prime}_{1},s^{\prime}}\;,\quad
j^{\prime}_{s_{1},s}b_{l,s_{1}}=b_{l,s}\;.$

Hence
$\hat{a}:=(\overline{[a_{s}]}_{s})_{s}\in\widehat{H}^{\bullet}J^{\prime}(\mathcal{F})$
is defined and $\hat{\iota}^{\prime}_{*}(\hat{a})=\hat{u}$.

####  6.5.4. Injectivity of $\bar{\iota}^{\prime}_{*}$

Let $\overline{[u]}\in\bar{H}^{r}J^{\prime}(\mathcal{F})$ such that
$\bar{\iota}^{\prime}_{*}(\overline{[u]})=0$. This means that there is a net
$v_{l}$ in $I^{\prime}(\mathcal{F})$ such that
$\iota^{\prime}u=\lim_{l}d_{\mathcal{F}}v_{l}$ in $I^{\prime}(\mathcal{F})$.
Write $u_{s}=j^{\prime}_{s}u\in ZK^{\prime\,(s)}(\mathcal{F})$ and
$v_{l,s}=j^{\prime}_{s}v_{l}\in I^{\prime\,(s)}(\mathcal{F})$. With the
notation of Section 6.5.3, let $b_{l,s}=F^{\prime}_{m}v_{l,s^{\prime}}\in
J^{\prime\,(s)}(\mathcal{F})$. By (6.5.1) and (6.5.4),

$u_{s}=j^{\prime}_{s^{\prime},s}u_{s^{\prime}}=F^{\prime}_{m}\iota^{\prime}_{s^{\prime}}u_{s^{\prime}}=\lim_{l}F^{\prime}_{m}d_{\mathcal{F}}v_{l,s^{\prime}}=\lim_{l}d_{\mathcal{F}}b_{l,s}\;.$

Like in Section 6.5.3, it can be shown that, taking $s\uparrow+\infty$,
$m\uparrow+\infty$ and $s^{\prime}\uparrow+\infty$ as above, the elements
$b_{l,s}\in J^{\prime\,(s)}(\mathcal{F})$ define elements
$b_{l}:=(b_{l,s})_{s}\in J^{\prime}(\mathcal{F})$, and we have
$u=\lim_{l}d_{\mathcal{F}}b_{l}$. Thus $\overline{[u]}=0$ in
$\bar{H}^{r}J^{\prime}(\mathcal{F})$.

######  Remark 6.5.3.

Like in Remark 6.5.1, we also get the injectivity of $\iota^{\prime}_{*}$.

######  Remark 6.5.4.

To prove the injectivity of $\hat{\iota}^{\prime}_{*}$, take any
$\hat{u}:=(\overline{[u_{s}]}_{s})_{s}\in\ker\hat{\iota}^{\prime}_{*}$ in
(6.2.1). Then there is some net $v_{l,s}$ in every
$I^{\prime\,(s)}(\mathcal{F})$ such that
$\iota^{\prime}_{s}u_{s}=\lim_{l}d_{\mathcal{F}}v_{l,s}$ in
$I^{\prime\,(s)}(\mathcal{F})$. Moreover,
$j^{\prime}_{s^{\prime},s}u_{s^{\prime}}=u_{s}+\lim_{l}d_{\mathcal{F}}g_{l,s,s^{\prime}}$
for some net $g_{l,s,s^{\prime}}$ in $J^{\prime\,(s)}(\mathcal{F})$. Take
$b_{l,s}$ as before. The above argument shows that

$u_{s}+\lim_{l}d_{\mathcal{F}}g_{l,s^{\prime},s}=\lim_{l}d_{\mathcal{F}}b_{l,s}\;.$

So $\hat{u}=0$ in $\widehat{H}^{\bullet}J^{\prime}(\mathcal{F})$.

####  6.5.5. Surjectivity of $\bar{R}^{\prime}_{*}$

Take any $[u]\in H^{\bullet}K(\mathcal{F})$, and write
$u_{s}=j^{\prime}_{s}u\in ZK^{(s)}(\mathcal{F})$. With the notation of Section
6.5.2, we have $v_{s}:=E^{\prime}_{m}v_{s^{\prime}}\in
ZI^{(s^{\prime})}(\mathcal{F})$ by (6.5.7), and
$R^{\prime}_{s}v_{s}=j^{\prime}_{s^{\prime},s}u_{s^{\prime}}=u_{s}$ by
(6.5.6).

Now consider the above notation for greater real numbers $s_{1}$, $m_{1}$ and
$s^{\prime}_{1}$, satisfying the same properties as $s$, $m$ and $s^{\prime}$.
By (6.5.9),

$j^{\prime}_{s_{1},s}v_{s_{1}}=j^{\prime}_{s_{1},s}E^{\prime}_{m_{1}}v_{s^{\prime}_{1}}=E^{\prime}_{m}j^{\prime}_{s^{\prime}_{1},s^{\prime}}v_{s^{\prime}_{1}}=E^{\prime}_{m}v_{s^{\prime}}=v_{s}\;.$

So $v:=(v_{s})_{s}\in ZI^{\prime}(\mathcal{F})$ satisfies $R^{\prime}v=u$, and
therefore $\bar{R}^{\prime}_{*}(\overline{[v]})=\overline{[u]}$.

######  Remark 6.5.5.

Using cohomology instead of reduced cohomology, the analogous argument gives
the surjectivity of $R^{\prime}_{*}$.

######  Remark 6.5.6.

To prove the surjectivity of $\widehat{R}^{\prime}_{*}$ in (6.2.1), for any
$[u]\in H^{\bullet}K(\mathcal{F})$, define $u_{s}$ and $v_{s}$ as above. We
also have $R^{\prime}_{s}v_{s}=u_{s}$ and
$j^{\prime}_{s_{1},s}v_{s_{1}}=v_{s}$. Thus
$\hat{v}:=(\overline{[v_{s}]}_{s})_{s}\in\widehat{H}^{\bullet}I^{\prime}(\mathcal{F})$
and $\widehat{R}^{\prime}_{*}\hat{v}=[u]$.

###  6.6. Functoriality and leafwise homotopy invariance

####  6.6.1. Pull-back of dual-conormal leafwise currents

Consider the notation and conditions of Section 5.7.3 (including the
conditions of Section 5.7.1). According to Section 2.3.5, the map (3.2.23) has
a continuous extension

(6.6.1) $\phi^{*}:I^{\prime}(\mathcal{F})\to
I^{\prime}(\mathcal{F}^{\prime})\;,$

defined as the composition

$I^{\prime}(\mathcal{F})\xrightarrow{\phi^{*}}I^{\prime}(M^{\prime},M^{\prime\,0};\phi^{*}\Lambda\mathcal{F})\xrightarrow{\phi^{*}}I^{\prime}(\mathcal{F}^{\prime})\;,$

like (5.7.2), using (2.3.11) with $E=\Lambda\mathcal{F}$. We can also describe
(6.6.1) as the restriction of (2.8.24) to dual-conormal currents of bidegree
$(0,\bullet)$, like in (3.2.33). The map (6.6.1) is also a restriction of
(3.2.33).

(2.3.11) with $E=\Lambda\mathcal{F}$

Similarly, the analogs of (2.2.20) with $E=\Lambda\mathcal{F}$ for (2.7.9) and
(2.7.10) induce continuous homomorphisms

(6.6.2) $\displaystyle\phi^{*}:K^{\prime}(\mathcal{F})\to
K^{\prime}(\mathcal{F}^{\prime})\;,$ (6.6.3)
$\displaystyle\phi^{*}:J^{\prime}(\mathcal{F})\to
J^{\prime}(\mathcal{F}^{\prime})\;.$

By passing to cohomology and reduced cohomology, we get continuous
homomorphisms,

(6.6.4)
$\left\\{\begin{gathered}\phi^{*}:H^{\bullet}K^{\prime}(\mathcal{F})\to
H^{\bullet}K^{\prime}(\mathcal{F}^{\prime})\;,\\\ \begin{aligned}
\phi^{*}:&H^{\bullet}I^{\prime}(\mathcal{F})\to
H^{\bullet}I^{\prime}(\mathcal{F}^{\prime})\;,&\quad\phi^{*}:&\bar{H}^{\bullet}I^{\prime}(\mathcal{F})\to\bar{H}^{\bullet}I^{\prime}(\mathcal{F}^{\prime})\;,\\\
\phi^{*}:&H^{\bullet}J^{\prime}(\mathcal{F})\to
H^{\bullet}J^{\prime}(\mathcal{F}^{\prime})\;,&\quad\phi^{*}:&\bar{H}^{\bullet}J^{\prime}(\mathcal{F})\to\bar{H}^{\bullet}J^{\prime}(\mathcal{F}^{\prime})\;.\end{aligned}\end{gathered}\right.$

The assignment of the homomorphisms (6.6.1)–(6.6.4) is functorial.

####  6.6.2. Description of $\phi^{*}:K^{\prime}(\mathcal{F})\to
K^{\prime}(\mathcal{F}^{\prime})$

Consider the notation and conditions of Section 5.7.2, and assume also that
$\phi$ is a submersion. By the density of the space of smooth forms in the
space of currents, we get from (5.7.6) that

(6.6.5) $\phi^{*}:C^{-\infty}(M^{0};\Lambda)\to
C^{-\infty}(M^{\prime\,0};\Lambda)$

is a cochain map for $d_{s\eta}$ and $d_{s\eta^{\prime}}$ ($s\in\mathbb{R}$),
and we get from (5.7.7) that

(6.6.6) $\phi^{*}:C^{-\infty}(M^{0};\Lambda\otimes\Omega^{s}NM^{0})\to
C^{-\infty}(M^{\prime\,0};\Lambda\otimes\Omega^{s}NM^{\prime\,0})$

is another cochain map for the de Rham differentials defined with the flat
bundle structures of $\Omega^{s}NM^{0}$ and $\Omega^{s}NM^{\prime\,0}$.

######  Proposition 6.6.1.

According to Corollary 6.3.1, the map (6.6.2) is given by

$\phi^{*}\equiv\prod_{k}\phi^{*}\equiv\prod_{k}\phi^{*}\;,$

where the terms of the first direct sum are given by (6.6.5), and the terms of
the second direct sum are given by (6.6.6), taking $s=k$.

###### Proof.

Apply Propositions 2.8.1, 2.8.2 and 5.7.4. ∎

####  6.6.3. Push-forward of dual-conormal leafwise currents

Consider the notation and conditions of Section 5.7.3 (containing those of
Section 5.7.1). Then the case of (3.2.33) on smooth leafwise forms has a
continuous extension

(6.6.7) $\phi_{*}:I^{\prime}_{{\text{\rm c}}/{\text{\rm
cv}}}(\mathcal{F}^{\prime})\to I^{\prime}_{{\text{\rm
c}}/{\cdot}}(\mathcal{F})\;.$

This map can be described as the restriction of the map (2.8.33) to dual-
conormal currents of bidegree $(0,\bullet)$, like (3.2.33) in Section 3.2.15.
We can also describe (6.6.7) as the composition

$I^{\prime}_{{\text{\rm c}}/{\text{\rm
cv}}}(M^{\prime},L^{\prime};\Lambda\mathcal{F})\xrightarrow{\pi_{\text{\rm
top}}}I^{\prime}_{{\text{\rm c}}/{\text{\rm
cv}}}(M^{\prime},L^{\prime};\phi^{*}\Lambda\mathcal{F}\otimes\Omega_{\text{\rm
fiber}})\xrightarrow{\phi_{*}}I^{\prime}_{{\text{\rm
c}}/{\cdot}}(M,L;\Lambda\mathcal{F})\;,$

like in (3.2.35), where $\phi_{*}$ is given by (2.3.10) with
$E=\Lambda\mathcal{F}$. The map (6.6.7) is also a restriction of the case of
(3.2.33) for leafwise currents.

According to Section 2.7.7, the map (6.6.7) induces homomorphisms

(6.6.8) $\displaystyle\phi_{*}:K^{\prime}(\mathcal{F}^{\prime})\to
K^{\prime}(\mathcal{F})\;,$ (6.6.9)
$\displaystyle\phi_{*}:J^{\prime}_{{\text{\rm c}}/{\text{\rm
cv}}}(\mathcal{F}^{\prime})\to J^{\prime}_{{\text{\rm
c}}/{\cdot}}(\mathcal{F})\;.$

Like in Section 6.6.1, we get induced continuous homomorphisms,

(6.6.10)
$\left\\{\begin{gathered}\phi_{*}:H^{\bullet}K^{\prime}(\mathcal{F}^{\prime})\to
H^{\bullet}K^{\prime}(\mathcal{F})\;,\\\ \begin{aligned}
\phi_{*}:&H^{\bullet}I^{\prime}_{\text{\rm c}}(\mathcal{F}^{\prime})\to
H^{\bullet}I^{\prime}_{\text{\rm
c}}(\mathcal{F})\;,&\quad\phi_{*}:&\bar{H}^{\bullet}I^{\prime}_{\text{\rm
c}}(\mathcal{F}^{\prime})\to\bar{H}^{\bullet}I^{\prime}_{\text{\rm
c}}(\mathcal{F})\;,\\\ \phi_{*}:&H^{\bullet}J^{\prime}_{\text{\rm
c}}(\mathcal{F}^{\prime})\to H^{\bullet}J^{\prime}_{\text{\rm
c}}(\mathcal{F})\;,&\quad\phi_{*}:&\bar{H}^{\bullet}J^{\prime}_{\text{\rm
c}}(\mathcal{F}^{\prime})\to\bar{H}^{\bullet}J^{\prime}_{\text{\rm
c}}(\mathcal{F})\;.\end{aligned}\end{gathered}\right.$

The assignments of homomorphisms (6.6.7)–(6.6.10) are clearly functorial.

####  6.6.4. Leafwise homotopy invariance

Consider the notation and conditions of Section 5.7.5, and assume that every
$H_{t}$ is a submersion. Like in Section 5.7.5, according to Sections 6.6.1
and 6.6.3, the corresponding leafwise homotopy operator
$\mathsf{h}:C^{\infty}(M;\Lambda\mathcal{F})\to
C^{\infty}(M^{\prime};\Lambda\mathcal{F}^{\prime})$ has continuous linear
extensions,

$\mathsf{h}:K^{\prime}(\mathcal{F})\to
K^{\prime}(\mathcal{F}^{\prime})\;,\quad\mathsf{h}:I^{\prime}(\mathcal{F})\to
I^{\prime}(\mathcal{F}^{\prime})\;,\quad\mathsf{h}:J^{\prime}(\mathcal{F})\to
J^{\prime}(\mathcal{F}^{\prime})\;.$

By continuity and according to Section 3.2.16, we have
$H_{1}^{*}-H_{0}^{*}=\mathsf{h}d_{\mathcal{F}}+d_{\mathcal{F}^{\prime}}\mathsf{h}$
with $H_{0}^{*}$ and $H_{1}^{*}$ given by (6.6.1), (6.6.2) and (6.6.3). Hence
we get the following.

######  Proposition 6.6.2.

Let $\phi,\psi:(M^{\prime},\mathcal{F}^{\prime})\to(M,\mathcal{F})$ be smooth
foliated maps transverse to $M^{0}$ with
$\phi^{-1}(M^{0})=\psi^{-1}(M^{0})=M^{\prime\,0}$. If $\phi$ is leafwise
homotopic to $\psi$, then $\phi$ and $\psi$ induce the same homomorphisms
(6.6.4).

###  6.7. Action of foliated flows on the dual-conormal sequence

Consider the notation and conditions of Section 5.8.

######  Proposition 6.7.1.

According to Corollaries 6.3.2 and 6.3.3,

$\phi^{t*}\equiv\prod_{k,L}e^{k\varkappa_{L}t}\equiv\prod_{k,L}e^{k\varkappa_{L}t}$

on $H^{\bullet}K(\mathcal{F})$, where $k$ runs in $\mathbb{N}_{0}$ and $L$ in
$\pi_{0}M^{0}$.

###### Proof.

Argue like in the proof of Proposition 5.8.1 and its previous observations,
using Corollary 6.3.1, Remark 6.3.3 and Proposition 6.6.2. ∎

Corollaries 6.3.1, 6.3.2, 6.3.3 and 6.7.1 show Theorem 1.3.4.

##  Chapter 7 Contribution from $M^{1}$

###  7.1. Operators on a suspension foliation

Consider again the notation of Section 4.2, where the case of a weakly simple
foliated flow $\phi=\\{\phi^{t}\\}$ on a suspension foliated manifold
$(M,\mathcal{F})$ was described. Equip $\mathring{M}_{\pm}$ with $g_{\pm}$,
obtaining that $\mathring{\mathcal{F}}_{\pm}$ is of bounded geometry
(Proposition 4.2.1). We can assume $\phi$ is of $\mathbb{R}$-local bounded
geometry on $\mathring{M}_{\pm}$ by Proposition 4.2.15 and according to
Section 2.4.7. Thus, on
$\mathring{\widetilde{M}}_{\pm}\equiv(\mathring{\widetilde{M}}_{\pm},\tilde{g}_{\pm}$),
$\mathring{\widetilde{\mathcal{F}}}_{\pm}$ is of bounded geometry and
$\tilde{\phi}$ is of $\mathbb{R}$-local bounded geometry. Consider the
leafwise perturbed operators for
$(\mathring{M}_{\pm},\mathring{\mathcal{F}}_{\pm})$ and
$(\mathring{\widetilde{M}}_{\pm},\mathring{\widetilde{\mathcal{F}}}_{\pm})$
defined by the leafwise-closed form $\eta_{0}$ and the leafwise-exact form
$\tilde{\eta}_{0}$ (Section 3.3.1 ). For any $\psi\in\mathcal{A}$, $f\in
C^{\infty}_{\text{\rm c}}(\mathbb{R})$ and $z\in\mathbb{C}$, the operator

(7.1.1)
$\mathring{P}_{\pm}=\int_{-\infty}^{+\infty}\phi^{t*}_{z}\,\psi(D_{\mathring{\mathcal{F}}_{\pm},z})\,f(t)\,dt$

on $H^{-\infty}(\mathring{M}_{\pm};\Lambda\mathring{\mathcal{F}}_{\pm})$ is a
version of (3.4.17) for $\phi^{t*}_{z}$ and
$D_{\mathring{\mathcal{F}}_{\pm},z}$, and therefore it is smoothing by the
corresponding analog of (3.4.18). Let
$\mathring{K}_{\pm}=K_{\mathring{P}_{\pm}}$.

By (4.2.1) and (4.2.2), for $\gamma\in\Gamma$ and $t\in\mathbb{R}$, the
equality
$\tilde{\phi}^{t*}_{z}T_{\gamma}^{*}=T_{\gamma}^{*}\tilde{\phi}^{t*}_{z}$
means that, for all $x\in\mathbb{R}$,

(7.1.2)
$\tilde{\phi}^{t*}_{x,z}T_{\gamma}^{*}=T_{\gamma}^{*}\tilde{\phi}^{t*}_{a_{\gamma}x,z}$

on $C^{\infty}(\widetilde{L},\Lambda)$ (Section 2.9.4). Consider also the
notation of Section 2.9.9 for the regular covering
$\pi=\pi_{L}:\widetilde{L}\to L$ used in the suspension construction; in
particular, recall the notation $\tilde{k}_{z}$. Recall that
$h_{\pm}(\gamma)=\varkappa^{-1}\ln a_{\gamma}$ for $\gamma\in\Gamma$, and
$D_{\pm}(x,\tilde{y})=\varkappa^{-1}\ln|x|$ for
$(x,\tilde{y})\in\mathring{\widetilde{M}}_{\pm}$ (Section 4.2.1). Thus, by the
version of (3.4.21) for the leafwise perturbed differential complex (Section
3.5), and by (4.2.1), (4.2.2) and (7.1.2), if $\hat{\psi}\in
C^{\infty}_{\text{\rm c}}(\mathbb{R})$, then

(7.1.3)
$\mathring{K}_{\pm}([x,\tilde{y}],[x^{\prime},\tilde{y}^{\prime}])\equiv\sum_{\gamma\in\Gamma}\mathring{K}_{\pm,\gamma}([x,\tilde{y}],[x^{\prime},\tilde{y}^{\prime}])$

for all
$(x,\tilde{y}),(x^{\prime},\tilde{y}^{\prime})\in\mathring{\widetilde{M}}_{\pm}$,
where

(7.1.4)
$\mathring{K}_{\pm,\gamma}([x,\tilde{y}],[x^{\prime},\tilde{y}^{\prime}])\\\
=\frac{1}{|\varkappa|}\tilde{\phi}_{x,z}^{\frac{1}{\varkappa}\ln\frac{x^{\prime}}{a_{\gamma}x}*}T_{\gamma}^{*}\tilde{k}_{z}\Big{(}\gamma\cdot\tilde{\phi}^{\frac{1}{\varkappa}\ln\frac{x^{\prime}}{a_{\gamma}x}}_{x}(\tilde{y}),\tilde{y}^{\prime}\Big{)}f\Big{(}\frac{1}{\varkappa}\ln\frac{x^{\prime}}{a_{\gamma}x}\Big{)}\Big{|}\frac{dx^{\prime}}{x^{\prime}}\Big{|}\;.$

According to Section 2.5.26, for the boundary-defining function $\rho$ on
$M_{\pm}$ (Section 4.2.5), let $\rho$ and $\rho^{\prime}$ denote its lifts to
$(M_{\pm})^{2}$ from the left and right factors, and let
$s=\rho/\rho^{\prime}:(M_{\pm})^{2}\to[0,\infty]$. We have corresponding
smooth functions $\rho$, $\rho^{\prime}$ and $s$ on $(M_{\pm})^{2}_{\text{\rm
b}}$. Similarly, let $\eta$ and $\eta^{\prime}$ denote the lifts of $\eta$
from the left and right factors. Using (4.2.13), we get

$(M_{\pm})^{2}\equiv[0,\infty)_{\rho}\times[0,\infty)_{\rho^{\prime}}\times
L^{2}\;,\quad(\mathring{M}_{\pm})^{2}\equiv(0,\infty)^{2}\times L^{2}\;.$

Then

$(M_{\pm})^{2}_{\text{\rm b}}\equiv[0,\infty)_{\rho}\times[0,\infty]_{s}\times
L^{2}\;,$

with boundary components $\operatorname{lb}=\\{s=0\\}$,
$\operatorname{rb}=\\{s=\infty\\}$ and $\operatorname{ff}=\\{\rho=0\\}$.
Moreover

$\Delta_{\text{\rm b}}\equiv\\{\,(\rho,1,y,y)\mid\rho\geq 0,\ y\in L\,\\}\;.$

With the above identities, the restriction of $\beta_{\text{\rm
b}}:(M_{\pm})^{2}_{\text{\rm b}}\to(M_{\pm})^{2}$ to the interior corresponds
to the diffeomorphism

$(0,\infty)^{2}\times L^{2}\to(0,\infty)^{2}\times
L^{2}\;,\quad(\rho,s,y,y^{\prime})\mapsto(\rho,\rho s^{-1},y,y^{\prime})\;.$

Similar observations apply to $\widetilde{M}_{\pm}$, using $\widetilde{L}$
instead of $L$, and using the lifts $\tilde{\rho}$, $\tilde{\rho}^{\prime}$
and $\tilde{s}$ instead of $\rho$, $\rho^{\prime}$ and $s$. The subscript
“$\pm$” will be added to the notation $\Delta_{\text{\rm b}}$ and
$\Delta_{\text{\rm b},0}=\Delta_{\text{\rm b}}\cap\operatorname{ff}$ if
needed.

Let $\mathring{\kappa}_{\pm}$ be the $C^{\infty}$ section of $\beta_{\text{\rm
b}}^{*}(\Lambda\mathcal{F}_{\pm}\boxtimes(\Lambda\mathcal{F}^{*}_{\pm}\otimes\Omega
M_{\pm}))$ on the interior of $(M_{\pm})^{2}_{\text{\rm b}}$ that corresponds
to $\mathring{K}_{\pm}$ via $\beta_{\text{\rm b}}$. If $\hat{\psi}\in
C^{\infty}_{\text{\rm c}}(\mathbb{R})$, then, using the changes of variables

$x=\pm e^{-F(\tilde{y})}\tilde{\rho}\;,\quad x^{\prime}=\pm
e^{-F(\tilde{y}^{\prime})}\tilde{\rho}^{\prime}\;,$

with

$\displaystyle\ln\frac{x^{\prime}}{x}=F(\tilde{y})-F(\tilde{y}^{\prime})-\ln\tilde{s}\;,\quad\frac{dx^{\prime}}{x^{\prime}}=-\tilde{\eta}^{\prime}+\frac{d\tilde{\rho}^{\prime}}{\tilde{\rho}^{\prime}}\;,\quad\frac{d\tilde{s}}{\tilde{s}}=-\frac{d\tilde{\rho}^{\prime}}{\tilde{\rho}^{\prime}}\;,$
$\displaystyle
x^{\prime}=0\Leftrightarrow\tilde{\rho}^{\prime}=0\Leftrightarrow\tilde{s}=\infty\;,\quad
x^{\prime}=\pm\infty\Leftrightarrow\tilde{\rho}^{\prime}=\infty\Leftrightarrow\tilde{s}=0\;,$

it follows from (7.1.3) and (7.1.4) that

(7.1.5)
$\displaystyle\mathring{K}_{\pm}(\rho,\rho^{\prime},[\tilde{y}],[\tilde{y}^{\prime}])$
$\displaystyle=\sum_{\gamma\in\Gamma}\mathring{K}_{\pm,\gamma}(\rho,\rho^{\prime},[\tilde{y}],[\tilde{y}^{\prime}])\;,$
(7.1.6)
$\displaystyle\mathring{\kappa}_{\pm}(\rho,s,[\tilde{y}],[\tilde{y}^{\prime}])$
$\displaystyle=\sum_{\gamma\in\Gamma}\mathring{\kappa}_{\pm,\gamma}(\rho,s,[\tilde{y}],[\tilde{y}^{\prime}])\;,$

where

(7.1.7)
$\mathring{K}_{\pm,\gamma}(\rho,\rho^{\prime},[\tilde{y}],[\tilde{y}^{\prime}])\\\
\begin{aligned} &=\frac{1}{|\varkappa|}\tilde{\phi}_{\pm
e^{-F(\tilde{y})}\rho,z}^{\frac{1}{\varkappa}(F(\tilde{y})-F(\tilde{y}^{\prime})+\ln\frac{\rho^{\prime}}{a_{\gamma}\rho})*}T_{\gamma}^{*}\,\tilde{k}_{z}\Big{(}\gamma\cdot\tilde{\phi}^{\frac{1}{\varkappa}(F(\tilde{y})-F(\tilde{y}^{\prime})+\ln\frac{\rho^{\prime}}{a_{\gamma}\rho})}_{\pm
e^{-F(\tilde{y})}\rho}(\tilde{y}),\tilde{y}^{\prime}\Big{)}\\\
&\phantom{=\text{}}\text{}\times
f\Big{(}\frac{1}{\varkappa}\Big{(}F(\tilde{y})-F(\tilde{y}^{\prime})+\ln\frac{\rho^{\prime}}{a_{\gamma}\rho}\Big{)}\Big{)}\Big{|}\frac{d\rho^{\prime}}{\rho^{\prime}}\Big{|}\;,\end{aligned}$

and

(7.1.8)
$\mathring{\kappa}_{\pm,\gamma}(\rho,s,[\tilde{y}],[\tilde{y}^{\prime}])\\\
\begin{aligned} &=\frac{1}{|\varkappa|}\tilde{\phi}_{\pm
e^{-F(\tilde{y})}\rho,z}^{\frac{1}{\varkappa}(F(\tilde{y})-F(\tilde{y}^{\prime})-\ln
a_{\gamma}s)*}\,T_{\gamma}^{*}\,\tilde{k}_{z}\Big{(}\gamma\cdot\tilde{\phi}^{\frac{1}{\varkappa}(F(\tilde{y})-F(\tilde{y}^{\prime})-\ln
a_{\gamma}s)}_{\pm
e^{-F(\tilde{y})}\rho}(\tilde{y}),\tilde{y}^{\prime}\Big{)}\\\
&\phantom{:=\text{}}\text{}\times
f\Big{(}\frac{1}{\varkappa}\big{(}F(\tilde{y})-F(\tilde{y}^{\prime})-\ln
a_{\gamma}s\big{)}\Big{)}\Big{|}\frac{ds}{s}\Big{|}\;.\end{aligned}$

Let us look for more general conditions on $\psi$ to get (7.1.6) by using the
Fréchet algebra and $\mathbb{C}[z]$-module $\mathcal{A}$ (Section 2.9.8).
Notice that every
$\mathring{\kappa}_{\pm,\gamma}(\rho,s,[\tilde{y}],[\tilde{y}^{\prime}])$ has
a $C^{\infty}$ extension to $\rho=0$.

######  Lemma 7.1.1.

If $\psi\in\mathcal{A}$, then, given any fundamental domain
$\boldsymbol{F}\subset\widetilde{L}$, the series in (7.1.6) converges with all
covariant derivatives, uniformly on $\rho\geq 0$, $0<s<\infty$ and
$\tilde{y},\tilde{y}^{\prime}\in\boldsymbol{F}$. Moreover its sum is
$\mathring{\kappa}_{\pm}(\rho,s,[\tilde{y}],[\tilde{y}^{\prime}])$ for
$\rho>0$.

###### Proof.

Since $\tilde{\phi}$ is of $\mathbb{R}$-local bounded geometry on
$\mathring{\widetilde{M}}_{\pm}$ with $\tilde{g}_{\pm}$ and
$\operatorname{supp}f$ is compact, we can take $R>0$ and
$\boldsymbol{K}\subset{\widetilde{L}}^{2}$ like in the proof of
LABEL:p:_d_FF(phi^t(p)_p)_ge_c_1^{-1}_|gamma|_-_c_2 with
$\operatorname{supp}f\subset I$ for any compact $I\subset\mathbb{R}$. Using
(2.9.19) with this $\boldsymbol{K}$, for any $W>0$, we get

$\big{|}\tilde{k}_{z}(\gamma\cdot\tilde{\phi}^{t}_{x}(\tilde{y}),\tilde{y}^{\prime})\big{|}\leq
C^{\prime}_{1}e^{-\frac{W}{c_{1}}\,|\gamma|}\,\|\psi\|_{\mathcal{A},W,N}$

for $\gamma\in\Gamma$, $x\in\mathbb{R}^{\pm}$, $t\in\operatorname{supp}f$ and
$\tilde{y},\tilde{y}^{\prime}\in\boldsymbol{F}$. Using again the
$\mathbb{R}$-local bounded geometry of $\tilde{\phi}$ on
$\mathring{\widetilde{M}}_{\pm}$ with $\tilde{g}_{\pm}$ and compactness of
$I$, it follows that there is some $C_{2}=C_{2}(z,W)>0$ such that

(7.1.9)
$\big{|}\mathring{\kappa}_{\pm,\gamma}(\rho,s,[\tilde{y}],[\tilde{y}^{\prime}])\big{|}\leq
C_{2}e^{-\frac{W}{c_{1}}\,|\gamma|}\,\|\psi\|_{\mathcal{A},W,N}\,\|f\|_{I,C^{0}}$

for $\gamma\in\Gamma$, $\rho\geq 0$, $s>0$ and
$\tilde{y},\tilde{y}^{\prime}\in\boldsymbol{F}$. By (2.9.20) and (7.1.9), if
$W>c_{1}W_{0}$, then the series in (7.1.6) converges uniformly on $\rho\geq
0$, $s>0$ and $\tilde{y},\tilde{y}^{\prime}\in\boldsymbol{F}$, and the norm of
its sum is${}\leq C\|\psi\|_{\mathcal{A},W,N}$ for some $C=C(z,W,N)>0$.

With more generality, by the $\mathbb{R}$-local bounded geometry of
$\tilde{\phi}$ on $\mathring{\widetilde{M}}_{\pm}$ and the compactness of $I$,
the higher order derivatives of $\tilde{\phi}^{t}_{x}(\tilde{y})$ with respect
to $x$, $t$ and $\tilde{y}$ (in normal coordinates) are also uniformly bounded
for $x\in\mathbb{R}^{\pm}$, $t\in I$ and $\tilde{y}\in\widetilde{L}$. Hence,
for every $m\in\mathbb{N}_{0}$, it follows from (2.9.19) that

$\Big{|}\nabla_{\tilde{y}}^{m_{1}}\nabla_{\tilde{y}^{\prime}}^{m_{2}}\tilde{k}_{z}(\gamma\cdot\tilde{\phi}^{t}_{x}(\tilde{y}),\tilde{y}^{\prime})\Big{|}\leq
C^{\prime}_{1}e^{-\frac{W}{c_{1}}\,|\gamma|}\,\|\psi\|_{\mathcal{A},W,N+m}$

for $\gamma\in\Gamma$, $x\in\mathbb{R}^{\pm}$, $t\in I$,
$\tilde{y},\tilde{y}^{\prime}\in\boldsymbol{F}$ and $m_{1}+m_{2}\leq m$.
Moreover, since $I$ and $\boldsymbol{F}$ are compact, there is some
$c_{3}\in\mathbb{R}$ such that, for all
$\tilde{y},\tilde{y}^{\prime}\in\boldsymbol{F}$,

$\ln
a_{\gamma}s>c_{3}\Rightarrow\varkappa^{-1}\big{(}F(\tilde{y})-F(\tilde{y}^{\prime})-\ln
a_{\gamma}s\big{)}\notin I\;.$

Thus we can assume $s^{-1}<e^{-c_{3}}a_{\gamma}$, yielding
$s^{-1}<e^{c_{0}|\gamma|-c_{3}}$ by (4.2.19). Hence there is some
$C_{3}=C_{3}(z,W,m)>0$ such that

(7.1.10)
$\big{|}\partial^{m_{1}}_{\rho}\partial^{m_{2}}_{s}\nabla^{m_{3}}_{[\tilde{y}]}\nabla^{m_{4}}_{[\tilde{y}^{\prime}]}\mathring{\kappa}_{\pm,\gamma}(\rho,s,[\tilde{y}],[\tilde{y}^{\prime}])\big{|}\\\
\leq
C_{3}e^{(mc_{0}-\frac{W}{c_{1}})|\gamma|}\,\|\psi\|_{\mathcal{A},W,N+m}\,\|f\|_{I,C^{m}}\;,$

for $\gamma\in\Gamma$, $\rho\geq 0$, $s>0$,
$\tilde{y},\tilde{y}^{\prime}\in\boldsymbol{F}$ and $m_{1}+\dots+m_{4}\leq m$.
By (2.9.20) and (7.1.10), if $W>c_{1}(mc_{0}+W_{0})$, then the series defined
by the covariant derivatives of order${}\leq m$ of the terms in (7.1.6) is
also convergent, uniformly on $\rho\geq 0$, $s>0$ and
$\tilde{y},\tilde{y}^{\prime}\in\boldsymbol{F}$, and the norm of its sum
is${}\leq C^{\prime}\|\psi\|_{\mathcal{A},W,N+m}\,\|f\|_{I,C^{m}}$ for some
$C^{\prime}=C^{\prime}(z,W,N,m)>0$.

We already know that the sum of the series in (7.1.6) is
$\mathring{\kappa}_{\pm}(\rho,s,[\tilde{y}],[\tilde{y}^{\prime}])$ for
$\rho>0$ if $\hat{\psi}\in C^{\infty}_{\text{\rm c}}(\mathbb{R})$. Then this
also holds when $\psi\in\mathcal{A}$, as follows by taking a convergent
sequence $\psi_{k}\to\psi$ in $\mathcal{A}$ with $\widehat{\psi_{k}}\in
C^{\infty}_{\text{\rm c}}(\mathbb{R})$, and using the above estimates of the
sum. ∎

######  Remark 7.1.2.

Like in Remark 2.9.3, Lemma 7.1.1 is true for any $\psi\in\SS$ since $\Gamma$
is abelian. But $\psi\in\mathcal{A}$ is needed for the estimates (7.1.9) and
(7.1.10), which will be used later.

######  Proposition 7.1.3.

If $\psi\in\mathcal{A}$, then $\mathring{\kappa}_{\pm}$ has a $C^{\infty}$
extension $\kappa_{\pm}$ to $(M_{\pm})^{2}_{\text{\rm b}}$, also given by
(7.1.6) and (7.1.8) using $C^{\infty}$ extensions $\kappa_{\pm,\gamma}$ of the
sections $\mathring{\kappa}_{\pm,\gamma}$ to $(M_{\pm})^{2}_{\text{\rm b}}$,
which vanishes to all orders at $\operatorname{lb}\cup\operatorname{rb}$.
Therefore $\kappa_{\pm}=\kappa_{P_{\pm}}$ for some $P_{\pm}\in\Psi_{\text{\rm
b}}^{-\infty}(M_{\pm};\Lambda\mathcal{F}_{\pm})$ induced by
$\mathring{P}_{\pm}$.

###### Proof.

By Lemma 7.1.1, $\mathring{\kappa}_{\pm}$ extends smoothly to
$\mathring{\operatorname{ff}}$ ($\rho=0$ and $0<s<\infty$).

Take any compact $I\subset\mathbb{R}$ containing $\operatorname{supp}f$.
According to (7.1.8), the sum in (7.1.6) can be taken for $\gamma\in\Gamma$
with

$\varkappa^{-1}(F(\tilde{y})-F(\tilde{y}^{\prime})-\ln a_{\gamma}s)\in I\;.$

Then, since $\tilde{y},\tilde{y}^{\prime}\in\boldsymbol{F}$, there exists
$R>0$ such that

$\ln s-R<\ln a_{\gamma}<\ln s+R\;.$

Combining this with (4.2.19), we get

(7.1.11) $c_{0}^{-1}(\pm\ln s-R)<|\gamma|\;.$

By (2.9.20), (7.1.9) and (7.1.11), for any $W>c_{1}W_{0}$, there is some
$C^{\prime}_{2}=C^{\prime}_{2}(z,W)>0$ such that, for $\rho\geq 0$, $s>0$ and
$y,y^{\prime}\in L$,

$\displaystyle\big{|}\mathring{\kappa}_{\pm}(\rho,s,y,y^{\prime})\big{|}$
$\displaystyle<C_{2}\sum_{|\gamma|>c_{0}^{-1}(\pm\ln
s-R)}e^{-c_{1}^{-1}W\,|\gamma|}\,\|\psi\|_{\mathcal{A},W,N}\,\|f\|_{I,C^{0}}$
$\displaystyle<C_{2}e^{-(c_{1}^{-1}W-W_{0})c_{0}^{-1}(\pm\ln
s-R)}\sum_{\gamma\in\Gamma}e^{-W_{0}|\gamma|}\,\|\psi\|_{\mathcal{A},W,N}\,\|f\|_{I,C^{0}}$
(7.1.12)
$\displaystyle<C^{\prime}_{2}s^{\mp(c_{1}^{-1}W-W_{0})c_{0}^{-1}}\,\|\psi\|_{\mathcal{A},W,N}\,\|f\|_{I,C^{0}}\;.$

Using (7.1.10) and (7.1.11), we similarly get that, for $m\in\mathbb{N}_{0}$,
if $W>c_{1}(mc_{0}+W_{0})$, then there is some
$C^{\prime}_{3}=C^{\prime}_{3}(z,W,m)>0$ such that

(7.1.13)
$\big{|}\partial^{m_{1}}_{\rho}\partial^{m_{2}}_{s}\nabla^{m_{3}}_{y}\nabla^{m_{4}}_{y^{\prime}}\mathring{\kappa}_{\pm}(\rho,s,y,y^{\prime})\big{|}\\\
<C^{\prime}_{3}s^{\mp(c_{1}^{-1}W-mc_{0}-W_{0})c_{0}^{-1}}\,\|\psi\|_{\mathcal{A},W,N+m}\,\|f\|_{I,C^{m}}$

for $\rho\geq 0$, $s>0$, $y,y^{\prime}\in L$ and $m_{1}+m_{2}+m_{3}+m_{4}\leq
m$. Since $W$ is arbitrarily large, it follows that $\mathring{\kappa}_{\pm}$
also extends smoothly to $\operatorname{lb}\cup\operatorname{rb}$
($s=0,\infty$), where it vanishes to all orders. ∎

######  Notation 7.1.4.

The subscripts “$\psi$”, “$f$” or “$z$” may be added to the notation
$\mathring{P}_{\pm}$, $\mathring{K}_{\pm}$, $\mathring{K}_{\pm,\gamma}$,
$\mathring{\kappa}_{\pm}$, $\kappa_{\pm}$ and $P_{\pm}$ if needed.

######  Proposition 7.1.5.

The bilinear map

$\mathcal{A}\times C^{\infty}_{\text{\rm c}}(\mathbb{R})\to
C^{\infty}\big{(}(M_{\pm})^{2}_{\text{\rm b}};\beta_{\text{\rm
b}}^{*}(\Lambda\mathcal{F}_{\pm}\boxtimes(\Lambda\mathcal{F}^{*}_{\pm}\otimes\Omega
M_{\pm}))\big{)}\;,\quad(\psi,f)\mapsto\kappa_{\pm,\psi,f}\;,$

is continuous.

###### Proof.

This is an additional consequence of (7.1.12) and (7.1.13). ∎

Recall the notation $\phi_{L}=\\{\phi_{L}^{t}\\}=\\{\phi_{0}^{t}\\}$ on
$M^{0}\equiv L$ and
$\tilde{\phi}_{\widetilde{L}}=\\{\tilde{\phi}_{\widetilde{L}}^{t}\\}=\\{\tilde{\phi}_{0}^{t}\\}$
on $\widetilde{M}^{0}\equiv\widetilde{L}$ (Section 4.2.2), and the
trivialization $\nu$ of ${}_{+}N\partial M_{\pm}$ (Section 4.2.5). Recall also
that the indicial family is defined in Section 2.5.28.

######  Proposition 7.1.6.

We have

$I_{\nu_{\pm}}(P_{\pm,z},\lambda)\equiv\int_{-\infty}^{+\infty}\phi_{L,z+i\lambda}^{t*}\,\psi(D_{L,z+i\lambda})\,e^{i\lambda\varkappa
t}f(t)\,dt\;.$

###### Proof.

By (2.5.63), it is enough to show that the Schwartz kernel of the smoothing
operator

$\int_{-\infty}^{+\infty}\phi_{L,z+i\lambda}^{t*}\,\psi(D_{L,z+i\lambda})\,e^{i\lambda\varkappa
t}f(t)\,dt$

on $C^{-\infty}(L;\Lambda)$ is given by

$\int_{0}^{\infty}s^{-i\lambda}\kappa_{\pm,z}(0,s,y,y^{\prime})\,\frac{ds}{s}\;,$

at every $(y,y^{\prime})\in L^{2}$. By Lemmas 7.1.1 and 7.1.3, for all
$\tilde{y},\tilde{y}^{\prime}\in\widetilde{L}$,

$\int_{0}^{\infty}s^{-i\lambda}\kappa_{\pm,z}(0,s,[\tilde{y}],[\tilde{y}^{\prime}])\,\frac{ds}{s}\\\
\begin{aligned}
&=\frac{1}{|\varkappa|}\sum_{\gamma\in\Gamma}\int_{0}^{\infty}s^{-i\lambda}\tilde{\phi}_{0,z}^{\frac{1}{\varkappa}(F(\tilde{y})-F(\tilde{y}^{\prime})-\ln
a_{\gamma}s)*}T_{\gamma}^{*}\\\
&\phantom{=\text{}}\text{}\circ\tilde{k}_{z}\Big{(}\gamma\cdot\tilde{\phi}^{\frac{1}{\varkappa}(F(\tilde{y})-F(\tilde{y}^{\prime})-\ln
a_{\gamma}s)}_{0}(\tilde{y}),\tilde{y}^{\prime}\Big{)}\\\
&\phantom{=\text{}}\text{}\times
f\Big{(}\frac{1}{\varkappa}\Big{(}F(\tilde{y})-F(\tilde{y}^{\prime})-\ln
a_{\gamma}s\Big{)}\Big{)}\frac{ds}{s}\\\
&=\sum_{\gamma\in\Gamma}e^{i\lambda(F(\tilde{y}^{\prime})-F(\tilde{y})-\ln
a_{\gamma})}\int_{-\infty}^{+\infty}\tilde{\phi}_{0,z}^{t*}T_{\gamma}^{*}\,\tilde{k}_{z}(\gamma\cdot\tilde{\phi}_{0}^{t}(\tilde{y}),\tilde{y}^{\prime})e^{i\lambda\varkappa
t}f(t)\,dt\;,\end{aligned}$

where we have used the change of variable

$t=\varkappa^{-1}(F(\tilde{y})-F(\tilde{y}^{\prime})-\ln s+\ln a_{\gamma})\;,$

with

$\displaystyle s=e^{F(\tilde{y})-F(\tilde{y}^{\prime})+\ln
a_{\gamma}-\varkappa t}\;,\quad dt=-\frac{ds}{\varkappa s}\;,$ $\displaystyle
s=0\Leftrightarrow t=\operatorname{sign}(\varkappa)\infty\;,\quad
s=\infty\Leftrightarrow t=-\operatorname{sign}(\varkappa)\infty\;.$

By Proposition 2.9.2,

$k_{z+i\lambda}([\tilde{y}],[\tilde{y}^{\prime}])=\sum_{\gamma\in\Gamma}T_{\gamma}^{*}\tilde{k}_{z+i\lambda}(\gamma\cdot\tilde{y},\tilde{y}^{\prime})\;.$

Moreover, by (2.9.4),

$\tilde{k}_{z+i\lambda}(\tilde{y},\tilde{y}^{\prime})=e^{i\lambda(F(\tilde{y}^{\prime})-F(\tilde{y}))}\tilde{k}_{z}(\tilde{y},\tilde{y}^{\prime})\;.$

So, by (4.2.2) and (4.2.4),

$\sum_{\gamma\in\Gamma}e^{i\lambda(F(\tilde{y}^{\prime})-F(\tilde{y})-\ln
a_{\gamma})}\tilde{\phi}_{0,z}^{t*}T_{\gamma}^{*}\,\tilde{k}_{z}(\gamma\cdot\tilde{\phi}_{0}^{t}(\tilde{y}),\tilde{y}^{\prime})\\\
\begin{aligned}
&=\sum_{\gamma\in\Gamma}e^{i\lambda(F(\tilde{y}^{\prime})-F(\gamma\cdot\tilde{y}))}\tilde{\phi}_{0,z}^{t*}T_{\gamma}^{*}\,\tilde{k}_{z}(\gamma\cdot\tilde{\phi}_{0}^{t}(\tilde{y}),\tilde{y}^{\prime})\\\
&=\sum_{\gamma\in\Gamma}e^{i\lambda(\tilde{\phi}_{0}^{t*}F-F)(\gamma\cdot\tilde{y})}\tilde{\phi}_{0,z}^{t*}e^{i\lambda(F(\tilde{y}^{\prime})-F(\gamma\cdot\tilde{\phi}_{0}^{t}(\tilde{y})))}T_{\gamma}^{*}\,\tilde{k}_{z}(\gamma\cdot\tilde{\phi}_{0}^{t}(\tilde{y}),\tilde{y}^{\prime})\\\
&=\tilde{\phi}_{0,z+i\lambda}^{t*}\sum_{\gamma\in\Gamma}T_{\gamma}^{*}\tilde{k}_{z+i\lambda}(\gamma\cdot\tilde{\phi}_{0}^{t}(\tilde{y}),\tilde{y}^{\prime})\equiv\phi_{0,z+i\lambda}^{t*}\,k_{z+i\lambda}(\phi_{0}^{t}([\tilde{y}]),[\tilde{y}^{\prime}])\;.\qed\end{aligned}$

######  Notation 7.1.7.

In Notation 7.1.4, we may also add the subscript “$u$” if we use a family of
functions $\psi_{u}\in\mathcal{A}$ depending on a parameter $u$. This also
applies to $k_{z}$ and $\tilde{k}_{z}$.

The identity element of $\Gamma$ is denoted by $e$.

######  Proposition 7.1.8.

If $\psi_{u}(x)=e^{-ux^{2}}$, then
$(\kappa_{\pm,u}-\kappa_{\pm,e,u})|_{\Delta_{\text{\rm b},\pm}}\to 0$ as
$u\downarrow 0$ in the $C^{\infty}$ topology.

###### Proof.

For $\gamma\in\Gamma$ and
$p=[x,\tilde{y}]=[a_{\gamma}x,\gamma\cdot\tilde{y}]\in\mathring{M}_{\pm}$, by
(4.2.1) and (4.2.2),

$\phi^{-h_{\pm}(\gamma)}(p)=\big{[}x,\phi^{-h_{\pm}(\gamma)}_{a_{\gamma}x}(\gamma\cdot\tilde{y})\big{]}=\big{[}x,\gamma\cdot\phi^{-h_{\pm}(\gamma)}_{x}(\tilde{y})\big{]}\;.$

Thus, using that $\pi_{M}$ defines an isometric diffeomorphism of
$\\{x\\}\times\widetilde{L}\equiv\widetilde{L}$ to $L_{p}$, it follows from
Corollary 4.2.8 that there is some $c_{3}>0$, independent of $p$ and $\gamma$,
such that, if $h_{\pm}(\gamma)\in\operatorname{supp}f$, then

$d_{\widetilde{L}}\big{(}\gamma\cdot\phi^{-h_{\pm}(\gamma)}_{x}(\tilde{y}),\tilde{y}\big{)}=d_{\mathcal{F}}\big{(}\phi^{-h_{\pm}(\gamma)}(p),p\big{)}\geq
c_{3}|\gamma|\;.$

Therefore, by (2.9.14) and since $\phi$ is of $\mathbb{R}$-local bounded
geometry, for $m_{1},m_{2}\in\mathbb{N}_{0}$, $0<u<u_{0}$, $\gamma\in\Gamma$,
$\rho>0$ and $\tilde{y}\in\widetilde{L}$, we get

$\big{|}\partial_{\rho}^{m_{1}}\nabla_{\tilde{y}}^{m_{2}}\tilde{k}_{u,z}\big{(}\gamma\cdot\phi^{-h_{\pm}(\gamma)}_{\pm
e^{-F(\tilde{y})}\rho}(\tilde{y}),\tilde{y}\big{)}\big{|}\leq
C_{1}u^{-(n-1+m_{1}+m_{2})/2}e^{-C_{2}c_{3}^{2}|\gamma|^{2}/u}\;,$

where $\tilde{k}_{u,z}$ is the Schwartz kernel of
$\psi_{u}(D_{\widetilde{L},z})=e^{-u\Delta_{\widetilde{L},z}}$. Using again
the $\mathbb{R}$-local bounded geometry of $\phi$ and the compactness of
$\operatorname{supp}f$, it follows that there is some $C_{3}>0$ such that

$|\partial_{\rho}^{m_{1}}\nabla_{y}^{m_{2}}\kappa_{\gamma,u}(\rho,1,y,y)|\leq
C_{3}u^{-(n-1+m_{1}+m_{2})/2}e^{-C_{2}c_{3}^{2}|\gamma|^{2}/u}$

for $m_{1},m_{2}\in\mathbb{N}_{0}$, $0<u<u_{0}$, $\gamma\in\Gamma$, $\rho>0$
and $y\in L$. So

$|\partial_{\rho}^{m_{1}}\nabla_{y}^{m_{2}}(\kappa_{\pm,u}-\kappa_{\pm,e,u})(\rho,1,y,y)|\leq
C_{3}u^{-(n-1+m_{1}+m_{2})/2}\sum_{\gamma\in\Gamma\setminus\\{e\\}}e^{-C_{2}c_{3}^{2}|\gamma|^{2}/u}\;,$

which converges to zero as $u\downarrow 0$. ∎

######  Corollary 7.1.9.

If $\psi_{u}(x)=e^{-ux^{2}}$ and $f(0)=0$, then
$\kappa_{\pm,u}|_{\Delta_{\text{\rm b},\pm}}\to 0$ as $u\downarrow 0$ in the
$C^{\infty}$ topology.

Recall the notation $e(\mathcal{F}_{\pm},g_{\mathcal{F}_{\pm}})$ if $n-1$ is
even (Sections 3.2.7 and 4.2.6), and also the notation
$C^{0,\infty}_{\varpi_{\pm}}(M_{\pm};{}^{\text{\rm b}}\Omega)$ (Section
3.1.8).

######  Corollary 7.1.10.

If $\psi_{u}(x)=e^{-ux^{2}}$, then

$\lim_{u\downarrow 0}\operatorname{str}(\kappa_{\pm,u}|_{\Delta_{\text{\rm
b},\pm}})\equiv\begin{cases}f(0)\,e(\mathcal{F}_{\pm},g_{\mathcal{F}_{\pm}})\,|\omega_{\pm}|&\text{if
$n-1$ is even}\\\ 0&\text{if $n-1$ is odd}\end{cases}$

in $C^{0,\infty}_{\varpi_{\pm}}(M_{\pm};{}^{\text{\rm b}}\Omega)$, using the
identity $\Delta_{\text{\rm b},\pm}\equiv M_{\pm}$.

###### Proof.

By Propositions 7.1.3 and 7.1.8, and (7.1.5), (7.1.7) and (4.2.17), for all
$\rho>0$ and $y=[\tilde{y}]\in L$ with $\tilde{y}\in\widetilde{L}$,

$\lim_{u\downarrow
0}\operatorname{str}\mathring{\kappa}_{\pm,u}(\rho,1,y,y)\\\ \begin{aligned}
&=\lim_{u\downarrow
0}\operatorname{str}\mathring{\kappa}_{\pm,e,u}(\rho,1,y,y)=\lim_{u\downarrow
0}\operatorname{str}\mathring{K}_{\pm,e,u}(\rho,\rho,y,y)\\\
&=\frac{f(0)}{|\varkappa|}\,\Big{|}\frac{d\rho}{\rho}\Big{|}\,\lim_{u\downarrow
0}\operatorname{str}\tilde{k}_{z,u}(\tilde{y},\tilde{y})=f(0)\,|\omega_{\pm}|(y)\,\lim_{u\downarrow
0}\operatorname{str}\tilde{k}_{z,u}(\tilde{y},\tilde{y})\;.\end{aligned}$

But, by Theorem 2.9.4,

$\lim_{u\downarrow
0}\operatorname{str}\tilde{k}_{z,u}(\tilde{y},\tilde{y})=e(\widetilde{L},g_{\widetilde{L}})(\tilde{y})=e(L,g_{L})(y)\equiv
e(\mathcal{F}_{\pm},g_{\mathcal{F}_{\pm}})(\rho,y)$

if $n-1$ is even, and

$\lim_{u\downarrow 0}\operatorname{str}\tilde{k}_{z,u}(\tilde{y},\tilde{y})=0$

if $n-1$ is odd. ∎

###  7.2. Operators on the components $M^{1}_{l}$

Consider the notation of Sections 4.3.3 and 4.3.4; in particular, consider the
boundary-defining function $\rho=\rho_{l}$ on every $M_{l}$ and the
trivialization $\nu=\nu_{l}$ of ${}_{+}N\partial M_{l}$. According to Section
2.5.26, consider also the lifts of $\rho$ to $M_{l}^{2}$ from the left and
right factors, $\rho$ and $\rho^{\prime}$, and the function
$s=s_{l}=\rho/\rho^{\prime}:M_{l}^{2}\to[0,\infty]$, as well as the
corresponding functions $\rho$, $\rho^{\prime}$ and $s$ on
$(M_{l})^{2}_{\text{\rm b}}$. Equip $\mathring{M}_{l}$ with the Riemannian
metric $g_{\text{\rm b},l}$, so that $\mathring{\mathcal{F}}_{l}$ becomes a
Riemannian foliation of bounded geometry (Section 3.4.1). Consider the
leafwise perturbed operators for
$(\mathring{M}_{l},\mathring{\mathcal{F}}_{l})$ defined by the leafwise-closed
form $\eta_{0}$, which agrees with $\eta$ on the collar neighborhood of the
boundaty we have fixed. For any $\psi\in\mathcal{A}$, $f\in
C^{\infty}_{\text{\rm c}}(\mathbb{R})$, $z\in\mathbb{C}$ and every index $l$,
the operator

$\mathring{P}_{l}=\int_{-\infty}^{+\infty}\phi_{l,z}^{t*}\,\psi(D_{\mathring{\mathcal{F}}_{l,z}})f(t)\,dt$

on $H^{-\infty}(\mathring{M}_{l};\Lambda\mathring{\mathcal{F}}_{l})$ is a
twisted version of (3.4.17), which is smoothing by the appropriate analog of
(3.4.18) (Section 3.5). Let

$\mathring{K}_{l}=K_{\mathring{P}_{l}}\in
C^{\infty}(\mathring{M}_{l}^{2};\Lambda\mathring{\mathcal{F}}_{l}\boxtimes(\Lambda\mathring{\mathcal{F}}_{l}^{*}\otimes\Omega\mathring{M}_{l}))\;.$

######  Lemma 7.2.1.

For any compact $I\subset\mathbb{R}$ containing $\operatorname{supp}f$, and
for all $k,m\in\mathbb{N}_{0}$ and $a\in\mathbb{R}$, there are some
$C^{\prime},C^{\prime\prime}>0$ and $N\in\mathbb{N}_{0}$, depending only on
$I$, $k$, $m$ and $a$, such that

$\big{\|}\mathring{P}_{l}\big{\|}_{\rho^{a}H^{k},\rho^{a}H^{m}}\leq
C^{\prime\prime}\,\|\psi\|_{\mathcal{A},C^{\prime},N}\,\|f\|_{I,C^{N}}\;.$

###### Proof.

By (2.9.4) and (4.3.1),
$D_{\mathring{\mathcal{F}}_{l},z}=\rho^{a}D_{\mathring{\mathcal{F}}_{l},z+a,z-a}\rho^{-a}$
(see Section 3.3.3). So the result follows from the analog of (3.4.20) for
$D_{\mathring{\mathcal{F}}_{l},z+a,z-a}$ (Section 3.5). ∎

######  Proposition 7.2.2.

The kernel $\mathring{K}_{l}$ has a $C^{\infty}$ extension to
$M_{l}^{2}\setminus(\partial M_{l})^{2}$ that vanishes to all orders on
$(\partial M_{l}\times\mathring{M}_{l})\cup(\mathring{M}_{l}\times\partial
M_{l})$.

###### Proof.

We will use the arguments from the proof of [Hör83, Theorem 5.2.6].

For any $q\in\mathring{M}_{l}$ and
$\alpha\in\Lambda_{q}\mathring{\mathcal{F}}_{l}\otimes\Omega^{-1}_{q}\mathring{M}_{l}$,
we have $\delta_{q}^{\alpha}\in
H^{k}(\mathring{M}_{l};\Lambda\mathring{\mathcal{F}}_{l})$ for any $k<-n/2$,
and $\|\delta_{q}^{\alpha}\|_{k}\leq C_{k}\,|\alpha|$, where $C_{k}>0$ is
independent of $q$ and $\alpha$ (Section 2.2.6). Therefore, by the definition
of weighted Sobolev spaces and the properties of Dirac sections at
submanifolds (Sections 2.1.12 and 2.2.6), for all $a\in\mathbb{R}$, we have
$\delta^{\alpha}_{q}\in\rho^{a}H^{k}(\mathring{M}_{l};\Lambda\mathring{\mathcal{F}}_{l})$
and

$\|\delta^{\alpha}_{q}\|_{\rho^{a}H^{k}}\leq C_{k}\rho(q)^{-a}|\alpha|\;.$

Moreover, for any $\alpha\in
C^{\infty}(\mathring{M}_{l};\Lambda\mathring{\mathcal{F}}_{l}\otimes\Omega^{-1}\mathring{M}_{l})$,
the map

$\mathring{M}_{l}\to\rho^{a}H^{k}(\mathring{M}_{l};\Lambda\mathring{\mathcal{F}}_{l})\;,\quad
q\mapsto\delta^{\alpha(q)}_{q}\;,$

is continuous by the continuity of (2.2.13).

Fix any compact $I\subset\mathbb{R}$ containing $\operatorname{supp}f$. By
Lemma 7.2.1, we have
$\mathring{P}_{l}\delta^{\alpha}_{q}\in\rho^{a}H^{m}(\mathring{M}_{l};\Lambda\mathring{\mathcal{F}}_{l})$
for any $m\in\mathbb{N}_{0}$, and

$\big{\|}\mathring{P}_{l}\delta^{\alpha}_{q}\big{\|}_{\rho^{a}H^{m}}\leq
C^{\prime}_{m}\rho(q)^{-a}\,\|\psi\|_{\mathcal{A},C^{\prime},N}\,\|f\|_{I,C^{N}}\,|\alpha|$

for $q\in\mathring{M}_{l}$ and
$\alpha\in\Lambda_{q}\mathring{\mathcal{F}}_{l}\otimes\Omega^{-1}_{q}\mathring{M}_{l}$,
where $C^{\prime}_{m}>0$ is independent of $a$, $q$ and $\alpha$. Moreover,
for any $\alpha\in
C^{\infty}(\mathring{M}_{l};\Lambda\mathring{\mathcal{F}}_{l}\otimes\Omega^{-1}\mathring{M}_{l})$,
the map

$\mathring{M}_{l}\to\rho^{a}H^{m}(\mathring{M}_{l};\Lambda\mathring{\mathcal{F}}_{l})\;,\quad
q\mapsto\mathring{P}_{l}\delta^{\alpha(q)}_{q}\;,$

is continuous. On the other hand, by (2.2.14), for all $q\in\mathring{M}_{l}$
and
$\alpha\in\Lambda_{q}\mathring{\mathcal{F}}_{l}\otimes\Omega^{-1}_{q}\mathring{M}_{l}$,

$\mathring{K}_{l}(\cdot,q)(\alpha)=\mathring{P}_{l}\delta_{q}^{\alpha}\in
C^{\infty}(\mathring{M}_{l};\Lambda\mathring{\mathcal{F}}_{l})\;.$

It follows that the map

$\mathring{M}_{l}\to\rho^{a}H^{m}(\mathring{M}_{l};\Lambda\mathring{\mathcal{F}}_{l}\otimes\Omega_{q}\mathring{M}_{l})\;,\quad
q\mapsto\mathring{K}_{l}(\cdot,q)\;,$

is continuous for any $a\in\mathbb{R}$ and $m\in\mathbb{N}_{0}$, with

$\big{\|}\mathring{K}_{l}(\cdot,q)\big{\|}_{\rho^{a}H^{m}}\leq
C^{\prime}_{m}\rho(q)^{-a}\,\|\psi\|_{\mathcal{A},C^{\prime},N}\,\|f\|_{I,C^{N}}$

for all $q\in\mathring{M}_{l}$. Using the Sobolev embedding theorem, we
conclude that $\mathring{K}_{l}$ is continuous on $\mathring{M}_{l}^{2}$, and

$\displaystyle\big{|}\mathring{K}_{l}(p,q)\big{|}$ $\displaystyle\leq
C\left(\frac{\rho(p)}{\rho(q)}\right)^{a}\,\|\psi\|_{\mathcal{A},C^{\prime},N},\|f\|_{I,C^{N}}$
$\displaystyle=C\,s(p,q)^{a}\,\|\psi\|_{\mathcal{A},C^{\prime},N}\,\|f\|_{I,C^{N}}\;,$

for all $a\in\mathbb{R}$ and $p,q\in\mathring{M}_{l}$, where $C,C^{\prime}>0$
and $N\in\mathbb{N}_{0}$ are independent of $a$, $p$ and $q$. So
$\mathring{K}_{l}$ extends to a continuous section on
$M_{l}^{2}\setminus(\partial M_{l})^{2}$, which vanishes on $(\partial
M_{l}\times\mathring{M}_{l})\cup(\mathring{M}_{l}\times\partial M_{l})$.

For any $D_{1},D_{2}\in\operatorname{Diff}_{\text{\rm
b}}^{k}(M_{l};\Lambda\mathcal{F}_{l})$, applying the above arguments to the
operator $D_{1}\mathring{P}_{l}D_{2}$ and using (2.1.28), it follows that, for
all $a\in\mathbb{R}$ and $p,q\in\mathring{M}_{l}$,

(7.2.1) $\big{|}D_{1,p}\,D_{2,q}^{\text{\rm
t}}\mathring{K}_{l}(p,q)\big{|}\leq
C\,s(p,q)^{a}\,\|\psi\|_{\mathcal{A},C^{\prime},N}\,\|f\|_{I,C^{N}}\;,$

where $C,C^{\prime}>0$ and $N\in\mathbb{N}_{0}$ are independent of $a$, $p$
and $q$. ∎

Let $\mathring{\kappa}_{l}$ be the $C^{\infty}$ section of $\beta_{\text{\rm
b}}^{*}(\Lambda\mathcal{F}_{l}\boxtimes(\Lambda\mathcal{F}_{l}^{*}\otimes\Omega
M_{l}))$ on the interior of $(M_{l})^{2}_{\text{\rm b}}$ that corresponds to
$\mathring{K}_{l}$ via $\beta_{\text{\rm b}}^{*}$, using the notation of
Section 2.5.26. Fix $0<\epsilon<\epsilon_{0}$ like in Section 4.3.3, and
consider the notation of Sections 4.3.2 and 4.3.3. Let $L$ be a boundary
component of $M_{l}$, which can be identified with a leaf of $\mathcal{F}$ in
$M^{0}$. For $0<\sigma\leq\epsilon_{0}$, via the identity
$\mathring{T}_{L,l,\sigma}\equiv\mathring{T}^{\prime}_{L,l,\sigma}$, the
sections in $C_{\text{\rm
c}}^{\infty}(\mathring{M}_{l};\Lambda\mathring{\mathcal{F}}_{l})$ and
$C_{\text{\rm
c}}^{\infty}(\mathring{M}_{l};\Lambda\mathring{\mathcal{F}}_{l}\otimes\Omega)$
supported in $\mathring{T}_{L,l,\sigma}$ can be identified with sections in
$C_{\text{\rm
c}}^{\infty}(\mathring{M}^{\prime}_{l};\Lambda\mathring{\mathcal{F}}^{\prime}_{l})$
and $C_{\text{\rm
c}}^{\infty}(\mathring{M}^{\prime}_{l};\Lambda\mathring{\mathcal{F}}^{\prime}_{l}\otimes\Omega)$
supported in $\mathring{T}^{\prime}_{L,l,\sigma}$. Similarly, according to
Section 7.1,

$\beta_{\text{\rm
b}}^{-1}(\mathring{T}_{L,l,\sigma}^{2})\equiv\beta_{\text{\rm
b}}^{-1}(\mathring{T}^{\prime\,2}_{L,l,\sigma})\equiv\\{\,(\rho,s,y,y^{\prime})\in(0,\infty)^{2}\times
L^{2}\mid\rho,\rho s^{-1}<\sigma\,\\}\;.$

The operator (7.1.1), studied in Section 7.1, is now expressed as

$\mathring{P}^{\prime}_{L,l}=\int_{-\infty}^{+\infty}\phi_{L,l,z}^{\prime\,t*}\,\psi(D_{\mathring{\mathcal{F}}^{\prime}_{L,l},z})f(t)\,dt\;.$

Let $\mathring{K}^{\prime}_{L,l}=K_{\mathring{P}^{\prime}_{L,l}}$, with lift
$\mathring{\kappa}^{\prime}_{L,l}$ to the interior of
$(M^{\prime}_{L,l})^{2}_{\text{\rm b}}$, and let
$\kappa^{\prime}_{L,l}=\kappa_{P^{\prime}_{L,l}}$ denote the extension of
$\mathring{\kappa}^{\prime}_{L,l}$ to $(M^{\prime}_{L,l})^{2}_{\text{\rm b}}$
given by Proposition 7.1.3.

The subscripts of Notations 7.1.4 and 7.1.7 may be also used with
$\mathring{P}_{l}$, $\mathring{K}_{l}$, $\mathring{\kappa}_{l}$,
$\mathring{P}^{\prime}_{L,l}$, $\mathring{K}^{\prime}_{L,l}$,
$\mathring{\kappa}^{\prime}_{L,l}$ and $\kappa^{\prime}_{L,l}$.

######  Proposition 7.2.3.

Given $\psi\in\mathcal{A}$ and $u>0$, take $\psi_{u}\in\mathcal{A}$ defined by
$\psi_{u}(x)=\psi(ux)$, and consider the restrictions of
$\mathring{\kappa}_{l,u}$ and $\mathring{\kappa}^{\prime}_{L,l,u}$ to
$\beta_{\text{\rm
b}}^{-1}(\mathring{T}_{L,l,\epsilon}^{2})\equiv\beta_{\text{\rm
b}}^{-1}(\mathring{T}^{\prime\,2}_{L,l,\epsilon})$. There is some
$0<\epsilon^{\prime}<\epsilon$ such that, for any $R>0$,
$m,N\in\mathbb{N}_{0}$ and $a\in\mathbb{R}$, there exist $\widehat{C},W>0$ and
$N^{\prime}\in\mathbb{N}_{0}$ so that, for $m_{1}+m_{2}+m_{3}+m_{4}\leq m$,
$0<u\leq 1$ and $(\rho,s,y,y^{\prime})\in\beta_{\text{\rm
b}}^{-1}(\mathring{T}_{L,l,\epsilon^{\prime}}^{2})$,

$\big{|}\partial^{m_{1}}_{\rho}\partial^{m_{2}}_{s}\nabla^{m_{3}}_{y}\nabla^{m_{4}}_{y^{\prime}}(\mathring{\kappa}_{l,u}-\mathring{\kappa}^{\prime}_{L,l,u})(\rho,s,y,y^{\prime})\big{|}\leq\widehat{C}e^{-\frac{R}{u}}\rho^{N}s^{a}\,\|\psi\|_{\mathcal{A},W,N^{\prime}}\,\|f\|_{I,C^{N^{\prime}}}\;.$

###### Proof.

Take $C>0$ and $c\geq 1$ like in Lemma 4.2.4 and Corollary 4.2.5, and take
$c^{\prime}>0$ like in Proposition 4.2.9, for the suspension foliation
$\mathcal{F}^{\prime}_{L,l}$ on $M^{\prime}_{L,l}$ and any compact
$I\subset\mathbb{R}$ containing $\operatorname{supp}f$.

######  Claim 7.2.4.

For $\alpha,\beta\in C^{\infty}_{\text{\rm
c}}(\mathring{M}_{L,l};\Lambda\mathring{\mathcal{F}}_{L,l})$,
$\alpha^{\prime},\beta^{\prime}\in C^{\infty}_{\text{\rm
c}}(\mathring{M}^{\prime}_{L,l};\Lambda\mathring{\mathcal{F}}^{\prime}_{L,l})$
and $\xi\in\mathbb{R}$, let

$\displaystyle\alpha(\xi)$ $\displaystyle=e^{i\xi
D_{\mathring{\mathcal{F}}_{L,l},z}}\alpha\;,$ $\displaystyle\quad\beta(\xi)$
$\displaystyle=e^{i\xi D_{\mathring{\mathcal{F}}_{L,l},z}}\beta\;,$
$\displaystyle\alpha^{\prime}(\xi)$ $\displaystyle=e^{i\xi
D_{\mathring{\mathcal{F}}^{\prime}_{L,l},z}}\alpha^{\prime}\;,$
$\displaystyle\quad\beta^{\prime}(\xi)$ $\displaystyle=e^{i\xi
D_{\mathring{\mathcal{F}}^{\prime}_{L,l},z}}\beta^{\prime}\;.$

The following properties hold for $0<\sigma,\tau<\epsilon$:

  1. (i)

If $\alpha$ and $\alpha^{\prime}$ are supported in
$\mathring{T}_{L,l,\sigma}\equiv\mathring{T}^{\prime}_{L,l,\sigma}$ and agree
there, then $\alpha(\xi)$ and $\alpha^{\prime}(\xi)$ are supported in
$\mathring{T}_{L,l,\epsilon}\equiv\mathring{T}^{\prime}_{L,l,\epsilon}$ and
agree there for $|\xi|<C\ln\frac{\epsilon}{c\sigma}$.

  2. (ii)

If $\beta$ and $\beta^{\prime}$ agree on
$\mathring{T}_{L,l,\epsilon}\equiv\mathring{T}^{\prime}_{L,l,\epsilon}$, then
$\beta(\xi)\equiv\beta^{\prime}(\xi)$ on
$\mathring{T}_{L,l,\tau}\equiv\mathring{T}^{\prime}_{L,l,\tau}$ for
$|\xi|<C\ln\frac{\epsilon}{c\tau}$.

This is a consequence of Corollary 4.2.5 and the leafwise twisted version of
(3.4.9) applied to the equation
$\partial_{\xi}\mu(\xi)=iD_{\mathring{\mathcal{F}}_{l},z}\mu(\xi)$ on
$\mathring{T}_{L,l,\epsilon}\equiv\mathring{T}^{\prime}_{L,l,\epsilon}$, where
$\mu(\xi)=\alpha(\xi)\equiv\alpha^{\prime}(\xi)$ in i, and
$\mu(\xi)=\beta(\xi)\equiv\beta^{\prime}(\xi)$ in ii.

######  Claim 7.2.5.

Let $\alpha$, $\alpha^{\prime}$, $\alpha(\xi)$ and $\alpha^{\prime}(\xi)$ be
defined like in Claim 7.2.4, and let $0<\sigma<\epsilon$ and
$0<\tau<\epsilon,\epsilon/c^{\prime}$. If $\alpha$ and $\alpha^{\prime}$ are
supported in
$\mathring{T}_{L,l,\sigma}\equiv\mathring{T}^{\prime}_{L,l,\sigma}$ and agree
there, then $\phi^{t*}\alpha(\xi)\equiv\phi^{\prime\,t*}\alpha^{\prime}(\xi)$
on $\mathring{T}_{L,l,\tau}\equiv\mathring{T}^{\prime}_{L,l,\tau}$ for any
$t\in I$ and
$|\xi|<C(\ln\frac{\epsilon}{c\sigma}+\ln\frac{\epsilon}{cc^{\prime}\tau})$.

By Claim 7.2.4 i, if $\xi<C\ln\frac{\epsilon}{c\sigma}$, then $\alpha(\xi)$
and $\alpha^{\prime}(\xi)$ are supported in
$\mathring{T}_{L,l,\epsilon}\equiv\mathring{T}^{\prime}_{L,l,\epsilon}$ and
agree there. Thus, by Claim 7.2.4 ii, if
$|\zeta|<C\ln\frac{\epsilon}{cc^{\prime}\tau}$, then
$\alpha(\xi+\zeta)\equiv\alpha^{\prime}(\xi+\zeta)$ on
$\mathring{T}_{L,l,c^{\prime}\tau}\equiv\mathring{T}^{\prime}_{L,l,c^{\prime}\tau}$.
Hence
$\phi^{t*}\alpha(\xi+\zeta)\equiv\phi^{\prime\,t*}\alpha^{\prime}(\xi+\zeta)$
on $\mathring{T}_{L,l,\tau}\equiv\mathring{T}^{\prime}_{L,l,\tau}$ for all
$t\in I$ since
$\phi^{t}(\mathring{T}_{L,l,\tau})\subset\mathring{T}_{L,l,c^{\prime}\tau}$ by
Proposition 4.2.9. This shows Claim 7.2.5.

Take any $\mu\in C^{\infty}(\mathbb{R})$ such that $0\leq\mu\leq 1$,
$\operatorname{supp}\mu\subset(-\infty,0]$, and $\mu=1$ on $(-\infty,-\ln 2]$.
For $0<\sigma<\epsilon$, let $\chi_{\sigma}=\mu(\ln\rho-\ln\sigma)\in
C^{\infty}_{\text{\rm ub}}(\mathring{M}_{l})$. We have $\chi_{\sigma}\geq 0$,
$\operatorname{supp}\chi_{\sigma}\subset\mathring{T}_{L,l,\sigma}$, and
$\chi_{\sigma}=1$ on $\mathring{T}_{L,l,\sigma/2}$. Moreover $\chi_{\sigma}\in
C^{\infty}_{\text{\rm ub}}(\mathring{M}_{l})$ and
$\|\chi_{\sigma}\|_{C^{m}_{\text{\rm ub}}}$ is independent of $\sigma$ for
$m\in\mathbb{N}_{0}$ because $d(\ln\rho)\in C^{\infty}_{\text{\rm
ub}}(\mathring{M}_{l};T^{*}\mathring{M}_{l})$ (Section 4.3.3). Let also
$0<\tau<\epsilon,\epsilon/c^{\prime}$ and define $\chi_{\tau}$ as above. Then
the operator
$\chi_{\tau}(\mathring{P}_{l,u}-\mathring{P}^{\prime}_{L,l,u})\chi_{\sigma}$
is well defined on
$H^{-\infty}(\mathring{M}_{l};\Lambda\mathring{\mathcal{F}}_{l})$ via the
identity
$\mathring{T}_{L,l,\epsilon}\equiv\mathring{T}^{\prime}_{L,l,\epsilon}$.

Let $\alpha\in C_{\text{\rm
c}}^{\infty}(\mathring{M}_{l};\Lambda\mathring{\mathcal{F}}_{l})$ and
$\beta\in C_{\text{\rm
c}}^{\infty}(\mathring{M}_{l};\Lambda\mathring{\mathcal{F}}_{l}^{*}\otimes\Omega)$.
By Claim 7.2.5 and the version of (3.4.11) for $\xi uD_{\mathcal{F}_{l},z}$
and $\xi uD_{\mathcal{F}^{\prime}_{L,l},z}$ instead of $tD_{0}$ (Section 3.5),

$\big{\langle}\chi_{\tau}(\mathring{P}_{l,u}-\mathring{P}^{\prime}_{L,l,u})\chi_{\sigma}\alpha,\beta\big{\rangle}=\frac{1}{2\pi}\int_{|\xi|>\frac{C}{u}\ln\frac{\epsilon^{2}}{c^{2}c^{\prime}\sigma\tau}}\int_{-\infty}^{+\infty}\hat{\psi}(\xi)A_{l,z,u}(t,\xi)f(t)\,d\xi\,dt\;,$

where

$A_{l,z,u}(t,\xi)=\Big{\langle}\Big{(}\phi_{l,z}^{t*}\,e^{i\xi
uD_{\mathring{\mathcal{F}}_{l},z}}-\phi_{L,l,z}^{\prime\,t*}\,e^{i\xi
uD_{\mathring{\mathcal{F}}^{\prime}_{L,l},z}}\Big{)}\chi_{\sigma}\alpha,\chi_{\tau}\beta\Big{\rangle}\;.$

Then, by the version of (3.4.10) for $\xi uD_{\mathcal{F}_{l},z}$ and $\xi
uD_{\mathcal{F}^{\prime}_{L,l},z}$ instead of $tD_{0}$ (Section 3.5), since
$\phi_{l}$ and $\phi^{\prime}_{L,l}$ are of $\mathbb{R}$-local bounded
geometry, and using that $\|\chi_{\sigma}\|_{C^{k}_{\text{\rm ub}}}$ and
$\|\chi_{\tau}\|_{C^{k}_{\text{\rm ub}}}$ are finite and independent of
$\sigma$ and $\tau$ for all $k\in\mathbb{N}_{0}$, we get that, for all
$m\in\mathbb{R}$,

$\displaystyle|A_{l,z,u}(t,\xi)|$
$\displaystyle\leq\Big{\|}\Big{(}\phi_{l,z}^{t*}\,e^{i\xi
uD_{\mathring{\mathcal{F}}_{l},z}}-\phi_{L,l,z}^{\prime\,t*}\,e^{i\xi
uD_{\mathring{\mathcal{F}}^{\prime}_{L,l},z}}\Big{)}\chi_{\sigma}\alpha\Big{\|}_{m}\|\chi_{\tau}\beta\|_{-m}$
$\displaystyle\leq
C^{\prime}e^{C_{m}|\xi|}\|\chi_{\sigma}\alpha\|_{m}\|\chi_{\tau}\beta\|_{-m}\leq
C^{\prime\prime}e^{C_{m}|\xi|}\|\alpha\|_{m}\|\beta\|_{-m}\;,$

for some $C_{m},C^{\prime},C^{\prime\prime}>0$ independent of $\alpha$,
$\beta$, $\sigma$, $\tau$, and $u\in(0,1]$. So, for all $W>0$,

$\big{|}\big{\langle}\chi_{\tau}(\mathring{P}_{l,u}-\mathring{P}^{\prime}_{L,l,u})\chi_{\sigma}\alpha,\beta\big{\rangle}\big{|}\\\
\begin{aligned}
&\leq\frac{1}{2\pi}\int_{|\xi|>\frac{C}{u}(\ln\frac{\epsilon}{c\sigma}+\ln\frac{\epsilon}{cc^{\prime}\tau})}\int_{-\infty}^{+\infty}\big{|}\hat{\psi}(\xi)\big{|}\,|A_{l,z,u}(t,\xi)|\,|f(t)|\,d\xi\,dt\\\
&\leq
C^{\prime\prime}\|\alpha\|_{m}\|\beta\|_{-m}\|f\|_{L^{1}}\int_{|\xi|>\frac{C}{u}(\ln\frac{\epsilon}{c\sigma}+\ln\frac{\epsilon}{cc^{\prime}\tau})}e^{C_{m}|\xi|}\,\big{|}\hat{\psi}(\xi)\big{|}\,d\xi\\\
&\leq
C^{\prime\prime}\|\alpha\|_{m}\|\beta\|_{-m}\|f\|_{L^{1}}e^{-\frac{CW}{u}(\ln\frac{\epsilon}{c\sigma}+\ln\frac{\epsilon}{cc^{\prime}\tau})}\int_{-\infty}^{\infty}e^{(W+C_{m})|\xi|}\,\big{|}\hat{\psi}(\xi)\big{|}\,d\xi\;,\end{aligned}$

for some $C_{m},C^{\prime},C^{\prime\prime}>0$ independent of $\alpha$,
$\beta$, $\sigma$, $\tau$, and $u\in(0,1]$. Now, assume

(7.2.2)
$\sigma<\frac{\epsilon}{ce}\;,\quad\tau<\frac{\epsilon}{cc^{\prime}e}\;.$

Thus $\ln\frac{\epsilon}{c\sigma},\ln\frac{\epsilon}{cc^{\prime}\tau}>1$,
obtaining

$\displaystyle
e^{-\frac{CW}{u}(\ln\frac{\epsilon}{c\sigma}+\ln\frac{\epsilon}{cc^{\prime}\tau})}$
$\displaystyle\leq
e^{-\frac{CW}{u}(1+\frac{1}{2}(\ln\frac{\epsilon}{c\sigma}+\ln\frac{\epsilon}{cc^{\prime}\tau}))}\leq
e^{-\frac{CW}{u}}e^{-\frac{CW}{2}(\ln\frac{\epsilon}{c\sigma}+\ln\frac{\epsilon}{cc^{\prime}\tau})}$
$\displaystyle=e^{-\frac{CW}{u}}e^{-\frac{CW}{2}(\ln\frac{\epsilon}{c}+\ln\frac{\epsilon}{cc^{\prime}})}(\sigma\tau)^{\frac{CW}{2}}\leq
e^{-\frac{CW}{u}}(\sigma\tau)^{\frac{CW}{2}}\;.$

Hence

$\big{|}\big{\langle}\chi_{\tau}(\mathring{P}_{l,u}-\mathring{P}^{\prime}_{L,l,u})\chi_{\sigma}\alpha,\beta\big{\rangle}\big{|}\\\
\leq
C^{\prime\prime\prime}e^{-\frac{CW}{u}}(\sigma\tau)^{\frac{CW}{2}}\,\|\psi\|_{\mathcal{A},W+C_{m},0}\,\|f\|_{I,C^{0}}\,\|\alpha\|_{m}\,\|\beta\|_{-m}\;,$

for some $C^{\prime\prime\prime}>0$ independent of $\alpha$, $\beta$,
$\sigma$, $\tau$, and $u\in(0,1]$, but involving the length of $I$. Thus, for
any $R>0$, $N\in\mathbb{N}_{0}$ and $m\in\mathbb{R}$, there are some
$\widehat{C},W>0$, such that, for all $\sigma$ and $\tau$ as in (7.2.2), and
every $u\in(0,1]$,

$\big{\|}\chi_{\tau}(\mathring{P}_{l,u}-\mathring{P}^{\prime}_{L,l,u})\chi_{\sigma}\big{\|}_{m}\leq\widehat{C}e^{-\frac{R}{u}}\sigma^{N}\tau^{N}\|\psi\|_{\mathcal{A},W,0}\,\|f\|_{I,C^{0}}\;.$

Using the arguments of the proof of (7.2.1), we similarly get that, for any
$R>0$, $N\in\mathbb{N}_{0}$ and $m,m^{\prime}\in\mathbb{R}$, there are
$\widehat{C},W>0$ and $N^{\prime}\in\mathbb{N}_{0}$ such that, for all
$\sigma$ and $\tau$ as in (7.2.2), and every $u\in(0,1]$,

$\big{\|}\chi_{\tau}(\mathring{P}_{l,u}-\mathring{P}^{\prime}_{L,l,u})\chi_{\sigma}\big{\|}_{m,m^{\prime}}\leq\widehat{C}e^{-\frac{R}{u}}\sigma^{N}\tau^{N}\|\psi\|_{\mathcal{A},W,N^{\prime}}\,\|f\|_{I,C^{N^{\prime}}}\;.$

Moreover, for any $a\in\mathbb{R}$, replacing $\alpha$ with $\rho^{-a}\alpha$
and $\beta$ with $\rho^{a}\beta$ in the above argument, we also get

$\big{\langle}\chi_{\tau}(\mathring{P}_{l,u}-\mathring{P}^{\prime}_{L,l,u})\chi_{\sigma}\rho^{-a}\alpha,\rho^{a}\beta\big{\rangle}\\\
=\frac{1}{2\pi}\int_{|\xi|u>C(\ln\frac{\epsilon}{c\sigma}+\ln\frac{\epsilon}{cc^{\prime}\tau})}\int_{-\infty}^{+\infty}\hat{\psi}(\xi)B_{l,z}(t,\xi,a)f(t)\,d\xi\,dt\;,$

where

$\displaystyle B_{l,z}(t,\xi,a)$
$\displaystyle=\Big{\langle}\Big{(}\phi_{l,z}^{t*}\,e^{i\xi
uD_{\mathring{\mathcal{F}}_{l},z}}-\phi_{L,l,z}^{\prime\,t*}\,e^{i\xi
uD_{\mathring{\mathcal{F}}^{\prime}_{L,l},z}}\Big{)}\chi_{\sigma}\rho^{-a}\alpha,\chi_{\tau}\rho^{a}\beta\Big{\rangle}$
$\displaystyle=\Big{\langle}\Big{(}\rho^{a}\phi_{l,z}^{t*}\,e^{i\xi
uD_{\mathring{\mathcal{F}}_{l},z}}\rho^{-a}-\rho^{a}\phi_{L,l,z}^{\prime\,t*}\,e^{i\xi
uD_{\mathring{\mathcal{F}}^{\prime}_{L,l},z}}\rho^{-a}\Big{)}\chi_{\sigma}\alpha,\chi_{\tau}\beta\Big{\rangle}$
$\displaystyle=\Big{\langle}\Big{(}\phi_{l,z-a}^{t*}e^{i\xi
uD_{\mathring{\mathcal{F}}_{l},z-a,z+a}}-\phi_{L,l,z-a}^{\prime\,t*}e^{i\xi
uD_{\mathring{\mathcal{F}}^{\prime}_{L,l},z-a,z+a}}\Big{)}\chi_{\sigma}\alpha,\chi_{\tau}\beta\Big{\rangle}\;.$

Then, proceeding as above, we obtain

$\big{\|}\chi_{\tau}(\mathring{P}_{l,u}-\mathring{P}^{\prime}_{L,l,u})\chi_{\sigma}\big{\|}_{\rho^{a}H^{m},\rho^{a}H^{m^{\prime}}}\leq\widehat{C}e^{-\frac{R}{u}}\sigma^{N}\tau^{N}\|\psi\|_{\mathcal{A},W,N^{\prime}}\,\|f\|_{I,C^{N^{\prime}}}\;,$

for some $\widehat{C},W>0$ and $N^{\prime}\in\mathbb{N}_{0}$, depending only
on $R$, $N$, $m$, $m^{\prime}$ and $a$. Using the Sobolev embedding theorem as
in Proposition 7.2.2, it follows that, for any $a\in\mathbb{R}$, $R>0$ and
$N\in\mathbb{N}_{0}$, there are some $\widehat{C},W>0$ and
$N^{\prime}\in\mathbb{N}_{0}$ such that

$\big{|}\chi_{\tau}(p)(\mathring{K}_{l,u}(p,q)-\mathring{K}^{\prime}_{L,l,u}(p,q))\chi_{\sigma}(q)\big{|}\\\
\leq\widehat{C}e^{-\frac{R}{u}}\Big{(}\frac{\rho(p)}{\rho(q)}\Big{)}^{a}\sigma^{N}\tau^{N}\,\|\psi\|_{\mathcal{A},W,N^{\prime}}\,\|f\|_{I,C^{N^{\prime}}}\;,$

for all $p,q\in\mathring{T}_{L,l,\epsilon}$ and $u\in(0,1]$, and every
$\sigma$ and $\tau$ as in (7.2.2). Put

$\epsilon^{\prime}=\min\left(\frac{\epsilon}{4ce},\frac{\epsilon}{4cc^{\prime}e}\right)\;.$

For $p,q\in\mathring{T}_{L,l,\epsilon^{\prime}}$, we set $\tau=3\rho(p)$ and
$\sigma=3\rho(q)$. It is clear that $\sigma$ and $\tau$ satisfy (7.2.2) and
$\chi_{\tau}(p)=\chi_{\sigma}(q)=1$ (since $\rho(p)<\tau/2,\rho(q)<\sigma/2$).
Therefore, by the above estimate, we get

$\displaystyle\big{|}(\mathring{K}_{l,u}-\mathring{K}^{\prime}_{L,l,u})(p,q)\big{|}$
$\displaystyle\leq
9^{N}\widehat{C}e^{-\frac{R}{u}}s(p,q)^{a}\rho(p)^{N}\rho(q)^{N}\,\|\psi\|_{\mathcal{A},W,N^{\prime}}\,\|f\|_{I,C^{N^{\prime}}}$
$\displaystyle=9^{N}\widehat{C}e^{-\frac{R}{u}}s(p,q)^{a-N}\rho(p)^{a+N}\|\psi\|_{\mathcal{A},W,N^{\prime}}\,\|f\|_{I,C^{N^{\prime}}}\;,$

for all $p,q\in\mathring{T}_{L,l,\epsilon^{\prime}}$ and $u\in(0,1]$.

For any $k\in\mathbb{N}_{0}$, taking arbitrary operators
$D_{1},D_{2}\in\operatorname{Diff}_{\text{\rm
b}}^{k}(M_{l};\Lambda\mathcal{F}_{l})$ and
$D^{\prime}_{1},D^{\prime}_{2}\in\operatorname{Diff}_{\text{\rm
b}}^{k}(M^{\prime}_{L,l};\Lambda\mathcal{F}^{\prime}_{L,l})$ with $D_{i}\equiv
D^{\prime}_{i}$ on
$\mathring{T}_{L,l,\epsilon}\equiv\mathring{T}^{\prime}_{L,l,\epsilon}$
($i=1,2$), and applying the above arguments to the operators
$D_{1}\mathring{P}_{l,u}D_{2}$ and
$D^{\prime}_{1}\mathring{P}^{\prime}_{l,u}D^{\prime}_{2}$, we obtain that, for
all $a\in\mathbb{R}$ and $N\in\mathbb{N}_{0}$, there are some
$\widehat{C},W>0$ and $N^{\prime}\in\mathbb{N}_{0}$ such that, for all
$p,q\in\mathring{T}_{L,l,\epsilon^{\prime}}$ and $u\in(0,1]$,

(7.2.3) $\big{|}D_{1,p}\,D_{2,q}^{\text{\rm
t}}(\mathring{K}_{l,u}-\mathring{K}^{\prime}_{L,l,u})(p,q)\big{|}\leq\widehat{C}e^{-\frac{R}{u}}s(p,q)^{a}\rho(p)^{N}\|\psi\|_{\mathcal{A},W,N^{\prime}}\,\|f\|_{I,C^{N^{\prime}}}\;.$

Consider the vector bundle
$S=\Lambda\mathcal{F}_{l}\boxtimes(\Lambda\mathcal{F}_{l}^{*}\otimes\Omega
M_{l})$ over $M_{l}^{2}$. Recall that $\operatorname{Diff}_{\text{\rm
b}}^{k}((M_{l})^{2}_{\text{\rm b}};\beta_{\text{\rm b}}^{*}S)$ is
$C^{\infty}((M_{l})^{2}_{\text{\rm b}})$-spanned by the lift of
$\operatorname{Diff}_{\text{\rm b}}^{k}(M_{l}^{2};S)$, and
$\operatorname{Diff}_{\text{\rm b}}^{k}(M_{l}^{2};S)$ is
$C^{\infty}(M_{l}^{2})$-spanned by the lift of $\operatorname{Diff}_{\text{\rm
b}}^{k}(M_{l};\Lambda\mathcal{F}_{l})$ from the left-factor projection and the
lift of $\operatorname{Diff}_{\text{\rm
b}}^{k}(M_{l};\Lambda\mathcal{F}_{l}^{*}\otimes\Omega)$ from right-factor
projection (Section 2.5.26). Then it follows from (7.2.3) that, for all
$A\in\operatorname{Diff}_{\text{\rm b}}^{k}((M_{l})^{2}_{\text{\rm
b}};\beta_{\text{\rm b}}^{*}S)$, $a\in\mathbb{R}$ and $N\in\mathbb{N}_{0}$,
there are some $\widehat{C},W>0$ and $N^{\prime}\in\mathbb{N}_{0}$ such that,
on $\beta_{\text{\rm b}}^{-1}(\mathring{T}_{L,l,\epsilon^{\prime}}^{2})$,

$\big{|}A(\mathring{\kappa}_{l,u}-\mathring{\kappa}^{\prime}_{L,l,u})\big{|}\leq\widehat{C}e^{-\frac{R}{u}}s^{a}\rho^{N}\|\psi\|_{\mathcal{A},W,N^{\prime}}\,\|f\|_{I,C^{N^{\prime}}}\;.$

Since $a$ and $N$ are arbitrary, this indeed holds with
$A\in\operatorname{Diff}^{k}((M_{l})^{2}_{\text{\rm b}};\beta_{\text{\rm
b}}^{*}S)$, after possibly increasing $\widehat{C}$, obtaining the stated
inequality. ∎

Proposition 7.2.3 means that
$\mathring{\kappa}_{l,u}-\mathring{\kappa}^{\prime}_{L,l,u}$ has a
$\dot{C}^{\infty}$ extension on the open subset
$(T_{L,l,\epsilon})^{2}_{\text{\rm b}}\subset(M_{l})^{2}_{\text{\rm b}}$ over
$T_{\epsilon^{\prime}}\equiv T^{\prime}_{\epsilon^{\prime}}$.

Recall that $\phi_{L}=\\{\phi_{L}^{t}\\}$ denotes the restriction of
$\phi_{l}$, or of $\phi$, to any boundary leaf $L$ of $\mathcal{F}_{l}$.

######  Corollary 7.2.6.

The section $\mathring{\kappa}_{l}$ has a $C^{\infty}$ extension $\kappa_{l}$
to $(M_{l})^{2}_{\text{\rm b}}$, which vanishes to all orders at
$\operatorname{lb}\cup\operatorname{rb}$, and therefore $\mathring{P}_{l}$
defines an operator $P_{l}\in\Psi_{\text{\rm
b}}^{-\infty}(M_{l};\Lambda\mathcal{F}_{l})$. Moreover

$\displaystyle I_{\nu_{l}}(P_{l},\lambda)$
$\displaystyle\equiv\bigoplus_{L}\int_{-\infty}^{+\infty}\phi_{L,z+i\lambda}^{t*}\,\psi(D_{L,z+i\lambda})\,e^{i\lambda\varkappa_{L}t}f(t)\,dt$
$\displaystyle\in\Psi^{-\infty}(\partial
M_{l};\Lambda)\equiv\bigoplus_{L}\Psi^{-\infty}(L;\Lambda)\;,$

where $L$ runs in $\pi_{0}(\partial M_{l})$.

###### Proof.

This follows from Propositions 7.1.3, 7.1.6, 7.2.2 and 7.2.3. ∎

The subscripts of Notations 7.1.4 and 7.1.7 may be also used with $P_{l}$ and
$\kappa_{l}$. If needed, the subscript “$l$” is also added to the notation of
the b-diagonal $\Delta_{\text{\rm b}}$ of $(M_{l})^{2}_{\text{\rm b}}$, and to
$\Delta_{\text{\rm b},0}=\Delta_{\text{\rm b}}\cap\operatorname{ff}$.

######  Corollary 7.2.7.

The bilinear map

$\mathcal{A}\times C^{\infty}_{\text{\rm c}}(\mathbb{R})\to
C^{\infty}\big{(}(M_{l})^{2}_{\text{\rm b}};\beta_{\text{\rm
b}}^{*}(\Lambda\mathcal{F}_{l}\boxtimes(\Lambda\mathcal{F}^{*}_{l}\otimes\Omega
M_{l}))\big{)}\;,\quad(\psi,f)\mapsto\kappa_{l,\psi,f}\;,$

is continuous.

###### Proof.

Apply (7.2.1) and Propositions 7.1.5 and 7.2.3. ∎

######  Corollary 7.2.8.

If $\psi_{u}(x)=e^{-ux^{2}}$ ($u>0$) and $f(0)=0$, then there is some
$0<\epsilon^{\prime}<\epsilon$ such that $\kappa_{l,u}\to 0$ on
$\Delta_{\text{\rm b},l}\cap\beta_{\text{\rm
b}}^{-1}(T_{l,\epsilon^{\prime}}^{2})\equiv T_{l,\epsilon^{\prime}}$, in the
$C^{\infty}$ topology, as $u\downarrow 0$.

###### Proof.

This is a consequence of Corollary 7.1.9 and Proposition 7.2.3. ∎

######  Corollary 7.2.9.

If $\psi_{u}(x)=e^{-ux^{2}}$, then there is some
$0<\epsilon^{\prime}<\epsilon$ such that

$\lim_{u\downarrow 0}\operatorname{str}(\kappa_{l,u}|_{\Delta_{\text{\rm
b},l}})\equiv\begin{cases}f(0)\,e(\mathcal{F}_{l},g_{\mathcal{F}_{l}})\,|\omega_{\text{\rm
b},l}|&\text{if $n-1$ is even}\\\ 0&\text{if $n-1$ is odd}\end{cases}$

in $C^{0,\infty}_{\varpi_{l}}(T_{l,\epsilon^{\prime}};{}^{\text{\rm
b}}\Omega)$, using the identity $\Delta_{\text{\rm b},l}\cap\beta_{\text{\rm
b}}^{-1}(T_{l,\epsilon^{\prime}}^{2})\equiv T_{l,\epsilon^{\prime}}$.

###### Proof.

This is a consequence of Corollary 7.1.10 and Proposition 7.2.3. ∎

######  Proposition 7.2.10.

We have

$d_{\mathcal{F}_{l},z}\in\operatorname{Diff}_{\text{\rm
b}}^{1}(M_{l};\Lambda\mathcal{F}_{l})\;,\quad
I_{\nu_{l}}(d_{\mathcal{F}_{l},z},\lambda)=d_{\partial M_{l},z+i\lambda}\;.$

###### Proof.

By (3.2.28),
$d_{\mathcal{F}_{l},z}\in\operatorname{Diff}^{1}(\mathcal{F}_{l};\Lambda\mathcal{F}_{l})\subset\operatorname{Diff}_{\text{\rm
b}}^{1}(M_{l};\Lambda\mathcal{F}_{l})$. By (2.5.62) and (4.3.1),

$I_{\nu_{l}}(d_{\mathcal{F}_{l},z},\lambda)=(\rho^{-i\lambda}d_{\mathcal{F}_{l},z}\rho^{i\lambda})_{\partial}=(d_{\mathcal{F}_{l},z}+i\lambda\rho^{-1}d_{\mathcal{F}_{l}}\rho\wedge)_{\partial}=d_{\partial
M_{l},z+i\lambda}\;.$

Alternatively, we can use (2.5.64) and (3.2.28) to describe
$I_{\nu_{l}}(d_{\mathcal{F}_{l},z},\lambda)$. ∎

Recall that $\boldsymbol{M}\equiv\bigsqcup_{l}M_{l}$ and $\boldsymbol{\nu}$ is
the combination of the sections $\nu_{l}$ (Section 4.3.3). This boldface
notation of Sections 4.2.2, 4.3.2 and 4.3.3 allows to simplify the notation of
direct sums of section spaces, cohomologies and operators defined on the
manifolds $M_{l}$. For instance, we get the operators

$\displaystyle\boldsymbol{P}\equiv\bigoplus_{l}P_{l}\in{\textstyle\Psi_{\text{\rm
b}}^{-\infty}(\boldsymbol{M};\Lambda\boldsymbol{\mathcal{F}})}\equiv\bigoplus_{l}{\textstyle\Psi_{\text{\rm
b}}^{-\infty}(M_{l};\Lambda\mathcal{F}_{l})}\;,$ $\displaystyle
d_{\boldsymbol{\mathcal{F}},z}\equiv\bigoplus_{l}d_{\mathcal{F}_{l},z}\in{\textstyle\Psi_{\text{\rm
b}}^{1}(\boldsymbol{M};\Lambda\boldsymbol{\mathcal{F}})}\equiv\bigoplus_{l}{\textstyle\Psi_{\text{\rm
b}}^{1}(M_{l};\Lambda\mathcal{F}_{l})}\;,$

whose indicial operators are

$\displaystyle
I_{\boldsymbol{\nu}}(\boldsymbol{P},\lambda)\equiv\bigoplus_{l}I_{\nu_{l}}(P_{l},\lambda)\in\Psi^{-\infty}(\partial\boldsymbol{M};\Lambda)\equiv\bigoplus_{L}\Psi^{-\infty}(L;\Lambda)\;,$
$\displaystyle
I_{\boldsymbol{\nu}}(d_{\boldsymbol{\mathcal{F}},z},\lambda)\equiv\bigoplus_{l}I_{\nu_{l}}(d_{\mathcal{F}_{l},z},\lambda)\in\Psi^{1}(\partial\boldsymbol{M};\Lambda)\equiv\bigoplus_{L}\Psi^{1}(L;\Lambda)\;,$

where $L$ runs in
$\pi_{0}(\partial\boldsymbol{M})\equiv\pi_{0}M^{0}\sqcup\pi_{0}M^{0}$. On the
other hand, according to Proposition 7.2.10,
$I_{\boldsymbol{\nu}}(d_{\boldsymbol{\mathcal{F}},z},\lambda)=d_{\partial\boldsymbol{M},z+i\lambda}$.
Let also $\kappa=\kappa_{\boldsymbol{P}}$ on $\boldsymbol{M}^{2}_{\text{\rm
b}}\equiv\bigsqcup_{l}(M_{l})^{2}_{\text{\rm b}}$, which is the combination of
the sections $\kappa_{l}$. In $\boldsymbol{M}^{2}_{\text{\rm b}}$, we have
$\Delta_{\text{\rm b}}\equiv\bigsqcup_{l}\Delta_{\text{\rm b},l}$ and
$\Delta_{\text{\rm b},0}\equiv\bigsqcup_{l}\Delta_{\text{\rm b},0,l}$. The
subscripts of Notations 7.1.4 and 7.1.7 may be also used with $\boldsymbol{P}$
and $\kappa$.

When $z=\mu\in\mathbb{R}$ and $\psi(x)=\psi_{u}(x)=e^{-ux^{2}}$ ($u>0$), the
above $\boldsymbol{P}=\boldsymbol{P}_{u}$ is the operator
$\boldsymbol{P}_{\mu,u,f}$ of Section 1.3.4. Thus Theorem 1.3.7 is a
consequence of Corollaries 2.5.10, 7.2.6 and 7.2.7.

###  7.3. The limit of ${}^{\text{\rm
b}}\\!\operatorname{Str}(\boldsymbol{P}_{u})$ as $u\downarrow 0$

With the notation of Section 4.1.1, let $\mathcal{C}=\mathcal{C}(\phi)$,
$\mathcal{P}=\mathcal{P}(\phi)$, $\mathcal{C}_{l}=\mathcal{C}(\phi_{l})$ and
$\mathcal{P}_{l}=\mathcal{P}(\phi_{l})$. For any leafwise density $\alpha\in
C^{\infty}(M_{l};\Omega\mathcal{F}_{l})$, we can consider
$\alpha\,|\omega_{\text{\rm b},l}|\in C^{\infty}(M_{l};{}^{\text{\rm
b}}\Omega)$. In particular, if $n-1$ is even, the leafwise Euler density
$e(\mathcal{F}_{l},g_{\mathcal{F}_{l}})\in
C^{\infty}(M_{l};\Omega\mathcal{F}_{l})$ (Section 3.2.7) gives rise to the
b-density $e(\mathcal{F}_{l},g_{\mathcal{F}_{l}})\,|\omega_{\text{\rm
b},l}|\in C^{\infty}(M_{l};{}^{\text{\rm b}}\Omega)$, whose b-integral,

${}^{\text{\rm b}}\\!\chi_{|\omega_{\text{\rm
b},l}|}(\mathcal{F}_{l})=\\!\\!\\!\\!\\!{\phantom{\int}}^{\nu_{l}}\\!\\!\\!\\!\int_{M_{l}}e(\mathcal{F}_{l},g_{\mathcal{F}_{l}})\,|\omega_{\text{\rm
b},l}|\;,$

can be called the _b-Connes $|\omega_{\text{\rm b},l}|$-Euler characteristic_
of $\mathcal{F}_{l}$. This is a b-normalized version of the Connes
$|\omega_{\text{\rm b},l}|$-Euler characteristic, where $|\omega_{\text{\rm
b},l}|$ is considered as an invariant transverse measure of
$\mathcal{F}^{1}_{l}$. The usual Connes $|\omega_{\text{\rm b},l}|$-Euler
characteristic is not defined because $M^{1}_{l}$ is not compact. If $n-1$ is
odd, let ${}^{\text{\rm b}}\\!\chi_{|\omega_{\text{\rm
b},l}|}(\mathcal{F}_{l})=0$.

Recall the operator $\boldsymbol{P}$ defined in Section 7.2.

######  Theorem 7.3.1.

If $\psi_{u}(x)=e^{-ux^{2}}$ ($u>0$), then

$\lim_{u\downarrow 0}{}^{\text{\rm
b}}\\!\operatorname{Str}(\boldsymbol{P}_{u})=\sum_{l}{}^{\text{\rm
b}}\\!\chi_{|\omega_{\text{\rm b},l}|}(\mathcal{F}_{l})\cdot
f(0)+\sum_{c\in\mathcal{C}}\ell(c)\sum_{k\in\mathbb{Z}^{\times}}\epsilon_{c}(k)\cdot
f(k\ell(c))\;.$

To prove this theorem, we consider every $P_{l,u}$, separately. Recall that
$\mathring{\kappa}_{l,u,z}$ corresponds to $\mathring{k}_{l,z,u}$ via the
restriction of $\beta_{\text{\rm b}}:(M_{l})^{2}_{\text{\rm b}}\to M_{l}^{2}$
to the interiors. Thus we are going to study the asymptotic behaviour of
$\mathring{k}_{l,z,u}$ as $u\downarrow 0$. The identities
$\mathring{M}_{l}\equiv M^{1}_{l}$,
$\mathring{\mathcal{F}}_{l}\equiv\mathcal{F}^{1}_{l}$ and
$\mathring{T}_{l,\epsilon^{\prime}}\equiv T^{1}_{l,\epsilon^{\prime}}$
($0<\epsilon^{\prime}\leq\epsilon$) will be used without further comment. With
the notation of Sections 4.3.4 and 4.3.5, and adapting the notation of Section
3.4.7, let $\mathfrak{G}_{l}=\operatorname{Hol}\mathcal{F}^{1}_{l}$ and
$\widetilde{\mathfrak{G}}_{l}=\operatorname{Hol}\widetilde{\mathcal{F}}^{1}_{l}$,
with source and target projections,
$\boldsymbol{s},\boldsymbol{r}:\mathfrak{G}_{l}\to M^{1}_{l}$ and
$\boldsymbol{s},\boldsymbol{r}:\widetilde{\mathfrak{G}}_{l}\to\widetilde{M}^{1}_{l}$.
The pairs $(\boldsymbol{r},\boldsymbol{s})$ define identities
$\mathfrak{G}_{l}\equiv\mathcal{R}_{l}:=\mathcal{R}_{\mathcal{F}^{1}_{l}}$ and
$\widetilde{\mathfrak{G}}_{l}\equiv\widetilde{\mathcal{R}}_{l}:=\mathcal{R}_{\widetilde{\mathcal{F}}^{1}_{l}}$.
Let $\Delta_{l}\subset\mathcal{R}_{l}$ denote the diagonal. Consider also the
vector bundles

$S_{l}=\boldsymbol{s}^{*}\Lambda\mathcal{F}^{1}_{l}\otimes\boldsymbol{r}^{*}(\Lambda\mathcal{F}_{l}^{1*}\otimes\Omega\mathcal{F}^{1}_{l})\;,\quad\widetilde{S}_{l}=\boldsymbol{s}^{*}\Lambda\widetilde{\mathcal{F}}^{1}_{l}\otimes\boldsymbol{r}^{*}\big{(}\Lambda\widetilde{\mathcal{F}}_{l}^{1*}\otimes\Omega\widetilde{\mathcal{F}}^{1}_{l}\big{)}\;,$

over $\mathfrak{G}_{l}$ and $\widetilde{\mathfrak{G}}_{l}$, and the leafwise
Schwartz kernel $\tilde{k}_{l,z,u}$ defined by the Schwartz kernels of the
operators $e^{-u\Delta_{\widetilde{L}^{\prime},z}}$ on the leaves
$\widetilde{L}^{\prime}$ of $\widetilde{\mathcal{F}}^{1}_{l}$, for
$z\in\mathbb{C}$ (Section 3.4.7). By (3.4.21) and since
$\tilde{\omega}_{\text{\rm b},l}=D_{l}^{*}dx$ (Section 4.3.4), for
$\tilde{p}\in\widetilde{M}^{1}_{l}$ and $p=[\tilde{p}]\in M^{1}_{l}$,

(7.3.1)
$\mathring{k}_{l,z,u}(p,p)\equiv\sum_{\gamma\in\Gamma_{l}}\tilde{\phi}_{l,z}^{-h_{l}(\gamma)*}\,T_{\gamma}^{*}\,\tilde{k}_{l,z,u}\big{(}T_{\gamma}\tilde{\phi}_{l}^{-h_{l}(\gamma)}(\tilde{p}),\tilde{p}\big{)}\,f(-h_{l}(\gamma))\,|\omega_{\text{\rm
b},l}|(p)\;,$

using that $\widetilde{S}_{(\gamma\cdot\tilde{p},\tilde{p})}\equiv S_{(p,p)}$.
This defines a convergent series in $C^{\infty}_{\text{\rm
ub}}(\Delta_{l};S_{l})$.

Any leaf of $\widetilde{\mathcal{F}}^{1}_{l}$ is of the form
$\widetilde{L}^{\prime}=\\{x\\}\times L_{l}\equiv L_{l}$ for some
$x\in\mathbb{R}$. Then the restriction of $\tilde{g}_{\text{\rm b},l}$ to
$\widetilde{L}^{\prime}$ is identified with a metric $\tilde{g}_{l,x}$ on
$L_{l}$, $\Delta_{\widetilde{L}^{\prime},z}$ is identified with the twisted
Laplacian $\Delta_{l,x,z}$ on $(L_{l},\tilde{g}_{l,x})$ defined by the
restriction of $\tilde{\eta}_{0}$, and $\tilde{k}_{l,z,u}$ on
$\widetilde{L}^{\prime\,2}$ is identified with the Schwartz kernel
$\tilde{k}_{l,x,u,z}$ of $e^{-u\Delta_{l,x,z}}$, defined on $L_{l}^{2}$.

Theorem 7.3.1 follows from the following result.

######  Proposition 7.3.2.

Let $I\subset\mathbb{R}$ be a compact interval with
$\operatorname{supp}f\subset I$. Then the following properties hold:

  1. (i)

If $I\subset\mathbb{R}^{\times}$ and $I\cap\mathcal{P}_{l}=\emptyset$, then

$\lim_{u\downarrow 0}{}^{\text{\rm b}}\\!\operatorname{Str}(P_{l,u})=0\;.$

  2. (ii)

If $I\subset\mathbb{R}^{\times}$ and $I\cap\mathcal{P}_{l}=\\{t_{0}\\}$, then

$\lim_{u\downarrow 0}{}^{\text{\rm
b}}\\!\operatorname{Str}(P_{l,u})=f(t_{0})\sum_{c\in\mathcal{C}_{l,t_{0}}}\ell(c)\,\epsilon_{c}(t_{0}/\ell(c))\;,$

where $\mathcal{C}_{l,t_{0}}$ consists of the orbits $c\in\mathcal{C}_{l}$
with period $t_{0}$.

  3. (iii)

If $0\in I$ and $I\cap\mathcal{P}_{l}=\emptyset$, then

$\lim_{u\downarrow 0}{}^{\text{\rm
b}}\\!\operatorname{Str}(P_{l,u})=f(0)\,{}^{\text{\rm
b}}\\!\chi_{|\omega_{\text{\rm b},l}|}(\mathcal{F}_{l})\;.$

###### Proof.

Choose some $0<\epsilon^{\prime}<\epsilon$ satisfying the statements of
Corollaries 7.2.8 and 7.2.9, and take some
$0<\epsilon^{\prime\prime}<\epsilon^{\prime}$. Take some $C_{3}\geq 1$
satisfying (4.3.5). Since $\tilde{\phi}_{l}$ is of $\mathbb{R}$-local bounded
geometry (Section 2.4.7), there is some $R\geq 0$ such that
$\tilde{d}_{l}(\tilde{\phi}_{l}^{t}(\tilde{p}),\tilde{p})\leq R$ for all
$\tilde{p}\in\widetilde{M}^{1}_{l}$ and $t\in I$. So, by (4.3.5) and the
triangle inequality, for all
$\tilde{p}\in\widetilde{M}^{1}_{l,\epsilon^{\prime\prime}}$ and
$\gamma\in\Gamma_{l}$ with $-h_{l}(\gamma)\in I$, we get

(7.3.2)
$C_{3}^{-1}|\gamma|-R\leq\tilde{d}_{l}\big{(}\gamma\cdot\tilde{\phi}_{l}^{-h_{l}(\gamma)}(\tilde{p}),\tilde{p}\big{)}\leq
C_{3}|\gamma|+R\;,$

using also that
$\gamma\cdot\tilde{\phi}_{l}^{-h_{l}(\gamma)}(\tilde{p})=\tilde{\phi}_{l}^{-h_{l}(\gamma)}(\gamma\cdot\tilde{p})$
with $\gamma\cdot\tilde{p}\in\widetilde{M}^{1}_{l,\epsilon^{\prime\prime}}$.