$X=\overline{\mathcal{X}}_{\mathbb{G}}$ and that $\delta_{X}$ restricts to a
co-action of $\mathbb{G}$ on $\mathcal{X}_{\mathbb{G}}$, which we denote by
$\delta_{X,\mathbb{G}}$. In particular, we have
$(id_{\mathcal{X}_{\mathbb{G}}}\otimes\varepsilon_{\mathbb{G}})\circ\delta_{X,\mathbb{G}}=id_{\mathcal{X}_{\mathbb{G}}}$.

By a standard argument, both $\delta_{X,\mathbb{H}}$ and
$\delta_{X,\mathbb{G}}$ extends to actions of $\mathbb{G}$ on $X$ (for
instance, consider the transformation on $\mathfrak{X}_{\mathbb{H}}\odot
Pol(\mathbb{H})$ given by $\xi\otimes
b\mapsto\delta_{X,\mathbb{H}}(\xi)(1\otimes b)$, for all
$\xi\in\mathfrak{X}_{\mathbb{H}}$ and all $b\in Pol(\mathbb{H})$; and notice
that it extends to a unitary on $X\otimes C(\mathbb{G})$ thanks to the above
density property). By construction these extensions are both equal to
$\delta_{X}$. Since moreover we have
$\mathfrak{X}_{\mathbb{H},\epsilon}=X^{\delta_{X}}=\mathcal{X}_{\epsilon}$
because $\mathscr{R}ep(\mathbb{H})$ is a full subcategory of
$\mathscr{R}ep(\mathbb{G})$ containing the trivial representation, then we can
conclude that $\delta_{X,\mathbb{H}}$ and $\delta_{X,\mathbb{G}}$ coincide at
the algebraic level (compare with Theorem 2.3.6). In other words, we have
obtained that
$\mathfrak{X}_{\mathbb{H}}=\mathcal{X}_{\mathbb{H}}=\mathcal{X}_{\mathbb{G}}$
following the notations of Proposition 2.3.12 so that
$\delta_{X}:X\longrightarrow X\otimes C(\mathbb{H})\subset X\otimes
C(\mathbb{G})$ and the original $\mathbb{G}$-equivariant module can be
regarded as $(X,(id_{X}\otimes\iota)\circ\delta_{X})$.

In conclusion, we have showed that the map
$\varphi:R(\mathbb{G})\underset{R(\mathbb{H})}{\odot}Fus(\mathscr{N}_{\beta})\longrightarrow
Fus\Big{(}\mathscr{M}_{\widetilde{\beta}}\Big{)}$ such that
$\varphi(\mathbbold{1}_{t}\odot(Y,\delta_{Y}))=(Ind^{\mathbb{G}}_{\mathbb{H}}(Y),\widetilde{\delta}_{Y})_{t}$,
for all $t\in\Omega$ and all $(Y,\delta_{Y})\in Irr(\mathscr{N}_{\beta})$
defines an isomorphism of based $R(\mathbb{G})$-modules as soon as
$\widehat{\mathbb{H}}$ is divisible in $\widehat{\mathbb{G}}$. ∎

In view of the examples discussed in the begining of this section, when the
discrete quantum subgroup $\widehat{\mathbb{H}}$ is not divisible in
$\widehat{\mathbb{G}}$, a non-trivial torsion action of $\mathbb{H}$ can be
realized as the trivial torsion action of $\mathbb{G}$ even when
$\widehat{\mathbb{G}}$ is torsion-free, which prevents $\widehat{\mathbb{H}}$
to be torsion-free. Under the divisibility assumption we claim that the
correspondence of Proposition 3.4 preserves non-trivial actions, which will
yield our stability result.

######  3.6 Theorem.

Let $\mathbb{G}$ and $\mathbb{H}$ be two compact quantum groups such that
$\widehat{\mathbb{H}}<\widehat{\mathbb{G}}$ is divisible. If
$\widehat{\mathbb{G}}$ is torsion-free, then $\widehat{\mathbb{H}}$ is
torsion-free.

###### Proof.

Let $(B,\beta)$ be a torsion action of $\mathbb{H}$ and consider the
corresponding torsion $\mathscr{R}$ep$(\mathbb{H})$-module $C^{*}$-category
$\mathscr{N}_{\beta}$ given by Theorem 2.3.25. Consider the torsion
$\mathscr{R}$ep$(\mathbb{G})$-module $C^{*}$-category
$\mathscr{M}_{\widetilde{\beta}}$ associated to the induced torsion action
$(Ind^{\mathbb{G}}_{\mathbb{H}}(B),\widetilde{\beta})$. By Lemma 3.5 we have
that:

$R(\mathbb{G})\underset{R(\mathbb{H})}{\odot}Fus(\mathscr{N}_{\beta})=Ind^{R(\mathbb{G})}_{R(\mathbb{H})}\Big{(}Fus(\mathscr{N}_{\beta})\Big{)}\cong
Fus\big{(}\mathscr{M}_{\widetilde{\beta}}\big{)}$

as based $R(\mathbb{G})$-modules. Since $\widehat{\mathbb{G}}$ is torsion-free
by assumption, we have that
$\mathscr{M}_{\widetilde{\beta}}\cong\mathscr{R}ep(\mathbb{G})$. Hence,
$Fus(\mathscr{M}_{\widetilde{\beta}})\cong R(\mathbb{G})$ as based
$R(\mathbb{G})$-modules. Since $\widehat{\mathbb{H}}$ is divisible in
$\widehat{\mathbb{G}}$, then we have
$R(\mathbb{G})=\underset{\Omega}{\bigoplus}R(\mathbb{H})$ as based
$R(\mathbb{H})$-modules.

In this situation, we can apply verbatim the proof of Proposition $1.28$ in
[3] concluding that $Fus(\mathscr{N}_{\beta})\cong R(\mathbb{H})$ as
$R(\mathbb{H})$-modules, which yields that
$\mathscr{N}_{\beta}\cong\mathscr{R}ep(\mathbb{H})$. In other words, that
$\widehat{\mathbb{H}}$ is torsion-free.

Namely, if $J$ denotes the basis for the $R(\mathbb{H})$-module
$Fus(\mathscr{N}_{\beta})$, then the basis for
$Ind^{R(\mathbb{G})}_{R(\mathbb{H})}\Big{(}Fus(\mathscr{N}_{\beta})\Big{)}$
taking into account the divisibility condition is given by
$\widetilde{J}:=\\{\alpha\otimes j\ |\ \alpha\in I_{0}\mbox{ and }j\in
J\\}\mbox{,}$ where $I_{0}$ denotes the set of elements in $Irr(\mathbb{G})$
corresponding to the units of the components $R(\mathbb{H})$ in the
decomposition $R(\mathbb{G})=\oplus R(\mathbb{H})$.

By the above discussion we have
$Ind^{R(\mathbb{G})}_{R(\mathbb{H})}\Big{(}Fus(\mathscr{N}_{\beta})\Big{)}\cong
R(\mathbb{G})$ as based $R(\mathbb{G})$-modules. Hence, take
$\alpha_{0}\otimes j_{0}\in\widetilde{J}$ corresponding to $\epsilon\in
Irr(\mathbb{G})$. Then $\alpha\otimes(\alpha_{0}\otimes
j_{0})\in\widetilde{J}$, for each $\alpha\in Irr(\mathbb{G})$ because the
isomorphism
$Ind^{R(\mathbb{G})}_{R(\mathbb{H})}\Big{(}Fus(\mathscr{N}_{\beta})\Big{)}\cong
R(\mathbb{G})$ sends basis to basis. In particular,
$\alpha\otimes\alpha_{0}\in Irr(\mathbb{G})$, for each $\alpha\in
Irr(\mathbb{G})$ and so $\overline{\alpha}_{0}\otimes\alpha_{0}=\epsilon$. We
may thus assume $\alpha_{0}=\epsilon$ and we find that
$Fus(\mathscr{N}_{\beta})\cong
R(\mathbb{H})\underset{R(\mathbb{H})}{\odot}Fus(\mathscr{N}_{\beta})\cong
R(\mathbb{H})$ as based $R(\mathbb{H})$-modules as we wanted to show. ∎

######  3.7 Proposition.

The following properties hold.

  1. i)

Let $\Gamma$ be a discrete group, $\mathbb{G}$ a compact quantum group. Assume
that $\Gamma$ acts on $\mathbb{G}$ by quantum automorphisms, $\alpha$. Then
both $\Gamma$ and $\widehat{\mathbb{G}}$ are divisible discrete quantum
subgroups in $\widehat{\Gamma\underset{\alpha}{\ltimes}\mathbb{G}}$.

  2. ii)

Let $\Gamma$ be a discrete group, $G$ a compact group. Assume that
$(\Gamma,G)$ is a matched pair with actions
$\Gamma\overset{\alpha}{\curvearrowright}G$ and
$\Gamma\overset{\beta}{\curvearrowleft}G$. Then $\widehat{G}\mbox{ is a
divisible in
}\widehat{\Gamma{{}_{\alpha}}\bowtie_{\beta}G}\Leftrightarrow\beta\mbox{ is
trivial.}$ In this case, $\Gamma$ is also a divisible discrete quantum
subgroup in $\widehat{\Gamma{{}_{\alpha}}\bowtie_{\beta}G}$.

  3. iii)

Let $\mathbb{G}$ and $\mathbb{H}$ be two compact quantum groups. Then both
$\widehat{\mathbb{G}}$ and $\widehat{\mathbb{H}}$ are divisible discrete
quantum subgroups in $\widehat{\mathbb{G}\times\mathbb{H}}$.

  4. iv)

Let $\mathbb{G}$ and $\mathbb{H}$ be two compact quantum groups. Then both
$\widehat{\mathbb{G}}$ and $\widehat{\mathbb{H}}$ are divisible discrete
quantum subgroups in $\widehat{\mathbb{G}\ast\mathbb{H}}$.

###### Proof.

  1. i)

The description of the representation theory of
$\mathbb{F}:=\Gamma\underset{\alpha}{\ltimes}\mathbb{G}$ recalled in Section
2.1 yields that $\widehat{\mathbb{G}}$ and $\Gamma$ are divisible in
$\widehat{\mathbb{F}}$. Namely, take an irreducible representation
$y:=(\gamma,x)\in Irr(\mathbb{F})$ with $\gamma\in\Gamma$ and $x\in
Irr(\mathbb{G})$. Then $\gamma=(\gamma,\epsilon_{\mathbb{G}})\in[y]$ in
$Irr(\mathbb{F})/Irr(\mathbb{G})$ because
$(\gamma^{-1},\epsilon_{\mathbb{G}})\otop(\gamma,x)=(e,x)=x\in
Irr(\mathbb{G})$. Likewise, we have that $x=(e,x)\in[y]$ in $\Gamma\backslash
Irr(\mathbb{F})$ because
$(\gamma,x)\otop(e,\overline{x})=(\gamma,\epsilon_{\mathbb{G}})=\gamma\in\Gamma$.

Consequently, $\widehat{\mathbb{G}}$ is divisible in $\widehat{\mathbb{F}}$
because for all $s\in Irr(\mathbb{G})$ we have that
$(\gamma,\epsilon_{\mathbb{G}})\otop
s=(\gamma,\epsilon_{\mathbb{G}})\otop(e,s)=(\gamma,s)\in Irr(\mathbb{F})$.
Likewise, $\Gamma$ is divisible in $\widehat{\mathbb{F}}$ because for all
$s\in\Gamma$ we have that
$s\otop(e,x)=(s,\epsilon_{\mathbb{G}})\otop(e,x)=(s,x)\in Irr(\mathbb{F})$.

  2. ii)

On the one hand, if $\Gamma\overset{\beta}{\curvearrowleft}G$ is the trivial
action, then we simply have
$\Gamma{{}_{\alpha}}\bowtie_{\beta}G=\Gamma\underset{\alpha}{\ltimes}G$; so
that property $(i)$ above yields that both $\Gamma$ and $\widehat{G}$ are
divisible discrete quantum subgroups in
$\widehat{\Gamma\underset{\alpha}{\ltimes}G}$.

On the other hand, in order to show that divisibility of $\widehat{G}$ implies
triviality of $\beta$, we are going to use the description of the
representation theory of $\mathbb{F}:=\Gamma{{}_{\alpha}}\bowtie_{\beta}G$
recalled in Section 2.1. Let $y=\gamma(x)\in Irr(\mathbb{F})$ an irreducible
representation of the bicrossed product with $\gamma\in\Gamma$ and $x\in
Irr(G_{\gamma})$. If $y^{\prime}=\gamma^{\prime}(x^{\prime})\in
Irr(\mathbb{F})$ is another irreducible representation of $\mathbb{F}$ with
$\gamma^{\prime}\in\Gamma$ and $x^{\prime}\in Irr(G_{\gamma^{\prime}})$, then,
by virtue of the fusion rules of the bicrossed product (recall Equation
(2.3)), $y^{\prime}\in[y]$ in $Irr(\mathbb{F})/Irr(G)$ if and only if
$y^{\prime}=\gamma(x^{\prime})$ for some $x^{\prime}\in Irr(G_{\gamma})$
(recall in addition that $\\#[e]=1$ in the orbit space $\Gamma/G$ and that
$(\gamma\cdot G)^{-1}=\gamma^{-1}\cdot G$).

Now, assume that $\widehat{G}$ is divisible in $\widehat{\mathbb{F}}$. This
means that given $[y]=[\gamma(x)]\in Irr(\mathbb{F})/Irr(G)$ we can find a
representative, say $\gamma(x^{\prime})\in[y]$ for some $x^{\prime}\in Irr(G)$
such that for all irreducible $s\in Irr(G)$, $\gamma(x^{\prime})\otop
s=\gamma(x^{\prime})\otop e(s)$ is an irreducible representation of
$\mathbb{F}$. For this recall again the fusion rules for a bicrossed product
(Equation (2.3)): given $\mu(z)\in Irr(\mathbb{F})$ with $\mu\in\Gamma$ and
$z\in Irr(G_{\mu})$, we have:

${\rm dim}_{\mathbb{F}}(\mu(z),\gamma(x^{\prime})\otop
e(s))=\left\\{\begin{array}[]{ll}\underset{r\in\mu\cdot G\cap\gamma\cdot
G}{\sum}\frac{1}{|r\cdot G|}{\rm
dim}_{G_{r}}(z\circ\psi^{\mu}_{r,r},x^{\prime}\underset{r}{\otimes}s)&\text{
if }\mu\cdot G\cap\gamma\cdot G\neq\emptyset,\\\ 0&\text{
otherwise.}\end{array}\right.$

Then, in order to $\gamma(x^{\prime})\otop e(s)$ to be irreducible, it is
necessary that $|\mu\cdot G\cap\gamma\cdot G|=1$, for all
$\mu,\gamma\in\Gamma$. This implies that all classes in the orbit space
$\Gamma/G$ are trivial, that is, $\beta$ is the trivial action as we wanted to
show. Observe moreover that in this case, a representative of $y=\gamma(x)$ in
$Irr(\mathbb{F})/Irr(G)$ can be chosen as $\gamma(\epsilon_{G})$ and the above
fusion rules become:

${\rm dim}_{\mathbb{F}}(\mu(z),\gamma(\epsilon_{G})\otop
e(s))=\left\\{\begin{array}[]{ll}{\rm
dim}_{G}(z\circ\psi^{\mu}_{\gamma,\gamma},s)&\text{ if }\mu=\gamma,\\\
0&\text{ otherwise,}\end{array}\right.$

where recall that $\psi^{\gamma}_{\gamma,\gamma}=id$. In other words,
$\gamma(\epsilon_{G})\otop e(s)=\gamma(s)$; recovering then the same
computations as done for a quantum semi-direct product in $(i)$ above.

  3. iii)

The description of the representation theory of
$\mathbb{F}:=\mathbb{G}\times\mathbb{H}$ recalled in Section 2.1 yields that
$\widehat{\mathbb{G}}$ and $\widehat{\mathbb{H}}$ are divisible in
$\widehat{\mathbb{F}}$. Namely, take an irreducible representation
$y:=(x,z)\in Irr(\mathbb{F})$ with $x\in Irr(\mathbb{G})$ and $z\in
Irr(\mathbb{H})$. Then $x=(x,\epsilon_{\mathbb{H}})\in[y]$ in
$Irr(\mathbb{H})\backslash Irr(\mathbb{F})$ because
$(x,z)\otop(\overline{x},\epsilon_{\mathbb{H}})=(\epsilon_{\mathbb{G}},z)=z\in
Irr(\mathbb{H})$. Likewise, we have that $z=(\epsilon_{\mathbb{G}},z)\in[y]$
in $Irr(\mathbb{G})\backslash Irr(\mathbb{F})$ because
$(x,z)\otop(\epsilon_{\mathbb{G}},\overline{z})=(x,\epsilon_{\mathbb{H}})=x\in
Irr(\mathbb{G})$.

Consequently, $\widehat{\mathbb{G}}$ is divisible in $\widehat{\mathbb{F}}$
because for all $s\in Irr(\mathbb{G})$ we have that
$s\otop(\epsilon,z)=(s,\epsilon_{\mathbb{H}})\otop(\epsilon_{\mathbb{G}},z)=(s,z)\in
Irr(\mathbb{F})$. Likewise, $\widehat{\mathbb{H}}$ is divisible in
$\widehat{\mathbb{F}}$ because for all $s\in Irr(H)$ we have that
$(x,\epsilon_{\mathbb{H}})\otop
s=(x,\epsilon_{\mathbb{H}})\otop(\epsilon_{\mathbb{G}},s)=(x,s)\in
Irr(\mathbb{F})$.

  4. iv)

The description of the representation theory of
$\mathbb{F}:=\mathbb{G}\ast\mathbb{H}$ recalled in Section 2.1 yields that
both $\widehat{\mathbb{G}}$ and $\widehat{\mathbb{H}}$ are divisible in
$\widehat{\mathbb{F}}$. Let us show that $\widehat{\mathbb{G}}$ is divisible
in $\widehat{\mathbb{F}}$ (the proof for $\widehat{\mathbb{H}}$ is analogous).
Take any irreducible representation of $\mathbb{F}$, which is given by an
alternating word in $Irr(\mathbb{G})$ and $Irr(\mathbb{H})$, say
$y:=x_{i_{1}}z_{i_{2}}\ldots x_{i_{n-1}}z_{i_{n}}$. By definition, a
representative of $y$ in $Irr(\mathbb{G})\backslash Irr(\mathbb{F})$ (resp. in
$Irr(\mathbb{F})/Irr(\mathbb{G})$) is an irreducible representation
$y^{\prime}\in Irr(\mathbb{F})$ such that $y\otop\overline{y^{\prime}}$ (resp.
$\overline{y^{\prime}}\otop y$) contains an irreducible representation of
$\mathbb{G}$ (inside $\mathbb{F}$). The latter is possible if and only if the
tensor product $y\otop\overline{y^{\prime}}$ (resp.
$\overline{y^{\prime}}\otop y$) reduces to a single letter in
$Irr(\mathbb{G})$.

Assume that $y$ starts in $Irr(\mathbb{G})$, then it is enough to put
$y^{\prime}:=z_{i_{2}}\ldots x_{i_{n-1}}z_{i_{n}}$. The fusion rules of a
quantum free product yield that $y\otop\overline{y^{\prime}}=x_{i_{1}}\in
Irr(\mathbb{G})$.

In this situation, given any $s\in Irr(\mathbb{G})$ we have to prove that
$s\otop y^{\prime}\in Irr(\mathbb{F})$. Since $y$ starts in $Irr(\mathbb{G})$,
then $y^{\prime}$ starts in $Irr(\mathbb{H})$, so that the fusion rules of a
quantum free product yield that $s\otop y^{\prime}=sy^{\prime}\in
Irr(\mathbb{F})$.

Assume that $y$ starts in $Irr(\mathbb{H})$ and ends in $Irr(\mathbb{G})$,
then it is enough to put $y^{\prime}:=x_{i_{1}}z_{i_{2}}\ldots x_{i_{n-1}}$.
The fusion rules of a quantum free product yield that
$\overline{y^{\prime}}\otop y=z_{i_{n}}\in Irr(\mathbb{G})$. Since $y$ ends in
$Irr(\mathbb{G})$, then $y^{\prime}$ ends in $Irr(\mathbb{H})$ and the fusion
rules of a quantum free product yield that $y^{\prime}\otop s=y^{\prime}s\in
Irr(\mathbb{F})$, for every $s\in Irr(\mathbb{G})$.

Assume that $y$ starts and ends in $Irr(\mathbb{H})$, then we can not choose
any representative $y^{\prime}$ of $y$ either in $Irr(\mathbb{G})\backslash
Irr(\mathbb{F})$ or in $Irr(\mathbb{F})/Irr(\mathbb{G})$ such that either
$y\otop\overline{y^{\prime}}$ or $\overline{y^{\prime}}\otop y$ reduces to a
single letter in $Irr(\mathbb{G})$. In other words, the class of $[y]$ is
formed only by $y$ itself (notice that
$y\otimes\overline{y}=\epsilon_{\mathbb{H}}\cong\epsilon_{\mathbb{G}}$ in
$Irr(\mathbb{F})$). In this case, it is obvious that $s\otop
y^{\prime}=sy^{\prime}\in Irr(\mathbb{F})$, for every $s\in Irr(\mathbb{G})$.

∎

######  3.8 Corollary.

The following stability properties for torsion-freeness hold.

  1. i)

Let $\Gamma$ be a discrete group, $\mathbb{G}$ a compact quantum group. Assume
that $\Gamma$ acts on $\mathbb{G}$ by quantum automorphisms, $\alpha$. Then
$\widehat{\Gamma\underset{\alpha}{\ltimes}\mathbb{G}}\mbox{ is torsion-free
}\Leftrightarrow\Gamma\mbox{ and }\widehat{\mathbb{G}}\mbox{ are torsion-
free.}$

  2. ii)

Let $\Gamma$ be a discrete group, $G$ a compact group. Assume that
$(\Gamma,G)$ is a matched pair with actions
$\Gamma\overset{\alpha}{\curvearrowright}G$ and
$G\overset{\beta}{\curvearrowright}\Gamma$. Then
$\widehat{\Gamma{{}_{\alpha}}\bowtie_{\beta}G}\mbox{ is torsion-free
}\Rightarrow\Gamma\mbox{ and }\widehat{G}\mbox{ are torsion-free.}$

  3. iii)

Let $\mathbb{G}$ and $\mathbb{H}$ be two compact quantum groups. Then
$\widehat{\mathbb{G}\times\mathbb{H}}\mbox{ is torsion-free
}\Leftrightarrow\widehat{\mathbb{G}}\mbox{ and }\widehat{\mathbb{H}}\mbox{ are
torsion-free.}$

  4. iv)

Let $\mathbb{G}$ and $\mathbb{H}$ be two compact quantum groups. Then
$\widehat{\mathbb{G}\ast\mathbb{H}}\mbox{ is torsion-free
}\Leftrightarrow\widehat{\mathbb{G}}\mbox{ and }\widehat{\mathbb{H}}\mbox{ are
torsion-free.}$

###### Proof.

  1. i)

Implication $\Rightarrow$ is a consequence of Proposition 3.7 and Theorem 3.6.
Implication $\Leftarrow$ is contained in [26, Theorem 3.1.1].

  2. ii)

Implication $\Rightarrow$ is a consequence of Proposition $4.2$ in [26] (by
which we know that if $\widehat{\Gamma{{}_{\alpha}}\bowtie_{\beta}G}$ is
torsion-free, then $\beta$ is trivial) and property $(i)$ above.

  3. iii)

Implication $\Rightarrow$ is a consequence of Proposition 3.7 and Theorem 3.6.
Implication $\Leftarrow$ is contained in [3, Theorem 3.17].

  4. iv)

Implication $\Rightarrow$ is a consequence of Proposition 3.7 and Theorem 3.6.
Implication $\Leftarrow$ is contained in [3, Theorem 3.16].

∎

##  Baum-Connes conjecture for discrete quantum subgroups

The stability of torsion-freeness (in the sense of Meyer-Nest) by divisible
discrete quantum subgroups will allow to complete the picture of elementary
stability properties of the Baum-Connes property for torsion-free discrete
quantum groups initiated in [26]. To begin with, let
$\widehat{\mathbb{H}}<\widehat{\mathbb{G}}$ be a discrete quantum subgroup of
$\widehat{\mathbb{G}}$. We have two relevant functors: restriction, which is
obvious, and induction, which has been studied by S. Vaes in [37] in the
framework of quantum groups,
$Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}:\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}}\longrightarrow\mathscr{K}\mathscr{K}^{\widehat{\mathbb{H}}}\mbox{
and
}Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}:\mathscr{K}\mathscr{K}^{\widehat{\mathbb{H}}}\longrightarrow\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}}$.
It is well-known that restriction and induction are _triangulated_ functors by
virtue of the universal property of the Kasparov category (see [33] for more
details). Assume that both $\widehat{\mathbb{G}}$ and $\widehat{\mathbb{H}}$
are torsion-free. Denote by $(L^{\prime},N^{\prime})$ the canonical
triangulated functors associated to the complementary pair
$(\mathscr{L}_{\widehat{\mathbb{H}}},\mathscr{N}_{\widehat{\mathbb{H}}})$ and
by $F^{\prime}$ the homological functor defining the quantum Baum-Connes
assembly map for $\widehat{\mathbb{H}}$. We recall the following facts
appearing already in [26].

######  4.1 Lemma.

Let $\mathbb{G}$, $\mathbb{H}$ be two compact quantum groups. If
$\widehat{\mathbb{G}}$ is torsion-free and
$\widehat{\mathbb{H}}<\widehat{\mathbb{G}}$ is a divisible torsion-free
discrete quantum subgroup, then the following properties hold.

  1. i)

$Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(\mathscr{L}_{\widehat{\mathbb{G}}})\subset\mathscr{L}_{\widehat{\mathbb{H}}}$
and
$Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(\mathscr{N}_{\widehat{\mathbb{G}}})\subset\mathscr{N}_{\widehat{\mathbb{H}}}$.
Hence, we have the following natural isomorphisms
$Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}\circ L\cong L^{\prime}\circ
Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}$ and
$Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}\circ N\cong N^{\prime}\circ
Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}$.

  2. ii)

$Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(\mathscr{L}_{\widehat{\mathbb{H}}})\subset\mathscr{L}_{\widehat{\mathbb{G}}}$
and
$Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(\mathscr{N}_{\widehat{\mathbb{H}}})\subset\mathscr{N}_{\widehat{\mathbb{G}}}$.
Hence, we have the following natural isomorphisms
$Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}\circ L^{\prime}\cong L\circ
Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}$ and
$Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}\circ N^{\prime}\cong N\circ
Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}$.

Consequently, $Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}$ transforms
the assembly map for $\widehat{\mathbb{G}}$ into the assembly map for
$\widehat{\mathbb{H}}$ and $Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}$
transforms the assembly map for $\widehat{\mathbb{H}}$ into the assembly map
for $\widehat{\mathbb{G}}$.

######  4.2 Proposition.

Let $\mathbb{G}$, $\mathbb{H}$ be two compact quantum groups. Assume that
$\widehat{\mathbb{G}}$ is torsion-free. $\widehat{\mathbb{G}}$ satisfies the
quantum Baum-Connes property if and only if every divisible torsion-free
discrete quantum subgroup $\widehat{\mathbb{H}}<\widehat{\mathbb{G}}$
satisfies the quantum Baum-Connes property.

######  4.3 Remark.

Observe that by virtue of Theorem 3.6 of the preceding section we can simplify
the assumptions of the above statements. Namely, it suffices to assume that
$\widehat{\mathbb{G}}$ is a torsion-free discrete quantum group and that
$\widehat{\mathbb{H}}$ a divisible discrete quantum subgroup of
$\widehat{\mathbb{G}}$; it will be automatically torsion-free.

One wishes to remove the torsion-freeness and divisibility assumptions in
Lemma 4.1 in order to improve the permanence of the (quantum) Baum-Connes
property by discrete quantum subgroups of Proposition 4.2. Let
$\widehat{\mathbb{G}}$ be a discrete quantum group and
$\widehat{\mathbb{H}}<\widehat{\mathbb{G}}$ a discrete quantum subgroup. Both
$\widehat{\mathbb{G}}$ and $\widehat{\mathbb{H}}$ are not necessarily torsion-
free and $\widehat{\mathbb{H}}$ is not necessarily divisible in
$\widehat{\mathbb{G}}$. In that case, consider the localizing subcategories:

$\mathscr{L}_{\widehat{\mathbb{G}}}:=\langle\\{Ind^{\widehat{\mathbb{G}}}_{\mathbb{E}}(C)=c_{0}(\widehat{\mathbb{G}})\otimes
C\ |\ C\in
Obj.(\mathscr{K}\mathscr{K})\\}\cup\\{\mathbb{G}\underset{r}{\ltimes}T\otimes
C\ |\ C\in Obj.(\mathscr{K}\mathscr{K})\mbox{,
}T\in\mathbb{T}\mbox{or}(\widehat{\mathbb{G}})\\}\rangle,$
$\mathscr{L}_{\widehat{\mathbb{H}}}:=\langle\\{Ind^{\widehat{\mathbb{H}}}_{\mathbb{E}}(D)=c_{0}(\widehat{\mathbb{H}})\otimes
D\ |\ D\in
Obj.(\mathscr{K}\mathscr{K})\\}\cup\\{\mathbb{H}\underset{r}{\ltimes}S\otimes
D\ |\ D\in Obj.(\mathscr{K}\mathscr{K})\mbox{,
}S\in\mathbb{T}\mbox{or}(\widehat{\mathbb{H}})\\}\rangle,$

introduced in Section 2.4. As explained previously, the only candidate for
their complementaries are the corresponding right orthogonals. Namely:

$\mathscr{N}_{\widehat{\mathbb{G}}}:=\mathscr{L}^{\dashv}_{\widehat{\mathbb{G}}}=\\{A\in
Obj(\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}})\ |\
KK^{\widehat{\mathbb{G}}}(L,A)=(0)\mbox{, $\forall$ $L\in
Obj(\mathscr{L}_{\widehat{\mathbb{G}}})$}\\},$
$\mathscr{N}_{\widehat{\mathbb{H}}}:=\mathscr{L}^{\dashv}_{\widehat{\mathbb{H}}}=\\{B\in
Obj(\mathscr{K}\mathscr{K}^{\widehat{\mathbb{H}}})\ |\
KK^{\widehat{\mathbb{H}}}(L^{\prime},B)=(0)\mbox{, $\forall$ $L^{\prime}\in
Obj(\mathscr{L}_{\widehat{\mathbb{H}}})$}\\}.$

######  4.4 Lemma.

Let $\widehat{\mathbb{G}}$ be a discrete quantum group and
$\widehat{\mathbb{H}}<\widehat{\mathbb{G}}$ a discrete quantum subgroup (both
$\widehat{\mathbb{G}}$ and $\widehat{\mathbb{H}}$ are not necessarily torsion-
free and $\widehat{\mathbb{H}}$ is not necessarily divisible in
$\widehat{\mathbb{G}}$): $i)$
$Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(\mathscr{N}_{\widehat{\mathbb{G}}})\subset\mathscr{N}_{\widehat{\mathbb{H}}}$
if and only if
$Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(\mathscr{L}_{\widehat{\mathbb{H}}})\subset\mathscr{L}_{\widehat{\mathbb{G}}}$;
$ii)$
$Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(\mathscr{N}_{\widehat{\mathbb{H}}})\subset\mathscr{N}_{\widehat{\mathbb{G}}}$
if and only if
$Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(\mathscr{L}_{\widehat{\mathbb{G}}})\subset\mathscr{L}_{\widehat{\mathbb{H}}}$.

###### Proof.

This is a consequence of the orthogonality condition of the complementary
pairs. Indeed, recall that the functors
$Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}$ and
$Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}$ are adjoint meaning that
$KK^{\widehat{\mathbb{G}}}(A,Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(B))\cong
KK^{\widehat{\mathbb{H}}}(Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(A),B)$
and
$KK^{\widehat{\mathbb{G}}}(Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(B),A)\cong
KK^{\widehat{\mathbb{H}}}(B,Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(A)),$
for all $A\in Obj(\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}})$ and all
$B\in Obj(\mathscr{K}\mathscr{K}^{\widehat{\mathbb{H}}})$ (we refer to [33] or
[38] for a proof). These relations yield the following: $i)$ given
$N\in\mathscr{N}_{\widehat{\mathbb{G}}}$,
$Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(N)\in\mathscr{N}_{\widehat{\mathbb{H}}}\Leftrightarrow(0)=KK^{\widehat{\mathbb{H}}}(L^{\prime},Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(N))\cong
KK^{\widehat{\mathbb{G}}}(Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(L^{\prime}),N)\
\forall L^{\prime}\in\mathscr{L}_{\widehat{\mathbb{H}}}\Leftrightarrow
Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(\mathscr{L}_{\widehat{\mathbb{H}}})\subset\mathscr{L}_{\widehat{\mathbb{G}}}$;
and $ii)$ given $N^{\prime}\in\mathscr{N}_{\widehat{\mathbb{H}}}$,
$Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(N^{\prime})\in\mathscr{N}_{\widehat{\mathbb{G}}}\Leftrightarrow(0)=KK^{\widehat{\mathbb{G}}}(L,Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(N^{\prime}))\cong
KK^{\widehat{\mathbb{H}}}(Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(L),N^{\prime})\
\forall L\in\mathscr{L}_{\widehat{\mathbb{G}}}\Leftrightarrow
Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(\mathscr{L}_{\widehat{\mathbb{G}}})\subset\mathscr{L}_{\widehat{\mathbb{H}}}.$
∎

######  4.5 Proposition.

Let $\widehat{\mathbb{G}}$ be a discrete quantum group and
$\widehat{\mathbb{H}}<\widehat{\mathbb{G}}$ a discrete quantum subgroup (both
$\widehat{\mathbb{G}}$ and $\widehat{\mathbb{H}}$ are not necessarily torsion-
free and $\widehat{\mathbb{H}}$ is not necessarily divisible in
$\widehat{\mathbb{G}}$). The inclusions
$Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(\mathscr{L}_{\widehat{\mathbb{H}}})\subset\mathscr{L}_{\widehat{\mathbb{G}}}$
and
$Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(\mathscr{L}_{\widehat{\mathbb{G}}})\subset\mathscr{L}_{\widehat{\mathbb{H}}}$
hold.

###### Proof.

On the one hand, as showed in Proposition 3.4, the restriction to $\mathbb{H}$
of any torsion action of $\mathbb{G}$ decomposes as a direct sum of torsion
actions of $\mathbb{H}$. Moreover, the restriction functor is triangulated and
compatible with countable direct sums. Thus the inclusion
$Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(\mathscr{L}_{\widehat{\mathbb{G}}})\subset\mathscr{L}_{\widehat{\mathbb{H}}}$
follows. On the other hand, in order to show the inclusion
$Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(\mathscr{L}_{\widehat{\mathbb{H}}})\subset\mathscr{L}_{\widehat{\mathbb{G}}}$
we are going to show that the following diagram is commutative:

$\begin{split}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
24.21362pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mbox{$\mathscr{K}\mathscr{K}^{\mathbb{H}}$}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.10583pt\hbox{$\scriptstyle{\mbox{$\mathbb{H}\underset{r}{\ltimes}\cdot$}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-45.19135pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
11.03181pt\raise 6.97522pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22522pt\hbox{$\scriptstyle{\mbox{$Ind^{\mathbb{G}}_{\mathbb{H}}(\cdot)$}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 37.02504pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
37.02504pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mbox{$\mathscr{K}\mathscr{K}^{\mathbb{G}}$}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
50.00006pt\raise-28.95677pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.10583pt\hbox{$\scriptstyle{\mbox{$\mathbb{G}\underset{r}{\ltimes}\cdot$}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
50.00006pt\raise-45.19135pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mbox{$\mathscr{K}\mathscr{K}^{\widehat{\mathbb{H}}}$}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
10.34764pt\raise-66.23297pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{\mbox{$Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(\cdot)$}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
37.00003pt\raise-57.91354pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
37.00003pt\raise-57.91354pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mbox{$\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}}$}}$}}}}}}}{\hbox{\kern
87.00009pt\raise-57.91354pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$
(4.1)

Indeed, then given a torsion action of $\mathbb{H}$,
$(B,\beta)\in\mathbb{T}or(\widehat{\mathbb{H}})$, we have explained in
Proposition 3.4 that $(Ind^{\mathbb{G}}_{\mathbb{H}}(B),\widetilde{\beta})$ is
a torsion action of $\mathbb{G}$ (it might be the trivial one; recall the
discussion of Section 3). Hence, by definition we have
$\mathbb{G}\underset{r}{\ltimes}Ind^{\mathbb{G}}_{\mathbb{H}}(B,\beta)\in\mathscr{L}_{\widehat{\mathbb{G}}}$.
Also by definition we have that
$\mathbb{H}\underset{r}{\ltimes}B\in\mathscr{L}_{\widehat{\mathbb{H}}}$. If
Diagram (4.1) commutes, then we will have
$Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}\Big{(}\mathbb{H}\underset{r}{\ltimes}B\Big{)}\cong\mathbb{G}\underset{r}{\ltimes}Ind^{\mathbb{G}}_{\mathbb{H}}(B,\beta)\in\mathscr{L}_{\widehat{\mathbb{G}}}$
(where the isomorphism is in the Kasparov category
$\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}}$), which would yield then the
desired inclusion. To prove commutativity of Diagram (4.1) observe that
$\mathscr{K}\mathscr{K}^{\mathbb{H}}\overset{\mathbb{H}\ltimes\cdot}{\cong}\mathscr{K}\mathscr{K}^{\widehat{\mathbb{H}}}$
and
$\mathscr{K}\mathscr{K}^{\mathbb{G}}\overset{\mathbb{G}\ltimes\cdot}{\cong}\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}}$
by virtue of Baaj-Skandalis duality. Therefore, it is enough to show that
$\widehat{\mathbb{G}}\underset{r}{\ltimes}Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}\Big{(}\mathbb{H}\underset{r}{\ltimes}B\Big{)}\cong\widehat{\mathbb{G}}\underset{r}{\ltimes}\Big{(}\mathbb{G}\underset{r}{\ltimes}Ind^{\mathbb{G}}_{\mathbb{H}}(B,\beta)\Big{)}$
as objects in $\mathscr{K}\mathscr{K}^{\mathbb{G}}$. In other words, by
applying the quantum version of Green’s imprimitivity theorem (see [37] for
more details), we have to show that
$\widehat{\mathbb{H}}\underset{r}{\ltimes}\Big{(}\mathbb{H}\underset{r}{\ltimes}B\Big{)}\cong\widehat{\mathbb{G}}\underset{r}{\ltimes}\Big{(}\mathbb{G}\underset{r}{\ltimes}Ind^{\mathbb{G}}_{\mathbb{H}}(B,\beta)\Big{)}\Leftrightarrow
B\cong Ind^{\mathbb{G}}_{\mathbb{H}}(B,\beta)$ as objects in
$\mathscr{K}\mathscr{K}^{\mathbb{G}}$, which is true. ∎

The first direct consequence from these claims is a generalization of Lemma
4.1.

######  4.6 Corollary.

Let $\mathbb{G}$, $\mathbb{H}$ be two compact quantum groups such that
$\widehat{\mathbb{H}}<\widehat{\mathbb{G}}$. The following properties hold.

  1. i)

$Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(\mathscr{L}_{\widehat{\mathbb{G}}})\subset\mathscr{L}_{\widehat{\mathbb{H}}}$
and
$Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(\mathscr{N}_{\widehat{\mathbb{G}}})\subset\mathscr{N}_{\widehat{\mathbb{H}}}$.
Hence, we have the following natural isomorphisms
$Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}\circ L\cong L^{\prime}\circ
Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}$ and
$Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}\circ N\cong N^{\prime}\circ
Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}$.

  2. ii)

$Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(\mathscr{L}_{\widehat{\mathbb{H}}})\subset\mathscr{L}_{\widehat{\mathbb{G}}}$
and
$Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(\mathscr{N}_{\widehat{\mathbb{H}}})\subset\mathscr{N}_{\widehat{\mathbb{G}}}$.
Hence, we have the following natural isomorphisms
$Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}\circ L^{\prime}\cong L\circ
Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}$ and
$Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}\circ N^{\prime}\cong N\circ
Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}$.

Consequently, $Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}$ transforms
the assembly map for $\widehat{\mathbb{G}}$ into the assembly map for
$\widehat{\mathbb{H}}$ and $Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}$
transforms the assembly map for $\widehat{\mathbb{H}}$ into the assembly map
for $\widehat{\mathbb{G}}$.

The second consequence is a generalization of Proposition 4.2.

######  4.7 Corollary.

Let $\mathbb{G}$, $\mathbb{H}$ be two compact quantum groups such that
$\widehat{\mathbb{H}}<\widehat{\mathbb{G}}$.

  1. i)

$\widehat{\mathbb{G}}$ satisfies the quantum Baum-Connes property if and only
if every discrete quantum subgroup $\widehat{\mathbb{H}}<\widehat{\mathbb{G}}$
satisfies the quantum Baum-Connes property.

  2. ii)

If $\widehat{\mathbb{G}}$ satisfies the
$\mathscr{L}_{\widehat{\mathbb{G}}}$-strong Baum-Connes property, then
$\widehat{\mathbb{H}}$ satisfies the
$\mathscr{L}_{\widehat{\mathbb{H}}}$-strong Baum-Connes property.

###### Proof.

  1. i)

This is done verbatim the proof appearing in [26] of Proposition 4.2 by
applying now Corollary 4.6 above.

  2. ii)

As showed in Proposition 4.5 above, we always have
$Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(\mathscr{L}_{\widehat{\mathbb{G}}})\subset\mathscr{L}_{\widehat{\mathbb{H}}}$.
Assume that $\widehat{\mathbb{G}}$ satisfies the
$\mathscr{L}_{\widehat{\mathbb{G}}}$-strong Baum-Connes property, which means
that
$\mathscr{L}_{\widehat{\mathbb{G}}}=\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}}$.
Given $(B,\beta)\in Obj(\mathscr{K}\mathscr{K}^{\widehat{\mathbb{H}}})$, we
have thus
$Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(B)\in\mathscr{K}\mathscr{K}^{\widehat{\mathbb{G}}}=\mathscr{L}_{\widehat{\mathbb{G}}}$.
Hence,
$Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}\Big{(}Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(B)\Big{)}\in\mathscr{L}_{\widehat{\mathbb{H}}}$.
To conclude recall that $B$ is a retract of
$Res^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}\Big{(}Ind^{\widehat{\mathbb{G}}}_{\widehat{\mathbb{H}}}(B)\Big{)}$
and that localization subcategories are closed under retracts, which implies
that $B\in\mathscr{L}_{\widehat{\mathbb{H}}}$. So $\widehat{\mathbb{H}}$
satisfies the $\mathscr{L}_{\widehat{\mathbb{H}}}$-strong Baum-Connes
property.

∎

######  4.8 Remark.

It is well-known that the _strong_ Baum-Connes property is preserved by
_divisible_ discrete quantum subgroups whenever $\widehat{\mathbb{G}}$ is
torsion-free (see [38, Lemma 6.7] for a proof). The above corollary
generalizes this property for any discrete quantum subgroup (not necessarily
divisible and without any torsion-freeness assumption), which has been
possible thanks to the more abstract approach in terms of fusion rings and
module $C^{*}$-categories. Moreover, it is important to mention that property
$(ii)$ of the previous corollary appears already in [15, Remark 3.20]. The
argument appearing there has been developed in the present paper with
Corollary 3.3 and Proposition 3.4. However, the property $(i)$ of the previous
corollary concerning the usual Baum-Connes property is new and legitimate to
consider now thanks to [4], [7].

The generalization of [26] reached with Corollary 4.7 above allows to improve
some stability results for the quantum Baum-Connes conjecture with respect to
some relevant constructions, which already appear in [26], by removing
torsion-freeness and divisibility assumptions.

######  4.9 Corollary.

The following stability properties for the quantum (resp. strong) Baum-Connes
property hold.

  1. i)

Let $\Gamma$ be a discrete group, $\mathbb{G}$ a compact quantum group. Assume
that $\Gamma$ acts on $\mathbb{G}$ by quantum automorphisms, $\alpha$. Then
$\Gamma$ and $\widehat{\mathbb{G}}$ satisfy the (resp. strong) Baum-Connes
property whenever $\widehat{\Gamma\underset{\alpha}{\ltimes}\mathbb{G}}$
satisfies the (resp. strong) Baum-Connes property.

  2. ii)

Let $\Gamma$ be a discrete group, $G$ a compact group. Assume that
$(\Gamma,G)$ is a matched pair with actions
$\Gamma\overset{\alpha}{\curvearrowright}G$ and
$G\overset{\beta}{\curvearrowright}\Gamma$. Then $\widehat{G}$ satisfies the
(resp. strong) Baum-Connes property whenever
$\widehat{\Gamma{{}_{\alpha}}\bowtie_{\beta}G}$ satisfies the (resp. strong)
Baum-Connes property.

  3. iii)

Let $\mathbb{G}$ and $\mathbb{H}$ be two compact quantum groups. Then
$\widehat{\mathbb{G}}$ and $\widehat{\mathbb{H}}$ satisfy the (resp. strong)
Baum-Connes property whenever $\widehat{\mathbb{G}\times\mathbb{H}}$ satisfies
the (resp. strong) Baum-Connes property.

  4. iv)

Let $\mathbb{G}$ and $\mathbb{H}$ be two compact quantum groups. Then
$\widehat{\mathbb{G}}$ and $\widehat{\mathbb{H}}$ satisfy the (resp. strong)
Baum-Connes property whenever $\widehat{\mathbb{G}\ast\mathbb{H}}$ satisfies
the (resp. strong) Baum-Connes property.

##  Alternative approach to torsion-freeness for divisible discrete quantum
subgroups

The first approach of the author, when the present paper was begun, to tackle
Theorem 3.6 and the questions discussed in Section 3 was more general by using
abstract $C^{*}$-tensor category tools. Technical issues concerning the
construction of relative tensor products of module $C^{*}$-categories lead to
a more direct approach as presented in Section 3. However, recent progresses
justify to review results of Section 3 by using this more general strategy,
which will be described in more detail in this section.

First of all, in the purely algebraic setting, the notion of relative tensor
product of module categories over tensor categories goes back to constructions
by P. Deligne [9] and D. Tambara [36] (see as well [30], [11], [25], [21] for
further related developments). The corresponding construction for module
$C^{*}$-categories is certainly known to experts or, at least, foreseen by
experts (see for instance [32, Section 4.1], [10] or [1]), but a formal and
general definition seemed to be elusive in the literature. The construction of
balanced tensor products of module categories over $C^{*}$-categories (without
any semisimplicity or rigidity assumption) has been established in a recent
work of J. Antoun and C. Voigt [2] as a byproduct of a more general category
related considerations.

Let us first introduce some terminology.

######  5.1 Definition.

Let $\mathscr{C}$ be a $C^{*}$-tensor category. Let
$(\mathscr{M},\bullet,\mu,e)$ be a right $\mathscr{C}$-$C^{*}$-module
category, $(\mathscr{N},\bullet,\mu^{\prime},e^{\prime})$ a left
$\mathscr{C}$-$C^{*}$-module category and $\mathscr{A}$ a $C^{*}$-category
(resp. $\mathbb{C}$-linear category). A $C^{*}$-bifunctor (resp.
$\mathbb{C}$-linear bifunctor)
$F:\mathscr{M}\times\mathscr{N}\longrightarrow\mathscr{A}$ is called
$\mathscr{C}$-balanced if there exists a natural equivalence
$b:F\circ(\bullet\times id_{\mathscr{N}})\longrightarrow
F\circ(id_{\mathscr{M}}\times\bullet)$ such that

  1. i)

the diagram:

$F(X\bullet(U\otimes V),Y)$$b_{X,U\otimes V,Y}$$F\circ(\mu_{X,U,V}\times
id_{Y})$$F(X,(U\otimes V)\bullet
Y)$$F\circ(id_{X}\times\mu^{\prime}_{U,V,Y})$$F((X\bullet U)\bullet
V,Y)$$b_{X\bullet U,V,Y}$$F(X,U\bullet(V\bullet Y))$$F(X\bullet U,V\bullet
Y)$$b_{X,U,V\bullet Y}$

is commutative for all objects $U,V\in Obj(\mathscr{C})$, $X\in
Obj(\mathscr{M})$ and $Y\in Obj(\mathscr{N})$.

  2. ii)

the diagram:

$F(X\bullet\mathbbold{1},Y)$$F\circ(e_{X}\times
id_{Y})$$b_{X,\mathbbold{1},Y}$$F(X,\mathbbold{1}\bullet
Y)$$F\circ(id_{X}\times e^{\prime}_{Y})$$F(X,Y)$

is commutative for all objects $X\in Obj(\mathscr{M})$ and $Y\in
Obj(\mathscr{N})$.

The collection of all $\mathscr{C}$-balanced functors from
$\mathscr{M}\times\mathscr{N}$ to $\mathscr{A}$ together with natural
transformations between them forms a $C^{*}$-category (resp. involutive
category) denoted by
$\mathscr{B}il_{\mathscr{C}}(\mathscr{M},\mathscr{N};\mathscr{A})$.

######  5.2 Remark.

On the one hand, for the sake of the presentation we omit the definition of
natural transformation between $\mathscr{C}$-balanced functors. On the other
hand, it is important to notice that the notion of $\mathscr{C}$-balanced
functor appearing in the algebraic context [36] must be adapted to the
$C^{*}$-level. Namely, if the $C^{*}$-categories involved are countably
additive, we will have to deal with _semi-categories_ , that is, categories
who do not necessarily contain identity homomorphisms. This entails the
definition of a _multiplier $C^{*}$-category_ together with a notion of _non-
degenerate $C^{*}$-functor_ by imitating the standard $C^{*}$-algebraic case.
This has been done in [2]. For our purpose we don’t need such general
considerations in view of assumptions of Note 2.2.17, so that the previous
definition is useful enough (compare with [2, Definition 5.8]).

Let $\mathscr{C}$ be a countably additive $C^{*}$-tensor category,
$\mathscr{M}$ a right $\mathscr{C}$-module $C^{*}$-category and $\mathscr{N}$
a left $\mathscr{C}$-module $C^{*}$-category. Then there exists a countably
additive $C^{*}$-category
$\mathscr{M}\underset{\mathscr{C}}{\boxtimes}\mathscr{N}$ together with a
$\mathscr{C}$-balanced functor
$Q_{\mathscr{C}}:\mathscr{M}\times\mathscr{N}\longrightarrow\mathscr{M}\underset{\mathscr{C}}{\boxtimes}\mathscr{N}$
such that composition with $Q_{\mathscr{C}}$ induces an equivalence of
$C^{*}$-categories
$\mathscr{F}un(\mathscr{M}\underset{\mathscr{C}}{\boxtimes}\mathscr{N},\mathscr{A})\cong\mathscr{B}il_{\mathscr{C}}(\mathscr{M},\mathscr{N};\mathscr{A})$,
for all $C^{*}$-category $\mathscr{A}$ (see Theorem 5.10 in [2] for a proof).
The category $\mathscr{M}\underset{\mathscr{C}}{\boxtimes}\mathscr{N}$ is
called _relative tensor product of $\mathscr{M}$ and $\mathscr{N}$ with
respect to $\mathscr{C}$_. In this way, if $\mathscr{D}$ is another countable
additive $C^{*}$-tensor category such that there exists a $C^{*}$-tensor
functor $J:\mathscr{D}\longrightarrow\mathscr{C}$ (so that $\mathscr{C}$ can
be considered as a right $\mathscr{D}$-module $C^{*}$-category, recall
Examples 2.2.16) and $\mathscr{N}$ is a left $\mathscr{D}$-module
$C^{*}$-category, we define the _induced $\mathscr{C}$-module $C^{*}$-category
of $\mathscr{N}$_ to be the following $\mathscr{C}$-bimodule $C^{*}$-category
$Ind^{\mathscr{C}}_{\mathscr{D}}(\mathscr{N}):=\mathscr{C}\underset{\mathscr{D}}{\boxtimes}\mathscr{N}$.
On the one hand, the $\mathscr{C}$-bimodule structure on
$Ind^{\mathscr{C}}_{\mathscr{D}}(\mathscr{N})$ is simply given by the one of
$\mathscr{C}$ itself. On the other hand, observe that thanks to [2, Lemma
5.11], we have $Ind^{\mathscr{D}}_{\mathscr{D}}(\mathscr{N})\cong\mathscr{N}$
as left $\mathscr{D}$-module $C^{*}$-categories.

In particular, let $\mathbb{G}$ and $\mathbb{H}$ be compact quantum groups
such that $\widehat{\mathbb{H}}<\widehat{\mathbb{G}}$. As already pointed out
in Examples 2.2.16(3), we have a fully faithful functor between $C^{*}$-tensor
categories,
$J:\mathscr{R}ep(\mathbb{H})\longrightarrow\mathscr{R}ep(\mathbb{G})$, given
by the natural inclusion of $\mathscr{R}ep(\mathbb{H})$ inside
$\mathscr{R}ep(\mathbb{G})$ as a full subcategory. In this way,
$\mathscr{R}ep(\mathbb{G})$ can be viewed as a right
$\mathscr{R}ep(\mathbb{H})$-module $C^{*}$-category. So, given a left
$\mathscr{R}\mbox{ep}(\mathbb{H})$-$C^{*}$-module category $\mathscr{N}$ we
put:

$Ind^{\mathscr{R}\mbox{ep}(\mathbb{G})}_{\mathscr{R}\mbox{ep}(\mathbb{H})}(\mathscr{N}):=\mathscr{R}\mbox{ep}(\mathbb{G})\underset{\mathscr{R}\mbox{ep}(\mathbb{H})}{\boxtimes}\mathscr{N}.$

In this setting, we can give an alternative approach to the results of Section
3. Firstly, the (resp. module) $C^{*}$-categories appearing in our purpose are
_semi-simple_ since this is the case for the considered categories associated
to compact quantum groups and its torsion actions (recall Note 2.2.17 and
Theorem 2.3.25).

Given a torsion action $(B,\beta)$ of $\mathbb{H}$ and $\mathscr{N}_{\beta}$
the corresponding $\mathscr{R}$ep$(\mathbb{H})$-module $C^{*}$-category,
denote by
$\mathcal{I}nd^{\mathscr{R}ep(\mathbb{G})}_{\mathscr{R}ep(\mathbb{H})}\Big{(}\mathscr{N}_{\beta}\Big{)}$
the $\mathscr{R}$ep$(\mathbb{G})$-module $C^{*}$-category whose objects are
simply the induced $\mathbb{G}$-equivariant Hilbert
$Ind^{\mathbb{G}}_{\mathbb{H}}(B,\beta)$-modules from $\mathbb{H}$-equivariant
Hilbert $B$-modules. Then, by our semisimplicity assumptions, the relative
tensor product defining
$Ind^{\mathscr{R}ep(\mathbb{G})}_{\mathscr{R}ep(\mathbb{H})}\Big{(}\mathscr{N}_{\beta}\Big{)}$
coincides with the usual relative tensor product of tensor categories of P.
Deligne and D. Tambara (see [2, Proposition 3.7] for more details).
Accordingly, we have
$\mathcal{I}nd^{\mathscr{R}ep(\mathbb{G})}_{\mathscr{R}ep(\mathbb{H})}\Big{(}\mathscr{N}_{\beta}\Big{)}\cong
Ind^{\mathscr{R}ep(\mathbb{G})}_{\mathscr{R}ep(\mathbb{H})}\Big{(}\mathscr{N}_{\beta}\Big{)}$
by construction. In particular,
$\mathcal{I}nd^{\mathscr{R}ep(\mathbb{G})}_{\mathscr{R}ep(\mathbb{H})}\Big{(}\mathscr{N}_{\beta}\Big{)}$
is again semi-simple. It follows, by using an analogous argument to the one of
Lemma 3.5 that:

$Ind^{R(\mathbb{G})}_{R(\mathbb{H})}(Fus(\mathscr{N}_{\beta}))\cong
Fus\Big{(}Ind^{\mathscr{R}\mbox{ep}(\mathbb{G})}_{\mathscr{R}\mbox{ep}(\mathbb{H})}(\mathscr{N}_{\beta})\Big{)}$

as based $R(\mathbb{G})$-modules, for all
$\mathscr{R}\mbox{ep}(\mathbb{H})$-$C^{*}$-module category $\mathscr{N}$ as
soon as $\widehat{\mathbb{H}}$ is divisible in $\widehat{\mathbb{G}}$. This
relation implies that if $\mathscr{N}$ is a torsion
$\mathscr{R}\mbox{ep}(\mathbb{H})$-$C^{*}$-module category, then
$Ind^{\mathscr{R}\mbox{ep}(\mathbb{G})}_{\mathscr{R}\mbox{ep}(\mathbb{H})}(\mathscr{N})$
is a torsion $\mathscr{R}\mbox{ep}(\mathbb{G})$-$C^{*}$-module category
whenever $\widehat{\mathbb{H}}$ is divisible in $\widehat{\mathbb{G}}$.

Finally, Theorem 3.6 can be regarded in the following way. It is clear that
$Irr\Big{(}\mathcal{I}nd^{\mathscr{R}ep(\mathbb{G})}_{\mathscr{R}ep(\mathbb{H})}\Big{(}\mathscr{N}_{\beta}\Big{)}\Big{)}\subset
Irr\big{(}\mathscr{M}_{\widetilde{\beta}}\big{)}$. Hence, the torsion action
obtained by the correspondence in [8] through any irreducible object in
$\mathcal{I}nd^{\mathscr{R}ep(\mathbb{G})}_{\mathscr{R}ep(\mathbb{H})}\Big{(}\mathscr{N}_{\beta}\Big{)}$
is $\mathbb{G}$-equivariantly Morita equivalent to
$(Ind^{\mathbb{G}}_{\mathbb{H}}(B),\widetilde{\beta})$ (whenever
$\widehat{\mathbb{G}}$ is torsion-free). So
$\mathcal{I}nd^{\mathscr{R}ep(\mathbb{G})}_{\mathscr{R}ep(\mathbb{H})}\Big{(}\mathscr{N}_{\beta}\Big{)}$
must be equivalent to $\mathscr{M}_{\widetilde{\beta}}$ as
$\mathscr{R}$ep$(\mathbb{G})$-module $C^{*}$-categories, which yields that
$Fus\Big{(}\mathcal{I}nd^{\mathscr{R}ep(\mathbb{G})}_{\mathscr{R}ep(\mathbb{H})}\Big{(}\mathscr{N}_{\beta}\Big{)}\Big{)}\cong
Fus\big{(}\mathscr{M}_{\widetilde{\beta}}\big{)}$ as based
$R(\mathbb{G})$-modules. Thus, Theorem 3.6 can be recovered again by using the
same argument. Notice that the difference here is that Lemma 3.5 has been
encoded with the more general category related considerations above.

## References

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Denmark.

E-mail address<EMAIL_ADDRESS>


# The Almost Schur Lemma in Quaternionic Contact Geometry

Stefan Ivanov  University of Sofia, Faculty of Mathematics and Informatics,
blvd. James Bourchier 5, 1164, Sofia, Bulgaria  and Institute of Mathematics
and Informatics, Bulgarian Academy of Sciences<EMAIL_ADDRESS>and
Alexander Petkov  University of Sofia, Faculty of Mathematics and Informatics,
blvd. James Bourchier 5, 1164, Sofia, Bulgaria<EMAIL_ADDRESS>
###### Abstract.

We establish quaternionic contact (qc) versions of the so called Almost Schur
Lemma, which give estimations of the qc scalar curvature on a compact qc
manifold to be a constant in terms of the norm of the $[-1]$-component and the
norm of the trace-free part of the $[3]$-component of the horizontal qc Ricci
tensor and the torsion endomorphism, under certain positivity conditions.

###### Key words and phrases:

Quaternionic contact structure, QC Lichnerowicz condition, QC Cordes estimate,
P-function

###### 2010 Mathematics Subject Classification:

53C21, 58J60, 53C17, 35P15, 53C25

###### Contents

  1. 1 Introduction
  2. 2 Quaternionic contact manifolds
    1. 2.1 Quaternionic contact structures and the Biquard connection
    2. 2.2 Invariant decompositions
    3. 2.3 The torsion tensor
    4. 2.4 Torsion and curvature
    5. 2.5 The Ricci identities
    6. 2.6 The $P-$form
  3. 3 The inequalities, main results

##  1\. Introduction

For a compact Riemannian manifold $(M^{n},g)$ of dimension $n\geq 3$ the
famous Schur lemma states that if $(M^{n},g)$ is Einstein, then it has
constant scalar curvature, $S=Const.$ The metric $g$ is said to be Einstein,
if the Ricci tensor is proportional to the metric, $Ric=\frac{S}{n}g.$ A
generalization of the Schur lemma is a recent result of De Lellis and Topping
[9] that states as follows.

######  Theorem 1.1.

[9, Almost Schur Lemma] Let $(M^{n},g)$ be a compact Riemannian manifold of
dimension $n\geq 3$ with non–negative Ricci tensor, $Ric\geq 0$. Then the
following inequality holds

$\int_{M}(S-\bar{S})^{2}vol_{g}\leq\frac{4n(n-1)}{(n-2)^{2}}\int_{M}\left|Ric-\frac{S}{n}g\right|_{g}^{2}vol_{g},$

where $\bar{S}$ means the average value of the scalar curvature $S$ of $g$.

The equality holds if and only if the manifold is Einstein.

It is also shown in [9] that the positivity condition $Ric\geq 0$ assumed on
the Ricci tensor is essential and can not be dropped.

In the CR case there are known two positivity conditions written in terms of
the Webster Ricci curvature and the pseudohermitian torsion; one is used for
obtaining a lower bound of the first egenvalue of the sub-Laplacian (see e.g.
[12]), while the other one appears in the CR Cordes type estimate (see [5]). A
CR version of the almost Schur lemma was first established in [6] under the
Greenleaf’s positivity condition. Recently, a CR version of the almost Schur
Lemma was presented in [15] under the CR Cordes positivity condition which
gives better estimate in the torsion-free (Sasakian) case.

The aim of this note is to present a quaternionic contact (qc) version of the
Almost Schur lemma. Quaternionic contact geometry is an example of
sub–Riemannian geometry and any quaternionic contact manifold admits a
canonical connection $\nabla$, namely the Biquard connection, whose role in qc
geometry is similar to that of Levi–Civita connection in Riemannian geometry
and Tanaka–Webster connection in CR geometry. We present in terms of the
Biquard connection two versions of quaternionic contact almost Schur lemma
depending on the positivity assumption, see Theorem 3.1 and Theorem 3.3 below.

######  Convention 1.2.

  1. a)

We shall use $X,Y,Z,U$ to denote horizontal vector fields, i.e. $X,Y,Z,U\in
H$.

  2. b)

$\\{e_{1},\dots,e_{4n}\\}$ denotes a local orthonormal basis of the horizontal
space $H$.

  3. c)

The triple $(i,j,k)$ denotes any cyclic permutation of $(1,2,3)$.

  4. d)

$s$ will be any number from the set $\\{1,2,3\\}$, $s\in\\{1,2,3\\}$.

Acknowledgments The research of both authors is partially supported by
Contract DH/12/3/12.12.2017, Contract 80-10-161/05.04.2021 with the Sofia
University “St. Kliment Ohridski”, and the National Science Fund of Bulgaria,
National Scientific Program “VIHREN”, Project No. KP-06-DV-7.

##  2\. Quaternionic contact manifolds

It is well known that the sphere at infinity of a non-compact symmetric space
of rank one carries a natural Carnot-Carathéodory structure, see [21, 22]. A
quaternionic contact (qc) structure, [3], appears naturally as the conformal
boundary at infinity of the quaternionic hyperbolic space.

Quaternionic contact manifolds were introduced in [3]. We also refer to [13]
and [19] for further results and background.

###  2.1. Quaternionic contact structures and the Biquard connection

A quaternionic contact (qc) manifold $(M,g,\mathbb{Q})$ is a
$4n+3$-dimensional manifold $M$ with a codimension three distribution $H$
equipped with an $Sp(n)Sp(1)$-structure. Explicitly, $H$ is the kernel of a
local 1-form $\eta=(\eta_{1},\eta_{2},\eta_{3})$ with values in
$\mathbb{R}^{3}$ together with a compatible Riemannian metric $g$ and a rank-
three bundle $\mathbb{Q}$ consisting of endomorphisms of $H$, locally
generated by three almost complex structures $I_{1},I_{2},I_{3}$ on $H$,
satisfying the identities of the imaginary unit quaternions. Thus, we have
$I_{1}I_{2}=-I_{2}I_{1}=I_{3},\quad I_{1}I_{2}I_{3}=-id_{|_{H}}$, which are
hermitian compatible with the metric $g(I_{s}.,I_{s}.)=g(.,.)$, and the
following compatibility conditions hold $2g(I_{s}X,Y)\ =\ d\eta_{s}(X,Y)$.

On a qc manifold of dimension $(4n+3)>7$ with a fixed metric $g$ on $H$ there
exists a canonical connection defined in [3]. Biquard also showed that there
is a unique connection $\nabla$ with torsion $T$ and a unique supplementary
subspace $V$ to $H$ in $TM$, such that:

  1. (i)

$\nabla$ preserves the splitting $H\oplus V$ and the $Sp(n)Sp(1)$-structure on
$H$, i.e., $\nabla g=0,\nabla\sigma\in\Gamma(\mathbb{Q})$ for a section
$\sigma\in\Gamma(\mathbb{Q})$, and its torsion on $H$ is given by
$T(X,Y)=-[X,Y]_{|V}$;

  2. (ii)

for $\xi\in V$, the endomorphism $T(\xi,.)_{|H}$ of $H$ lies in $(sp(n)\oplus
sp(1))^{\bot}\subset gl(4n)$;

  3. (iii)

the connection on $V$ is induced by the natural identification $\varphi$ of
$V$ with $\mathbb{Q}$, $\nabla\varphi=0$.

If the dimension of $M$ is at least eleven [3] also described the
supplementary _vertical distribution_ $V$, which is (locally) generated by the
so called _Reeb vector fields_ $\\{\xi_{1},\xi_{2},\xi_{3}\\}$ determined by

(2.1) $\displaystyle\eta_{s}(\xi_{k})=\delta_{sk},\qquad(\xi_{s}\lrcorner
d\eta_{s})_{|H}=0,\qquad(\xi_{s}\lrcorner d\eta_{k})_{|H}=-(\xi_{k}\lrcorner
d\eta_{s})_{|H},$

where $\lrcorner$ denotes the interior multiplication.

If the dimension of $M$ is seven Duchemin shows in [10] that if we assume, in
addition, the existence of Reeb vector fields as in (2.1), then the Biquard
result holds. _Henceforth, by a qc structure in dimension $7$ we shall mean a
qc structure satisfying (2.1)_. This implies the existence of the connection
with properties (i), (ii) and (iii) above.

The fundamental 2-forms $\omega_{s}$ of the quaternionic structure are defined
by

$2\omega_{s|H}\ =\ \,d\eta_{s|H},\qquad\xi\lrcorner\omega_{s}=0,\quad\xi\in
V.$

The torsion restricted to $H$ has the form
$T(X,Y)=-[X,Y]_{|V}=2\sum_{s=1}^{3}\omega_{s}(X,Y)\xi_{s}.$

###  2.2. Invariant decompositions

Any endomorphism $\Psi$ of $H$ can be decomposed with respect to the
quaternionic structure $(\mathbb{Q},g)$ uniquely into four $Sp(n)$-invariant
parts $\Psi=\Psi^{+++}+\Psi^{+--}+\Psi^{-+-}+\Psi^{--+},$ where $\Psi^{+++}$
commutes with all three $I_{i}$, $\Psi^{+--}$ commutes with $I_{1}$ and anti-
commutes with the others two, etc. The two $Sp(n)Sp(1)$-invariant components
are given by

$\Psi_{[3]}=\Psi^{+++},\quad\Psi_{[-1]}=\Psi^{+--}+\Psi^{-+-}+\Psi^{--+}$

with the following characterizing equations

$\displaystyle\Psi$ $\displaystyle=$ $\displaystyle\Psi_{[3]}\iff
3\Psi+I_{1}\Psi I_{1}+I_{2}\Psi I_{2}+I_{3}\Psi I_{3}=0,$ $\displaystyle\Psi$
$\displaystyle=$ $\displaystyle\Psi_{[-1]}\iff\Psi-I_{1}\Psi I_{1}-I_{2}\Psi
I_{2}-I_{3}\Psi I_{3}=0.$

These are the projections on the eigenspaces of the Casimir operator

$\Upsilon=\ I_{1}\otimes I_{1}\ +\ I_{2}\otimes I_{2}\ +\ I_{3}\otimes I_{3},$

corresponding, respectively, to the eigenvalues $3$ and $-1$, see [4]. Note
here that each of the three 2-forms $\omega_{s}$ belongs to the
[-1]-component, $\omega_{s}=\omega_{s[-1]}$, and constitute a basis of the Lie
algebra $sp(1)$.

If $n=1$ then the space of symmetric endomorphisms commuting with all $I_{s}$
is 1-dimensional, i.e., the [3]-component of any symmetric endomorphism $\Psi$
on $H$ is proportional to the identity, $\Psi_{[3]}=-\frac{tr\Psi}{4}Id_{|H}$.

###  2.3. The torsion tensor

The torsion endomorphism $T_{\xi}=T(\xi,\cdot):H\rightarrow H,\quad\xi\in V$
will be decomposed into its symmetric part $T_{\xi}^{0}$ and skew-symmetric
part $b_{\xi},T_{\xi}=T_{\xi}^{0}+b_{\xi}$. Biquard showed in [3] that the
torsion $T_{\xi}$ is completely trace-free, $tr\,T_{\xi}=tr\,(T_{\xi}\circ
I_{s})=0$, its symmetric part has the properties
$T_{\xi_{i}}^{0}I_{i}=-I_{i}T_{\xi_{i}}^{0},\quad
I_{2}(T_{\xi_{2}}^{0})^{+--}=I_{1}(T_{\xi_{1}}^{0})^{-+-},\quad
I_{3}(T_{\xi_{3}}^{0})^{-+-}=I_{2}(T_{\xi_{2}}^{0})^{--+},\quad
I_{1}(T_{\xi_{1}}^{0})^{--+}=I_{3}(T_{\xi_{3}}^{0})^{+--}$. The skew-symmetric
part can be represented as $b_{\xi_{i}}=I_{i}U$, where $U$ is a traceless
symmetric (1,1)-tensor on $H$ which commutes with $I_{1},I_{2},I_{3}$.
Therefore we have $T_{\xi_{i}}=T_{\xi_{i}}^{0}+I_{i}U$. When $n=1$ the tensor
$U$ vanishes identically, $U=0$, and the torsion is a symmetric tensor,
$T_{\xi}=T_{\xi}^{0}.$

The two $Sp(n)Sp(1)$-invariant trace-free symmetric 2-tensors on $H$

(2.2)
$T^{0}(X,Y)=g((T_{\xi_{1}}^{0}I_{1}+T_{\xi_{2}}^{0}I_{2}+T_{\xi_{3}}^{0}I_{3})X,Y)\
\text{ and }\ U(X,Y)=g(UX,Y)$

were introduced in [13] and enjoy the properties

(2.3) $\displaystyle
T^{0}(X,Y)+T^{0}(I_{1}X,I_{1}Y)+T^{0}(I_{2}X,I_{2}Y)+T^{0}(I_{3}X,I_{3}Y)=0,$
$\displaystyle U(X,Y)=U(I_{1}X,I_{1}Y)=U(I_{2}X,I_{2}Y)=U(I_{3}X,I_{3}Y).$

From [19, Proposition 2.3] we have

(2.4) $4T^{0}(\xi_{s},I_{s}X,Y)=T^{0}(X,Y)-T^{0}(I_{s}X,I_{s}Y),$

hence, taking into account (2.4) it follows

(2.5)
$T(\xi_{s},I_{s}X,Y)=T^{0}(\xi_{s},I_{s}X,Y)+g(I_{s}UI_{s}X,Y)=\frac{1}{4}\Big{[}T^{0}(X,Y)-T^{0}(I_{s}X,I_{s}Y)\Big{]}-U(X,Y).$

###  2.4. Torsion and curvature

Let $R=[\nabla,\nabla]-\nabla_{[\ ,\ ]}$ be the curvature tensor of $\nabla$
and the dimension is $4n+3$. We denote the curvature tensor of type (0,4) and
the torsion tensor of type (0,3) by the same letter,
$R(A,B,C,D):=g(R(A,B)C,D),\quad T(A,B,C):=g(T(A,B),C)$,
$A,B,C,D\in\Gamma(TM)$. The Ricci tensor, the normalized scalar curvature and
the Ricci $2$-forms of the Biquard connection, called _qc-Ricci tensor_ $Ric$,
_normalized qc-scalar curvature_ $S$ and _qc-Ricci forms_ $\rho_{s}$,
respectively, are defined by

(2.6) $\displaystyle Ric(A,B)=R(e_{a},A,B,e_{a}),\quad
S=\frac{1}{8n(n+2)}Ric(e_{a},e_{a}),\quad\rho_{s}(A,B)=\frac{1}{4n}R(A,B,e_{a},I_{s}e_{a}).$

######  Definition 2.1.

A qc structure is said to be qc-Einstein if the horizontal qc-Ricci tensor is
a scalar multiple of the metric,

$Ric(X,Y)=2(n+2)Sg(X,Y).$

The horizontal qc-Ricci tensor and the horizontal qc-Ricci 2-forms can be
expressed in terms of the torsion of the Biquard connection [13] (see also
[19]). We collect the necessary facts from [13, Theorem 1.3, Theorem 3.12,
Corollary 3.14, Proposition 4.3 and Proposition 4.4] with slight modification
presented in [19].

######  Theorem 2.2.

[13] On a $(4n+3)$-dimensional qc manifold $(M,\eta,\mathbb{Q})$ with a
normalized qc scalar curvature $S$ we have the following relations

(2.7) $\displaystyle Ric(X,Y)=(2n+2)T^{0}(X,Y)+(4n+10)U(X,Y)+2(n+2)Sg(X,Y),$
$\displaystyle\rho_{s}(X,I_{s}Y)=-\frac{1}{2}\Bigl{[}T^{0}(X,Y)+T^{0}(I_{s}X,I_{s}Y)\Bigr{]}-2U(X,Y)-Sg(X,Y),$
$\displaystyle T(\xi_{i},\xi_{j})=-S\xi_{k}-[\xi_{i},\xi_{j}]_{|H},\qquad
S=-g(T(\xi_{1},\xi_{2}),\xi_{3}).$

For $n=1$ the above formulas hold with $U=0$.

The qc-Einstein condition is equivalent to the vanishing of the torsion
endomorphism of the Biquard connection. In this case $S$ is constant and the
vertical distribution is integrable (see [14] for $n=1$).

Any 3-Sasakian manifold has zero torsion endomorphism, and the converse is
true if in addition the qc-scalar curvature is a positive constant [13].

The tensor $T^{0}$ determines the traceless $[-1]$-component of the horizontal
qc-Ricci tensor, while the tensor $U$ determines the traceless part of the
$[3]$-component of the horizontal qc-Ricci tensor [20] (see also [13, 19]):

(2.8) $T^{0}=\frac{1}{2n+2}Ric_{[-1]},\qquad U=\frac{1}{4n+10}Ric_{[3][0]}.$

A weaker condition than the qc-Einstein one is contained in the following

######  Definition 2.3.

[13, Definition 6.1] Let $(M,g,\mathbb{Q})$ be a quaternionic contact
manifold. We call M qc-pseudo-Einstein if the trace-free part of the
[3]-component of the qc-Ricci tensor vanishes, $U=0$. In dimension seven any
qc manifold is qc-pseudo-Einstein.

We also give the following

######  Definition 2.4.

Let $(M,g,\mathbb{Q})$ be a quaternionic contact manifold. We call M qc-
nearly-Einstein if the [-1]-component of the qc-Ricci tensor vanishes,
$T^{0}=0$.

We note that qc-nearly-Einstein manifolds are characterized by the condition
that the almost contact structure on the corresponding twistor space is
normal, see [7]. Examples of qc-nearly-Einstein manifolds are provided by sub-
Riemannian manifolds with transverse symmetry of qc type as well as by
Riemannian foliations with totally geodesic fibres of qc type (see [1, 2] and
references therein).

We use the contracted second Bianchi identity, established in [13], in the
form presented in [19, 20]:

(2.9)
$(n-1)(\nabla_{e_{a}}T^{0})(e_{a},X)+2(n+2)(\nabla_{e_{a}}U)(e_{a},X)-(n-1)(2n+1)dS(X)=0.$

Clearly, the contracted second Bianchi identity (2.9) shows that for $n>1$ any
qc-Einstein space has constant qc scalar curvature but for qc-pseudo-Einstein
and qc-nearly-Einstein spaces the qc scalar curvature could not be a constant.

###  2.5. The Ricci identities

We use repeatedly the Ricci identities of order two and three, see also [19].
Let $f$ be a smooth function on the qc manifold $M$ with horizontal gradient
$\nabla f$ defined by $g(\nabla f,X)=df(X)$. The sub-Laplacian of $f$ is, by
definition, $\triangle f=-\sum_{a=1}^{4n}\nabla^{2}f(e_{a},e_{a})$. We have
the following Ricci identities (see e.g. [13, 20])

(2.10)
$\displaystyle\nabla^{2}f(X,Y)-\nabla^{2}f(Y,X)=-2\sum_{s=1}^{3}\omega_{s}(X,Y)df(\xi_{s}),$
$\displaystyle\nabla^{2}f(X,\xi_{s})-\nabla^{2}f(\xi_{s},X)=T(\xi_{s},X,\nabla
f),$ $\displaystyle\nabla^{3}f(X,Y,Z)-\nabla^{3}f(Y,X,Z)=-R(X,Y,Z,\nabla
f)-2\sum_{s=1}^{3}\omega_{s}(X,Y)\nabla^{2}f(\xi_{s},Z).$

In view of (2.10) we have the decompositions

(2.11)
$\displaystyle(\nabla^{2}f)_{[3][0]}(X,Y)=(\nabla^{2}f)_{[3]}(X,Y)+\frac{1}{4n}\triangle
fg(X,Y),$
$\displaystyle|(\nabla^{2}f)_{[3][0]}|^{2}=|(\nabla^{2}f)_{[3]}|^{2}-\frac{1}{4n}(\Delta
f)^{2},$
$\displaystyle(\nabla^{2}f)_{[-1]}(X,Y)=(\nabla^{2}f)_{[-1][sym]}(X,Y)+(\nabla^{2}f)_{[-1][a]}(X,Y)$
$\displaystyle\qquad\qquad\qquad\quad=(\nabla^{2}f)_{[-1][sym]}(X,Y)-\sum_{s=1}^{3}\omega_{s}(X,Y)df(\xi_{s}),$
$\displaystyle|(\nabla^{2}f)_{[-1]}|^{2}=|(\nabla^{2}f)_{[-1][sym]}|^{2}+4n\sum_{s=1}^{3}df(\xi_{s})^{2}.$

We also need the qc-Bochner formula [16, (4.1)]

(2.12) $-\frac{1}{2}\triangle|\nabla
f|^{2}=|\nabla^{2}f|^{2}-g\left(\nabla(\triangle f),\nabla
f\right)+2(n+2)S|\nabla f|^{2}+2(n+2)T^{0}(\nabla f,\nabla f)\\\
+2(2n+2)U(\nabla f,\nabla f)+4\sum_{s=1}^{3}\nabla^{2}f(\xi_{s},I_{s}\nabla
f).$

Let $(M,g,\mathbb{Q})$ be a qc manifold of dimension $4n+3\geq 7$. For a fixed
local 1-form $\eta$ and a fixed $s\in\\{1,2,3\\}$ the form
$Vol_{\eta}=\eta_{1}\wedge\eta_{2}\wedge\eta_{3}\wedge\omega_{s}^{2n}$ is a
locally defined volume form. Note that $Vol_{\eta}$ is independent of $s$ as
well as the local one forms $\eta_{1},\eta_{2},\eta_{3}$. Hence, it is a
globally defined volume form. The (horizontal) divergence of a horizontal
vector field/one-form $\sigma\in\Lambda^{1}\,(H)$, defined by
$\nabla^{*}\sigma=-tr|_{H}\nabla\sigma=-\nabla\sigma(e_{a},e_{a}),$ supplies
the integration by parts formula [13], see also [23],

$\int_{M}(\nabla^{*}\sigma)\,\,Vol_{\eta}\ =\ 0.$

###  2.6. The $P-$form

We recall the definition of the P-form from [17]. Let $(M,g,\mathbb{Q})$ be a
compact quaternionic contact manifold of dimension $4n+3$.

For a smooth function $f$ on $M$ the $P-$form $P_{f}$ on $M$ is defined by
[17]

(2.13) $\displaystyle P_{f}(X)=$
$\displaystyle\nabla^{3}f(X,e_{a},e_{a})+\sum_{t=1}^{3}\nabla^{3}f(I_{t}X,e_{a},I_{t}e_{a})-4nSdf(X)+4nT^{0}(X,\nabla
f)$ $\displaystyle\hskip 202.35622pt-\frac{8n(n-2)}{n-1}U(X,\nabla
f),\quad\text{if $n>1$},$ $\displaystyle P_{f}(X)=$
$\displaystyle\nabla^{3}f(X,e_{a},e_{a})+\sum_{t=1}^{3}\nabla^{3}f(I_{t}X,e_{a},I_{t}e_{a})-4Sdf(X)+4T^{0}(X,\nabla
f),\ \text{if $n=1$}.$

We say that the $P-$function of $f$ is non-negative if the integral of
$P_{f}(\nabla f)$ exists and is non-positive:

(2.14) $-\int_{M}P_{f}(\nabla f)\,Vol_{\eta}\geq 0.$

If (2.14) holds for any smooth functon of compact support we say that the
$P$-function is non-negative. It turns out that the $P$-function is non-
negative on any compact qc manifold of dimension at least eleven [17]. Indeed,
[17, Theorem 3.3] asserts that on a compact qc manifold af dimension bigger
than seven, the next formula holds

(2.15) $\int_{M}P_{f}(\nabla
f)\,Vol_{\eta}=-\frac{4n}{n-1}\int_{M}|(\nabla^{2}f)_{[3][0]}|^{2}\,Vol_{\eta}.$

We recall also the following integral identities,

(2.16) $\int_{M}\sum_{s=1}^{3}\nabla^{2}f(\xi_{s},I_{s}\nabla
f)\,Vol_{\eta}\\\
=\int_{M}\Big{[}\frac{3}{4n}|(\nabla^{2}f)_{[3]}|^{2}-\frac{1}{4n}|(\nabla^{2}f)_{[-1]}|^{2}-\frac{n+2}{2n}T^{0}(\nabla
f,\nabla f)-\frac{3}{2}S|\nabla f|^{2}\Big{]}\,Vol_{\eta},$

proved in [16, Lemma 3.3], and

(2.17)
$\int_{M}|(\nabla^{2}f)_{[-1][a]}|^{2}\,Vol_{\eta}=4n\int_{M}\sum_{s=1}^{3}(df(\xi_{s}))^{2}\,Vol_{\eta}=\int_{M}\Big{[}\frac{1}{4n}P_{f}(\nabla
f)+\frac{1}{4n}(\triangle f)^{2}\\\ +S|\nabla f|^{2}-T^{0}(\nabla f,\nabla
f)+\frac{2n-4}{n-1}U(\nabla f,\nabla f)\Big{]}\,Vol_{\eta},$

established in [18, formula (4.11)].

######  Proposition 2.5.

For any smooth function $f$ on a compact qc manifold $(M,g,\mathbb{Q})$ of
dimension 4n+3 bigger than seven we have

(2.18) $0=\int_{M}\Big{[}\frac{2n+1}{2n}(\triangle
f)^{2}-|(\nabla^{2}f)_{[-1][sym]}|^{2}-\frac{n+2}{n-1}|(\nabla^{2}f)_{[3][0]}|^{2}-(2n+1)Lic(\nabla
f,\nabla f)\Big{]}\,Vol_{\eta},$

where

(2.19) $Lic(\nabla f,\nabla f)=S|\nabla f|^{2}+\frac{2n+3}{2n+1}T^{0}(\nabla
f,\nabla f)+\frac{2(n+2)(2n-1)}{(2n+1)(n-1)}U(\nabla f,\nabla f),$

and also

(2.20) $0=\int_{M}\Big{[}\frac{4n+3}{4n}(\triangle
f)^{2}-|(\nabla^{2}f)_{[-1]}|^{2}-\frac{n+3}{n-1}|(\nabla^{2}f)_{[3][0]}|^{2}-2nCor(\nabla
f,\nabla f)\Big{]}\,Vol_{\eta},$

where

(2.21) $Cor(\nabla f,\nabla f)=S|\nabla f|^{2}+\frac{n+2}{n}T^{0}(\nabla
f,\nabla f)+\frac{2(n+1)}{n-1}U(\nabla f,\nabla f).$

###### Proof.

Integrating the qc Bochner formula (2.12) over the compact $M$ and applying
(2.7) and (2.16), we get

(2.22) $0=\int_{M}\Big{[}|\nabla^{2}f|^{2}-(\triangle f)^{2}+Ric(\nabla
f,\nabla f)+2T^{0}(\nabla f,\nabla f)\\\ -6U(\nabla f,\nabla
f)+4\sum_{s=1}^{3}\nabla^{2}f(\xi_{s},I_{s}\nabla f)\Big{]}\,Vol_{\eta}\\\
=\int_{M}\Big{[}|\nabla^{2}f|^{2}-(\triangle f)^{2}+Ric(\nabla f,\nabla
f)+2T^{0}(\nabla f,\nabla f)-6U(\nabla f,\nabla f)\Big{]}\,Vol_{\eta}\\\
+\int_{M}\Big{[}\frac{3}{n}|(\nabla^{2}f)_{[3]}|^{2}-\frac{1}{n}|(\nabla^{2}f)_{[-1]}|^{2}-\frac{2(n+2)}{n}T^{0}(\nabla
f,\nabla f)-6S|\nabla f|^{2}\Big{]}\,Vol_{\eta}\\\
=\int_{M}\Big{[}|(\nabla^{2}f)_{[3][0]}|^{2}+\frac{1}{4n}(\triangle
f)^{2}+|(\nabla^{2}f)_{[-1]}|^{2}-(\triangle
f)^{2}+\frac{3}{n}|(\nabla^{2}f)_{[3][0]}|^{2}+\frac{3}{4n^{2}}(\triangle
f)^{2}-\frac{1}{n}|(\nabla^{2}f)_{[-1]}|^{2}\Big{]}\,Vol_{\eta}\\\
+\int_{M}\Big{[}Ric(\nabla f,\nabla f)-\frac{4}{n}T^{0}(\nabla f,\nabla
f)-6U(\nabla f,\nabla f)\ -6S|\nabla f|^{2}\Big{]}\,Vol_{\eta}\\\
=\int_{M}\Big{[}\frac{n+3}{n}|(\nabla^{2}f)_{[3][0]}|^{2}-\frac{(n-1)(4n+3)}{4n^{2}}(\triangle
f)^{2}+\frac{n-1}{n}|(\nabla^{2}f)_{[-1]}|^{2}+2(n-1)Cor(\nabla f,\nabla
f)\Big{]}\,\,Vol_{\eta},$

which proves (2.20).

The substitution of the fourth equality of (2.11) and (2.17) into (2.20) leads
to

$0=\int_{M}\Big{[}\frac{4n+3}{4n}(\triangle
f)^{2}-|(\nabla^{2}f)_{[-1]}|^{2}-\frac{n+3}{n-1}|(\nabla^{2}f)_{[3][0]}|^{2}-2nCor(\nabla
f,\nabla f)\Big{]}\,Vol_{\eta}\\\ =\int_{M}\Big{\\{}\frac{4n+3}{4n}(\triangle
f)^{2}-\frac{n+3}{n-1}|(\nabla^{2}f)_{[3][0]}|^{2}-|(\nabla^{2}f)_{[-1][sym]}|^{2}-\frac{1}{4n}\Big{[}P_{f}(\nabla
f)+(\triangle f)^{2}\Big{]}\\\ -\Big{[}S|\nabla f|^{2}-T^{0}(\nabla f,\nabla
f)+\frac{2n-4}{n-1}U(\nabla f,\nabla f)\ \Big{]}\\\
-2n\Big{[}\frac{n+2}{n}T^{0}(\nabla f,\nabla f)+\frac{2(n+1)}{n-1}U(\nabla
f,\nabla f)+S|\nabla f|^{2}\Big{]}\Big{\\}}\,Vol_{\eta},$

which proves (2.18) in view of (2.15). ∎

######  Remark 2.6.

Note that the expression in the formula (2.19) appears in the qc Lichnerowicz-
type positivity condition used to find a lower bound of the first eigenvalue
of the sub-Laplacian (see [16, formula (1.1)] and [17, formula (1.2)]). On the
other hand, the expression in (2.21) appears in the qc Cordes-type a priori
inequality between the (horizontal) Hessian and the sub-Laplacian of a
function, derived in [16, formula (1.2)].

Indeed, in view of (2.11), the equality (2.20) can be written in the form

(2.23) $\frac{n+1}{n}\int_{M}(\triangle f)^{2}\,Vol_{\eta}=\int_{M}|\nabla
f|^{2}\,Vol_{\eta}+\int_{M}\Big{[}\frac{4}{n-1}|(\nabla^{2}f)_{[3][0]}|^{2}+2nCor(\nabla
f,\nabla f)\Big{]}\,Vol_{\eta}.$

In this way we recovered the next result from [16], supplying more information
in the equality case:

######  Theorem 2.7.

[16] On a compact qc manifold of dimension bigger than seven one has the
inequality

(2.24) $\int_{M}(\triangle f)^{2}\,Vol_{\eta}\geq\frac{n}{n+1}\int_{M}|\nabla
f|^{2}\,Vol_{\eta}+\frac{2n^{2}}{n+1}\int_{M}Cor(\nabla f,\nabla
f)\,Vol_{\eta}.$

The equality in (2.24) can be achieved only for functions with vanishing
trace-free part of the $[3]$-component of the horizontal Hessian.

##  3\. The inequalities, main results

Denote by $\bar{S}$ the average value of the scalar curvature,

$\bar{S}=\int_{M}S\,Vol_{\eta}.$

For a smooth function $f$ the sub-Laplacian $\triangle f$ is a subelliptic
operator. According to [11] and [8], let $\varphi$ be the unique solution of
the following PDE:

(3.1) $\triangle\varphi=S-\bar{S},\quad\int_{M}\varphi\,Vol_{\eta}=0.$

We have

######  Theorem 3.1.

Let $(M,g,\mathbb{Q})$ be a compact qc manifold of dimension $(4n+3),\quad
n>1$.

  * a)

Suppose the next positivity condition is satisfied

(3.2)
$Lic(X,X)=Sg(X,X)+\frac{2n+3}{2n+1}T^{0}(X,X)+\frac{2(n+2)(2n-1)}{(n-1)(2n+1)}U(X,X)\geq
0,\quad X\in H.$

Then we have

(3.3)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}\leq\frac{n+2}{2n(n-1)(2n+5)^{2}(2n+1)}\int_{M}|Ric_{[3][0]}|^{2}\,Vol_{\eta}\\\
-\frac{2}{2n+1}\int_{M}T^{0}(e_{a},e_{b})(\nabla^{2}\varphi)_{[-1][sym]}(e_{a},e_{b})\,Vol_{\eta}\\\
=\frac{2(n+2)}{n(n-1)(2n+1)}\int_{M}|U|^{2}\,Vol_{\eta}-\frac{2}{2n+1}\int_{M}T^{0}(e_{a},e_{b})(\nabla^{2}\varphi)_{[-1][sym]}(e_{a},e_{b})\,Vol_{\eta}.$

If the equality in (3.3) holds then

(3.4)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}=\frac{n+2}{2n(n-1)(2n+5)^{2}(2n+1)}\int_{M}|Ric_{[3][0]}|^{2}\,Vol_{\eta}\\\
=\frac{2(n+2)}{n(n-1)(2n+1)}\int_{M}|U|^{2}\,Vol_{\eta}$

and the qc conformal structure $\bar{\eta}=\frac{n}{2}e^{-2\varphi}\eta$ will
be qc-pseudo-Einstein.

  * b)

Suppose the next positivity condition is satisfied

(3.5) $Cor(X,X)=Sg(X,X)+\frac{n+2}{n}T^{0}(X,X)+\frac{2(n+1)}{n-1}U(X,X)\geq
0,\quad X\in H.$

Then we have

(3.6)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}\leq\frac{(n+2)^{2}(4n+3)}{4n(n-1)(n+3)(2n+5)^{2}(2n+1)^{2}}\int_{M}|Ric_{[3][0]}|^{2}\,Vol_{\eta}\\\
-\frac{2}{2n+1}\int_{M}T^{0}(e_{a},e_{b})(\nabla^{2}\varphi)_{[-1][sym]}(e_{a},e_{b})\,Vol_{\eta}\\\
=\frac{(n+2)^{2}(4n+3)}{n(n-1)(n+3)(2n+1)^{2}}\int_{M}|U|^{2}\,Vol_{\eta}-\frac{2}{2n+1}\int_{M}T^{0}(e_{a},e_{b})(\nabla^{2}\varphi)_{[-1][sym]}(e_{a},e_{b})\,Vol_{\eta}.$

If the equality in (3.6) holds then

(3.7)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}=\frac{(n+2)^{2}(4n+3)}{4n(n-1)(n+3)(2n+5)^{2}(2n+1)^{2}}\int_{M}|Ric_{[3][0]}|^{2}\,Vol_{\eta}\\\
=\frac{(n+2)^{2}(4n+3)}{n(n-1)(n+3)(2n+1)^{2}}\int_{M}|U|^{2}\,Vol_{\eta}$

and the qc conformal structure
$\bar{\eta}=\frac{n(n+3)(2n+1)}{(n+2)(4n+3)}e^{-2\varphi}\eta$ will be qc-
pseudo-Einstein.

###### Proof.

The proof follows the approach of [9]. We have

(3.8)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}=\int_{M}(S-\bar{S})\triangle\varphi\,Vol_{\eta}=\int_{M}dS(\nabla\varphi)\,Vol_{\eta},$

where we used (3.1) and an integration by parts to achieve the last equality.

The equalities (2.8) and (2.9) imply

(3.9)
$dS(X)=\frac{1}{2n+1}(\nabla_{e_{a}}T^{0})(e_{a},X)+\frac{2n+4}{(n-1)(2n+1)}(\nabla_{e_{a}}U)(e_{a},X)\\\
=\frac{n+2}{(n-1)(2n+5)(2n+1)}(\nabla_{e_{a}}Ric_{[3][0]})(e_{a},X)+\frac{1}{2n+1}(\nabla_{e_{a}}T^{0})(e_{a},X)\\\
=\frac{1}{2(n+1)(2n+1)}(\nabla_{e_{a}}Ric_{[-1]})(e_{a},X)+\frac{2n+4}{(n-1)(2n+1)}(\nabla_{e_{a}}U)(e_{a},X).$

Using (3.9), we obtain from (3.8) and an integration by parts that

(3.10)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}=\int_{M}dS(\nabla\varphi)\,Vol_{\eta}\\\
=\frac{n+2}{(n-1)(2n+5)(2n+1)}\int_{M}(\nabla_{e_{a}}Ric_{[3][0]})(e_{a},\nabla\varphi)\,Vol_{\eta}+\frac{1}{2n+1}\int_{M}(\nabla_{e_{a}}T^{0})(e_{a},\nabla\varphi)\,Vol_{\eta}\\\
=-\frac{n+2}{(n-1)(2n+5)(2n+1)}\int_{M}(Ric_{[3][0]})(e_{a},e_{b})(\nabla^{2}\varphi)_{[3][0]}(e_{a},e_{b})\,Vol_{\eta}\\\
+\frac{1}{2n+1}\int_{M}(\nabla_{e_{a}}T^{0})(e_{a},\nabla\varphi)\,Vol_{\eta}.$

Applying the Young’s inequality

(3.11) $2pq\leq dp^{2}+d^{-1}q^{2},$

for a constant $d>0$, we get from (3.10) that

(3.12)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}\leq\frac{n+2}{2(n-1)(2n+5)(2n+1)}\int_{M}\Big{[}d|Ric_{[3][0]}|^{2}+d^{-1}|(\nabla^{2}\varphi)_{[3][0]}|^{2}\Big{]}\,Vol_{\eta}\\\
+\frac{1}{2n+1}\int_{M}(\nabla_{e_{a}}T^{0})(e_{a},\nabla\varphi)\,Vol_{\eta}.$

First we prove a). The equalities (2.18), (3.1) and the positivity condition
(3.2) help us to obtain from (3.12) the next inequality

(3.13)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}\leq\frac{d(n+2)}{2(n-1)(2n+5)(2n+1)}\int_{M}|Ric_{[3][0]}|^{2}\,Vol_{\eta}\\\
+\frac{1}{4nd(2n+5)}\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}+\frac{1}{2n+1}\int_{M}(\nabla_{e_{a}}T^{0})(e_{a},\nabla\varphi)\,Vol_{\eta}.$

Thus, (3.13) yields

(3.14)
$\Big{[}1-\frac{1}{4nd(2n+5)}\Big{]}\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}\leq\frac{d(n+2)}{2(n-1)(2n+5)(2n+1)}\int_{M}|Ric_{[3][0]}|^{2}\,Vol_{\eta}\\\
+\frac{1}{2n+1}\int_{M}(\nabla_{e_{a}}T^{0})(e_{a},\nabla\varphi)\,Vol_{\eta}.$

Set

(3.15) $d=\frac{1}{4n^{2}+10n}$

into (3.14) to get (3.3), which proves the first part of a).

If we have an equality in (3.3), we put the expression of
$|(\nabla^{2}f)_{[3][0]}|^{2}$ from (2.18) into (3.12), taken with the
constant $d$ given by (3.15), to get

$0\leq-\int_{M}\Big{[}|(\nabla^{2}\varphi)_{[-1][sym]}|^{2}+(2n+1)Lic(\nabla\varphi,\nabla\varphi)\Big{]}\,Vol_{\eta},$

which, in view of (3.2), yields

(3.16) $\displaystyle(\nabla^{2}\varphi)_{[-1][sym]}=0;$ $\displaystyle
Lic(\nabla\varphi,\nabla\varphi)=S|\nabla\varphi|^{2}+\frac{2n+3}{2n+1}T^{0}(\nabla\varphi,\nabla\varphi)+\frac{2(n+2)(2n-1)}{(n-1)(2n+1)}U(\nabla\varphi,\nabla\varphi)=0.$

The first equality in (3.16) and the equality in (3.3) give (3.4). Now, the
equalities (3.4), (2.18) and (3.16), together with (3.8), imply

(3.17)
$\displaystyle\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}=\frac{n+2}{2n(n-1)(2n+5)^{2}(2n+1)}\int_{M}|Ric_{[3][0]}|^{2}\,Vol_{\eta};$
$\displaystyle\int_{M}|(\nabla^{2}\varphi)_{[3][0]}|^{2}\,Vol_{\eta}=\frac{(2n+1)(n-1)}{2n(n+2)}\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}=\frac{1}{4n^{2}(2n+5)^{2}}\int_{M}|Ric_{[3][0]}|^{2}\,Vol_{\eta}.$

On the other hand, (3.10) and (3.17) yield

(3.18)
$\frac{n+2}{2n(n-1)(2n+5)^{2}(2n+1)}\int_{M}|Ric_{[3][0]}|^{2}\,Vol_{\eta}\\\
=-\frac{n+2}{(n-1)(2n+5)(2n+1)}\int_{M}(Ric_{[3][0]})(e_{a},e_{b})(\nabla^{2}\varphi)_{[3][0]}(e_{a},e_{b})\,Vol_{\eta}.$

The equalities (3.17) and (3.18) imply

(3.19)
$(\nabla^{2}\varphi)_{[3][0]}=-\frac{1}{2n(2n+5)}Ric_{[3][0]}=-\frac{1}{n}U.$

Taking the qc conformal change
$\bar{\eta}=\frac{1}{4d(2n+5)}e^{-2\varphi}\eta=\frac{n}{2}e^{-2\varphi}\eta$,
we get $\bar{U}=0$, according to [13, formula (5.6)], taken with
$2h=4d(2n+5)e^{2\varphi}=\frac{2}{n}e^{2\varphi}$ (cf. also [20, formula
(5.23)]), and (3.19). Hence, $\overline{Ric}_{[3][0]}=0$ and the qc conformal
qc structure $\bar{\eta}$ is qc-pseudo-Einstein, which completes the proof of
a).

Now we prove b). The equalities (2.20), (3.1) and the positivity condition
(3.5) help us to obtain from (3.12) the following inequality

(3.20)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}\leq\frac{d(n+2)}{2(n-1)(2n+5)(2n+1)}\int_{M}|Ric_{[3][0]}|^{2}\,Vol_{\eta}\\\
+\frac{(n+2)(4n+3)}{8nd(2n+5)(2n+1)(n+3)}\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}+\frac{1}{2n+1}\int_{M}(\nabla_{e_{a}}T^{0})(e_{a},\nabla\varphi)\,Vol_{\eta}.$

Thus, (3.20) yields

(3.21)
$\Big{[}1-\frac{(n+2)(4n+3)}{8nd(2n+5)(2n+1)(n+3)}\Big{]}\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}\\\
\leq\frac{d(n+2)}{2(n-1)(2n+5)(2n+1)}\int_{M}|Ric_{[3][0]}|^{2}\,Vol_{\eta}+\frac{1}{2n+1}\int_{M}(\nabla_{e_{a}}T^{0})(e_{a},\nabla\varphi)\,Vol_{\eta}.$

Set

(3.22) $d=\frac{(n+2)(4n+3)}{4n(2n+5)(2n+1)(n+3)}$

into (3.21) to get (3.6), which proves the first part of b).

If we have an equality in (3.6), we put the expression of
$|(\nabla^{2}f)_{[3][0]}|^{2}$ from (2.20) into (3.12), taken with the
constant $d$ given by (3.22), to get

$0\leq-\int_{M}\Big{[}|(\nabla^{2}\varphi)_{[-1]}|^{2}+2nCor(\nabla\varphi,\nabla\varphi)\Big{]}\,Vol_{\eta},$

which, in view of (3.5), yields

(3.23) $\displaystyle(\nabla^{2}\varphi)_{[-1]}=0;$ $\displaystyle
Cor(\nabla\varphi,\nabla\varphi)=Sg(\nabla\varphi,\nabla\varphi)+\frac{n+2}{n}T^{0}(\nabla\varphi,\nabla\varphi)+\frac{2(n+1)}{n-1}U(\nabla\varphi,\nabla\varphi)=0.$

The first equality in (3.23) and the equality in (3.6) imply (3.7). Now, the
equalities (3.7), (2.20) and (3.23) together with (3.8) imply

(3.24)
$\displaystyle\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}=\frac{(n+2)^{2}(4n+3)}{4n(n-1)(n+3)(2n+5)^{2}(2n+1)^{2}}\int_{M}|Ric_{[3][0]}|^{2}\,Vol_{\eta};$
$\displaystyle\int_{M}|\nabla^{2}\varphi_{[3][0]}|^{2}\,Vol_{\eta}=\frac{(4n+3)(n-1)}{4n(n+3)}\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}$
$\displaystyle\qquad\qquad\qquad\qquad\quad=\frac{(n+2)^{2}(4n+3)^{2}}{16n^{2}(n+3)^{2}(2n+5)^{2}(2n+1)^{2}}\int_{M}|Ric_{[3][0]}|^{2}\,Vol_{\eta}.$

On the other hand, (3.10) and (3.24) yield

(3.25)
$\frac{(n+2)^{2}(4n+3)}{4n(n-1)(n+3)(2n+5)^{2}(2n+1)^{2}}\int_{M}|Ric_{[3][0]}|^{2}\,Vol_{\eta}\\\
=-\frac{n+2}{(n-1)(2n+5)(2n+1)}\int_{M}(Ric_{[3][0]})(e_{a},e_{b})(\nabla^{2}\varphi)_{[3][0]}(e_{a},e_{b})\,Vol_{\eta}.$

The equalities (3.24) and (3.25) imply

(3.26)
$(\nabla^{2}\varphi)_{[3][0]}=-\frac{(n+2)(4n+3)}{4n(n+3)(2n+5)(2n+1)}Ric_{[3][0]}=-\frac{(n+2)(4n+3)}{2n(n+3)(2n+1)}U.$

Take the qc conformal change
$\bar{\eta}=\frac{1}{4d(2n+5)}e^{-2\varphi}\eta=\frac{n(n+3)(2n+1)}{(n+2)(4n+3)}e^{-2\varphi}\eta$
gives $\bar{U}=0$, according to [13, formula (5.6)], taken with
$2h=4d(2n+5)e^{2\varphi}=\frac{(n+2)(4n+3)}{n(n+3)(2n+1)}e^{2\varphi}$ (cf.
also [20, formula (5.23)]), and (3.26). Hence, $\overline{Ric}_{[3][0]}=0$ and
the qc conformal qc structure $\bar{\eta}$ is qc-pseudo-Einstein, which
completes the proof of b). ∎

An integration by parts of the last term in (3.3) yields the following

######  Corollary 3.2.

In addition to the same conditions as in Theorem 3.1, we assume

(3.27) $(\nabla_{e_{a}}\nabla_{e_{b}}T^{0})(e_{a},e_{b})=0.$

  * a)

If $Lic(X,X)\geq 0$ then

$\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}\leq\frac{n+2}{2n(n-1)(2n+5)^{2}(2n+1)}\int_{M}|Ric_{[3][0]}|^{2}\,Vol_{\eta}=\frac{2(n+2)}{n(n-1)(2n+1)}\int_{M}|U|^{2}\,Vol_{\eta}.$

  * b)

If $Cor(X,X)\geq 0$ then

$\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}\leq\frac{(n+2)^{2}(4n+3)}{4n(n-1)(n+3)(2n+5)^{2}(2n+1)^{2}}\int_{M}|Ric_{[3][0]}|^{2}\,Vol_{\eta}\\\
=\frac{(n+2)^{2}(4n+3)}{n(n-1)(n+3)(2n+1)^{2}}\int_{M}|U|^{2}\,Vol_{\eta}.$

In both cases, if M is qc-pseudo-Einstein then the qc scalar curvature $S$ is
a constant.

Further, in a similar way as the proof of Theorem 3.1, we obtain

######  Theorem 3.3.

Let $(M,g,\mathbb{Q})$ be a compact qc manifold of dimension $(4n+3),\quad
n>1$.

  * a)

Suppose the positivity condition (3.2) holds. Then we have

(3.28)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}\leq\frac{1}{8n(n+1)^{2}(2n+1)}\int_{M}|Ric_{[-1]}|^{2}\,Vol_{\eta}\\\
+\frac{4(n+2)}{(n-1)(2n+1)}\int_{M}(\nabla_{e_{a}}U)(e_{a},\nabla\varphi)\,Vol_{\eta}\\\
=\frac{1}{2n(2n+1)}\int_{M}|T^{0}|^{2}\,Vol_{\eta}-\frac{4(n+2)}{(n-1)(2n+1)}\int_{M}U(e_{a},e_{b})(\nabla^{2}\varphi)_{[3][0]}(e_{a},e_{b})\,Vol_{\eta}.$

If the equality holds then

(3.29)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}=\frac{1}{8n(n+1)^{2}(2n+1)}\int_{M}|Ric_{[-1]}|^{2}\,Vol_{\eta}=\frac{1}{2n(2n+1)}\int_{M}|T^{0}|^{2}\,Vol_{\eta}.$

  * b)

Suppose the positivity condition (3.5) holds. Then we have

(3.30)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}\leq\frac{4n+3}{16n(n+1)^{2}(2n+1)^{2}}\int_{M}|Ric_{[-1]}|^{2}\,Vol_{\eta}\\\
+\frac{4(n+2)}{(n-1)(2n+1)}\int_{M}(\nabla_{e_{a}}U)(e_{a},\nabla\varphi)\,Vol_{\eta}\\\
=\frac{4n+3}{4n(2n+1)^{2}}\int_{M}|T^{0}|^{2}\,Vol_{\eta}-\frac{4(n+2)}{(n-1)(2n+1)}\int_{M}U(e_{a},e_{b})(\nabla^{2}\varphi)_{[3][0]}(e_{a},e_{b})\,Vol_{\eta}.$

If the equality holds then

(3.31)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}=\frac{4n+3}{16n(n+1)^{2}(2n+1)^{2}}\int_{M}|Ric_{[-1]}|^{2}\,Vol_{\eta}=\frac{4n+3}{4n(2n+1)^{2}}\int_{M}|T^{0}|^{2}\,Vol_{\eta}.$

###### Proof.

Using (3.9), we obtain from (3.8) and an integration by parts that

(3.32)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}=\int_{M}dS(\nabla\varphi)\,Vol_{\eta}\\\
=\frac{1}{2(n+1)(2n+1)}\int_{M}\Big{[}(\nabla_{e_{a}}Ric_{[-1]})(e_{a},\nabla\varphi)+\frac{4(n+1)(n+2)}{n-1}(\nabla_{e_{a}}U)(e_{a},\nabla\varphi)\Big{]}\,Vol_{\eta}\\\
=-\frac{1}{2(n+1)(2n+1)}\int_{M}(Ric_{[-1]})(e_{a},e_{b})(\nabla^{2}\varphi)_{[-1][sym]}(e_{a},e_{b})\,Vol_{\eta}\\\
+\frac{2(n+2)}{(n-1)(2n+1)}\int_{M}(\nabla_{e_{a}}U)(e_{a},\nabla\varphi)\,Vol_{\eta}.$

First we prove a). Applying the Young’s inequality (3.11) for a constant
$d>0$, together with (2.18), we get from (3.32)

(3.33)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}\leq\frac{1}{4(n+1)(2n+1)}\Big{[}d\int_{M}|Ric_{[-1]}|^{2}\,Vol_{\eta}+d^{-1}\int_{M}|(\nabla^{2}\varphi)_{[-1][sym]}|^{2}\,Vol_{\eta}\Big{]}\\\
+\frac{2(n+2)}{(n-1)(2n+1)}\int_{M}(\nabla_{e_{a}}U)(e_{a},\nabla\varphi)\,Vol_{\eta}\\\
=\frac{d}{4(n+1)(2n+1)}\int_{M}|Ric_{[-1]}|^{2}\,Vol_{\eta}+\frac{2(n+2)}{(n-1)(2n+1)}\int_{M}(\nabla_{e_{a}}U)(e_{a},\nabla\varphi)\,Vol_{\eta}\\\
+\frac{1}{2d}\int_{M}\Big{[}\frac{1}{4n(n+1)}(\triangle\varphi)^{2}-\frac{n+2}{2(n^{2}-1)(2n+1)}|(\nabla^{2}\varphi)_{[3][0]}|^{2}-\frac{1}{2(n+1)}Lic(\nabla\varphi,\nabla\varphi)\Big{]}\,Vol_{\eta}.$

The equalities (3.1), the positivity condition (3.2) and (3.33) yield

(3.34)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}\leq\frac{d}{4(n+1)(2n+1)}\int_{M}|Ric_{[-1]}|^{2}\,Vol_{\eta}\\\
+\frac{1}{8nd(n+1)}\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}+\frac{2(n+2)}{(n-1)(2n+1)}\int_{M}(\nabla_{e_{a}}U)(e_{a},\nabla\varphi)\,Vol_{\eta}.$

Thus, (3.34) yields

(3.35)
$\Big{[}1-\frac{1}{8nd(n+1)}\Big{]}\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}\leq\frac{d}{4(n+1)(2n+1)}\int_{M}|Ric_{[-1]}|^{2}\,Vol_{\eta}\\\
+\frac{2(n+2)}{(n-1)(2n+1)}\int_{M}(\nabla_{e_{a}}U)(e_{a},\nabla\varphi)\,Vol_{\eta}.$

Set $d=\frac{1}{4n(n+1)}$ into (3.35) to get (3.28).

If we have an equality in (3.28), we use (2.18) to get from (3.33) that

$\begin{split}0\leq-\int_{M}\Big{[}\frac{n+2}{n-1}|(\nabla^{2}\varphi)_{[3][0]}|^{2}+(2n+1)Lic(\nabla\varphi,\nabla\varphi)\Big{]}\,Vol_{\eta},\end{split}$

which, in view of (3.2), yields

(3.36) $\begin{split}(\nabla^{2}\varphi)_{[3][0]}=0;\quad
S|\nabla\varphi|^{2}+\frac{2n+3}{2n+1}T^{0}(\nabla\varphi,\nabla\varphi)+\frac{2(n+2)(2n-1)}{(n-1)(2n+1)}U(\nabla\varphi,\nabla\varphi)=0.\end{split}$

Now, the equalities (3.28) and (3.36) imply (3.29), which proves a).

For b), using the Young’s inequality (3.11) for a constant $d>0$ together with
(2.20), we get from (3.32)

(3.37)
$\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}\leq\frac{1}{4(n+1)(2n+1)}\Big{[}d\int_{M}|Ric_{[-1]}|^{2}\,Vol_{\eta}+d^{-1}\int_{M}|(\nabla^{2}\varphi)_{[-1]}|^{2}\,Vol_{\eta}\Big{]}\\\
+\frac{2(n+2)}{(n-1)(2n+1)}\int_{M}(\nabla_{e_{a}}U)(e_{a},\nabla\varphi)\,Vol_{\eta}\\\
=\frac{d}{4(n+1)(2n+1)}\int_{M}|Ric_{[-1]}|^{2}\,Vol_{\eta}+\frac{2(n+2)}{(n-1)(2n+1)}\int_{M}(\nabla_{e_{a}}U)(e_{a},\nabla\varphi)\,Vol_{\eta}\\\
+\frac{1}{4d(n+1)(2n+1)}\int_{M}\Big{[}\frac{4n+3}{4n}(\triangle\varphi)^{2}-\frac{n+3}{n-1}|(\nabla^{2}\varphi)_{[3][0]}|^{2}-2nCor(\nabla\varphi,\nabla\varphi)\Big{]}\,Vol_{\eta}.$

The equality (3.1), the positivity condition (3.5) and (3.37) imply

(3.38)
$\Big{[}1-\frac{4n+3}{16nd(n+1)(2n+1)}\Big{]}\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}\leq\frac{d}{4(n+1)(2n+1)}\int_{M}|Ric_{[-1]}|^{2}\,Vol_{\eta}\\\
+\frac{2(n+2)}{(n-1)(2n+1)}\int_{M}(\nabla_{e_{a}}U)(e_{a},\nabla\varphi)\,Vol_{\eta}.$

Set $d=\frac{4n+3}{8n(n+1)(2n+1)}$ into (3.38) to get (3.30).

If we have an equality in (3.30), we use (2.20) to get from (3.37) that

$\begin{split}0\leq-\int_{M}\Big{[}\frac{n+3}{n-1}|(\nabla^{2}\varphi)_{[3][0]}|^{2}+2nCor(\nabla\varphi,\nabla\varphi)\Big{]}\,Vol_{\eta},\end{split}$

which, in view of (3.5), yields

(3.39) $\begin{split}(\nabla^{2}\varphi)_{[3][0]}=0;\quad
Cor(\nabla\varphi,\nabla\varphi)=Sg(\nabla\varphi,\nabla\varphi)+\frac{n+2}{n}T^{0}(\nabla\varphi,\nabla\varphi)+\frac{2(n+1)}{n-1}U(\nabla\varphi,\nabla\varphi)=0.\end{split}$

Now, the equalities (3.30) and (3.39) imply (3.31), which completes the proof.
∎

An integration by parts of the last term in (3.28) and in (3.30),
respectively, yields the following

######  Corollary 3.4.

In addition to the same conditions as in Theorem 3.3, we assume

$(\nabla_{e_{a}}\nabla_{e_{b}}U)(e_{a},e_{b})=0.$

  * a)

If $Lic(X,X)\geq 0$ then

$\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}\leq\frac{1}{8n(n+1)^{2}(2n+1)}\int_{M}|Ric_{[-1]}|^{2}\,Vol_{\eta}=\frac{1}{2n(2n+1)}\int_{M}|T^{0}|^{2}\,Vol_{\eta}.$

  * b)

If $Cor(X,X)\geq 0$ then

$\int_{M}(S-\bar{S})^{2}\,Vol_{\eta}\leq\frac{4n+3}{16n(n+1)^{2}(2n+1)^{2}}\int_{M}|Ric_{[-1]}|^{2}\,Vol_{\eta}=\frac{4n+3}{4n(2n+1)^{2}}\int_{M}|T^{0}|^{2}\,Vol_{\eta}.$

In both cases, if M is qc-nearly-Einstein then the qc scalar curvature $S$ is
a constant.

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