$\displaystyle|\langle ab,c\rangle_{\tilde{H}^{d}}|$
$\displaystyle\lesssim\|(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{d}{2}}(ab)\|_{L^{\frac{2Q}{Q+2}}}\|(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{d}{2}}c\|_{L^{\frac{2Q}{Q-2}}}\lesssim\sum_{\gamma\in[\\![1,2d]\\!]^{d}}\|\tilde{P}^{\gamma}(ab)\|_{L^{\frac{2Q}{Q+2}}}\|\nabla_{\mathbb{H}}c\|_{\tilde{H}^{d}}.$

Let $\gamma\in[\\![1,2d]\\!]^{d}$. Thanks to the Leibniz formula and the
Hölder estimates, we get

$\|\tilde{P}^{\gamma}(ab)\|_{L^{\frac{2Q}{Q+2}}}\lesssim\sum_{\ell=0}^{d}\sum_{\begin{subarray}{c}\alpha\in[\\![1,2d]\\!]^{\ell}\\\
\beta\in[\\![1,2d]\\!]^{d-\ell}\end{subarray}}\|\tilde{P}^{\alpha}a\|_{L^{\frac{2Q}{Q-2(d-\ell)}}}\|\tilde{P}^{\beta}b\|_{L^{\frac{2Q}{Q-2\ell}}}.$

Then, for all $\ell\in[\\![0,d]\\!]$, applying the Sobolev embedding
$\tilde{H}^{d-\ell}(\mathbb{H}^{d})\hookrightarrow
L^{\frac{2Q}{Q-2(d-\ell)}}(\mathbb{H}^{d})$ and
$\tilde{H}^{\ell}(\mathbb{H}^{d})\hookrightarrow
L^{\frac{2Q}{Q-2\ell}}(\mathbb{H}^{d})$ (recall Proposition 2.4.5 Item 3 and
Remark 2.4.6), it follows that for any $\alpha\in[\\![1,2d]\\!]^{\ell}$ and
$\beta\in[\\![1,2d]\\!]^{d-\ell}$,

$\|\tilde{P}^{\alpha}a\|_{L^{\frac{2Q}{Q-2(d-\ell)}}}\lesssim\|\tilde{P}^{\alpha}a\|_{\tilde{H}^{d-\ell}}\lesssim\|a\|_{\tilde{H}^{d}}\
\ \ \text{and}\ \ \
\|\tilde{P}^{\beta}b\|_{L^{\frac{2Q}{Q-2\ell}}}\lesssim\|\tilde{P}^{\beta}b\|_{\tilde{H}^{\ell}}\lesssim\|b\|_{\tilde{H}^{d}}.$

This concludes the proof of Lemma 6.1.3. ∎

####  6.1.2 Convergence

In this subsection we complete the proof of Theorem 6.1.1 by showing that the
sequence $(u_{k})$ converges in
$\mathcal{D}^{\prime}((0,+\infty)\times\mathbb{H}^{d})$ to a solution of
(1.5)-(1.6) satisfying the suitable energy estimates.

The first step is to get the convergence of the linear part. According to the
Lemma 6.1.2, using the weak compactness of spaces $L^{\infty}(\tilde{H}^{d})$
and $L^{2}(\tilde{H}^{d})$ and identifying the limits in
$\mathcal{D}^{\prime}(\mathbb{R}_{+}\times\mathbb{H}^{d})$, we deduce that, up
to extract a subsequence, $(u_{k})$ converges weakly-$\star$ to a function $u$
and $(\nabla_{\mathbb{H}}u_{k})$ converges weakly to $\nabla_{\mathbb{H}}u$ in
$L^{2}(\tilde{H}^{d})$. Moreover, thanks to the properties of the weak and the
weak-$\star$ convergence and the energy estimate from Lemma 6.1.2, for any
$T>0$, we have

$\|u\|_{L^{\infty}_{T}(\tilde{H}^{d})}^{2}+\|\nabla_{\mathbb{H}}u\|_{L_{T}^{2}(\tilde{H}^{d})}^{2}\leqslant
2\|u_{0}\|_{\tilde{H}^{d}}^{2}.$

Thanks to the weak convergence of $(u_{k})$ to $u$ and the continuity
properties of $\mathbb{P}$ (see Proposition 4.2.1, Item 9), we have the
following convergence in
$\mathcal{D}^{\prime}((0,+\infty)\times\mathbb{H}^{d})$

$\displaystyle\lim_{k\rightarrow+\infty}\partial_{t}u_{k}=\partial_{t}u,\ \,\
\
\lim_{k\rightarrow+\infty}\mathbb{P}\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}}u_{k}=\mathbb{P}\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}}u\
\ \text{and}\ \ \operatorname{\mathop{}div}_{\mathbb{H}}(u)=0.$

Moreover, the sequence $\tilde{\mathsf{J}}_{k}u_{0}$ converges to $u_{0}$ in
$\tilde{H}^{d}$ (see Proposition 4.1.4, Item 6). To conclude, it is sufficient
to show that
$(\mathbb{P}\mathsf{J}_{k}(u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k}))$
converges to $\mathbb{P}(u\cdot\nabla_{\mathbb{H}}u)$ in
$\mathcal{D}^{\prime}((0,+\infty)\times\mathbb{H}^{d})$. To this aim,
according to the classical strategy, we need to have a strong convergence on
the sequence $(u_{k})$, that we can get by bounding $(\partial_{t}u_{k})$ in a
suitable space, using an Aubin-Lions type lemma and a locally compact
embedding.

###### End of the proof of Theorem 6.1.1.

We shall show the local (strong) compactness of the sequence $(u_{k})$ in a
suitable space. In this proof the constant implied by the symbol $\lesssim$ is
assumed to be independent of the parameter $k\in\mathbb{N}$.

1) If $\varphi$ belongs to $\mathcal{D}((0,+\infty)\times\mathbb{H}^{d})$,
then $(\varphi u_{k})$ is bounded in
$H^{1}(\mathbb{R}_{+};W^{-1,Q}_{\mathbb{H}})$. By using the embedding
$L^{\frac{Q}{2}}(\mathbb{H}^{d})\hookrightarrow
W^{-1,Q}_{\mathbb{H}}(\mathbb{H}^{d})$ (see Proposition 2.4.5, Item 3), we
deduce that

$\|\varphi u_{k}\|_{L^{2}(W^{-1,Q}_{\mathbb{H}})}\lesssim\|\varphi
u_{k}\|_{L^{2}(L^{\frac{Q}{2}})}\lesssim\|\varphi\|_{L^{2}(L^{Q})}\|u_{k}\|_{L^{\infty}(L^{Q})}.$
(6.3)

Hence, according to the Sobolev embedding
$\tilde{H}^{d}(\mathbb{H}^{d})\hookrightarrow L^{Q}(\mathbb{H}^{d})$, we
obtain

$\|\varphi
u_{k}\|_{L^{2}(W^{-1,Q}_{\mathbb{H}})}\lesssim\|u_{k}\|_{L^{\infty}(\tilde{H}^{d})}.$

According to Lemma 6.1.2, it follows that

$\|\varphi
u_{k}\|_{L^{2}(W^{-1,Q}_{\mathbb{H}})}\lesssim\|u_{0}\|_{\tilde{H}^{d}}.$
(6.4)

Thus, $(\varphi u_{k})$ is bounded in $L^{2}(W^{-1,Q}_{\mathbb{H}})$. Let us
begin by remarking that, according to Lemma 4.2.3, we have

$\displaystyle\partial_{t}(\varphi u_{k})$
$\displaystyle=(\partial_{t}\varphi)u_{k}+\varphi\mathbb{P}\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}}u_{k}-\varphi\mathbb{P}\mathsf{J}_{k}(u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k})$
$\displaystyle=(\partial_{t}\varphi)u_{k}+\varphi\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}}u_{k}+\varphi\Pi_{\mathbb{H}}\partial_{s}u_{k}-\varphi\mathbb{P}\mathsf{J}_{k}(u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k}).$

Let us now estimate all the terms in $L^{2}(W^{-1,Q}_{\mathbb{H}})$ .
$\bullet$ Estimate on $(\partial_{t}\varphi)u_{k}$. Since
$\partial_{t}\varphi$ belongs to $L^{2}(L^{Q})$, as for (6.4), we obtain

$\|(\partial_{t}\varphi)u_{k}\|_{L^{2}(W^{-1,Q}_{\mathbb{H}})}\lesssim\|u_{0}\|_{\tilde{H}^{d}}.$

$\bullet$ Estimate on
$\varphi\mathbb{P}\mathsf{J}_{k}(u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k})$.
Using the embedding $L^{\frac{Q}{2}}(\mathbb{H}^{d})\hookrightarrow
W^{-1,Q}_{\mathbb{H}}(\mathbb{H}^{d})$, that $\varphi$ belongs to
$L^{\infty}(\mathbb{R}_{+}\times\mathbb{H}^{d})$ and the embedding
$\tilde{H}^{d-1}(\mathbb{H}^{d})\hookrightarrow
L^{\frac{Q}{2}}(\mathbb{H}^{d})$, we get

$\|\varphi\mathbb{P}\mathsf{J}_{k}(u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k})\|_{L^{2}(W^{-1,Q}_{\mathbb{H}})}\lesssim\|\mathbb{P}\mathsf{J}_{k}(u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k})\|_{L^{2}(\tilde{H}^{d-1})}.$
(6.5)

Since $\mathbb{P}$ belongs to $\mathcal{L}(\tilde{H}^{d-1})$ (see Proposition
4.2.1, Item 8) and $\mathsf{J}_{k}$ is bounded by $1$ in this space (see
Proposition 4.1.3, Item 1), we have

$\|\mathbb{P}\mathsf{J}_{k}(u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k})\|_{L^{2}(\tilde{H}^{d-1})}\lesssim\|u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k}\|_{L^{2}(\tilde{H}^{d-1})}.$

Then, by using the tame estimates of Proposition 2.4.7 for
$u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k}$ on
$\tilde{H}^{d-1}(\mathbb{H}^{d})$ with $u_{k}$ and
$\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k}$ in $\tilde{H^{d}}(\mathbb{H}^{d})\cap
L^{2}(\mathbb{H}^{d})$ (the fact that these two functions belong to
$L^{2}(\mathbb{H}^{d})$ follows from the continuity of the operator
$\tilde{\mathsf{J}}_{k}$ from $\tilde{H}^{d}(\mathbb{H}^{d})$ to
$L^{2}(\mathbb{H}^{d})$ from Proposition 4.1.4, Item 1), since
$\mathsf{J}_{k}$ is bounded by $1$ in $\mathcal{L}(\tilde{H}^{d})$, the energy
estimates from Lemma 6.1.2 yield

$\|\varphi\mathbb{P}\mathsf{J}_{k}(u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k})\|_{L^{2}(W^{-1,Q}_{\mathbb{H}})}\lesssim\|u_{0}\|_{\tilde{H}^{d}}^{2}.$

$\bullet$ Estimate on $\varphi\Pi_{\mathbb{H}}\partial_{s}u_{k}$. Similarly to
(6.5), we have

$\|\varphi\Pi_{\mathbb{H}}\partial_{s}u_{k}\|_{L^{2}(W^{-1,Q}_{\mathbb{H}})}\lesssim\|\Pi_{\mathbb{H}}\partial_{s}u_{k}\|_{L^{2}(\tilde{H}^{d-1})}.$

Let us recall that
$\Pi_{\mathbb{H}}:=8(\operatorname{Id}-\mathbb{P})\circ\mathfrak{S}$ (see
Lemma 4.2.3). Thus, since $\mathbb{P}$ belongs to
$\mathcal{L}(\tilde{H}^{d-1})$, we deduce that $\Pi_{\mathbb{H}}$ belongs to
$\mathcal{L}(\tilde{H}^{d-1})$. Accordingly, we have

$\|\Pi_{\mathbb{H}}\partial_{s}u_{k}\|_{L^{2}(\tilde{H}^{d-1})}\lesssim\|\partial_{s}u_{k}\|_{L^{2}(\tilde{H}^{d-1})}.$

Besides, according to Proposition 2.4.8 applied alternatively with
$\dot{H}^{1}(\mathbb{H}^{d})$ and $\tilde{H}^{1}(\mathbb{H}^{d})$, we have

$\|\partial_{s}u_{k}\|_{L^{2}(\tilde{H}^{d-1})}=\||D_{s}|^{\frac{1}{2}}|D_{s}|^{\frac{1}{2}}u_{k}\|_{L^{2}(\tilde{H}^{d-1})}\lesssim\||D_{s}|^{\frac{1}{2}}u_{k}\|_{L^{2}(\tilde{H}^{d})}\lesssim\|\nabla_{\mathbb{H}}u_{k}\|_{L^{2}(\tilde{H}^{d})}.$

Then, it follows from the energy estimates in Lemma 6.1.2 that

$\|\varphi\Pi_{\mathbb{H}}\partial_{s}u_{k}\|_{L^{2}(W^{-1,Q}_{\mathbb{H}})}\lesssim\|u_{0}\|_{\tilde{H}^{d}}.$

$\bullet$ Estimate on
$\varphi\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}}u_{k}$. Since
$\varphi$ and $u_{k}$ are smooth (this follows from Remark 4.1.5 and the third
line of (4.9), namely $\tilde{\mathsf{J}}_{k}u_{k}=u_{k}$), we can write

$\displaystyle\|\varphi\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}}u_{k}$
$\displaystyle\|_{L^{2}(W^{-1,Q}_{\mathbb{H}})}$
$\displaystyle\lesssim\|\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}}(\varphi
u_{k})\|_{L^{2}(W^{-1,Q}_{\mathbb{H}})}+\|(\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}}\varphi)u_{k}\|_{L^{2}(W^{-1,Q}_{\mathbb{H}})}+\|\nabla_{\mathbb{H}}\varphi\cdot\nabla_{\mathbb{H}}u_{k}\|_{L^{2}(W^{-1,Q}_{\mathbb{H}})}.$
(6.6)

By using the embedding $L^{\frac{Q}{2}}(\mathbb{H}^{d})\hookrightarrow
W^{-1,Q}_{\mathbb{H}}(\mathbb{H}^{d})$, the Hölder estimate and the embedding
$\tilde{H}^{d}(\mathbb{H}^{d})\hookrightarrow L^{Q}(\mathbb{H}^{d})$, we
deduce from Lemma 6.1.2 that

$\|(\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}}\varphi)u_{k}\|_{L^{2}(W^{-1,Q}_{\mathbb{H}})}+\|\nabla_{\mathbb{H}}\varphi\cdot\nabla_{\mathbb{H}}u_{k}\|_{L^{2}(W^{-1,Q}_{\mathbb{H}})}\lesssim\|u_{0}\|_{\tilde{H}^{d}}.$
(6.7)

Besides, using the Leibniz rules, the Sobolev embedding and the energy
estimate of Lemma 6.1.2, it follows that

$\displaystyle\|\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}}(\varphi
u_{k})\|_{L^{2}(W^{-1,Q}_{\mathbb{H}})}$
$\displaystyle\lesssim\|\nabla_{\mathbb{H}}(\varphi
u_{k})\|_{L^{2}(L^{Q})}\lesssim\|u_{0}\|_{\tilde{H}^{d}}.$

Combining this estimate with (6.6) and (6.7), we deduce that

$\|\varphi\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}}u_{k}\|_{L^{2}(W^{-1,Q}_{\mathbb{H}})}\lesssim\|u_{0}\|_{\tilde{H}^{d}}.$

We conclude that

$\|\partial_{t}(\varphi
u_{k})\|_{L^{2}(W^{-1,Q}_{\mathbb{H}})}\lesssim\|u_{0}\|_{\tilde{H}^{d}}+\|u_{0}\|_{\tilde{H}^{d}}^{2}.$

We conclude that $(\varphi u_{k})$ is bounded in
$H^{1}(W^{-1,Q}_{\mathbb{H}})$.

2) There exists a subsequence $(u_{k_{l}})$ of $(u_{k})$ such that for any
$\varphi\in\mathcal{D}(\mathbb{R}_{+}\times\mathbb{H}^{d})$, the
sequence $(\varphi u_{k_{l}})$ converges to $\varphi u$ in
$L^{2}(\mathbb{R}_{+};L^{Q})$. According to the Cantor diagonal extraction, we
are reduced to show that if
$\varphi\in\mathcal{D}(\mathbb{R}_{+}\times\mathbb{H}^{d})$, there is a
subsequence $(u_{k_{l}})$ of $(u_{k})$ such that $(\varphi u_{k_{l}})$
converge to $\varphi u$ in $L^{2}(L^{Q})$. Let
$\varphi\in\mathcal{D}((0,+\infty)\times\mathbb{H}^{d})$. Since $(\varphi
u_{k})$ is smooth in $\mathbb{R}_{+}\times\mathbb{H}^{d}$ and bounded in
$H^{1}(\mathbb{R}_{+};W^{-1,Q}_{\mathbb{H}})$, for any $\tau$ and
$\tau^{\prime}$ in $\mathbb{R}_{+}$, we have

$\|(\varphi u_{k})(\tau)-(\varphi
u_{k})(\tau^{\prime})\|_{W^{-1,Q}_{\mathbb{H}}}\lesssim|\tau-\tau^{\prime}|^{\frac{1}{2}}\|\partial_{t}(\varphi
u_{k})\|_{L^{2}(W^{-1,Q}_{\mathbb{H}})}\lesssim_{\varphi,u_{0}}|\tau-\tau^{\prime}|^{\frac{1}{2}}.$

We deduce that $(\varphi u_{k})$ is equicontinuous in $\mathbb{R}_{+}$ with
value in $W^{-1,Q}_{\mathbb{H}}$. From the energy estimate, the Hölder
estimate and the embedding $\tilde{H}^{d}\hookrightarrow L^{Q}$, it follows
that $(\varphi u_{k})$ is bounded in $L^{\infty}(L^{Q})$. Applying Proposition
C.0.1 to $(\varphi u_{k}(t))$ for $t\in\mathbb{R}_{+}$, we deduce that
$(\varphi u_{k}(t))$ is relatively compact in $W^{-1,Q}_{\mathbb{H}}$ for any
$t\in\mathbb{R}_{+}$. Thanks to the Arzela-Ascoli theorem, we conclude that,
up to extraction, that $(\varphi u_{k})$ converge to $\varphi u$ in
$\mathcal{C}(\mathbb{R}_{+},W^{-1,Q}_{\mathbb{H}})$. The next step is to show
that $(\varphi u_{k})$ converges in fact to $\varphi u$ in
$L^{2}(\mathbb{R}_{+};L^{Q})$. We will need the following lemma (the proof is
an adaptation of the proof [12, (A.17) p. 195] and is left to the reader).

######  Lemma 6.1.4.

Let $E$, $F\subset F^{\prime}$ three Banach spaces, $A\in\mathcal{L}(E,F)$ and
$B\in\mathcal{L}(F,F^{\prime})$ such that $R(A)\subset
N(\operatorname{Id}-B)$. Assume that $A$ is a compact operator. Then, for all
$\eta>0$, there exists $C_{\eta}>0$ such that for all $u\in E$, we have

$\|Au\|_{F}\leqslant\eta\|u\|_{E}+C_{\eta}\|B\circ Au\|_{F^{\prime}}.$

Choose a bounded open subset $\Omega$ of $\mathbb{H}^{d}$ such that
$supp(\varphi)\subset\mathbb{R}_{+}\times\Omega$. We now introduce
$\chi_{0}\in\mathcal{D}(\mathbb{H}^{d})$ and
$\chi_{1}\in\mathcal{D}(\mathbb{H}^{d})$ such that $\chi_{0}=1$ on $\Omega$,
and $\chi_{1}=1$ on the support of $\chi_{0}$, so that
$\chi_{1}\chi_{0}=\chi_{0}$. From Proposition C.0.1, the multiplication by
$\chi_{0}$ is a compact operator from $W^{1,Q}_{\mathbb{H}}$ to
$L^{Q}(\mathbb{H}^{d})$. Lemma 6.1.4 then provides, for any $\eta>0$, a
constant $C_{\eta}>0$ such that for any $v\in W^{1,Q}_{\mathbb{H}}$,

$\|\chi_{0}v\|_{L^{Q}}\leqslant\eta\|v\|_{W^{1,Q}_{\mathbb{H}}}+C_{\eta}\|\chi_{0}v\|_{W^{-1,Q}_{\mathbb{H}}}.$

It follows that for all $t>0$,

$\|\varphi(t,\cdot)v\|_{L^{Q}}\leqslant\eta\|v\|_{W^{1,Q}_{\mathbb{H}}}+C_{\eta}\|\varphi(t,\cdot)v\|_{W^{-1,Q}_{\mathbb{H}}}.$

Let us consider $T>0$ so that $supp(\varphi)\subset[0,T]\times\mathbb{H}^{d}$.
Let $\eta>0$. Since $(\varphi u_{k})$ converge in
$L^{\infty}(\mathbb{R}_{+};W^{-1,Q}_{\mathbb{H}})$, there is
$N_{\eta,T}\in\mathbb{N}$ such that for any $l,k\geqslant N_{\eta,T}$ we have

$\|\varphi u_{k}-\varphi
u_{l}\|_{L^{\infty}(W^{-1,Q}_{\mathbb{H}})}<\frac{\eta}{C_{\eta}T^{\frac{1}{2}}}.$

Then, using the Sobolev embedding and the energy estimate of Lemma 6.1.2, we
have, for any $k$ and $l$ larger than $N_{\eta,T}$,

$\|\varphi u_{k}-\varphi u_{l}\|_{L^{2}(L^{Q})}\leqslant\eta\|\varphi
u_{k}-\varphi
u_{l}\|_{L^{2}(W^{1,Q}_{\mathbb{H}})}+\eta\lesssim_{\varphi,u_{0}}\eta.$

This shows that $(\varphi u_{k})$ converges to $\varphi u$ in
$L^{2}(\mathbb{R}_{+};L^{Q})$.

3) Convergence of the nonlinear term. Let us begin by showing that, up to
extract a subsequence, $(u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k})$
converges to $u\cdot\nabla_{\mathbb{H}}u$ in
$\mathcal{D}^{\prime}(\mathbb{R}_{+}\times\mathbb{H}^{d})$. Let
$\varphi\in\mathcal{D}(\mathbb{R}_{+}\times\mathbb{H}^{d})$. We pick
$\chi\in\mathcal{D}(\mathbb{R}_{+}\times\mathbb{H}^{d})$ so that $\chi=1$ on
$supp(\varphi)$. Since $u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k}$
belongs to $L^{2}(L^{\frac{Q}{2}})$, we can write

$\langle
u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k},\varphi\rangle_{\mathcal{D}^{\prime},\mathcal{D}}=\int_{\mathbb{R}_{+}\times\mathbb{H}^{d}}(\varphi
u_{k})\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k}\,dtdx.$ (6.8)

Using the Hölder estimate and the Sobolev embedding
$\tilde{H}^{d}\hookrightarrow L^{Q}$, we deduce that $(\varphi\chi u_{k})$ is
strongly convergent to $\varphi u=\varphi\chi u$ in
$L^{2}(\mathbb{R}_{+};L^{\frac{Q}{Q-1}})=L^{2}(\mathbb{R}_{+};L^{Q})^{{}^{\prime}}$
(because $\varphi\in L^{\infty}(\mathbb{R}_{+};L^{Q/2})$) and
$(\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k})$ is weakly convergent to
$\nabla_{\mathbb{H}}u$ in $L^{2}(\mathbb{R}_{+};L^{Q})$. We deduce from (6.8)
and from the properties of the weak convergence (see [14, Proposition 3.5,
Item 4, p. 58]) that

$\lim_{k\rightarrow+\infty}\langle
u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k},\varphi\rangle_{\mathcal{D}^{\prime},\mathcal{D}}=\int_{\mathbb{R}_{+}\times\mathbb{H}^{d}}\varphi\left(u\cdot\nabla_{\mathbb{H}}u\right)dtdx.$

We have proved that $(u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k})$
converges to $u\cdot\nabla_{\mathbb{H}}u$ in
$\mathcal{D}^{\prime}(\mathbb{R}_{+}\times\mathbb{H}^{d})$.
Since the sequence $(u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k})$ is
bounded in $L^{2}(\tilde{H}^{d-1})$, we deduce from Proposition 4.1.3, Item
$6$, that
$(\mathsf{J}_{k}-\operatorname{Id})(u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k})$
converges weakly to $0$ in $L^{2}(\tilde{H}^{d-1})$ and, up to extract a
subsequence, that $(u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k})$
converges weakly to $u\cdot\nabla_{\mathbb{H}}u$ in $L^{2}(\tilde{H}^{d-1})$.
According to the continuity of $\mathbb{P}$ on
$\tilde{H}^{d-1}(\mathbb{H}^{d})$, this ensures that $u$ satisfies (1.5).

4) Continuity in time. We finish the proof by showing the continuity in time
for the solution, that is $u\in\mathcal{C}(\mathbb{R}_{+};\tilde{H}^{d})$.
Since
$(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{d}{2}}u$
belongs to $L^{2}_{loc}(\mathbb{R}_{+};H^{1})$ and
$\partial_{t}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{d}{2}}u\in
L^{2}_{loc}(\mathbb{R}_{+};H^{-1})$, we deduce by interpolation that $u$
belongs to $\mathcal{C}(\mathbb{R}_{+};\tilde{H}^{d})$. ∎

###  6.2 Stability and uniqueness

In this section our goal is to establish the stability of solutions of (1.5).

######  Theorem 6.2.1.

There exists a positive constant $C$ such that for any $T>0$ and for any
solutions $u$ and $v$ of (1.5) such that $u$ and $v$ belong to
$\mathcal{C}_{b}([0,T];\tilde{H}^{d})$ and $\nabla_{\mathbb{H}}u$ and
$\nabla_{\mathbb{H}}v$ belong to $L^{2}([0,T];\tilde{H}^{d})$, we have

$\displaystyle\|u-v\|_{L^{\infty}_{T}(\tilde{H}^{d})}^{2}+\|\nabla_{\mathbb{H}}(u-$
$\displaystyle v)\|_{L^{2}_{T}(\tilde{H}^{d})}^{2}$
$\displaystyle\leqslant\|u(0)-v(0)\|_{\tilde{H}^{d}}^{2}\exp{\left(C\|\nabla_{\mathbb{H}}v\|_{L^{2}_{T}(\tilde{H}^{d})}^{2}+C\|\nabla_{\mathbb{H}}u\|_{L^{2}_{T}(\tilde{H}^{d})}^{2}\right)}.$

From this result we deduce immediately the uniqueness of the solutions
constructed in Theorem 6.1.1. This thus concludes the proof of Theorem 1.2.4,
Items 1 and 2.

###### Proof.

In this proof the constant implied by $\lesssim$ is independent of $T$.
Without loss of generality, up to regularize $u$ and $v$ with respect to the
time and space variable, we can assume that $u$ and $v$ are smooth. Let us set
$w:=v-u$ and, for any $t\in[0,T]$,

$e(t):=\|w(t)\|_{\tilde{H}^{d}}^{2}+2\|\nabla_{\mathbb{H}}w\|_{L^{2}_{t}(\tilde{H}^{d})}^{2}.$

Let $t\in[0,T]$. By developing the expression of $e(t)$, we obtain

$e(t)=\|u(t)\|_{\tilde{H}^{d}}^{2}+2\|\nabla_{\mathbb{H}}u\|_{L^{2}_{t}(\tilde{H}^{d})}^{2}+\|v(t)\|_{\tilde{H}^{d}}^{2}+2\|\nabla_{\mathbb{H}}v\|_{L^{2}_{t}(\tilde{H}^{d})}^{2}-2E(t),$
(6.9)

where

$E(t):=\langle
v(t),u(t)\rangle_{\tilde{H}^{d}}+2\int_{0}^{t}\langle\nabla_{\mathbb{H}}v(\tau),\nabla_{\mathbb{H}}u(\tau)\rangle_{\tilde{H}^{d}}d\tau.$

Besides, by a direct calculus, using the equation satisfied by $u$ and $v$,
and keeping in mind that
$[(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{d}{2}},\nabla_{\mathbb{H}}]=0$
in order to cancel the pressure, we get

$\displaystyle\int_{0}^{t}\langle\nabla_{\mathbb{H}}v(\tau),\nabla_{\mathbb{H}}u(\tau)\rangle_{\tilde{H}^{d}}d\tau=$
$\displaystyle\langle v(0),u(0)\rangle_{\tilde{H}^{d}}-\langle
v(t),u(t)\rangle_{\tilde{H}^{d}}-\int_{0}^{t}\langle\nabla_{\mathbb{H}}v(\tau),\nabla_{\mathbb{H}}u(\tau)\rangle_{\tilde{H}^{d}}d\tau$
$\displaystyle-\int_{0}^{t}\langle
v(\tau)\cdot\nabla_{\mathbb{H}}v(\tau),u(\tau)\rangle_{\tilde{H}^{d}}d\tau-\int_{0}^{t}\langle
v(\tau),u(\tau)\cdot\nabla_{\mathbb{H}}u(\tau)\rangle_{\tilde{H}^{d}}d\tau,$

that is

$E(t)=\langle v(0),u(0)\rangle_{\tilde{H}^{d}}-\int_{0}^{t}\langle
v(\tau)\cdot\nabla_{\mathbb{H}}v(\tau),u(\tau)\rangle_{\tilde{H}^{d}}d\tau-\int_{0}^{t}\langle
v(\tau),u(\tau)\cdot\nabla_{\mathbb{H}}u(\tau)\rangle_{\tilde{H}^{d}}d\tau.$

In view of (6.9) and by computing the energy for $u$ and $v$ in
$\tilde{H}^{d}(\mathbb{H}^{d})$, it follows that

$\displaystyle e(t)$
$\displaystyle=\|u(0)\|_{\tilde{H}^{d}}^{2}+\|v(0)\|_{\tilde{H}^{d}}^{2}-2\langle
v(0),u(0)\rangle_{\tilde{H}^{d}}$ $\displaystyle\ \
-2\int_{0}^{t}\left(\langle
u(\tau)\cdot\nabla_{\mathbb{H}}u(\tau),u(\tau)\rangle_{\tilde{H}^{d}}+\langle
v(\tau)\cdot\nabla_{\mathbb{H}}v(\tau),v(\tau)\rangle_{\tilde{H}^{d}}\right)d\tau$
$\displaystyle\ \ +2\int_{0}^{t}\left(\langle
v(\tau)\cdot\nabla_{\mathbb{H}}v(\tau),u(\tau)\rangle_{\tilde{H}^{d}}+\langle
u(\tau)\cdot\nabla_{\mathbb{H}}u(\tau),v(\tau)\rangle_{\tilde{H}^{d}}\right)d\tau$
$\displaystyle=\|w(0)\|_{\tilde{H}^{d}}^{2}+2\int_{0}^{t}\left(\langle
w(\tau)\cdot\nabla_{\mathbb{H}}u(\tau),w(\tau)\rangle_{\tilde{H}^{d}}+\langle
v(\tau)\cdot\nabla_{\mathbb{H}}w(\tau),w(\tau)\rangle_{\tilde{H}^{d}}\right)d\tau.$
(6.10)

We have to estimate

$\int_{0}^{t}\langle
w(\tau)\cdot\nabla_{\mathbb{H}}u(\tau),w(\tau)\rangle_{\tilde{H}^{d}}d\tau+\int_{0}^{t}\langle
v(\tau)\cdot\nabla_{\mathbb{H}}w(\tau),w(\tau)\rangle_{\tilde{H}^{d}}d\tau.$
(6.11)

By applying Lemma 6.1.3, we have

$\left|\int_{0}^{t}\langle
w(\tau)\cdot\nabla_{\mathbb{H}}u(\tau),w(\tau)\rangle_{\tilde{H}^{d}}d\tau\right|\lesssim\int_{0}^{t}\|w(\tau)\|_{\tilde{H}^{d}}\|\nabla_{\mathbb{H}}w(\tau)\|_{\tilde{H}^{d}}\|\nabla_{\mathbb{H}}u(\tau)\|_{\tilde{H}^{d}}d\tau.$

In order to recover the estimate on the second term of (6.11), let us remark
that, if $d$ is even, then

$(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{d}=\mathcal{Z}^{2},\
\text{ where
}\mathcal{Z}:=(\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{d}{2}}=\sum_{i\in[\\![1,2d]\\!]^{\frac{d}{2}}}\prod_{j=1}^{\frac{d}{2}}\tilde{P}_{i_{j}}^{2},$

and if $d$ is odd, then we have

$(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{d}=-\sum_{h=1}^{2d}(\mathcal{M}\tilde{P}_{h})(\tilde{P}_{h}\mathcal{M}),\
\text{ where
}\mathcal{M}:=(\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{d-1}{2}}=\sum_{i\in[\\![1,2d]\\!]^{\frac{d-1}{2}}}\prod_{j=1}^{\frac{d-1}{2}}\tilde{P}_{i_{j}}^{2}.$

Note that $\mathcal{Z}$ and $\mathcal{M}$ are (unbounded) self-adjoint
operators in $L^{2}(\mathbb{H}^{d})$. At first, let us assume that $d$ is
even. Then, we have

$\langle
v\cdot\nabla_{\mathbb{H}}w,w\rangle_{\tilde{H}^{d}}=\langle\mathcal{Z}(v\cdot\nabla_{\mathbb{H}}w),\mathcal{Z}w\rangle_{L^{2}}=\langle
v\cdot\nabla_{\mathbb{H}}\mathcal{Z}w,\mathcal{Z}w\rangle_{L^{2}}+\langle[\mathcal{Z},v\cdot\nabla_{\mathbb{H}}]w,\mathcal{Z}w\rangle_{L^{2}}.$

However, if $a$ and $b$ are two smooth horizontal vector fields, such that
$\operatorname{\mathop{}div}_{\mathbb{H}}(a)=0$, then

$\langle a\cdot\nabla_{\mathbb{H}}b,b\rangle_{L^{2}}=0.$

Thus, we have

$\langle
v\cdot\nabla_{\mathbb{H}}w,w\rangle_{\tilde{H}^{d}}=\langle[\mathcal{Z},v\cdot\nabla_{\mathbb{H}}]w,\mathcal{Z}w\rangle_{L^{2}}.$

Let us now assume that $d$ is odd. Similarly to the case where $d$ is even, we
get

$\langle
v\cdot\nabla_{\mathbb{H}}w,w\rangle_{\tilde{H}^{d}}=\sum_{h=1}^{2d}\langle[\tilde{P}_{h}\mathcal{M},v\cdot\nabla_{\mathbb{H}}]w,\tilde{P}_{h}\mathcal{M}w\rangle_{L^{2}}.$

Furthermore, let us note that $\mathcal{Z}$ and $\tilde{P}_{h}\mathcal{M}$
with $h\in[\\![1,2d]\\!]$ commute with $P_{j}$, for any $j\in[\\![1,2d]\\!]$,
and are both a sum of terms of the form $\tilde{P}^{\gamma}$ where $\gamma$
belongs to $[\\![1,2d]\\!]^{d}$. Then, in order to get an estimate on $\langle
v\cdot\nabla_{\mathbb{H}}w,w\rangle_{\tilde{H}^{d}}$, we shall use the
following lemma.

######  Lemma 6.2.2.

There exists a constant $C>0$ such that for any smooth enough horizontal
vector fields $a$, $b$ and $c$ with
$\operatorname{\mathop{}div}_{\mathbb{H}}(a)=0$ and for all
$\gamma\in[\\![1,2d]\\!]^{d}$, we have

$|\langle[\tilde{P}^{\gamma},a\cdot\nabla_{\mathbb{H}}]b,c\rangle_{L^{2}}|\leqslant
C\|\nabla_{\mathbb{H}}a\|_{\tilde{H}^{d}}\|b\|_{\tilde{H}^{d}}\|\nabla_{\mathbb{H}}c\|_{L^{2}}.$

We will give the proof of this lemma after the proof of Theorem 6.2.1.
By using Lemma 6.2.2, we deduce that

$\left|\int_{0}^{t}\langle
v(\tau)\cdot\nabla_{\mathbb{H}}w(\tau),w(\tau)\rangle_{\tilde{H}^{d}}d\tau\right|\lesssim\int_{0}^{t}\|\nabla_{\mathbb{H}}v(\tau)\|_{\tilde{H}^{d}}\|w(\tau)\|_{\tilde{H}^{d}}\|\nabla_{\mathbb{H}}w(\tau)\|_{\tilde{H}^{d}}d\tau.$

and

$\displaystyle\bigg{|}\int_{0}^{t}\langle
w(\tau)\cdot\nabla_{\mathbb{H}}u(\tau),w(\tau)\rangle_{\tilde{H}^{d}}$
$\displaystyle+\langle
v(\tau)\cdot\nabla_{\mathbb{H}}w(\tau),w(\tau)\rangle_{\tilde{H}^{d}}d\tau\bigg{|}$
$\displaystyle\lesssim\int_{0}^{t}\|w(\tau)\|_{\tilde{H}^{d}}\|\nabla_{\mathbb{H}}w(\tau)\|_{\tilde{H}^{d}}\left(\|\nabla_{\mathbb{H}}u(\tau)\|_{\tilde{H}^{d}}+\|\nabla_{\mathbb{H}}v(\tau)\|_{\tilde{H}^{d}}\right)d\tau.$

Thus, from (6.10) and the Young estimate, it follows that

$e(t)\leqslant\|w(0)\|_{\tilde{H}^{d}}^{2}+C_{\star}\int_{0}^{t}\|w(\tau)\|_{\tilde{H}^{d}}^{2}\left(\|\nabla_{\mathbb{H}}u(\tau)\|_{\tilde{H}^{d}}^{2}+\|\nabla_{\mathbb{H}}v(\tau)\|_{\tilde{H}^{d}}^{2}\right)d\tau+\int_{0}^{t}\|\nabla_{\mathbb{H}}w(\tau)\|_{\tilde{H}^{d}}^{2}d\tau,$

for some positive constant $C_{\star}$, which is independent of $t$, $u$ and
$v$. We conclude that the following estimate

$\displaystyle\|w(t)\|_{\tilde{H}^{d}}^{2}+\int_{0}^{t}\|\nabla_{\mathbb{H}}$
$\displaystyle w(\tau)\|_{\tilde{H}^{d}}^{2}d\tau$
$\displaystyle\leqslant\|w(0)\|_{\tilde{H}^{d}}^{2}+C_{\star}\int_{0}^{t}\|w(\tau)\|_{\tilde{H}^{d}}^{2}\left(\|\nabla_{\mathbb{H}}u(\tau)\|_{\tilde{H}^{d}}^{2}+\|\nabla_{\mathbb{H}}v(\tau)\|_{\tilde{H}^{d}}^{2}\right)d\tau,$

holds for any $t\in[0,T]$. Finally, Theorem 6.2.1 follows from the Gronwall
estimate. ∎

###### Proof of Lemma 6.2.2.

Let $\gamma\in[\\![1,2d]\\!]^{d}$ and $a$, $b$ and $c$, three smooth enough
horizontal vector fields on $\mathbb{H}^{d}$ with
$\operatorname{\mathop{}div}_{\mathbb{H}}(a)=0$. Then, according to (2.8),
$[\tilde{P}^{\gamma},a\cdot\nabla_{\mathbb{H}}]b$ is the sum of terms of the
form

$\tilde{P}^{\alpha}a\cdot\nabla_{\mathbb{H}}\tilde{P}^{\beta}b,$

with $\alpha\in[\\![1,2d]\\!]^{\ell}$ and $\beta\in[\\![1,2d]\\!]^{d-\ell}$
where $\ell$ belongs to $[\\![1,d]\\!]$, according to
$\operatorname{\mathop{}div}_{\mathbb{H}}(a)=0$, we have

$\tilde{P}^{\alpha}a\cdot\nabla_{\mathbb{H}}\tilde{P}^{\beta}b=\operatorname{\mathop{}div}_{\mathbb{H}}(\tilde{P}^{\beta}b\otimes\tilde{P}^{\alpha}a).$

Hence, we get

$\displaystyle\langle\tilde{P}^{\alpha}a\cdot\nabla_{\mathbb{H}}\tilde{P}^{\beta}b,c\rangle_{L^{2}}$
$\displaystyle=-\sum_{i,j\in[\\![1,2d]\\!]}\langle\tilde{P}^{\alpha}a_{i},\tilde{P}^{\beta}b_{j}P_{i}c_{j}\rangle_{L^{2}}$
$\displaystyle\lesssim\|\tilde{P}^{\alpha}a\|_{\dot{W}^{1,\frac{2Q}{Q-2(d-\ell)}}_{\mathbb{H}}}\|\tilde{P}^{\beta}b\nabla_{\mathbb{H}}c\|_{\dot{W}^{-1,\frac{2Q}{Q+2(d-\ell)}}_{\mathbb{H}}}$
$\displaystyle\lesssim\|\nabla_{\mathbb{H}}\tilde{P}^{\alpha}a\|_{L^{\frac{2Q}{Q-2(d-\ell)}}}\|\tilde{P}^{\beta}b\nabla_{\mathbb{H}}c\|_{\dot{W}^{-1,\frac{2Q}{Q+2(d-\ell)}}_{\mathbb{H}}}.$

Furthermore, since $\ell\neq 0$, we have the Sobolev embedding
$L^{\frac{Q}{Q-\ell}}(\mathbb{H}^{d})\hookrightarrow\dot{W}_{\mathbb{H}}^{-1,\frac{2Q}{Q+2(d-\ell)}}(\mathbb{H}^{d})$
(see Proposition 2.4.5, Item 3), and then thanks to the Hölder estimate, we
deduce that

$\|\tilde{P}^{\beta}b\nabla_{\mathbb{H}}c\|_{\dot{W}^{-1,\frac{2Q}{Q-2(d-\ell)}}}\lesssim\|\tilde{P}^{\beta}b\nabla_{\mathbb{H}}c\|_{L^{\frac{Q}{Q-\ell}}}\lesssim\|\tilde{P}^{\beta}b\|_{L^{\frac{2Q}{Q-2\ell}}}\|\nabla_{\mathbb{H}}c\|_{L^{2}}.$

We conclude the proof of Lemma 6.2.2 by using the Sobolev embedding
$\tilde{H}^{d-\ell}(\mathbb{H}^{d})\hookrightarrow
L^{\frac{2Q}{Q-2(d-\ell)}}(\mathbb{H}^{d})$ and
$\tilde{H}^{\ell}(\mathbb{H}^{d})\hookrightarrow
L^{\frac{2Q}{Q-2\ell}}(\mathbb{H}^{d})$. ∎

##  7 Smoothing effects: proof of Theorem 1.2.5

###  7.1 Vertical regularity

Let us begin by establishing some basic facts about the characterization of
analyticity with respect to the vertical variable $s$ by using the Fourier
transform on the Heisenberg group.

####  7.1.1 Vertical analyticity and Fourier transform on $\mathbb{H}^{d}$

Let $\zeta$ be a positive real number. Let $f\in\mathcal{S}(\mathbb{H}^{d})$
so that the following map

$\mathsf{T}_{\zeta}:(n,m,\lambda)\in\tilde{\mathbb{H}}^{d}\mapsto
e^{\zeta|\lambda|}\mathcal{F}_{\mathbb{H}}(f)(n,m,\lambda),$

belongs to $L^{2}(\tilde{\mathbb{H}}^{d})$. When this definition makes sense,
we write

$e^{\zeta|D_{s}|}f:=\mathcal{F}_{\mathbb{H}}^{-1}(\mathsf{T}_{\zeta}f).$

As suggested by the Fourier inversion formula on $\mathbb{H}^{d}$, we aim to
make the link between the Fourier multiplier $e^{\zeta|D_{s}|}$ on
$\mathbb{H}^{d}$ and the Euclidean Fourier multiplier $e^{\zeta|\Lambda|}$ of
symbol $e^{\zeta|\xi_{s}|}$ on $\mathbb{R}^{2d+1}$. It is well known that the
operator $e^{\zeta|\Lambda|}$ acts on a space of analytic functions with
respect to the vertical variable $s$.

######  Remark 7.1.1.

Let $p\in(1,+\infty)$ and $\zeta\geqslant 0$. We define the two following
spaces

$\mathcal{A}^{p,\zeta}_{\mathbb{R}}:=\left\\{f\in
L^{p}(\mathbb{H}^{d})~{}\middle|~{}e^{\zeta|\Lambda|}f\in
L^{p}(\mathbb{H}^{d})\right\\}\ \text{and}\ \
\mathcal{A}^{p,\zeta}_{\mathbb{H}}:=\left\\{f\in
L^{p}(\mathbb{H}^{d})~{}\middle|~{}e^{\zeta|D_{s}|}f\in
L^{p}(\mathbb{H}^{d})\right\\}.$

These two spaces are equal and we denote them by
$\mathcal{A}^{p,\zeta}(\mathbb{H}^{d})$. In fact, we have, for any
$f\in\mathcal{A}^{p,\zeta}(\mathbb{H}^{d})$,

$e^{\zeta|D_{s}|}f=e^{\zeta|\Lambda|}f.$ (7.1)

###### Proof of Remark 7.1.1.

With $W$ defined by the formula (2.21), if $f$ belongs to
$L^{1}(\mathbb{H}^{d})$, then for any
$(n,m,\lambda)\in\tilde{\mathbb{H}}^{d}$, we have

$\mathcal{F}_{\mathbb{H}}(f)(n,m,\lambda)=\int_{\mathbb{R}^{2d}}W(n,m,\lambda,Y)\mathcal{F}_{\mathbb{R}}(f(Y,\cdot))(\lambda)dY.$

Thus, if $f\in\mathcal{S}(\mathbb{H}^{d})$, then $e^{-\zeta|\Lambda|}f\in
L^{1}(\mathbb{H}^{d})\cap L^{2}(\mathbb{H}^{d})$ and we have

$\displaystyle\mathcal{F}_{\mathbb{H}}(e^{-\zeta|\Lambda|}f)(n,m,\lambda)=\int_{\mathbb{R}^{2d}}W(n,m,\lambda,Y)e^{-\zeta|\lambda|}\mathcal{F}_{\mathbb{R}}(f(Y,\cdot))(\lambda)dY=e^{-\zeta|\lambda|}\mathcal{F}_{\mathbb{H}}(f)(n,m,\lambda).$

We deduce that $e^{\zeta|D_{s}|}e^{-\zeta|\Lambda|}=\operatorname{Id}$ on
$\mathcal{S}(\mathbb{H}^{d})$. We conclude by using the density of
$\mathcal{S}(\mathbb{H}^{d})$ in $L^{p}(\mathbb{H}^{d})$. ∎

####  7.1.2 Analytic smoothing effects in the vertical variable

Continuing our study of the Cauchy theory for solutions of (1.5)–(1.6) for
initial data in the critical space $\tilde{H}^{d}(\mathbb{H}^{d})$, we now
show that the solutions of (1.5) are instantaneously smoothed with respect to
the variable $s$. More precisely, we aim to show the following theorem.

######  Theorem 7.1.2.

Let $\sigma\in(0,4d)$. There exists $\varepsilon_{\sigma}>0$ such that for any
initial data $u_{0}\in\tilde{H}^{d}(\mathbb{H}^{d})$ satisfying
$\operatorname{\mathop{}div}_{\mathbb{H}}(u_{0})=0$ and
$\|u_{0}\|_{\tilde{H}^{d}}<\varepsilon_{\sigma}$, there exists a unique
solution $u$ of (1.5) satisfying
$u\in\mathcal{C}_{b}(\mathbb{R}_{+};\tilde{H}^{d})$ and
$\nabla_{\mathbb{H}}u\in L^{2}(\mathbb{R}_{+};\tilde{H}^{d})$ and the
following energy estimate

$\|e^{\sigma
t|D_{s}|}u\|_{L^{\infty}(\tilde{H}^{d})}^{2}+\left(1-\frac{\sigma}{4d}\right)\|e^{\sigma
t|D_{s}|}\nabla_{\mathbb{H}}u\|_{L^{2}(\tilde{H}^{d})}^{2}\leqslant\|u_{0}\|_{\tilde{H}^{d}}^{2}.$
(7.2)

Moreover, there is a positive constant $A$, which is independent of $\sigma$,
such that $\varepsilon_{\sigma}=A(1-\sigma/4d)$.

In other words, for any $t>0$, the radius of analyticity (see (1.9) for the
definition) of $u(t)$ with respect to the variable $s$ is bounded from below
by $\sigma t$.

###### Proof.

The uniqueness of such solution follows from Theorem 6.2.1. Let
$k\in\mathbb{N}$, $\sigma\in(0,4d)$ and $u_{0}\in\tilde{H}^{d}$. Let us
consider the solution $u_{k}$ of the approximate problem (4.9)-(4.10). In this
proof the constant implied by $\lesssim$ is independent of $k$ and the time
variable. By using the Plancherel theorem on $\mathbb{H}^{d}$, we deduce that

$\frac{1}{2}\frac{d}{dt}\|e^{\sigma
t|D_{s}|}u_{k}\|_{\tilde{H}^{d}}^{2}-\sigma\||D_{s}|^{\frac{1}{2}}e^{\sigma
t|D_{s}|}u_{k}\|_{\tilde{H}^{d}}^{2}=\langle e^{\sigma
t|D_{s}|}\partial_{t}u_{k},e^{\sigma t|D_{s}|}u_{k}\rangle_{\tilde{H}^{d}}.$

Besides, by computing the right-hand side of the above estimate thanks to
(4.9), we obtain

$\displaystyle\frac{1}{2}\frac{d}{dt}\|e^{\sigma
t|D_{s}|}u_{k}\|_{\tilde{H}^{d}}^{2}-\sigma\|$
$\displaystyle|D_{s}|^{\frac{1}{2}}e^{\sigma
t|D_{s}|}u_{k}\|_{\tilde{H}^{d}}^{2}$ $\displaystyle=-\|e^{\sigma
t|D_{s}|}\nabla_{\mathbb{H}}u_{k}\|_{\tilde{H}^{d}}^{2}-\langle e^{\sigma
t|D_{s}|}(u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k}),e^{\sigma
t|D_{s}|}\mathsf{J}_{k}u_{k}\rangle_{\tilde{H}^{d}}.$

Due to the following estimate

$\||D_{s}|^{\frac{1}{2}}e^{\sigma
t|D_{s}|}u_{k}\|_{\tilde{H}^{d}}^{2}\leqslant\frac{1}{4d}\|(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{\frac{1}{2}}e^{\sigma
t|D_{s}|}u_{k}\|_{\tilde{H}^{d}}^{2}=\frac{1}{4d}\|\nabla_{\mathbb{H}}e^{\sigma
t|D_{s}|}u_{k}\|_{\tilde{H}^{d}}^{2},$

we get

$\frac{1}{2}\frac{d}{dt}\|e^{\sigma
t|D_{s}|}u_{k}\|_{\tilde{H}^{d}}^{2}+\left(1-\frac{\sigma}{4d}\right)\|e^{\sigma
t|D_{s}|}\nabla_{\mathbb{H}}u_{k}\|_{\tilde{H}^{d}}^{2}\leqslant|\langle
e^{\sigma
t|D_{s}|}(u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k}),e^{\sigma
t|D_{s}|}\mathsf{J}_{k}u_{k}\rangle_{\tilde{H}^{d}}|.$

Thus, we have

$\displaystyle\|e^{\sigma\tau|D_{s}|}u_{k}$
$\displaystyle\|_{L^{\infty}_{t}(\tilde{H}^{d})}^{2}+2\left(1-\frac{\sigma}{4d}\right)\|e^{\sigma\tau|D_{s}|}\nabla_{\mathbb{H}}u_{k}\|_{L^{2}_{t}(\tilde{H}^{d})}^{2}$
$\displaystyle\leqslant\|\tilde{\mathsf{J}}_{k}u_{0}\|_{\tilde{H}^{d}}^{2}+\int_{0}^{t}|\langle
e^{\sigma\tau|D_{s}|}(u_{k}(\tau)\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k}(\tau)),e^{\sigma\tau|D_{s}|}\mathsf{J}_{k}u_{k}(\tau)\rangle_{\tilde{H}^{d}}|d\tau.$
(7.3)

We now derive an estimate on the nonlinear term by using the following lemma
that we shall temporarily assume (The proof will be given after the proof of
Theorem 7.1.2).

######  Lemma 7.1.3.

There exists a constant $C_{d}$ such that for any positive real numbers
$\zeta$ and for every function $a$, $b$ and $c$ such that $e^{\zeta|D_{s}|}a$,
$e^{\zeta|D_{s}|}b$ and $e^{\zeta|D_{s}|}\nabla_{\mathbb{H}}c$ belong to
$\tilde{H}^{d}(\mathbb{H}^{d})$, we have

$|\langle
e^{\zeta|D_{s}|}(ab),e^{\zeta|D_{s}|}c\rangle_{\tilde{H}^{d}}|\leqslant
C_{d}\|e^{\zeta|D_{s}|}a\|_{\tilde{H}^{d}}\|e^{\zeta|D_{s}|}b\|_{\tilde{H}^{d}}\|e^{\zeta|D_{s}|}\nabla_{\mathbb{H}}c\|_{\tilde{H}^{d}}.$

Note that the constant $C_{d}$ is independent of $\zeta$, which is essential
in order to get a suitable energy estimate (7.2) for global solutions.

We are now able to complete the proof of Theorem 7.1.2. By applying Lemma
7.1.3, Proposition 4.1.3, Items 1 and 3, and using that
$\tilde{\mathsf{J}}_{k}$ is bounded by $1$ in $\mathcal{L}(\tilde{H}^{d})$
(see Proposition 4.1.4, Item 1), we deduce from (7.3) that for any $t\geqslant
0$,

$\displaystyle\|e^{\sigma\tau|D_{s}|}u_{k}\|_{L^{\infty}_{t}(\tilde{H}^{d})}^{2}+2\left(1-\frac{\sigma}{4d}\right)$
$\displaystyle\|e^{\sigma\tau|D_{s}|}\nabla_{\mathbb{H}}u_{k}\|_{L^{2}_{t}(\tilde{H}^{d})}^{2}$
$\displaystyle\leqslant\|u_{0}\|_{\tilde{H}^{d}}^{2}+B\|e^{\sigma\tau|D_{s}|}u_{k}\|_{L^{\infty}_{t}(\tilde{H}^{d})}\|e^{\sigma\tau|D_{s}|}\nabla_{\mathbb{H}}u_{k}\|_{L^{2}_{t}(\tilde{H}^{d})}^{2},$
(7.4)

where $B$ is a positive constant which is independent of $\sigma$.

We can now perform the standard bootstrap argument. Let us define

$T_{\star}:=\sup\left\\{T\geqslant 0~{}\middle|~{}\|e^{\sigma
t|D_{s}|}u_{k}\|_{L^{\infty}_{T}(\tilde{H}^{d})}^{2}\leqslant
2\|u_{0}\|_{\tilde{H}^{d}}^{2}\right\\}.$

We shall first show that $T_{\star}>0$. Since
$u_{k}=\tilde{\mathsf{J}}_{k}u_{k}$, we have, for any $(n,m,\lambda)$ in
$\mathbb{N}^{d}\times\mathbb{N}^{d}\times\mathbb{R}^{*}$ and $t\geqslant 0$,

$e^{\sigma|\lambda|t}|\mathcal{F}_{\mathbb{H}}(u_{k}(t))(n,m,\lambda)|\leqslant\mathbf{1}_{\\{|\lambda|\leqslant
2^{k}\\}}e^{\sigma|\lambda|t}|\mathcal{F}_{\mathbb{H}}(u_{k}(t))(n,m,\lambda)|\leqslant
e^{\sigma 2^{k}t}|\mathcal{F}_{\mathbb{H}}(u_{k}(t))(n,m,\lambda)|.$

Then, according to the Plancherel formula on $\mathbb{H}^{d}$ (see Proposition
2.2.1), we deduce that

$\|e^{\sigma t|D_{s}|}u_{k}(t)\|_{\tilde{H}^{d}}\leqslant e^{\sigma
t2^{k}}\|u_{k}(t)\|_{\tilde{H}^{d}}.$

Furthermore, by using the continuity of $t\mapsto u_{k}(t)$ from
$\mathbb{R}_{+}$ into $\tilde{H}^{d}(\mathbb{H}^{d})$, we deduce that there
exists $t_{k}>0$, such that $e^{\sigma 2^{k+1}t_{k}}\leqslant\frac{3}{2}$ and
$\|u_{k}(t_{k})\|_{\tilde{H}^{d}}^{2}\leqslant\frac{4}{3}\|\tilde{\mathsf{J}}_{k}u_{0}\|_{\tilde{H}^{d}}^{2}$.
Then, since $\|\tilde{\mathsf{J}}_{k}\|_{\mathcal{L}(\tilde{H}^{d})}\leqslant
1$, we deduce that $T_{\star}\geqslant t_{k}>0$. Let us now assume that
$T_{\star}<+\infty$ and show a contradiction for a suitable choice of
$\varepsilon_{\sigma}$. Let $0<T<T_{\star}$. By definition of $T_{\star}$, we
have

$\|e^{\sigma
t|D_{s}|}u_{k}\|_{L^{\infty}_{T}(\tilde{H}^{d})}\leqslant\sqrt{2}\|u_{0}\|_{\tilde{H}^{d}}.$

Thus it follows from (7.4) and the smallness condition on
$\|u_{0}\|_{\tilde{H}^{d}}$, that

$\displaystyle\|e^{\sigma
t|D_{s}|}u_{k}\|_{L^{\infty}_{T}(\tilde{H}^{d})}^{2}+2\left(1-\frac{\sigma}{4d}\right)\|e^{\sigma
t|D_{s}|}$
$\displaystyle\nabla_{\mathbb{H}}u_{k}\|_{L^{2}_{T}(\tilde{H}^{d})}^{2}$
$\displaystyle\leqslant\|u_{0}\|_{\tilde{H}^{d}}^{2}+\varepsilon_{\sigma}B\sqrt{2}\|e^{\sigma
t|D_{s}|}\nabla_{\mathbb{H}}u_{k}\|_{L_{T}^{2}(\tilde{H}^{d})}^{2}.$ (7.5)

Then, if we chose
$\varepsilon_{\sigma}:=\frac{1-\frac{\sigma}{4d}}{\sqrt{2}B}$, thanks to
(7.1.2), we obtain

$\|e^{\sigma
t|D_{s}|}u_{k}\|_{L^{\infty}_{T}(\tilde{H}^{d})}^{2}+\left(1-\frac{\sigma}{4d}\right)\|e^{\sigma
t|D_{s}|}\nabla_{\mathbb{H}}u_{k}\|_{L^{2}_{T}(\tilde{H}^{d})}^{2}\leqslant\|u_{0}\|_{\tilde{H}^{d}}^{2}.$

In view of the definition of $T_{\star}$, by the classical continuity argument
this shows that $T_{\star}=+\infty$, and then the above estimate holds with
$T=+\infty$ for any $k\in\mathbb{N}$. Then we finish the proof by passing to
the limit $k\to+\infty$ as in the proof of Theorem 6.1.1. ∎

We now prove Lemma 7.1.3. The proof relies on the multilinear Calderon-Zygmund
theory (see [30, Lemma 24.8, p. 252]) to recover an estimate on the nonlinear
term
$(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{d}{2}}e^{\sigma
t|D_{s}|}(ab)$. More precisely, in this paper we use an anisotropic version of
[30, Lemma 24.8, p. 252], that we will now establish. We aim to study
continuity properties of the following bilinear operators, defined for
$\zeta\geqslant 0$, $A$ and $B$ in $\mathcal{S}(\mathbb{H}^{d})$ and
$(Y,s)\in\mathbb{H}^{d}$ by

$\displaystyle M_{\zeta}(A,B)(Y,s)$
$\displaystyle:=e^{\zeta|D_{s}|}\left((e^{-\zeta|D_{s}|}A)(e^{-\zeta|D_{s}|}B)\right)(Y,s)$
$\displaystyle=\frac{1}{4\pi^{2}}\int_{\mathbb{R}}\int_{\mathbb{R}}e^{is(\lambda+\mu)}e^{\zeta(|\lambda+\mu|-|\lambda|-|\mu|)}\mathcal{F}_{\mathbb{R}}(A(Y,\cdot))(\lambda)\mathcal{F}_{\mathbb{R}}(B(Y,\cdot))(\mu)d\lambda
d\mu.$

######  Lemma 7.1.4.

Let $p$, $p_{1}$ and $p_{2}$ in $(1,+\infty)$ satisfying
$1/p=1/p_{1}+1/p_{2}$. Then, there exists a constant $K$ such that for any
$\zeta\geqslant 0$ and for all $A$ in $L^{p_{1}}(\mathbb{H}^{d})$ and $B$ in
$L^{p_{2}}(\mathbb{H}^{d})$, we have
$e^{\zeta|D_{s}|}\left((e^{-\zeta|D_{s}|}A)(e^{-\zeta|D_{s}|}B)\right)\in
L^{p}(\mathbb{H}^{d})$ and

$\|e^{\zeta|D_{s}|}\left((e^{-\zeta|D_{s}|}A)(e^{-\zeta|D_{s}|}B)\right)\|_{L^{p}}\leqslant
K\|A\|_{L^{p_{1}}}\|B\|_{L^{p_{2}}}.$ (7.6)

Let us give the main arguments of the proof of Lemma 7.1.4 (which is an
adaptation of the proof of [30, Lemma 24.8, p. 252]) for the sake of clarity,
since we use both Fourier transforms on $\mathbb{R}$ and on $\mathbb{H}^{d}$
in our analysis.

###### Sketch of the proof of Lemma 7.1.4..

For any $f\in\mathcal{S}(\mathbb{R})$ and $s\in\mathbb{R}$, we define

$K_{1}f(s):=\frac{1}{2\pi}\int_{0}^{+\infty}e^{is\lambda}\mathcal{F}_{\mathbb{R}}(f)(\lambda)d\lambda\
\ \ \text{and}\ \ \
K_{-1}f(s):=\frac{1}{2\pi}\int_{-\infty}^{0}e^{is\lambda}\mathcal{F}_{\mathbb{R}}(f)(\lambda)d\lambda.$

Moreover, for any $\zeta>0$ and any $f\in\mathcal{S}(\mathbb{R})$, we
introduce

$L_{\zeta,1}f:=f\ \ \ \text{and}\ \ \
L_{\zeta,-1}f:=\frac{1}{2\pi}\int_{\mathbb{R}}e^{is\lambda}e^{-2\zeta|\lambda|}\mathcal{F}_{\mathbb{R}}(f)(\lambda)d\lambda.$

Finally, we set for any $\zeta>0$ and $\alpha,\beta\in\\{-1,1\\}$

$\mathsf{K}_{\alpha}:=\operatorname{Id}_{\mathcal{S}^{{}^{\prime}}(\mathbb{R}^{2d})}\otimes
K_{\alpha}\ \ \text{and}\ \
\mathsf{Z}_{\zeta,\alpha,\beta}:=\mathsf{K}_{\alpha}\big{(}\operatorname{Id}_{\mathcal{S}^{{}^{\prime}}(\mathbb{R}^{2d})}\otimes
L_{\zeta,\alpha\beta}\big{)},$

where we denote by
$\operatorname{Id}_{\mathcal{S}^{{}^{\prime}}(\mathbb{R}^{2d})}$ the identity
operator on $\mathcal{S}^{{}^{\prime}}(\mathbb{R}^{2d})$. Then, we have

$M_{\zeta}(A,B)=\sum_{(\gamma,\beta,\alpha)\in\\{-1,1\\}^{3}}\mathsf{K}_{\alpha}\left(\mathsf{Z}_{\zeta,\alpha,\beta}A\cdot\mathsf{Z}_{\zeta,\alpha,\gamma}B\right).$

Operators $\mathsf{K}_{\alpha}$ and $\mathsf{Z}_{\zeta,\alpha,\beta}$ are
Fourier multipliers of order zero, hence they are bounded in any
$L^{p}(\mathbb{H}^{d})$ with $p\in(1,+\infty)$. The main point is the
following: the family $\mathsf{Z}_{\zeta,\alpha,\beta}$ is uniformly bounded
with respect to the parameter $\zeta>0$ in any $L^{p}(\mathbb{H}^{d})$ with
$p\in(1,+\infty)$. Indeed, the kernel of $L_{\zeta,-1}$ is
$\frac{1}{\pi}\frac{2\zeta}{s^{2}+4\zeta^{2}}:=k_{\zeta}(s)$ and we have
$\|k_{\zeta}\|_{L^{1}(\mathbb{R})}=\|k_{1}\|_{L^{1}(\mathbb{R})}$, so that by
Young’s estimate with respect to the $s$ variable, we deduce that for any
$p\in(1,+\infty)$ and for any $f\in L^{p}(\mathbb{H}^{d})$,

$\|(\operatorname{Id}_{\mathcal{S}^{{}^{\prime}}(\mathbb{R}^{2d})}\otimes
L_{\zeta,-1})f\|_{L^{p}(\mathbb{H}^{d})}\leqslant\|k_{1}\|_{L^{1}(\mathbb{R})}\|f\|_{L^{p}(\mathbb{H}^{d})}.$

Then Lemma 7.1.4 follows from the Hölder estimate. ∎

We are now able to prove Lemma 7.1.3.

###### Proof of Lemma 7.1.3.

Let us begin by setting $A:=e^{\zeta|D_{s}|}a$ and $B:=e^{\zeta|D_{s}|}b$ so
that $A$ and $B$ belong to $\tilde{H}^{d}(\mathbb{H}^{d})$. Assume that $A$
and $B$ belong to $\mathcal{S}(\mathbb{H}^{d})$. Then,
$(e^{-\zeta|D_{s}|}A)(e^{-\zeta|D_{s}|}B)$ belongs to
$\mathcal{S}(\mathbb{H}^{d})$ and for any $\gamma\in[\\![1,2d]\\!]^{d}$, by
using the Leibniz formula, we get

$\left|e^{\zeta|D_{s}|}\tilde{P}^{\gamma}\left((e^{-\zeta|D_{s}|}A)(e^{-\zeta|D_{s}|}B)\right)\right|\lesssim\sum_{\ell=0}^{d}\\!\\!\sum_{\begin{subarray}{c}\alpha\in[\\![1,2d]\\!]^{\ell}\\\
\beta\in[\\![1,2d]\\!]^{d-\ell}\end{subarray}}\left|e^{\zeta|D_{s}|}\left(e^{-\zeta|D_{s}|}(\tilde{P}^{\alpha}A)e^{-\zeta|D_{s}|}(\tilde{P}^{\beta}B)\right)\right|.$
(7.7)

If we pick $\ell\in[\\![0,d]\\!]$, $\alpha\in[\\![1,2d]\\!]^{\ell}$,
$\beta\in[\\![1,2d]\\!]^{d-\ell}$ and $\gamma\in[\\![1,2d]\\!]^{d}$, applying
Lemma 7.1.4, we deduce that

$\left|\\!\left|e^{\zeta|D_{s}|}\left(e^{-\zeta|D_{s}|}(\tilde{P}^{\alpha}A)e^{-\zeta|D_{s}|}(\tilde{P}^{\beta}B)\right)\right|\\!\right|_{L^{\frac{2Q}{Q+2}}}\leqslant
C_{1}\|\tilde{P}^{\alpha}A\|_{L^{\frac{2Q}{Q-2(d-\ell)}}}\|\tilde{P}^{\beta}B\|_{L^{\frac{2Q}{Q-2\ell}}},$

where $C_{1}$ is a positive constant which is independent of $\zeta$. Applying
the Sobolev embeddings $\tilde{H}^{d-\ell}(\mathbb{H}^{d})\hookrightarrow
L^{\frac{2Q}{Q-2(d-\ell)}}(\mathbb{H}^{d})$ and
$\tilde{H}^{\ell}(\mathbb{H}^{d})\hookrightarrow
L^{\frac{2Q}{Q-2\ell}}(\mathbb{H}^{d})$, and using
$\alpha\in[\\![1,2d]\\!]^{\ell}$ and $\beta\in[\\![1,2d]\\!]^{d-\ell}$, we
obtain that

$\displaystyle\|\tilde{P}^{\alpha}A\|_{L^{\frac{2Q}{Q-2(d-\ell)}}}\lesssim\|A\|_{\tilde{H}^{d}}\
\ \text{and}\ \
\|\tilde{P}^{\beta}B\|_{L^{\frac{2Q}{Q-2\ell}}}\lesssim\|B\|_{\tilde{H}^{d}}.$

In view of (7.7), it follows that

$\left|\\!\left|(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{d}{2}}e^{\zeta|D_{s}|}\left(e^{-\zeta|D_{s}|}Ae^{-\zeta|D_{s}|}B\right)\right|\\!\right|_{L^{\frac{2Q}{Q+2}}}\leqslant
C_{2}\|A\|_{\tilde{H}^{d}}\|B\|_{\tilde{H}^{d}},$ (7.8)

where $C_{2}$ depend only on the constant $C_{1}$, $d$ and the constants of
the Sobolev embeddings. By density of $\mathcal{S}(\mathbb{H}^{d})$ in
$\tilde{H}^{d}(\mathbb{H}^{d})$, thanks to (7.8), we conclude that (7.8) holds
for any $A$ and $B$ in $\tilde{H}^{d}(\mathbb{H}^{d})$. In addition, by
applying the Hölder estimate, we deduce that

$\left|\langle
e^{\zeta|D_{s}|}(ab),e^{\zeta|D_{s}|}c\rangle_{\tilde{H}^{d}}\right|\leqslant
C_{2}\|e^{\zeta|D_{s}|}a\|_{\tilde{H}^{d}}\|e^{\zeta|D_{s}|}b\|_{\tilde{H}^{d}}\|(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{d}{2}}e^{\zeta|D_{s}|}c\|_{L^{\frac{2Q}{Q-2}}}.$

We conclude by using the Sobolev embedding
$\dot{H}^{1}(\mathbb{H}^{d})\hookrightarrow
L^{\frac{2Q}{Q-2}}(\mathbb{H}^{d})$. ∎

We have proved that the analytic smoothing effect holds at least in the
vertical variable. In the next section, we will improve the natural smoothing
effect from the left-invariant sub-Laplacian provided by the control of the
horizontal gradient of the solution in Theorem 6.1.1.

###  7.2 Horizontal regularity

We now analyze the smoothness of the solutions $u$ of (1.5)–(1.6) given by
Theorem 7.1.2 with respect to the horizontal vector fields
$(P_{j})_{j\in[\\![1,2d]\\!]}$, thus with respect to all variables.

######  Corollary 7.2.1.

Let $u$ be a solution of (1.5) constructed in Theorem 7.1.2. Then for any
nonnegative integers $\ell$, $\alpha\in[\\![1,2d]\\!]^{\ell}$,
$\beta\in\mathbb{N}$ and $t>0$, we have

$P^{\alpha}\partial_{s}^{\beta}u(t)\in\tilde{H}^{d}.$ (7.9)

Furthermore, $u$ and $p$ belong to
$\mathcal{C}^{\infty}((0,+\infty)\times\mathbb{R}^{2d+1})$. Accordingly, such
solution $u$ of (1.5)-(1.6) is a strong solution of (1.5).

For any $\sigma\in(0,4d)$, we set

$\sigma_{r}:=\frac{\sigma}{2^{r}},\ \ \ r\in\mathbb{N}.$

For any $\ell\in\mathbb{R}$ and $\ell^{\prime}\in[0,Q/2)$, we introduce the
space $H_{\ell,\ell^{\prime}}(\mathbb{H}^{d})$ by setting

$H_{\ell,\ell^{\prime}}(\mathbb{H}^{d}):=\left\\{f\in\mathcal{S}^{\prime}(\mathbb{H}^{d})~{}\middle|~{}(\operatorname{Id}-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{\frac{\ell}{2}}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{\ell^{\prime}}{2}}f\in
L^{2}(\mathbb{H}^{d})\right\\},$

and we denote by $\|\cdot\|_{H_{\ell,\ell^{\prime}}}$ the corresponding norms
and by $\langle\cdot,\cdot\rangle_{H_{\ell,\ell^{\prime}}}$ the corresponding
scalar product.

######  Lemma 7.2.2.

Let $u$ be a solution of (1.5)-(1.6) constructed in Theorem 7.1.2 with
corresponding radius of analyticity bounded from below by $t\mapsto\sigma t$
for $\sigma\in(0,4d)$, and $(u_{k})$ the sequence of solutions of the
approximate problems (4.9)-(4.10) which converges to $u$. Let $T$ and
$T_{\star}$ be two real numbers such that $T>T_{\star}>0$ and
$r\in\mathbb{N}$. Assume that there exist $T_{r}\in(0,T_{\star})$ and a
positive constant $C_{r}$ such that for any $k\in\mathbb{N}$, we have

$\|e^{\sigma_{r}t|D_{s}|}u_{k}\|^{2}_{L^{\infty}([T_{r},T];H_{r,d})}+\int_{T_{r}}^{T}\|e^{\sigma_{r}t|D_{s}|}u_{k}(t)\|_{H_{r+1,d}}^{2}dt\leqslant
C_{r}.$ (7.10)

Then there exist $T_{r+1}\in[T_{r},T_{\star})$ and a positive constant
$C_{r+1}$ such that for any $k\in\mathbb{N}$,

$\|e^{\sigma_{r+1}t|D_{s}|}u_{k}\|_{L^{\infty}([T_{r+1},T];H_{r+1,d})}^{2}+\int_{T_{r+1}}^{T}\|e^{\sigma_{r+1}t|D|}u_{k}(t)\|_{H_{r+2,d}}^{2}dt\leqslant
C_{r+1}.$ (7.11)

###### Proof of Lemma 7.2.2.

Energy estimate: Thanks to (7.10), we deduce that

$\forall k\in\mathbb{N},\ \exists
t_{k}\in\left(T_{r},\frac{T_{r}+T_{\star}}{2}\right)\ \ \
\|e^{\sigma_{r+1}t_{k}|D_{s}|}u_{k}(t_{k})\|_{H_{r+1,d}}^{2}\leqslant\frac{4C_{r}}{(T_{\star}-T_{r})}.$
(7.12)

Indeed, if this is not the case, there exists $k\in\mathbb{N}$ such that for
any $t\in\left(T_{r},(T_{\star}+T_{r})/2\right)$, we have

$\|e^{\sigma_{r+1}t|D_{s}|}u_{k}(t)\|_{H_{r+1,d}}^{2}>\frac{4C_{r}}{(T_{\star}-T_{r})}.$

This would entail, since $(T_{\star}+T_{r})/2<T_{\star}<T$, that

$\int_{T_{r}}^{T}\|e^{\sigma_{r+1}t_{k}|D_{s}|}u_{k}(t)\|_{H_{r+1,d}}^{2}dt>\frac{4C_{r}}{(T_{\star}-T_{r})}\times\frac{T_{\star}-T_{r}}{2}=2C_{r},$

which would contradict (7.10).

Now, let us consider two integers $a$ and $k$. Recall that
$\mathsf{J}_{a}^{2}=\mathsf{J}_{a}$ and that $\mathsf{J}_{a}$ is self-adjoint
(see Proposition 4.1.3 Items 5 and 7). Thanks to the commutation properties of
$\mathsf{J}_{a}$ (see Proposition 4.1.3 Items 3 and 4) and using that
$\mathsf{J}_{a}$ is bounded by $1$ in $\mathcal{L}(L^{2})$, for any
$t\in\left((T_{r}+T_{\star})/2,T_{\star}\right)$, we have

$\displaystyle\|e^{\sigma_{r+1}t|D_{s}|}\mathsf{J}_{a}u_{k}(t)$
$\displaystyle\|_{H_{r+1,d}}^{2}+2\left(1-\frac{\sigma_{r+1}}{4d}\right)\int_{t_{k}}^{t}\|e^{\sigma_{r+1}\tau|D_{s}|}(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{\frac{1}{2}}\mathsf{J}_{a}u_{k}(\tau)\|_{H_{r+1,d}}^{2}d\tau$
$\displaystyle\leqslant\|e^{\sigma_{r+1}t_{k}|D_{s}|}u_{k}(t_{k})\|_{H_{r+1,d}}^{2}$
$\displaystyle\ \ \ +\int_{t_{k}}^{t}|\langle
e^{\sigma_{r+1}\tau|D_{s}|}(\operatorname{Id}-\mathbb{P})\circ(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})u_{k}(\tau),e^{\sigma_{r+1}\tau|D_{s}|}\mathsf{J}_{a}u_{k}(\tau)\rangle_{H_{r+1,d}}|d\tau$
$\displaystyle\ \ \ +\int_{t_{k}}^{t}|\langle
e^{\sigma_{r+1}\tau|D_{s}|}\mathbb{P}\circ\mathsf{J}_{k}(u_{k}(\tau)\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k}(\tau)),e^{\sigma_{r+1}\tau|D_{s}|}\mathsf{J}_{a}u_{k}(\tau)\rangle_{H_{r+1,d}}|d\tau.$

Let us take $t\in[t_{k},T]$. According to (7.12), we obtain that

$\displaystyle\|e^{\sigma_{r+1}t|D_{s}|}\mathsf{J}_{a}u_{k}(t)\|_{H_{r+1,d}}^{2}+2$
$\displaystyle\left(1-\frac{\sigma_{r+1}}{4d}\right)\int_{t_{k}}^{t}\|e^{\sigma_{r+1}\tau|D_{s}|}(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{\frac{1}{2}}\mathsf{J}_{a}u_{k}(\tau)\|_{H_{r+1,d}}^{2}d\tau$
$\displaystyle\leqslant\frac{4C_{r}}{(T_{\star}-T_{r})}+I_{1}^{a,k}(t)+I_{2}^{a,k}(t),$
(7.13)

where

$I_{1}^{a,k}(t):=\int_{t_{k}}^{t}|\langle
e^{\sigma_{r+1}\tau|D_{s}|}(\operatorname{Id}-\mathbb{P})\circ(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})u_{k}(\tau),e^{\sigma_{r+1}\tau|D_{s}|}\mathsf{J}_{a}u_{k}(\tau)\rangle_{H_{r+1,d}}|d\tau$

and

$I_{2}^{a,k}(t):=\int_{t_{k}}^{t}|\langle
e^{\sigma_{r+1}\tau|D_{s}|}\mathbb{P}\circ\mathsf{J}_{k}(u_{k}(\tau)\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k}(\tau)),e^{\sigma_{r+1}\tau|D_{s}|}\mathsf{J}_{a}u_{k}(\tau)\rangle_{H_{r+1,d}}|d\tau.$

Estimate on $I_{1}^{a,k}(t)$. Let us recall that the Leray projector
$\mathbb{P}$ (defined in (3.2)) is of order $0$. Accordingly,
$(\operatorname{Id}-\mathbb{P})\circ(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})$
is of order $2$ which implies that to estimate $I_{1}^{a,k}(t)$, we need to
control $e^{\sigma_{r+1}\tau|D_{s}|}u_{k}$ at least in
$L^{2}((t_{k},T);H_{r+2,d})$. Unfortunately, the condition (7.10) gives only a
control of $e^{\sigma_{r}\tau|D_{s}|}u_{k}$ in $L^{2}((t_{k},T);H_{r+1,d})$.
We thus need to gain one additional regularity level with respect to the
horizontal derivative, to the price of possibly losing some regularity in the
vertical variable, since $\sigma_{r}>\sigma_{r+1}$. The key idea to overcome
this difficulty is to take advantage of the equation
$\operatorname{\mathop{}div}_{\mathbb{H}}(u_{k})=0$, so that we can use Lemma
4.2.3 and the identity
$(\operatorname{Id}-\mathbb{P})\circ(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})u_{k}=\Pi_{\mathbb{H}}\circ\partial_{s}u_{k}$,
where $\Pi_{\mathbb{H}}$ is of order $0$ and commutes with $|D_{s}|$. This
yields

$\displaystyle I_{1}^{a,k}(t)$ $\displaystyle=\int_{t_{k}}^{t}|\langle
e^{\sigma_{r+1}\tau|D_{s}|}\partial_{s}\circ\Pi_{\mathbb{H}}u_{k}(\tau),e^{\sigma_{r+1}\tau|D_{s}|}\mathsf{J}_{a}u_{k}(\tau)\rangle_{H_{r+1,d}}|d\tau$
$\displaystyle\leqslant\int_{t_{k}}^{t}\|e^{\sigma_{r+1}\tau|D_{s}|}|D_{s}|^{\frac{1}{2}}\Pi_{\mathbb{H}}u_{k}(\tau)\|_{H_{r+1,d}}\|e^{\sigma_{r+1}\tau|D_{s}|}|D_{s}|^{\frac{1}{2}}\mathsf{J}_{a}u_{k}(\tau)\|_{H_{r+1,d}}d\tau$
$\displaystyle\lesssim\int_{t_{k}}^{T}\|e^{\sigma_{r+1}\tau|D_{s}|}|D_{s}|^{\frac{1}{2}}u_{k}(\tau)\|_{H_{r+1,d}}^{2}d\tau.$
(7.14)

In order to bound $e^{\sigma_{r+1}\tau|D_{s}|}|D_{s}|^{\frac{1}{2}}u_{k}$ in
$L^{2}((t_{k},T);H_{r+1,d})$ we use Hypothesis (7.10): Indeed, for any
$\lambda\in\mathbb{R}^{*}$ and $\tau$ larger than $t_{k}$, we have

$e^{2\sigma_{r+1}\tau|\lambda|}|\lambda|\leqslant\frac{e^{2\sigma_{r+1}\tau|\lambda|}(2\sigma_{r+1}\tau|\lambda|)}{2\sigma_{r+1}t_{k}}\leqslant\frac{e^{4\sigma_{r+1}\tau|\lambda|}}{2\sigma_{r+1}t_{k}}=\frac{e^{2\sigma_{r}\tau|\lambda|}}{\sigma_{r}t_{k}}\leqslant\frac{e^{2\sigma_{r}\tau|\lambda|}}{\sigma_{r}T_{r}}.$

Thus, using the Plancherel formula on $\mathbb{H}^{d}$, we get

$\|e^{\sigma_{r+1}\tau|D_{s}|}|D_{s}|^{\frac{1}{2}}u_{k}(\tau)\|^{2}_{H_{r+1,d}}\leqslant\frac{1}{\sigma_{r+1}T_{r}}\|e^{\sigma_{r}\tau|D_{s}|}u_{k}(\tau)\|_{H_{r+1,d}}^{2}.$

Then, it follows from (7.14) and (7.10) that there exists a constant
$C_{I_{1}}$ independent of $k$ such that for any $t\in[t_{k},T]$,

$I_{1}^{a,k}(t)\lesssim\int_{T_{r}}^{T}\|e^{\sigma_{r}\tau|D_{s}|}u_{k}(\tau)\|_{H_{r+1,d}}^{2}d\tau\leqslant
C_{I_{1}}.$ (7.15)

Regularity of $u_{k}$. Let us first remark that, according to the properties
of $\mathsf{J}_{a}$, we have

$\displaystyle I_{2}^{a,k}(t)$ $\displaystyle\leqslant\int_{0}^{T}|\langle
e^{\sigma_{r+1}\tau|D_{s}|}(\operatorname{Id}-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{r+1}\circ\mathbb{P}\circ\mathsf{J}_{k}(u_{k}(\tau)\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k}(\tau)),e^{\sigma_{r+1}\tau|D_{s}|}\mathsf{J}_{a}u_{k}(\tau)\rangle_{\tilde{H}^{d}}|d\tau$
$\displaystyle\leqslant
T^{\frac{1}{2}}\|e^{\sigma_{r+1}\tau|D_{s}|}(\operatorname{Id}-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{r+1}\circ\mathbb{P}\circ\mathsf{J}_{k}(u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k})\|_{L^{2}_{T}(\tilde{H}^{d})}\|e^{\sigma_{r+1}\tau|D_{s}|}u_{k}\|_{L^{\infty}(\tilde{H}^{d})}.$

Since $\mathbb{P}$ commutes with
$(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{d}{2}}$
and $e^{\sigma_{r+1}\tau|D_{s}|}$, and belongs to $\mathcal{L}(H^{r+1})$, we
deduce that

$\displaystyle\|e^{\sigma_{r+1}\tau|D_{s}|}(\operatorname{Id}-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{r+1}\circ\mathbb{P}\circ\mathsf{J}_{k}($
$\displaystyle
u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k})\|_{L^{2}_{T}(\tilde{H}^{d})}$
$\displaystyle\leqslant\|\mathbb{P}\|_{\mathcal{L}(H^{r+1})}\|e^{\sigma_{r+1}\tau|D_{s}|}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{d}{2}}\mathsf{J}_{k}(u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k})\|_{L^{2}_{T}(H^{r+1})}.$

Furthermore, it follows from Proposition 4.1.3, Item $1$, that
$\mathsf{J}_{k}e^{\sigma_{r+1}\tau|D_{s}|}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{d}{2}}(u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k})$
belongs to $L^{2}_{T}(H^{r+1})$. Then, for any $k\in\mathbb{N}$, there exists
a constant $C_{k}$ such that

$\sup_{a\in\mathbb{N},t\in[t_{k},T]}\\{I^{a,k}_{2}(t)\\}\leqslant C_{k}.$

Thus, according to (7.10) and (7.15), it follows from (7.13) that the two
quantities

$\sup_{a\in\mathbb{N}}\\{\|e^{\sigma_{r+1}t|D_{s}|}\mathsf{J}_{a}u_{k}\|_{L^{\infty}(t_{k},T;H_{r+1,d})}^{2}\\}$

and

$\sup_{a\in\mathbb{N}}\\{\|e^{\sigma_{r+1}t|D_{s}|}(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{\frac{1}{2}}\mathsf{J}_{a}u_{k}\|_{L^{2}(t_{k},T;H_{r+1,d})}^{2}\\}$

are finite. Let us set

$F_{k}(t):=e^{\sigma_{r+1}t|D_{s}|}(\operatorname{Id}-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{r+1}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{d}{2}}u_{k}(t),$

with $t\in[t_{k},T]$. By using that $\mathsf{J}_{a}$ commute with
$e^{\sigma_{r+1}t|D_{s}|}(\operatorname{Id}-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{r+1}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{d}{2}}$
and $(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{\frac{1}{2}}$ and a
weak compactness argument, we deduce that, up to extract a subsequence and by
identifying the limits in
$\mathcal{D}^{\prime}((t_{k},T)\times\mathbb{H}^{d})$, the sequences
$(\mathsf{J}_{a}F_{k})_{a\in\mathbb{N}}$ and
$(\mathsf{J}_{a}(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{\frac{1}{2}}F_{k})_{a\in\mathbb{N}}$
converge, respectively, in $L^{\infty}(t_{k},T;L^{2})$ for the weak-$\star$
topology to $F_{k}$, and in $L^{2}(t_{k},T;L^{2})$ for the weak topology to
$(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{\frac{1}{2}}F_{k}$.
Moreover, it follows from $\eqref{III 31/07/2023}$ and $\eqref{V 31/07/2023}$
that for any $t\in(t_{k},T)$, we have

$\displaystyle\liminf_{a\rightarrow+\infty}\|e^{\sigma_{r+1}t|D_{s}|}\mathsf{J}_{a}u_{k}$
$\displaystyle\|_{L^{\infty}(t_{k},t;H_{r+1,d})}^{2}$
$\displaystyle+2\left(1-\frac{\sigma_{r+1}}{4d}\right)\liminf_{a\rightarrow+\infty}\|e^{\sigma_{r+1}t|D_{s}|}(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{\frac{1}{2}}\mathsf{J}_{a}u_{k}\|_{L^{2}(t_{k},t;H_{r+1,d})}^{2}$
$\displaystyle\leqslant\frac{4C_{r}}{(T_{\star}-T_{r})}+C_{I_{1}}+\sup_{a\in\mathbb{N}}\\{I^{a,k}_{2}(t)\\}.$

Thus according to the properties of the weak and the weak-$\star$ convergence,
we deduce that for any $t\in(t_{k},T)$, we have

$\displaystyle\|e^{\sigma_{r+1}t|D_{s}|}u_{k}(t)\|_{H_{r+1,d}}^{2}+2$
$\displaystyle\left(1-\frac{\sigma_{r+1}}{4d}\right)\int_{t_{k}}^{t}\|e^{\sigma_{r+1}\tau|D_{s}|}(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{\frac{1}{2}}u_{k}(\tau)\|_{H_{r+1,d}}^{2}d\tau$
$\displaystyle\leqslant\frac{4C_{r}}{(T_{\star}-T_{r})}+C_{I_{1}}+\sup_{a\in\mathbb{N}}\\{I^{a,k}_{2}(t)\\}.$
(7.16)

We are thus reduced to estimating $I^{a,k}_{2}(t)$ uniformly with respect to
the parameters $a\in\mathbb{N}$ and $k\in\mathbb{N}$.

Estimate on $I_{2}^{a,k}(t)$. Let us begin by remarking that
$\operatorname{\mathop{}div}_{\mathbb{H}}(u_{k})=0$ implies that

$u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k}=\operatorname{\mathop{}div}_{\mathbb{H}}(\mathsf{J}_{k}u_{k}\otimes
u_{k}).$ (7.17)

Besides, let us write

$\displaystyle(\operatorname{Id}-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{\frac{r+1}{2}}\circ\mathbb{P}\circ\mathsf{J}_{k}\circ\operatorname{\mathop{}div}_{\mathbb{H}}$
$\displaystyle=(\operatorname{Id}-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{-\frac{r+1}{2}}\circ\Gamma_{k}\circ(\operatorname{Id}-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{r+1},$
(7.18)

with

$\Gamma_{k}:=(\operatorname{Id}-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{r+1}\circ\mathbb{P}\circ\mathsf{J}_{k}\circ\operatorname{\mathop{}div}_{\mathbb{H}}\circ(\operatorname{Id}-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{-(r+1)}.$

Note that $\Gamma_{k}$ is an operator of order $1$ with respect to the left-
invariant sub-Laplacian, which maps $2d\times 2d$ matrix value functions to
horizontal vector fields, and thus its adjoint

$\Gamma_{k}^{*}=-(\operatorname{Id}-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{-(r+1)}\circ\nabla_{\mathbb{H}}\circ\mathsf{J}_{k}\circ\mathbb{P}\circ(\operatorname{Id}-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{r+1},$

is also of order 1 with respect to the left-invariant sub-Laplacian and maps
horizontal vector fields to $2d\times 2d$ matrix value functions. Then, using
(7.17), (7.18) and the binomial expansion on
$(\operatorname{Id}-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{r+1}$,
we get

$\displaystyle|\langle
e^{\sigma_{r+1}\tau|D_{s}|}\mathbb{P}\circ\mathsf{J}_{k}(u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}$
$\displaystyle
u_{k}),e^{\sigma_{r+1}\tau|D_{s}|}\mathsf{J}_{a}u_{k}\rangle_{H_{r+1,d}}|$
$\displaystyle=|\langle
e^{\sigma_{r+1}\tau|D_{s}|}\mathbb{P}\circ\mathsf{J}_{k}\circ\operatorname{\mathop{}div}_{\mathbb{H}}(\mathsf{J}_{k}u_{k}\otimes
u_{k}),e^{\sigma_{r+1}\tau|D_{s}|}\mathsf{J}_{a}u_{k}\rangle_{H_{r+1,d}}|$
$\displaystyle=|\langle
e^{\sigma_{r+1}\tau|D_{s}|}(\operatorname{Id}-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{r+1}(\mathsf{J}_{k}u_{k}\otimes
u_{k}),e^{\sigma_{r+1}\tau|D_{s}|}\Gamma_{k}^{*}\mathsf{J}_{a}u_{k}\rangle_{\tilde{H}^{d}}|$
$\displaystyle\lesssim\sum_{\ell=0}^{r+1}\sum_{\gamma,\gamma^{\prime}\in[\\![1,2d]\\!]^{\ell}}|\langle
e^{\sigma_{r+1}\tau|D_{s}|}P^{\gamma}(\mathsf{J}_{k}u_{k}\otimes
u_{k}),e^{\sigma_{r+1}\tau|D_{s}|}P^{\gamma^{\prime}}\Gamma_{k}^{*}\mathsf{J}_{a}u_{k}\rangle_{\tilde{H}^{d}}|.$

Furthermore, if we take $\ell\in[\\![0,r+1]\\!]$ and
$\gamma,\gamma^{\prime}\in[\\![1,2d]\\!]^{\ell}$, we have

$\displaystyle|\langle e^{\sigma_{r+1}\tau|D_{s}|}P^{\gamma}($
$\displaystyle\mathsf{J}_{k}u_{k}\otimes
u_{k}),e^{\sigma_{r+1}\tau|D_{s}|}P^{\gamma^{\prime}}\Gamma_{k}^{*}\mathsf{J}_{a}u_{k}\rangle_{\tilde{H}^{d}}|$
$\displaystyle\lesssim\|e^{\sigma_{r+1}\tau|D_{s}|}P^{\gamma}(u\otimes
u)\|_{\tilde{H}^{d}}\|e^{\sigma_{r+1}\tau|D_{s}|}\Gamma^{*}_{k}\mathsf{J}_{a}u_{k}\|_{H_{\ell,d}}$
$\displaystyle\lesssim\left(\sum_{\tilde{\gamma}\in[\\![1,2d]\\!]^{d}}\|e^{\sigma_{r+1}\tau|D_{s}|}\tilde{P}^{\tilde{\gamma}}P^{\gamma}(\mathsf{J}_{k}u_{k}\otimes
u_{k})\|_{L^{2}}\right)\|e^{\sigma_{r+1}\tau|D_{s}|}\Gamma_{k}^{*}\mathsf{J}_{a}u_{k}\|_{H_{r+1,d}}.$

On one hand, using that $\mathbb{P}$ commutes with
$(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{d}{2}}$,
we have

$\displaystyle\|e^{\sigma_{r+1}\tau|D_{s}|}\Gamma_{k}^{*}\mathsf{J}_{a}u_{k}\|_{H_{r+1,d}}$
$\displaystyle=\|\nabla_{\mathbb{H}}\circ\mathsf{J}_{k}\circ\mathbb{P}\circ(\operatorname{Id}-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{r+1}e^{\sigma_{r+1}\tau|D_{s}|}\mathsf{J}_{a}u_{k}\|_{\tilde{H}^{d}}$
$\displaystyle=\|(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{\frac{1}{2}}\circ\mathsf{J}_{k}\circ\mathbb{P}\circ\mathsf{J}_{a}\circ(\operatorname{Id}-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{r+1}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{d}{2}}e^{\sigma_{r+1}\tau|D_{s}|}u_{k}\|_{L^{2}}.$

Then, using that $\mathsf{J}_{k}$ and $\mathsf{J}_{a}$ are bounded by $1$ in
$\mathcal{L}(\dot{H}^{1})$, we deduce that

$\displaystyle\|e^{\sigma_{r+1}\tau|D_{s}|}\Gamma_{k}^{*}\mathsf{J}_{a}u_{k}\|_{H_{r+1,d}}$
$\displaystyle\leqslant\|\mathbb{P}\|_{\mathcal{L}(\dot{H}^{1})}\|(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{\frac{1}{2}}(\operatorname{Id}-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{r+1}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{d}{2}}e^{\sigma_{r+1}\tau|D_{s}|}u_{k}\|_{\tilde{H}^{d}}$
$\displaystyle=\|\mathbb{P}\|_{\mathcal{L}(\dot{H}^{1})}\|e^{\sigma_{r+1}\tau|D_{s}|}(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{\frac{1}{2}}u_{k}\|_{H_{r+1,d}}.$

On the other hand, by applying the Leibniz formula for $P^{\gamma}$ with
$\gamma\in[\\![1,2d]\\!]^{\ell}$ and for $\tilde{P}^{\tilde{\gamma}}$ with
$\tilde{\gamma}\in[\\![1,2d]\\!]^{d}$, we deduce that

$\|e^{\sigma_{r+1}\tau|D_{s}|}P^{\gamma}\tilde{P}^{\tilde{\gamma}}(\mathsf{J}_{k}u_{k}\otimes
u_{k})\|_{L^{2}}\lesssim\sum_{\begin{subarray}{c}0\leqslant i\leqslant\ell\\\
0\leqslant j\leqslant
d\end{subarray}}\sum_{\begin{subarray}{c}\alpha\in[\\![1,2d]\\!]^{i}\\\
\beta\in[\\![1,2d]\\!]^{\ell-i}\end{subarray}}\sum_{\begin{subarray}{c}\tilde{\alpha}\in[\\![1,2d]\\!]^{j}\\\
\tilde{\beta}\in[\\![1,2d]\\!]^{d-j}\end{subarray}}\|e^{\sigma_{r+1}\tau|D_{s}|}(\tilde{P}^{\tilde{\alpha}}P^{\alpha}\mathsf{J}_{k}u_{k}\otimes\tilde{P}^{\tilde{\beta}}P^{\beta}u_{k})\|_{L^{2}}.$

Then we have

$\displaystyle|\langle e^{\sigma_{r+1}\tau|D_{s}|}$
$\displaystyle\mathbb{P}\circ\mathsf{J}_{k}(u_{k}\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u_{k}),e^{\sigma_{r+1}\tau|D_{s}|}\mathsf{J}_{a}u_{k}\rangle_{H_{r+1,d}}|$
$\displaystyle\lesssim\|e^{\sigma_{r+1}\tau|D_{s}|}\nabla_{\mathbb{H}}u_{k}\|_{H_{r+1,d}}$
$\displaystyle\ \ \ \ \
\times\sum_{\ell=0}^{r+1}\sum_{\begin{subarray}{c}0\leqslant i\leqslant\ell\\\
0\leqslant j\leqslant
d\end{subarray}}\sum_{\begin{subarray}{c}\alpha\in[\\![1,2d]\\!]^{i}\\\
\beta\in[\\![1,2d]\\!]^{\ell-i}\end{subarray}}\sum_{\begin{subarray}{c}\tilde{\alpha}\in[\\![1,2d]\\!]^{j}\\\
\tilde{\beta}\in[\\![1,2d]\\!]^{d-j}\end{subarray}}\|e^{\sigma_{r+1}\tau|D_{s}|}(\tilde{P}^{\tilde{\alpha}}P^{\alpha}\mathsf{J}_{k}u_{k}\otimes\tilde{P}^{\tilde{\beta}}P^{\beta}u_{k})\|_{L^{2}}.$
(7.19)

Let us point out that the Calderón-Zygmund theory using in Lemma 7.1.4 cannot
be applied in order to obtain a suitable estimate on $I_{2}^{a,k}(t)$ in the
cases $(i,j)=(r+1,d)$ and $(i,j)=(0,0)$. Indeed, in these cases, we would like
to choose one of the indexes $p_{1}$ or $p_{2}$ in Lemma 7.1.4 as $1$ or
$+\infty$, which is not allowed. In order to work around this difficulty, we
will use interpolation inequalities. Let us fix $\theta\in(0,1)$. Let
$\ell\in[\\![0,r+1]\\!]$, $i\in[\\![0,\ell]\\!]$, $j\in[\\![0,d]\\!]$,
$\alpha\in[\\![1,2d]\\!]^{i}$, $\tilde{\alpha}\in[\\![1,2d]\\!]^{j}$,
$\beta\in[\\![1,2d]\\!]^{\ell-i}$ and $\tilde{\beta}\in[\\![1,2d]\\!]^{d-j}$.
Then, using Lemma 7.1.4, we get

$\displaystyle\|e^{\sigma_{r+1}\tau|D_{s}|}(\tilde{P}^{\tilde{\alpha}}P^{\alpha}\mathsf{J}_{k}u_{k}\otimes\tilde{P}^{\tilde{\beta}}$
$\displaystyle P^{\beta}u_{k})\|_{L^{2}}$
$\displaystyle\leqslant\|e^{\sigma_{r+1}\tau|D_{s}|}\tilde{P}^{\tilde{\alpha}}P^{\alpha}\mathsf{J}_{k}u_{k}\|_{L^{\frac{2Q}{Q-2(d-j+\theta)}}}\|e^{\sigma_{r+1}\tau|D_{s}|}\tilde{P}^{\tilde{\beta}}P^{\beta}u_{k}\|_{L^{\frac{2Q}{Q-2(j+1-\theta)}}}.$

Thanks to the Sobolev embeddings
$\tilde{W}^{d-j,\frac{2Q}{Q-2\theta}}_{\mathbb{H}}\hookrightarrow
L^{\frac{2Q}{Q-2(d-j+\theta)}}$ and
$\tilde{W}^{j,\frac{2Q}{Q-2(1-\theta)}}_{\mathbb{H}}\hookrightarrow
L^{\frac{2Q}{Q-2(j+1-\theta)}}$, we obtain

$\displaystyle\|e^{\sigma_{r+1}\tau|D_{s}|}\tilde{P}^{\tilde{\alpha}}P^{\alpha}\mathsf{J}_{k}u_{k}\|_{L^{\frac{2Q}{Q-2(d-j+\theta)}}}\lesssim\|e^{\sigma_{r+1}\tau|D_{s}|}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{d}{2}}(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{\frac{i}{2}}\mathsf{J}_{k}u_{k}\|_{L^{\frac{2Q}{Q-2\theta}}}$

and

$\displaystyle\|e^{\sigma_{r+1}\tau|D_{s}|}\tilde{P}^{\tilde{\beta}}P^{\beta}u_{k}\|_{L^{\frac{2Q}{Q-2(j+1-\theta)}}}\lesssim\|e^{\sigma_{r+1}\tau|D_{s}|}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{d}{2}}(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{\frac{\ell-i}{2}}u_{k}\|_{L^{\frac{2Q}{Q-2(1-\theta)}}}.$

Furthermore, using interpolation, the Sobolev embedding
$\dot{H}^{1}(\mathbb{H}^{d})\hookrightarrow
L^{\frac{2Q}{Q-2}}(\mathbb{H}^{d})$ and the properties of $\mathsf{J}_{k}$, we
deduce that

$\displaystyle\|e^{\sigma_{r+1}\tau|D_{s}|}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{d}{2}}(-$
$\displaystyle\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{\frac{i}{2}}\mathsf{J}_{k}u_{k}\|_{L^{\frac{2Q}{Q-2\theta}}}$
$\displaystyle\leqslant\|e^{\sigma_{r+1}\tau|D_{s}|}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{d}{2}}(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{\frac{i}{2}}u_{k}\|_{L^{2}}^{1-\theta}\|e^{\sigma_{r+1}\tau|D_{s}|}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{d}{2}}(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{\frac{i}{2}}\mathsf{J}_{k}u_{k}\|^{\theta}_{L^{\frac{2Q}{Q-2}}}$
$\displaystyle\lesssim\|e^{\sigma_{r+1}\tau|D_{s}|}u_{k}\|_{H_{i,d}}^{1-\theta}\|e^{\sigma_{r+1}\tau|D_{s}|}\nabla_{\mathbb{H}}u_{k}\|_{H_{i,d}}^{\theta}$

and

$\displaystyle\|e^{\sigma_{r+1}\tau|D_{s}|}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}}$
$\displaystyle)^{\frac{d}{2}}(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{\frac{\ell-i}{2}}u_{k}\|_{L^{\frac{2Q}{Q-2(1-\theta)}}}$
$\displaystyle\leqslant\|e^{\sigma_{r+1}\tau|D_{s}|}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{d}{2}}(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{\frac{\ell-i}{2}}u_{k}\|_{L^{2}}^{\theta}\|e^{\sigma_{r+1}\tau|D_{s}|}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\frac{d}{2}}(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{\frac{\ell-i}{2}}u_{k}\|_{L^{\frac{2Q}{Q-2}}}^{1-\theta}$
$\displaystyle\lesssim\|e^{\sigma_{r+1}\tau|D_{s}|}u_{k}\|_{H_{\ell-i,d}}^{\theta}\|e^{\sigma_{r+1}\tau|D_{s}|}\nabla_{\mathbb{H}}u_{k}\|_{H_{\ell-i,d}}^{1-\theta}.$

Then, it follows that

$\displaystyle\|$ $\displaystyle
e^{\sigma_{r+1}\tau|D_{s}|}(\tilde{P}^{\tilde{\alpha}}P^{\alpha}\mathsf{J}_{k}u_{k}\otimes\tilde{P}^{\tilde{\beta}}P^{\beta}u_{k})\|_{L^{2}}$
$\displaystyle\lesssim\|e^{\sigma_{r+1}\tau|D_{s}|}u_{k}\|_{H_{i,d}}^{1-\theta}\|e^{\sigma_{r+1}\tau|D_{s}}\nabla_{\mathbb{H}}u_{k}\|_{H_{i,d}}^{\theta}\|e^{\sigma_{r+1}\tau|D_{s}|}u_{k}\|_{H_{\ell-i,d}}^{\theta}\|e^{\sigma_{r+1}\tau|D_{s}|}\nabla_{\mathbb{H}}u_{k}\|^{1-\theta}_{H_{\ell-i,d}}.$
(7.20)

We have to distinguish three cases in order to estimate the right-hand side of
(7.20).

$\bullet$ Case $\ell<r+1$ or $\ell=r+1$ and $0<i<r+1$. In this case, we have
$i\leqslant r$ and $\ell-i\leqslant r$. It then follows that

$\displaystyle\|e^{\sigma_{r+1}\tau|D_{s}|}u_{k}\|_{H_{i,d}}^{1-\theta}\|e^{\sigma_{r+1}\tau|D_{s}|}\nabla_{\mathbb{H}}u_{k}\|_{H_{i,d}}^{\theta}\|e^{\sigma_{r+1}\tau|D_{s}|}u_{k}\|_{H_{\ell-i,d}}^{\theta}\|$
$\displaystyle
e^{\sigma_{r+1}\tau|D_{s}|}\nabla_{\mathbb{H}}u_{k}\|_{H_{\ell-i,d}}^{1-\theta}$
$\displaystyle\leqslant\|e^{\sigma_{r+1}\tau|D_{s}|}u_{k}\|^{2}_{H_{r+1,d}}.$

$\bullet$ Case $\ell=r+1$ and $i=0$. In this case, since
$\sigma_{r+1}<\sigma_{r}$, according to (7.10), we get

$\displaystyle\|e^{\sigma_{r+1}\tau|D_{s}|}u_{k}\|_{\tilde{H}^{d}}^{1-\theta}\|e^{\sigma_{r+1}\tau|D_{s}|}$
$\displaystyle\nabla_{\mathbb{H}}u_{k}\|_{\tilde{H}^{d}}^{\theta}\|e^{\sigma_{r+1}\tau|D_{s}|}u_{k}\|_{H_{r+1,d}}^{\theta}\|e^{\sigma_{r+1}\tau|D_{s}|}\nabla_{\mathbb{H}}u_{k}\|_{H_{r+1,d}}^{1-\theta}$
$\displaystyle\leqslant
C_{r}^{1-\theta}\|e^{\sigma_{r+1}\tau|D_{s}|}u_{k}\|_{H_{r+1,d}}^{2\theta}\|e^{\sigma_{r+1}\tau|D_{s}|}\nabla_{\mathbb{H}}u_{k}\|_{H_{r+1,d}}^{1-\theta}.$

$\bullet$ Case $i=\ell=r+1$. Similarly as above, it follows from (7.10) that

$\displaystyle\|e^{\sigma_{r+1}\tau|D_{s}|}u_{k}\|_{H_{r+1,d}}^{1-\theta}\|e^{\sigma_{r+1}\tau|D_{s}|}\nabla_{\mathbb{H}}u_{k}$
$\displaystyle\|_{H_{r+1,d}}^{\theta}\|e^{\sigma_{r+1}\tau|D_{s}|}u_{k}\|_{\tilde{H}^{d}}^{\theta}\|e^{\sigma_{r+1}\tau|D_{s}|}\nabla_{\mathbb{H}}u_{k}\|_{\tilde{H}^{d}}^{1-\theta}$
$\displaystyle\leqslant
C_{r}^{\theta}\|e^{\sigma_{r+1}\tau|D_{s}|}\nabla_{\mathbb{H}}u_{k}\|_{H_{r+1,d}}^{\theta}\|e^{\sigma_{r+1}\tau|D_{s}|}u_{k}\|_{H_{r+1,d}}^{2(1-\theta)}.$

Thus, according to (7.19) and (7.20), we deduce that there exists a positive
constant $C_{\star}$ independent of $k$ and $a$ such that

$\displaystyle I_{2}^{a,k}(t)\leqslant$ $\displaystyle
C_{\star}\int_{t_{k}}^{t}\|e^{\sigma_{r+1}\tau|D_{s}|}\nabla_{\mathbb{H}}u_{k}(\tau)\|_{H_{r+1,d}}\|e^{\sigma_{r+1}\tau|D_{s}|}u_{k}(\tau)\|_{H_{r+1,d}}^{2}d\tau$
$\displaystyle+C_{\star}\int_{t_{k}}^{t}\|e^{\sigma_{r+1}\tau|D_{s}|}u_{k}(\tau)\|_{H_{r+1,d}}^{2\theta}\|e^{\sigma_{r+1}\tau|D_{s}|}\nabla_{\mathbb{H}}u_{k}(\tau)\|_{H_{r+1,d}}^{2-\theta}d\tau$
$\displaystyle+C_{\star}\int_{t_{k}}^{t}\|e^{\sigma_{r+1}\tau|D_{s}|}\nabla_{\mathbb{H}}u_{k}(\tau)\|_{H_{r+1,d}}^{1+\theta}\|e^{\sigma_{r+1}\tau|D_{s}|}u_{k}(\tau)\|_{H_{r+1,d}}^{2(1-\theta)}d\tau.$

In view of $\theta\in(0,1)$, by using the Young estimates with $1/2+1/2=1$,
$\theta/2+(2-\theta)/2=1$ and $(1+\theta)/2+(1-\theta)/2=1$ we deduce that
there exists a positive constant $\tilde{C}_{*}$, such that

$\displaystyle I_{2}^{a,k}(t)\leqslant$
$\displaystyle\left(1-\frac{\sigma_{r+1}}{4d}\right)\int_{t_{k}}^{t}\|e^{\sigma_{r+1}\tau|D_{s}|}\nabla_{\mathbb{H}}u_{k}(\tau)\|^{2}_{H_{r+1,d}}d\tau$
$\displaystyle+\tilde{C}_{*}\int^{t}_{t_{k}}\|e^{\sigma_{r+1}\tau|D_{s}|}u_{k}(\tau)\|_{H_{r+1,d}}^{4}d\tau.$
(7.21)

Conclusion. By combining (7.14) and (7.21), according to (7.13), we conclude
that for all $t\in(t_{k},T)$,

$\displaystyle\|e^{\sigma_{r+1}t|D_{s}|}u_{k}(t)$
$\displaystyle\|_{H_{r+1,d}}^{2}+\left(1-\frac{\sigma_{r+1}}{4d}\right)\int_{t_{k}}^{t}\|e^{\sigma_{r+1}\tau|D_{s}|}\nabla_{\mathbb{H}}u_{k}(\tau)\|_{H_{r+1,d}}^{2}d\tau$
$\displaystyle\leqslant\frac{4C_{r}}{(T_{\star}-T_{r})}+C_{I_{1}}+\tilde{C}_{*}\int_{t_{k}}^{t}\|e^{\sigma_{r+1}\tau|D_{s}|}u_{k}(\tau)\|_{H_{r+1,d}}^{4}d\tau.$

We deduce that there exists a positive constant $C^{\star}$ independent of $k$
such that for all $t\in(t_{k},T)$,

$\displaystyle\|e^{\sigma_{r+1}t|D_{s}|}u_{k}(t)\|_{H_{r+1,d}}^{2}+\int_{t_{k}}^{t}\|e^{\sigma_{r+1}\tau|D_{s}|}$
$\displaystyle\nabla_{\mathbb{H}}u_{k}(\tau)\|_{H_{r+1,d}}^{2}d\tau$
$\displaystyle\leqslant
C^{\star}+C^{\star}\int_{t_{k}}^{t}\|e^{\sigma_{r+1}\tau|D_{s}|}u_{k}(\tau)\|^{4}_{H_{r+1,d}}d\tau.$
(7.22)

By applying the Gronwall lemma, we deduce from (7.22), (7.10),
$\sigma_{r+1}<\sigma_{r}$ and $t_{k}>T_{r}$ that

$\|e^{\sigma_{r+1}t|D_{s}|}u_{k}\|_{L^{\infty}([t_{k},T];H_{r+1,d})}^{2}\leqslant
C^{\star}e^{C^{\star}\int_{t_{k}}^{T}\|e^{\sigma_{r+1}\tau|D_{s}|}u_{k}(\tau)\|_{H_{r+1,d}}^{2}d\tau}\leqslant
C^{\star}e^{C^{\star}C_{r}}.$

Thus, according to (7.22), we conclude that for any $t\in(t_{k},T)$

$\displaystyle\|e^{\sigma_{r+1}t|D_{s}|}u_{k}(t)\|_{H_{r+1,d}}^{2}+\int_{t_{k}}^{t}\|e^{\sigma_{r+1}\tau|D_{s}|}\nabla_{\mathbb{H}}u_{k}(\tau)\|_{H_{r+1,d}}^{2}d\tau\leqslant
C^{\star}+C^{\star}e^{C^{\star}C_{r}(T-t_{k})}.$

Then, by setting

$T_{r+1}:=\sup\\{t_{k}\\}\in\left(T_{r},T_{\star}\right),$

since $T_{r}<t_{k}\leqslant T_{r+1}$, we deduce that

$\displaystyle\|e^{\sigma_{r+1}t|D_{s}|}u_{k}\|_{L^{\infty}([T_{r+1},T];H_{r+1,d})}^{2}+\int_{T_{r+1}}^{T}\|e^{\sigma_{r+1}\tau|D_{s}|}\nabla_{\mathbb{H}}u_{k}(\tau)\|_{H_{r+1,d}}^{2}d\tau\leqslant
C^{\star}+C^{\star}e^{C^{\star}C_{r}(T-T_{r})}.$

This complete the proof of the lemma. ∎

######  Remark 7.2.3.

The proof of Lemma 7.2.2 contains two main difficulties: the main difficulty
due to the term
$(\operatorname{Id}-\mathbb{P})\circ(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})$
appears in the estimate of the term $I^{a,k}_{1}$; in order to handle
$I_{2}^{a,k}$, we have to use the operators $\mathsf{J}_{a}$ to show that each
$u_{k}$ belongs to the appropriate space of regularity with respect to the
left invariant sub-Laplacian
$\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}}$, so that it can be used to
absorb a part of the nonlinear term with the diffusive term, see (7.21).

Now, we are able to show Corollary 7.2.1.

###### Proof of Corollary 7.2.1..

We shall perform a bootstrap argument in order to show that the solution is
smooth. Let us consider two real numbers $T_{\star}$ and $T^{\star}$ such that
$T^{\star}>T_{\star}>0$. According to (7.2), we deduce that there exists a
positive real number $C_{0}$ such that

$\|e^{\sigma
t|D_{s}|}u\|_{L^{\infty}([0,T^{\star}];\tilde{H}^{d})}+\int_{0}^{T^{\star}}\|e^{\sigma\tau|D_{s}|}\nabla_{\mathbb{H}}u(\tau)\|_{\tilde{H}^{d}}^{2}d\tau\leqslant
C_{0}.$

By reasoning by induction using Lemma 7.2.2 and passing to the limit in
$(u_{k})$, up to extract a subsequence, we deduce that for any
$r\in\mathbb{N}$ and $\beta\in\mathbb{N}$

$\|u\|_{L^{\infty}([T_{\star},T^{\star}];H_{r,d})}\lesssim 1.$ (7.23)

Let $\alpha\in\mathbb{N}^{2d+1}$. There exists a family of polynomial
function, on the horizontal variable $Y$,
$(\mu_{\beta,\gamma})_{(\beta,\gamma)\in\Gamma_{\alpha}}$ where
$\Gamma_{\alpha}$ is a subset of
$\left(\bigcup_{i=0}^{|\alpha|}[\\![1,2d]\\!]^{i}\right)\times[\\![1,|\alpha|]\\!]$,
such that

$\partial^{\alpha}=\sum_{(\beta,\gamma)\in\Gamma_{\alpha}}\mu_{\beta,\gamma}P^{\beta}\partial_{s}^{\gamma}.$

Thus, for any open bounded subset $\Omega$ of $\mathbb{R}^{2d+1}$ and for any
$t\in[T_{\star},T^{\star}]$, we have

$\displaystyle\|\partial^{\alpha}u(t)\|_{L^{\infty}(\Omega)}\lesssim\sum_{i=0}^{|\alpha|}\sum_{\begin{subarray}{c}\beta\in[\\![1,2d]\\!]^{i}\\\
\gamma\in[\\![1,|\alpha|]\\!]\end{subarray}}\|P^{\beta}\partial_{s}^{\gamma}u(t)\|_{L^{\infty}(\Omega)}.$
(7.24)

We have a continuous embedding from $W^{2,Q}_{\mathbb{H}}(\mathbb{H}^{d})$
into $\mathcal{C}(\mathbb{R}^{2d+1})\cap L^{\infty}(\mathbb{R}^{2d+1})$. By
using the Sobolev embedding of $\tilde{H}^{d}(\mathbb{H}^{d})$ into
$L^{Q}(\mathbb{H}^{d})$, we conclude that $H_{2,d}(\mathbb{H}^{d})$ is
continuously embedding in $\mathcal{C}_{b}(\mathbb{R}^{2d+1})$. Thus, we
deduce from (7.24) and Proposition 2.4.8, that for any
$t\in[T_{\star},T^{\star}]$, we have

$\displaystyle\|\partial^{\alpha}u(t)\|_{L^{\infty}(\Omega)}$
$\displaystyle\lesssim\sum_{i=0}^{|\alpha|}\sum_{\begin{subarray}{c}\alpha\in[\\![1,2d]\\!]^{i}\\\
\gamma\in[\\![1,|\alpha|]\\!]\end{subarray}}\|P^{\beta}\partial_{s}^{\gamma}u(t)\|_{H_{2,d}}\lesssim\|u(t)\|_{H_{3|\alpha|+2,d}}.$

It follows from (7.23) that for any $t\in[T_{\star},T^{\star}]$, we have
$\partial^{\alpha}u(t)\in L^{\infty}(\Omega)$. Since $T_{\star}<T^{\star}$,
$\alpha$ and $\Omega$ are arbitrarily chosen, this shows that for any $t>0$,
$u(t)$ belongs to $\mathcal{C}^{\infty}(\mathbb{R}^{2d+1})$. Since $p$ is
explicitly given by (3.1), by hypoellipticity (see [27]) of the left-invariant
sub-Laplacian $\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}}$, we also
have that for all $t>0$, $p(t)\in\mathcal{C}^{\infty}(\mathbb{R}^{2d+1})$.
Besides, for any $\ell\in\mathbb{N}^{*}$, we have

$\partial_{t}^{\ell}u=\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}}\partial_{t}^{\ell-1}u-\sum_{k=0}^{\ell-1}\binom{\ell-1}{k}\partial_{t}^{k}u\cdot\nabla_{\mathbb{H}}\partial_{t}^{\ell-1-k}u-\nabla_{\mathbb{H}}\partial_{t}^{\ell-1}p.$

By induction on $\ell$, we deduce that for any $t>0$ and $\ell\in\mathbb{N}$,
$\partial_{t}^{\ell}u(t)$, and then, according to the hypoellipticity of
$\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}}$,
$\partial_{t}^{\ell}p(t)$, belong to
$\mathcal{C}^{\infty}(\mathbb{R}^{2d+1})$. We conclude that $u$ and $p$
belongs to $\mathcal{C}^{\infty}((0,+\infty)\times\mathbb{R}^{2d+1})$. ∎

##  8 Long time existence in $\dot{H}^{d}(\mathbb{H}^{d})$

In this subsection, we discuss the existence of solution for (1.5) in
$\dot{H}^{d}(\mathbb{H}^{d})$, and explain how to obtain long time existence
results in this framework.

######  Theorem 8.0.1.

Let $T_{\star}>0$. Then there exist $a>0$ and $\varepsilon>0$ such that for
any horizontal vector field $u_{0}\in\dot{H}^{d}(\mathbb{H}^{d})$ with
$\operatorname{\mathop{}div}_{\mathbb{H}}(u_{0})=0$ and
$\|e^{a|D_{s}|}u_{0}\|_{\dot{H}^{d}}<\varepsilon$, there exists a solution $u$
of (1.5)-(1.6) satisfying $u\in\mathcal{C}_{b}([0,T_{\star}];\dot{H}^{d})$ and
$\nabla_{\mathbb{H}}u\in L^{2}((0,T_{\star});\dot{H}^{d})$.

Note that Theorem 8.0.1 requires the initial datum to have some analyticity
properties with respect to the vertical variable. Also note that the
uniqueness of the solution of (1.5)-(1.6) satisfying
$u\in\mathcal{C}_{b}([0,T_{\star}];\dot{H}^{d})$ and $\nabla_{\mathbb{H}}u\in
L^{2}((0,T_{\star});\dot{H}^{d})$ is an open problem.

###### Proof.

We only give the a priori estimates on smooth solution. The the convergence is
left to the reader. Let $a>0$, which will be chosen later and $u_{0}$ be a
horizontal vector field in $\dot{H}^{d}(\mathbb{H}^{d})$ so that

$\operatorname{\mathop{}div}_{\mathbb{H}}(u_{0})=0\ \ \text{and}\ \
\|e^{a|D_{s}|}u_{0}\|_{\dot{H}^{d}}<\varepsilon,$

for some $\varepsilon>0$ small enough which we will fix later in the proof.
Let us consider a decreasing positive function $\delta$ in
$\mathcal{C}^{1}(\mathbb{R}_{+})$ (which depend on the time variable $t$). Let
$u$ a solution of (4.9)-(4.10) for some index $k\in\mathbb{N}$. We have

$\frac{1}{2}\frac{d}{dt}\|e^{\delta|D_{s}|}u\|_{\dot{H}^{d}}^{2}-\dot{\delta}\|e^{\delta|D_{s}|}|D_{s}|^{\frac{1}{2}}u\|_{\dot{H}^{d}}^{2}+\|e^{\delta|D_{s}|}\nabla_{\mathbb{H}}u\|_{\dot{H}^{d}}^{2}\\\
=-\langle
e^{\delta|D_{s}|}\mathsf{J}_{k}(\operatorname{Id}-\mathbb{P})\circ(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})u,e^{\delta|D_{s}|}u\rangle_{\dot{H}^{d}}-\langle
e^{\delta|D_{s}|}\mathsf{J}_{k}\mathbb{P}(u\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u),e^{\delta|D_{s}|}u\rangle_{\dot{H}^{d}}.$
(8.1)

Since
$(\operatorname{Id}-\mathbb{P})\circ(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})u=\Pi_{\mathbb{H}}\partial_{s}u$
and thanks to Proposition 4.1.3 Items 3, 4 and 5, we deduce that there exists
a constant $C_{\star}$ which does not depend on $k\in\mathbb{N}$ such that

$|\langle
e^{\delta|D_{s}|}\mathsf{J}_{k}(\operatorname{Id}-\mathbb{P})\circ(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})u,e^{\delta|D_{s}|}u\rangle_{\dot{H}^{d}}|\leqslant
C_{\star}\|e^{\delta|D_{s}|}|D_{s}|^{\frac{1}{2}}u\|_{\dot{H}^{d}}^{2}.$

Then, it follows from Proposition 2.4.8 and (8.1) that

$\frac{1}{2}\frac{d}{dt}\|e^{\delta|D_{s}|}u\|_{\dot{H}^{d}}^{2}+\left(1-\frac{(\dot{\delta}+C_{\star})}{4d}\right)\|e^{\delta|D_{s}|}\nabla_{\mathbb{H}}u\|_{\dot{H}^{d}}^{2}\leqslant-\langle
e^{\delta|D_{s}|}\mathsf{J}_{k}\mathbb{P}(u\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u),e^{\delta|D_{s}|}u\rangle_{\dot{H}^{d}}.$
(8.2)

Using Proposition 4.1.3 Items 3, 4 and 5 and performing the same argument as
in the proof of Lemma 6.1.3 we deduce that

$\displaystyle|\langle
e^{\delta|D_{s}|}\mathsf{J}_{k}\mathbb{P}(u\cdot\nabla_{\mathbb{H}}\mathsf{J}_{k}u),e^{\delta|D_{s}|}u\rangle_{\dot{H}^{d}}|\leqslant
C^{\star}\|e^{\delta|D_{s}|}u\|_{\dot{H}^{d}}\|e^{\delta|D_{s}|}\nabla_{\mathbb{H}}u\|_{\dot{H}^{d}}^{2}.$
(8.3)

Accordingly, using (8.2) and (8.3), we get

$\frac{1}{2}\frac{d}{dt}\|e^{\delta|D_{s}|}u\|_{\dot{H}^{d}}^{2}+\left(1-\frac{(\dot{\delta}+C_{\star})}{4d}\right)\|e^{\delta|D_{s}|}\nabla_{\mathbb{H}}u\|_{\dot{H}^{d}}^{2}\leqslant
C^{\star}\|e^{\delta|D_{s}|}u\|_{\dot{H}^{d}}\|e^{\delta|D_{s}|}\nabla_{\mathbb{H}}u\|_{\dot{H}^{d}}^{2}.$
(8.4)

We now choose $T_{\star}>0$, $a$ and $\delta$ a positive function in
$\mathcal{C}^{1}(\mathbb{R}_{+})$ such that

$C_{\star}+\max_{t\in[0,T_{\star}]}\\{\dot{\delta}(t)\\}<4d\ \ \text{and}\ \ \
\delta(0)=a.$

This can be done for instance by choosing $\delta_{1}\in\mathbb{R}$ such that
$C_{\star}+\delta_{1}<4d$, $a\geqslant\max\\{0,-\delta_{1}T_{\star}\\}$, and
$\delta(t)=a+\delta_{1}t$.

Then, if we set
$\delta_{\star}:=(C_{\star}+\max_{t\in[0,T_{\star}]}\\{\dot{\delta}(t)\\})/4d$,
we deduce that for any $T$ in $[0,T_{\star}]$, we have

$\displaystyle\|e^{\delta|D_{s}|}u\|_{L^{\infty}_{T}(\dot{H}^{d})}^{2}+2(1-\delta_{\star})$
$\displaystyle\|e^{\delta|D_{s}|}\nabla_{\mathbb{H}}u\|_{L^{2}_{T}(\dot{H}^{d})}^{2}$
$\displaystyle\leqslant\|e^{a|D_{s}|}u_{0}\|_{\dot{H}^{d}}^{2}+2C^{\star}\|e^{\delta|D_{s}|}u\|_{L^{\infty}_{T}(\dot{H}^{d})}\|e^{\delta|D_{s}|}\nabla_{\mathbb{H}}u\|_{L^{2}_{T}(\dot{H}^{d})}^{2}.$

We deduce from bootstrap type arguments that for $\varepsilon>0$ chosen small
enough, the following inequality holds

$\|u\|_{L^{\infty}_{T_{\star}}(\dot{H}^{d})}^{2}+\|\nabla_{\mathbb{H}}u\|_{L^{2}_{T_{\star}}(\dot{H}^{d})}^{2}\leqslant\|e^{\delta|D_{s}|}u\|_{L^{\infty}_{T_{\star}}(\dot{H}^{d})}^{2}+\|e^{\delta|D_{s}|}\nabla_{\mathbb{H}}u\|_{L^{2}_{T_{\star}}(\dot{H}^{d})}^{2}\leqslant\|e^{a|D_{s}|}u_{0}\|_{\dot{H}^{d}}^{2}.$

Using a compactness argument, we deduce from the above energy inequality that
there is a solution $u$ in $\mathcal{C}([0,T_{\star}];\dot{H}^{d})$ so that
$u_{|_{t=0}}=u_{0}$ and the above inequality holds. ∎

The proof of Theorem 8.0.1 suggests the following important remark. In view of
(8.4), if the constants $C_{\star}$ is strictly smaller than $4d$, the
analyticity assumption on the initial datum is not needed: in such case, one
could then take $a=0$ and $\delta_{1}\in(0,4d-C_{\star})$, and
$\delta(t)=\delta_{1}t$ in the above proof. In this sense, the size of
$C_{\star}$ measures the loss of regularity (or dissipation) generated by the
term
$(\operatorname{Id}-\mathbb{P})\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}}u$
in $\dot{H}^{d}(\mathbb{H}^{d})$. However, even if $C_{\star}<4d$, it is not
clear that we can obtain an analog of the stability estimate of Theorem 6.2.1
in the $\dot{H}^{d}(\mathbb{H}^{d})$ framework. Indeed, we cannot obtain a
commutator estimate similar to Lemma 6.2.2 with $P^{\alpha}$ instead
$\tilde{P}^{\alpha}$, essentially due to the fact that the commutators
$[X_{j},\Xi_{j}]=-4\partial_{s}$, with $j\in[\\![1,d]\\!]$, are homogeneous
left-invariant operators of order two (see Lemma 2.4.8).

##  9 Sub-Riemannian Euler and Navier-Stokes systems for general left-
invariant structures

Let $G$ be a stratified Lie group of step $r$ and dimension $N$ (which we
identify to $\mathbb{R}^{N}$ with a suitable group law) and let us denote by
$\mathfrak{g}$ its Lie algebra of left-invariant vector fields. We fix a
stratification $\mathfrak{g}=\bigoplus_{j=1}^{r}\mathfrak{g}_{j}$ and we
consider the Jacobian generators (see [13, Definition 1.4.1, p. 56])
$Z_{1},\dots,Z_{N^{\prime}}$, with
$N^{\prime}:=\dim(\mathfrak{g}_{1})\leqslant N$, of $\mathfrak{g}$, that is:
$Z_{1},\dots,Z_{N^{\prime}}$ are generators of $\mathfrak{g}$ such that
$\mathfrak{g}_{1}=\mathrm{Span}(Z_{1},\dots,Z_{N^{\prime}})$. Then, for any
$j\in[\\![1,N^{\prime}]\\!]$, we have
$Z_{j}=\sum_{k=1}^{N}b_{k}^{j}\partial_{k}$, with $b_{j}^{j}=1$, $b_{k}^{j}=0$
for any $k\in[\\![1,N^{\prime}]\\!]\setminus\\{j\\}$ and, if $N^{\prime}<N$,
for any $k\in[\\![N^{\prime}+1,N]\\!]$, the function $b_{k}^{j}$ is polynomial
and independent of $x_{k}$. Let us set
$\mathcal{R}_{\mathfrak{g}_{1}}:=(b_{k}^{j})_{1\leqslant k\leqslant
N,1\leqslant j\leqslant N^{\prime}}$. Then
$\mathcal{R}_{\mathfrak{g}_{1}}:\mathbb{R}^{N}\rightarrow\mathcal{L}(\mathbb{R}^{N^{\prime}},\mathbb{R}^{N})$.
Now, if we consider the left-invariant sub-Riemannian structure on $G$, then
$v=\\!^{t}(v_{1},\dots,v_{N})$ is a horizontal vector field if and only if
$\mathcal{R}_{\mathfrak{g}_{1}}\\!^{t}(v_{1},\dots,v_{N^{\prime}})=v$. We also
say that $\\!{}^{t}(v_{1},\dots,v_{N^{\prime}})$ is a horizontal vector field.
Generalizing the strategy of Section 1.1 to $G$ (in such context
$\mathcal{R}_{\mathfrak{g}_{1}}$ plays the same role as $\mathcal{R}$) for the
sub-Riemannian Euler and Navier-Stokes system on the Heisenberg group, we
obtain the following system:

$\begin{cases}\partial_{t}u+u\cdot\nabla_{G}u-\nu\mathop{}\\!\mathbin{\bigtriangleup}_{G}u=-\nabla_{G}p,\\\
\operatorname{\mathop{}div}_{G}(u)=0,\end{cases}$ (9.1)

where $\nu$ belongs to $[0,+\infty)$,
$\nabla_{G}:=\\!^{t}\mathcal{R}_{\mathfrak{g}_{1}}\nabla$,
$\operatorname{\mathop{}div}_{G}:=\operatorname{\mathop{}div}(\mathcal{R}_{\mathfrak{g}_{1}}\cdot)$
and
$\mathop{}\\!\mathbin{\bigtriangleup}_{G}:=\operatorname{\mathop{}div}(\mathcal{R}_{\mathfrak{g}_{1}}\\!^{t}\mathcal{R}_{\mathfrak{g}_{1}}\nabla\cdot)$.
Let us remark that if $G=\mathbb{H}^{d}$, then
$\mathcal{R}_{\mathfrak{h}_{1}}=\mathcal{R}$. Furthermore, if $N^{\prime}=N$,
then $G=(\mathbb{R}^{N},+)$, $\mathcal{R}_{\mathfrak{g}_{1}}$ is the identity
matrix and thus (9.1) is the incompressible Euler system on $\mathbb{R}^{N}$
when $\nu=0$, or the incompressible Navier-Stokes system on $\mathbb{R}^{N}$
when $\nu>0$. Moreover, for any smooth enough solution $(u,p)$ of (9.1), we
have

$\frac{d}{dt}\|u\|_{L^{2}}^{2}+\nu\|\nabla_{G}u\|_{L^{2}}^{2}=0.$

##### Existence of weak solutions.

In this paragraph, we show that we can generalize Theorem 1.2.3 to the case of
System (9.1) with $\nu>0$. Let us first give the definition of a global weak
solution for the Cauchy problem (9.1) (which reduces to Definition 1.2.2 when
$G$ is the Heisenberg group and to the usual Leray solutions if
$G=\mathbb{R}^{N}$).

######  Definition 9.0.1.

Let $\nu>0$, and $u_{0}\in L^{2}(G)^{N^{\prime}}$ be a horizontal vector field
satisfying $\operatorname{\mathop{}div}_{G}(u_{0})=0$. We say that $u\in
L^{2}_{loc}(\mathbb{R}_{+}\times G)^{N^{\prime}}$ is a global weak solution of

$\begin{cases}\partial_{t}u-\nu\mathop{}\\!\mathbin{\bigtriangleup}_{G}u+u\cdot\nabla_{G}u=-\nabla_{G}p\
&\text{in}\ \ (0,+\infty)\times G,\\\ \operatorname{\mathop{}div}_{G}(u)=0\
&\text{in}\ \ (0,+\infty)\times G\end{cases}$ (9.2)

and

$u_{|_{t=0}}=u_{0}\ \text{in}\ \ G,$ (9.3)

if

  1. 1.

(Integrability conditions) $u$ belongs to
$\mathcal{C}_{w}([0,+\infty);L^{2})\cap L^{\infty}(\mathbb{R}_{+};L^{2})\cap
L^{2}(\mathbb{R}_{+};\dot{H}^{1}(G))$,

  2. 2.

(Initial condition) $\lim_{t\rightarrow 0^{+}}u(t)=u_{0}$ for the weak
topology of $L^{2}(G)$,

  3. 3.

(Momentum equation) for any $t^{\prime}\leqslant t$ in $[0,+\infty)$ and for
any $\varphi\in\mathcal{D}((0,+\infty)\times G)^{N^{\prime}}$ such that
$\operatorname{\mathop{}div}_{G}(\varphi)=0$, we have

$\int_{G}u(t)\cdot\varphi(t)dx-\int_{t^{\prime}}^{t}\int_{G}\left(u\cdot\partial_{t}\varphi+\nu
u\cdot\mathop{}\\!\mathbin{\bigtriangleup}_{G}\varphi+(u\otimes
u)\cdot\nabla_{G}\varphi\right)dxd\tau=\int_{G}u(t^{\prime})\cdot\varphi(t^{\prime})dx,$

  4. 4.

(Continuity equation) For all $t>0$, we have
$\operatorname{\mathop{}div}_{G}(u(t))=0$ in
$\mathcal{D}^{\prime}(G)^{N^{\prime}}$.

By adapting the proof of Theorem 1.2.3 (the proof is left to the reader), we
obtain the following theorem.

######  Theorem 9.0.2.

Let $\nu>0$ and $u_{0}$ be a horizontal vector field belonging to $L^{2}(G)$
and satisfying $\operatorname{\mathop{}div}_{G}(u_{0})=0$. Then there exists a
global weak solution $u$ of (9.2)-(9.3), satisfying the following energy
estimate

$\|u\|_{L^{\infty}(L^{2})}^{2}+2\nu\|\nabla_{G}u\|_{L^{2}(L^{2})}^{2}\leqslant\|u_{0}\|_{L^{2}}^{2}.$

By setting
$\mathbb{P}^{G}:=\operatorname{Id}+\nabla_{G}\circ(-\mathop{}\\!\mathbin{\bigtriangleup}_{G})^{-1}\circ\operatorname{\mathop{}div}_{G}$,
we can rewrite System (9.2) as follows

$\begin{cases}\partial_{t}u-\nu\mathbb{P}^{G}\mathop{}\\!\mathbin{\bigtriangleup}_{G}u+\mathbb{P}^{G}(u\cdot\nabla_{G}u)=0,\\\
\operatorname{\mathop{}div}_{G}(u)=0.\end{cases}$ (9.4)

Let us now give the main elements allowing to prove Theorem 9.0.2.

##### Analysis on stratified Lie group.

Let us denote by $\widehat{G}$ the unitary dual of $G$. For any
$\pi\in\widehat{G}$ we denote by $H_{\pi}$ the associated representation space
and by $\mathscr{F}_{G}$ the Fourier transform on $G$. Note that $H_{\pi}$ is
a Hilbert space and, thanks to the Kirillov theory (see [28, Theorem 3, p.
103]), can be realized as the space of square integrable functions on a
Euclidean space. Now, for any $X\in\mathfrak{g}$, we define the right-
invariant vector field $\tilde{X}$ by $\tilde{X}f:=-X\check{f}$, where for any
$w\in G$, $\check{f}(w):=f(w^{-1})$ and $w^{-1}$ denotes the inverse of $w$
for the law of $G$. We also denote by
$\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{G}$ the right-invariant sub-
Laplacian on $G$ define by
$\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{G}=\sum_{j=1}^{N}\tilde{Z}_{j}^{2}$.
Let
$T\in\\{-\mathop{}\\!\mathbin{\bigtriangleup}_{G},Z_{1},\dots,Z_{N^{\prime}}\\}$
and
$\tilde{T}\in\\{-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{G},\tilde{Z}_{1},\dots,\tilde{Z}_{N^{\prime}}\\}$.
Then $T$ and $\tilde{T}$ commute. Moreover, if $X\in\mathfrak{g}$, then we
have $\mathscr{F}_{G}(Xf)(\pi)=\mathscr{F}_{G}(f)(\pi)\circ\pi(X)$ and
$\mathscr{F}_{G}(\tilde{X}f)(\pi)=\pi(X)\circ\mathscr{F}_{G}(f)(\pi)$. Also,
for any $\pi\in\widehat{G}$, the operator
$\pi(-\mathop{}\\!\mathbin{\bigtriangleup}_{G})$ is a self-adjoint compact
operator on a separable Hilbert space $H_{\pi}$ and its spectrum
$\\{E(m,\pi)\\}_{m\in\mathbb{N}}$ lies in $(0,+\infty)$. Thus if we denote by
$(h_{m}^{\pi})_{m\in\mathbb{N}}$ an orthonormal basis of $H_{\pi}$ of
eigenvectors of $\pi(-\mathop{}\\!\mathbin{\bigtriangleup}_{G})$, then for any
$f\in L^{1}(G)$, we can set

$\mathcal{F}_{G}(f)(n,m,\pi):=\langle\mathscr{F}_{G}(f)(\pi)h_{m}^{\pi},h_{n}^{\pi}\rangle_{H_{\pi}},$

for any $(n,m,\pi)\in\mathbb{N}\times\mathbb{N}\times\widehat{G}$. (Note that,
by using the Kirillov theory, we can parametrize $\widehat{G}$ by the orbit of
the coadjoint action from $G$ on $\mathfrak{g}^{{}^{\prime}}$ in order to
obtain a more concrete description of the unitary dual, see [28]). Similarly
as in Sections 2.3 and 2.4, for any $\ell\in\mathbb{R}$, we define the
operators
$(-\mathop{}\\!\mathbin{\bigtriangleup}_{G})^{\ell/2}:\mathrm{Dom}((-\mathop{}\\!\mathbin{\bigtriangleup}_{G})^{\ell/2})\subset
L^{2}(G)\rightarrow L^{2}(G)$ and we denote by $\dot{H}^{\ell}(G)$ the closure
of
$\mathcal{S}(G)\cap\mathrm{Dom}((-\mathop{}\\!\mathbin{\bigtriangleup}_{G})^{\ell/2}))$
for the norm
$\|(-\mathop{}\\!\mathbin{\bigtriangleup}_{G})^{\ell}\cdot\|_{L^{2}}$. By the
same way, we define
$(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{G})^{\ell/2}$ and
$\tilde{H}^{\ell}(G)$, with $\ell\in\mathbb{R}$. In particular, Proposition
B.0.1, Lemma B.0.3 and the identities involving the sub-Laplacian in
Proposition 2.3.1 can be adapted to $G$ (the details are left to the reader).
Thanks to classical results, Propositions 2.4.5 and 2.4.7 also hold on $G$
(see [19, Section 4]).

##### Approximate problem.

Our strategy in this article, based on a priori estimates on suitable
approximate problems, can be generalized to $G$ as follows. We first introduce
for any $k\in\mathbb{N}$ the operators $\mathsf{J}_{k}^{G}$ and
$\tilde{\mathsf{J}}_{k}^{G}$ by setting for any $f\in\mathcal{S}(G)$ and
$(n,m,\pi)\in\mathbb{N}\times\mathbb{N}\times\widehat{G}$

$\mathcal{F}_{G}(\mathsf{J}_{k}^{G}f)(n,m,\pi):=\mathbf{1}_{\\{\frac{1}{2^{k+1}}\leqslant
E(n,\pi)\leqslant 2^{k}\\}}(n,m,\pi)\mathbf{1}_{\\{\frac{1}{2^{k+1}}\leqslant
E(m,\pi)\leqslant 2^{k}\\}}(n,m,\pi)\mathcal{F}_{G}(f)(n,m,\pi)$

and

$\mathcal{F}_{G}(\tilde{\mathsf{J}}_{k}^{G}f)(n,m,\pi):=\mathbf{1}_{\\{\frac{1}{2^{k+1}}\leqslant
E(n,\pi)\leqslant 2^{k}\\}}(n,m,\pi)\mathcal{F}_{G}(f)(n,m,\pi).$

Thus Proposition 4.1.3, Items 1, 2, 3, 5, 6 and 7, Proposition 4.1.4, Items 1,
2, 3, 5, 6 and 7, and Proposition 4.2.1, Items 1, 2, 3, 4, 5, 7, 8 and 9 can
be generalized to $\mathsf{J}_{k}^{G}$, $\tilde{\mathsf{J}}_{k}^{G}$ and
$\mathbb{P}^{G}$. Furthermore, we can adapt the strategy of Section 4 to
System (9.4) in order to derive the following approximate problems depending
on $k\in\mathbb{N}$:

$\begin{cases}\partial_{t}u_{k}-\nu\mathbb{P}^{G}\mathop{}\\!\mathbin{\bigtriangleup}_{G}u_{k}+\mathbb{P}^{G}\mathsf{J}_{k}^{G}(u_{k}\cdot\nabla_{G}\mathsf{J}_{k}^{G}u_{k})=0\
&\text{in}\ \ (0,+\infty)\times G,\\\
\operatorname{\mathop{}div}_{G}(u_{k})=0\ &\text{in}\ \ (0,+\infty)\times
G,\\\ \tilde{\mathsf{J}}_{k}^{G}u_{k}=u_{k}\ &\text{in}\ \ (0,+\infty)\times
G\\\ \end{cases}$ (9.5)

and

$u_{k|_{t=0}}=\tilde{\mathsf{J}}_{k}^{G}u_{0}\ \text{in}\ \ G,$ (9.6)

where $u_{0}$ is a horizontal vector field belonging to
$L^{2}(G)^{N^{\prime}}$ or $\tilde{H}^{\ell}(G)^{N^{\prime}}$ for
$\ell\in\mathbb{R}$ and $k\in\mathbb{N}$. Thus, the solution $u_{k}$ belongs
to $L^{2}(L^{2})\cap
L^{2}(\dot{H}^{1}(G))\cap\mathcal{C}_{b}(\tilde{H}^{\ell^{\prime}}(G))$ with
$\nabla_{G}u_{k}\in L^{2}(\tilde{H}^{\ell^{\prime}}(G))$, for any
$\ell^{\prime}\in\mathbb{R}$ and satisfies

$\|u_{k}\|_{L^{\infty}(L^{2})}^{2}+2\nu\|\nabla_{G}u_{k}\|_{L^{2}(L^{2})}^{2}\leqslant\|\tilde{\mathsf{J}}_{k}^{G}u_{0}\|_{L^{2}}^{2}.$
(9.7)

##### Open problems.

We end this section with some open questions related to (9.2)-(9.3).

A first open problem concerns the global well-posedness in the case of a
general stratified Lie group $G$ for initial data in the critical space
$\tilde{H}^{Q/2-1}(G)$. Indeed, when the homogeneous dimension is odd, this
imposes to work with fractional order Sobolev spaces. In such case, estimating
the nonlinear terms is challenging because we need to simultaneously address
the regularity associated with both the right-invariant and left-invariant
sub-Laplacians on $G$ (in this article, since $Q/2-1=d\in\mathbb{N}$ when
$G=\mathbb{H}^{d}$, we only need to use the Leibniz formula and the Sobolev’s
embedding.)

A second question concerns the regularity of the solution. In particular, when
the step of $G$ is strictly larger than $2$, the interplay between the
different stratums of vector fields is not so clear. Therefore, the
regularization properties of the solutions of (9.2)-(9.3) become much more
difficult to analyze. Indeed, looking at the proof of Theorem 1.2.5, we
compensate the loss of derivatives due to the degenerate diffusion operator
$(\operatorname{Id}-\mathbb{P}^{G})\circ\mathop{}\\!\mathbin{\bigtriangleup}_{G}$
by a regularity of infinite order in the other directions. Such strategy is
made possible in the case of the Heisenberg group by the crucial facts that a
parametrization of $\widehat{G}$ is simply given by $\mathbb{R}^{*}$, and that
we have a spectral gap in the sense of Proposition 2.4.8, revealed by the
analysis of the symbol of the sub-Laplacian (see also [5] in the context of
the Engel group).

##  Appendix A A result of anisotropic analytic hypoellipticity of the
fractional power of the sub-Laplacian on $\mathbb{H}^{d}$

In this section we state and prove a result which interprets the radius of
analyticity with respect to the vertical variable in the framework of the
homogeneous Sobolev-type space $\tilde{H}^{\ell}(\mathbb{H}^{d})$ (the same
result holds if we replace
$\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}}$ by
$\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}}$, with the suitable
modifications):

######  Theorem A.0.1.

Let $\ell>0$. Let $f\in\tilde{H}^{\ell}(\mathbb{H}^{d})$ and $\sigma>0$. If
$e^{\sigma|D_{s}|}f\in\tilde{H}^{\ell}(\mathbb{H}^{d})$, then there exist two
functions $g$ and $h$ satisfying

$f=g+h$

such that

  1. 1.

$g$ belongs to
$\tilde{H}^{\ell}(\mathbb{H}^{d})\cap\mathcal{C}^{\infty}(\mathbb{R}^{2d+1})$
and for any $Y\in\mathbb{R}^{2d}$, the function $g(Y,\cdot)$ can be extended
to a holomorphic function on $\mathbb{C}$,

  2. 2.

$h$ belongs to $L^{2}(\mathbb{H}^{d})\cap\tilde{H}^{\ell}(\mathbb{H}^{d})$ and
for any $Y\in\mathbb{R}^{2d}$, the function $h(Y,\cdot)$ can be extended to a
holomorphic function in
$\mathsf{S}_{\sigma}:=\left\\{z\in\mathbb{C}~{}\middle|~{}|\Im(z)|<\sigma\right\\}$.

In particular, $f$ is analytic with respect to the vertical variable $s$.

Let us remark that this theorem implies partial (and global) analytic
hypoellipticity properties for
$(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\ell}$. Indeed,
if $f$ is a complex value smooth function on $\mathbb{H}^{d}$, such that
$(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\ell}f$ belongs
to $L^{2}(\mathbb{H}^{d})$ and, for any $Y\in\mathbb{R}^{2d}$, the function
$(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\ell}f(Y,\cdot)$
extends to a holomorphic function on $\mathsf{S}_{\sigma}$ satisfying

$\sup_{|\eta|<\sigma}\\{\|(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\ell}f(Y,\cdot+i\eta)\|_{L^{2}(\mathbb{R})}\\}<+\infty,$

then we have
$e^{\sigma|D_{s}|}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{\ell}f\in
L^{2}(\mathbb{H}^{d})$ (this can be done for instance by adapting the proof of
[2, Theorem 1.1] to the case of $\mathbb{R}$). It then follows from Theorem
A.0.1 that $f(Y,\cdot)$ is analytic on $\mathbb{R}$ and extends to a
holomorphic function on $\mathsf{S}_{\sigma}$.

Additionally, let us point out that, in the case of $\mathbb{H}^{d}$, the
exponential decay of the Fourier transform does not imply analyticity. Indeed,
there exists a smooth function $\psi\in L^{2}(\mathbb{H}^{d})$ such that the
support of $\mathcal{F}_{\mathbb{H}}(f)$ is included in the set

$\left\\{(n,m,\lambda)~{}\middle|~{}4|\lambda|(2|n|+d)\leqslant R\right\\}$

for some $R>0$, and which cannot be extended into an analytic function on
$\mathbb{C}^{2d+1}$ (see [6]).

###### Proof of Theorem A.0.1.

Let $\psi_{1}$ be a smooth function on $\mathbb{R}$ with value in $[0,1]$ and
satisfying $supp(\psi_{1})\subset(-2,2)$ and $\psi_{1}=1$ on $[-1,1]$. Let us
set $\psi_{2}:=1-\psi_{1}$. Since $\psi_{1}$ and $\psi_{2}$ are bounded,
thanks to Proposition B.0.1, we can write

$f=\psi_{1}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})f+\psi_{2}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})f.$

Because $\psi_{1}$ is compactly supported smooth function, we have
$\psi_{1}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})f\in\mathcal{C}^{\infty}(\mathbb{R}^{2d+1})$
(use Proposition B.0.1 to ensure that this function belongs to
$\tilde{H}^{\ell^{\prime}}(\mathbb{H}^{d})$ for any
$\ell^{\prime}\in[\ell,+\infty)$ and the Sobolev embedding in order to recover
the continuity of all derivative with respect to the right-invariant vector
fields). Besides, according to the Hulanicki theorem (see [19, Corollary
4.5.2, p. 252]), the operators
$\psi_{1}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})$ maps
$\mathcal{S}(\mathbb{H}^{d})$ to $\mathcal{S}(\mathbb{H}^{d})$. Assume that
$f\in\mathcal{S}(\mathbb{H}^{d})$. In this case
$\psi_{1}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})f$
belongs to $\mathcal{S}(\mathbb{H}^{d})$ and we have (see Formula (2.20))

$\displaystyle\mathcal{F}_{\mathbb{H}}\left(\psi_{1}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})f\right)(n,m,\lambda)$
$\displaystyle=\mathbf{1}_{\\{|\lambda|\leqslant
2\\}}(n,m,\lambda)\mathcal{F}_{\mathbb{H}}\left(\psi_{1}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})f\right)(n,m,\lambda)$
$\displaystyle=\mathcal{F}_{\mathbb{H}}\left(\mathbf{1}_{\\{|D_{s}|\leqslant
2\\}}\psi_{1}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})f\right)(n,m,\lambda).$

According to the Fourier inversion formula in Proposition 2.2.1, we thus get

$\psi_{1}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})f=\mathbf{1}_{\\{|D_{s}|\leqslant
2\\}}\psi_{1}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})f.$

Since $\psi_{1}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})$
and $\mathbf{1}_{\\{|D_{s}|\leqslant
2\\}}\psi_{1}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})$ are
bounded operators on $\tilde{H}^{\ell}(\mathbb{H}^{d})$, by using the density
of $\mathcal{S}(\mathbb{H}^{d})$ in $\tilde{H}^{\ell}(\mathbb{H}^{d})$, we
deduce that the above formula holds for any
$f\in\tilde{H}^{\ell}(\mathbb{H}^{d})$. Then for any $Y\in\mathbb{R}^{2d}$,
the Fourier transform on $\mathbb{R}$ of
$\psi_{1}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})f(Y,\cdot)$
is compactly supported in $\mathbb{R}$. Then according to the Paley-Wiener
theorem, we deduce that
$\psi_{1}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})f(Y,\cdot)$
extends to a holomorphic function on $\mathbb{C}$.

On the other hand, since $supp(\psi_{2})\subset\mathbb{R}\setminus[-1,1]$ and
$\|\psi_{2}\|_{L^{\infty}}\leqslant 1$, for any
$(n,m,\lambda)\in\mathbb{N}^{d}\times\mathbb{N}^{d}\times\mathbb{R}^{*}$, we
have

$|\mathcal{F}_{\mathbb{H}}\left(\psi_{2}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})e^{\sigma|D_{s}|}f\right)(n,m,\lambda)|\leqslant(4|\lambda|(2|m|+d))^{\ell}|\mathcal{F}_{\mathbb{H}}\left(e^{\sigma|D_{s}|}f\right)(n,m,\lambda)|.$

Thus, using that $e^{\sigma|D_{s}|}f\in\tilde{H}^{\ell}(\mathbb{H}^{d})$, we
deduce that
$e^{\sigma|D_{s}|}\psi_{2}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})f=\psi_{2}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})e^{\sigma|D_{s}|}f$
belongs to $L^{2}(\mathbb{H}^{d})$. Then, we deduce that for any
$Y\in\mathbb{R}^{2d}$, the function
$\psi_{2}(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})f(Y,\cdot)$
can be extended to a holomorphic function with respect to the variable $s$ in
$\mathsf{S}_{\sigma}$. ∎

##  Appendix B Symbol of the negative powers of the sub-Laplacian

Our first goal is to prove the following proposition, which is deduced by
suitably combining several results and arguments scattered in [19]:

######  Proposition B.0.1.

For any $\lambda\in\mathbb{R}^{*}$, the infinitesimal representation
$\mathrm{U}^{\lambda}(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})$ of
$-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}}$ is given by

$\mathrm{U}^{\lambda}(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})=-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda}\
\text{ where
}-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda}:=4\left(\sum_{j=1}^{d}-\partial_{x_{j}}^{2}+|\lambda|^{2}x_{j}^{2}\right),$

and the space of smooth vector fields of
$(\mathrm{U}^{\lambda},L^{2}(\mathbb{R}^{d}))$ is
$\mathcal{S}(\mathbb{R}^{d})$. Moreover, for $f\in L^{2}(\mathbb{H}^{d})$ and
$\varphi\in L^{\infty}(\mathbb{R}_{+};\mathbb{R})$, we have

$\displaystyle\mathscr{F}_{\mathbb{H}}(\varphi(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})f)(\mathrm{U}^{\lambda})=\mathscr{F}_{\mathbb{H}}(f)(\mathrm{U}^{\lambda})\circ\varphi(-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda})$
(B.1)
$\displaystyle\mathscr{F}_{\mathbb{H}}(\varphi(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})f)(\mathrm{U}^{\lambda})=\varphi(-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda})\circ\mathscr{F}_{\mathbb{H}}(f)(\mathrm{U}^{\lambda}).$
(B.2)

where $\varphi(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})$ and
$\varphi(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})$ are
bounded self-adjoint operators on $L^{2}(\mathbb{H}^{d})$ and

$\varphi(-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda}):=\sum_{m\in\mathbb{N}^{d}}\varphi(4|\lambda|(2|m|+d))\mathcal{P}_{m,\lambda},$
(B.3)

where
$\mathcal{P}_{m,\lambda}:=\langle\cdot,h_{m,\lambda}\rangle_{L^{2}(\mathbb{R}^{d})}h_{m,\lambda}$
is a bounded self-adjoint operator on $L^{2}(\mathbb{R}^{d})$.

If $\varphi$ is a measurable function on $\mathbb{R}_{+}$ such that there
exist positive constants $C$ and $\ell\in[0,+\infty)$ such that

$\forall\mu\in(0,\infty),\quad|\varphi(\mu)|\leqslant C(1+|\mu|)^{\ell},$
(B.4)

then formula (B.1) and (B.2) hold for $f\in\mathcal{S}(\mathbb{H}^{d})$.

######  Remark B.0.2.

Let us point out that in [19] (see [19, Proposition 1.7.6] where it is given
for $\varphi(x)=x$), the formula (B.1) and (B.2) are inverted due to the
difference of convention: in [19] the authors define the Fourier transform as
$\int_{\mathbb{H}^{d}}f(w)\mathrm{U}_{-w}^{\lambda}dw$ while, we use
$\int_{\mathbb{H}^{d}}f(w)\mathrm{U}_{w}^{\lambda}dw$.

###### Proof.

Let $\lambda\in\mathbb{R}^{*}$. For any
$\varphi\in\mathcal{S}(\mathbb{H}^{d})$ and $j\in[\\![1,d]\\!]$, we have
$\mathrm{U}^{\lambda}_{\exp_{\mathbb{H}^{d}}(tX_{j})}\varphi(x)=\varphi(x-2te_{j})$
and
$\mathrm{U}^{\lambda}_{\exp_{\mathbb{H}^{d}}(t\Xi_{j})}\varphi(x)=e^{-2i\lambda
tx_{j}}\varphi(x)$, we deduce that the values of the infinitesimal
representation (see [19, Proposition 1.7.3, p. 38]) associated to
$\mathrm{U}^{\lambda}$ in $X_{j}$ and $\Xi_{j}$ are respectively
$\mathrm{U}^{\lambda}(X_{j})=-2\partial_{x_{j}}$ and
$\mathrm{U}^{\lambda}(\Xi_{j})=-2i\lambda x_{j}$. We thus deduce that, for any
$\lambda\in\mathbb{R}^{*}$, we have

$\mathrm{U}^{\lambda}(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})=-\sum_{j=1}^{d}\left(\mathrm{U}^{\lambda}(X_{j})^{2}+\mathrm{U}^{\lambda}(\Xi_{j})^{2}\right)=-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda}.$
(B.5)

If $\varphi$ belongs to $L^{\infty}(\mathbb{R}_{+})$, then $\varphi$ is
bounded on the spectrum
$\mathrm{Sp}(-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda})=\\{4|\lambda|(2|m|+d)\\}_{m\in\mathbb{N}^{d}}$
of $-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda}$, and consequently
Formula (B.3) defines a bounded self-adjoint operator on
$L^{2}(\mathbb{R}^{d})$. Let $\varphi$ be a real-valued function on
$L^{\infty}(\mathbb{R}_{+})$. We now prove (B.1) and (B.2). Thanks to the
spectral theorem for unbounded self-adjoint operators on
$L^{2}(\mathbb{H}^{d})$, the operators
$\varphi(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})$ and
$\varphi(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})$ are
bounded self-adjoint operators since $\varphi$ is real valued and bounded on
the spectrum of $-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}}$ and
$-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}}$. Moreover, it is
easy to check that
$\varphi(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})$ and
$\varphi(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})$ are
respectively left-invariant and right-invariant operators on
$L^{2}(\mathbb{H}^{d})$. Thus according to the Schwartz kernel theorem on Lie
group (see for instance [19, Corollary 3.2.1, p. 133]), we deduce that there
exist $\kappa_{\varphi}^{r}$ and $\kappa_{\varphi}^{\ell}$ in
$\mathcal{S}^{\prime}(\mathbb{H}^{d})$ such that

$\varphi(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})f=f\star\kappa_{\varphi}^{\ell}\
\text{and}\
\varphi(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})f=\kappa_{\varphi}^{r}\star
f,$ (B.6)

where $\star$ denotes the (non-commutative) convolution product on
$\mathbb{H}^{d}$ extended to
$\mathcal{S}^{\prime}(\mathbb{H}^{d})\times\mathcal{S}(\mathbb{H}^{d})$ and
define for any functions $a$ and $b$ in $L^{1}(\mathbb{H}^{d})$ by

$(a\star
b)(Y,s):=\int_{\mathbb{H}^{d}}a(Y-Y^{\prime},s-s^{\prime}-\langle\mathfrak{S}Y,Y^{\prime}\rangle_{\mathbb{R}^{2d}})b(Y^{\prime},s^{\prime})dY^{\prime}ds^{\prime},\quad\text{
for }(Y,s)\in\mathbb{H}^{d}.$

For any $a\in\mathcal{S}^{\prime}(\mathbb{H}^{d})$, we denote
$\check{a}:=a(-\cdot)$ (in the sense of distributions). Since
$\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}}f=(\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}}\check{f})(-\cdot)$
for any $f\in\mathcal{S}(\mathbb{H}^{d})$, we deduce that the spectral
measures $E$ and $\tilde{E}$ corresponding to
$\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}}$ and
$\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}}$ are linked by the
relation $\tilde{E}f=(E\check{f})(-\cdot)$, by the uniqueness of the spectral
measure of $-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}}$. Then,
up to a suitable approximation, we deduce that for any $\varphi\in
L^{\infty}(\mathbb{R}_{+})$ ($\varphi$ is real valued) and $f\in
L^{2}(\mathbb{H}^{d})$, that

$\varphi(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})f=(\varphi(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})\check{f})(-\cdot).$

Thus, thanks to (B.6), we deduce from the above identity that
$\kappa_{\varphi}^{r}\star
f=(\check{f}\star\kappa_{\varphi}^{\ell})(-\cdot)=\check{\kappa}_{\varphi}^{\ell}\star
f$, and by the uniqueness of the Schwartz kernel theorem, we deduce that

$\kappa_{\varphi}^{r}=\check{\kappa}_{\varphi}^{\ell}.$ (B.7)

By applying the Fourier transform on the two identities in (B.6), thanks to
the abstract Plancherel theorem on $\mathbb{H}^{d}$ (see for instance [19,
Theorem 1.8.11, p. 52]) (keeping in mind that $f\mapsto
f\star\kappa_{\varphi}^{\ell}$ and $f\mapsto\kappa_{\varphi}^{r}\star f$ are
bounded operators on $L^{2}(\mathbb{H}^{d})$), we deduce that for any
$\lambda\in\mathbb{R}^{*}$, we have

$\displaystyle\mathscr{F}_{\mathbb{H}}(\varphi(-{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})f)(\mathrm{U}^{\lambda})=\mathscr{F}_{\mathbb{H}}(f)(\mathrm{U}^{\lambda})\circ\mathcal{F}_{\mathbb{H}}(\kappa_{\varphi}^{\ell})(\mathrm{U}^{\lambda}),$
(B.8)
$\displaystyle\mathscr{F}_{\mathbb{H}}(\varphi(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})f)(\mathrm{U}^{\lambda})=\mathscr{F}_{\mathbb{H}}(\check{\kappa}_{\varphi}^{\ell})(\mathrm{U}^{\lambda})\circ\mathscr{F}_{\mathbb{H}}(f)(\mathrm{U}^{\lambda}).$
(B.9)

Following the same lines as in the proof of [19, Corollary 4.1.16, Identity
(4.5), p. 179], we deduce the formula

$\mathscr{F}_{\mathbb{H}}(\check{\kappa}_{\varphi}^{\ell})(\mathrm{U}^{\lambda})=\varphi(-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda}).$

The first step of this proof consists of applying formula (B.9) up to a
suitable approximation $\varphi=\operatorname{Id}_{\mathbb{R}_{+}}$ and
$f\in\mathcal{S}(\mathbb{H}^{d})$. Then, according to the Dixmier-Malliavin
theorem (see [19, Theorem 1.7.8, p. 42]), the uniqueness of the spectral
measure of
$\mathrm{U}^{\lambda}(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})$ and
the identity
$\mathrm{U}^{\lambda}(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})=-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda}$
in (B.5), we deduce the above identity.

Then, since
$\mathscr{F}_{\mathbb{H}}(\kappa_{\varphi}^{\ell})(\mathrm{U}^{\lambda})^{*}=\mathscr{F}_{\mathbb{H}}(\check{\kappa}_{\varphi}^{\ell})(\mathrm{U}^{\lambda})$
and $\varphi(-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda})$ is self-
adjoint, we deduce from the above identity that

$\mathscr{F}_{\mathbb{H}}(\kappa_{\varphi}^{\ell})(\mathrm{U}^{\lambda})=\varphi(-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda})=\mathscr{F}_{\mathbb{H}}(\check{\kappa}_{\varphi}^{\ell})(\mathrm{U}^{\lambda})=\mathscr{F}_{\mathbb{H}}(\kappa_{\varphi}^{r})(\mathrm{U}^{\lambda}).$
(B.10)

The identities (B.1) and (B.2) then follow from (B.8)-(B.9) and (B.10). The
fact that these identities can be extended to any $\varphi$ satisfying (B.4)
can be done as in [19, Example 5.1.27, p. 291]. ∎

According to Proposition B.0.1, if $\varphi(\mu)=\mu^{\ell}$ for
$\ell\geqslant 0$, then (B.1) and (B.2) hold for every
$f\in\mathcal{S}(\mathbb{H}^{d})$. This is no longer the case when $\ell<0$,
which deserves some special attention:

######  Lemma B.0.3.

Let $\ell>0$. If $f$ belongs to
$\mathcal{S}(\mathbb{H}^{d})\cap\mathrm{Dom}((-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{-\ell})$,
respectively
$f\in\mathcal{S}(\mathbb{H}^{d})\cap\mathrm{Dom}((-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{-\ell})$,
then for almost every $\lambda\in\mathbb{R}^{*}$ we have

$\mathscr{F}_{\mathbb{H}}((-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{-\ell}f)(\mathrm{U}^{\lambda})=\mathscr{F}_{\mathbb{H}}(f)(\mathrm{U}^{\lambda})\circ(-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda})^{-\ell},$

and if $f$ belongs to
$\mathcal{S}(\mathbb{H}^{d})\cap\mathrm{Dom}((-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{-\ell})$,
then for almost every $\lambda\in\mathbb{R}^{*}$ we have

$\mathscr{F}_{\mathbb{H}}((-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{-\ell}f)(\mathrm{U}^{\lambda})=(-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda})^{-\ell}\circ\mathscr{F}_{\mathbb{H}}(f)(\mathrm{U}^{\lambda}).$

###### Proof.

Let $\ell>0$ and
$f\in\mathcal{S}(\mathbb{H}^{d})\cap\mathrm{Dom}((-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{-\ell})$.
For any $j\in\mathbb{N}$, we set
$\chi_{j}:=\mathbf{1}_{\\{\mu^{-\ell}\leqslant 2^{j}\\}}$ and we define the
function $\varphi_{j}$ by setting, for any $\mu>0$,

$\varphi_{j}(\mu):=\chi_{j}(\mu)\mu^{-\ell}.$

Since $\varphi_{j}$ is a bounded measurable function, according to Proposition
B.0.1, we have

$\mathscr{F}_{\mathbb{H}}(\varphi_{j}(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})f)(\mathrm{U}^{\lambda})=\mathscr{F}_{\mathbb{H}}(f)(\mathrm{U}^{\lambda})\circ\varphi_{j}(-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda}).$
(B.11)

Moreover,

$\displaystyle\|\mathscr{F}_{\mathbb{H}}(f)(\mathrm{U}^{\lambda})\circ$
$\displaystyle\varphi_{j}(-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda})-\mathscr{F}_{\mathbb{H}}(f)(\mathrm{U}^{\lambda})\circ(-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda})^{-\ell}\|_{\mathrm{HS}(L^{2}(\mathbb{R}^{d}))}^{2}$
$\displaystyle=\sum_{(m,n)\in\mathbb{N}^{2d}}|\langle\mathscr{F}_{\mathbb{H}}(f)(\mathrm{U}^{\lambda})\circ\left(\varphi_{j}(-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda})-(-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda})^{-\ell}\right)h_{m,\lambda},h_{n,\lambda}\rangle_{L^{2}}|^{2}.$

Besides, thanks to the functional calculus of
$-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda}$ (see Proposition
B.0.1), for any $m\in\mathbb{N}^{d}$, we have

$\varphi_{j}(-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda})h_{m,\lambda}=\varphi_{j}(4|\lambda|(2|m|+d))h_{m,\lambda}\text{
and
}(-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda})^{-\ell}h_{m,\lambda}=(4|\lambda|(2|m|+d))^{-\ell}h_{m,\lambda}.$

Thus

$\displaystyle\|\mathscr{F}_{\mathbb{H}}(f)(\mathrm{U}^{\lambda})\circ$
$\displaystyle\varphi_{j}(-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda})-\mathscr{F}_{\mathbb{H}}(f)(\mathrm{U}^{\lambda})\circ(-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda})^{-\ell}\|_{\mathrm{HS}(L^{2}(\mathbb{R}^{d}))}^{2}$
$\displaystyle\leqslant\sum_{(m,n)\in\mathbb{N}^{2d}}\left(\varphi_{j}(4|\lambda|(2|m|+d))-(4|\lambda|(2|m|+d))^{-\ell}\right)^{2}|\langle\mathscr{F}_{\mathbb{H}}(f)(\mathrm{U}^{\lambda})h_{m,\lambda},h_{n,\lambda}\rangle_{L^{2}}|^{2}.$

Furthermore, for any $j\in\mathbb{N}^{*}$ and
$(m,\lambda)\in\mathbb{N}^{d}\times\mathbb{R}^{*}$, we have

$\displaystyle\left(\varphi_{j}(4|\lambda|(2|m|+d))-(4|\lambda|(2|m|+d))^{-\ell}\right)^{2}$
$\displaystyle=\mathbf{1}_{(0,2^{-j/\ell})}\left(4|\lambda|(2|m|+d)\right)(4|\lambda|(2|m|+d))^{-2\ell}$
$\displaystyle\leqslant\frac{2^{-\frac{j}{\ell}}}{(4|\lambda|(2|m|+d))^{2\ell+1}}\leqslant\frac{2^{-\frac{j}{\ell}}}{|\lambda|^{2\ell+1}}.$

Thus, we obtain

$\displaystyle\|\mathscr{F}_{\mathbb{H}}(f)(\mathrm{U}^{\lambda})\circ\varphi_{j}(-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda})-\mathscr{F}_{\mathbb{H}}(f)(\mathrm{U}^{\lambda})\circ(-$
$\displaystyle\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda})^{-\ell}\|_{\mathrm{HS}(L^{2}(\mathbb{R}^{d}))}^{2}$
$\displaystyle\leqslant\frac{2^{-j/\ell}}{|\lambda|^{2\ell+1}}\|\mathscr{F}_{\mathbb{H}}(f)(\mathrm{U}^{\lambda})\|_{\mathrm{HS}(L^{2}(\mathbb{R}^{d}))}^{2}.$
(B.12)

The quantity in the right-hand side of the above inequality is finite because
$f\in\mathcal{S}(\mathbb{H}^{d})$ implies that
$\mathscr{F}_{\mathbb{H}}(f)(\mathrm{U}^{\lambda})\in\mathrm{HS}(L^{2}(\mathbb{R}^{d}))$
(see Proposition 2.2.1). Combining (B.1) and (B.2), we deduce that

$\forall\lambda\in\mathbb{R}^{*},\ \
\lim_{j\rightarrow+\infty}\mathscr{F}_{\mathbb{H}}(\varphi_{j}(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})f)(\mathrm{U}^{\lambda})=\mathscr{F}_{\mathbb{H}}(f)(\mathrm{U}^{\lambda})\circ(-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda})^{-\ell}.$
(B.13)

Moreover, on one hand, since
$f\in\mathrm{Dom}((-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{-\ell})$,
we have $(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{-\ell}f\in
L^{2}(\mathbb{H}^{d})$. On the other hand, using that $f\in
L^{2}(\mathbb{H}^{d})$ and $\varphi_{j}\in L^{\infty}(\mathbb{R}_{+})$, we get
$\varphi_{j}(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})f\in
L^{2}(\mathbb{H}^{d})$. Then, we deduce from the Plancherel theorem on
$\mathbb{H}^{d}$ (see Proposition 2.2.1) and the functional calculus for
$-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}}$ that

$\displaystyle\int_{\mathbb{R}^{*}}\|\mathscr{F}_{\mathbb{H}}(\varphi_{j}(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})f)(\mathrm{U}^{\lambda})-$
$\displaystyle\mathscr{F}_{\mathbb{H}}((-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{-\ell}f)(\mathrm{U}^{\lambda})\|_{\mathrm{HS}(L^{2}(\mathbb{R}^{d}))}^{2}|\lambda|^{d}d\lambda$
$\displaystyle=\frac{\pi^{d+1}}{2^{d-1}}\|\mathbf{1}_{(0,2^{-j/\ell})}\left(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}}\right)\circ(-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{-\ell}f\|_{L^{2}(\mathbb{H}^{d})}^{2}$
$\displaystyle=\frac{\pi^{d+1}}{2^{d-1}}\int_{0}^{+\infty}\mathsf{1}_{(0,2^{-j/\ell})}(\mu)|\mu|^{-2\ell}d\langle
E(\mu)f,f\rangle_{L^{2}(\mathbb{H}^{d})}.$

Since
$f\in\mathrm{Dom}((-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{-\ell})$,
we have $|\cdot|^{-2\ell}\in L^{1}\left(\mathbb{R}_{+},d\langle
Ef,f\rangle_{L^{2}(\mathbb{H}^{d})}\right)$. Then, according to the dominated
convergence theorem, the right-hand side of the above identity converges to
$0$ when $j$ goes to $+\infty$. Hence, we deduce that

$a.\ e.\ \lambda\in\mathbb{R}^{*},\ \ \
\mathscr{F}_{\mathbb{H}}((-\mathop{}\\!\mathbin{\bigtriangleup}_{\mathbb{H}})^{-\ell}f)(\mathrm{U}^{\lambda})=\mathscr{F}_{\mathbb{H}}(f)(\mathrm{U}^{\lambda})\circ(-\mathop{}\\!\mathbin{\bigtriangleup}_{osc}^{\lambda})^{-\ell},$

in $\mathrm{HS}(L^{2}(\mathbb{R}^{d}))$ (the class of Hilbert-Schmidt
operators in $L^{2}(\mathbb{R}^{d})$). The case of
$(-\tilde{\mathop{}\\!\mathbin{\bigtriangleup}}_{\mathbb{H}})^{-\ell}$ follows
similarly (see Proposition B.0.1). ∎

##  Appendix C Compactness results for sub-elliptic Sobolev spaces

We have the following Rellich-type theorem.

######  Proposition C.0.1.

Let $\ell$ and $\ell^{\prime}$ be two real numbers such that
$\ell<\ell^{\prime}$ and $p\in(1,+\infty)$. The multiplication by any element
of $\mathcal{D}(\mathbb{H}^{d})$ is a compact operator from
$W^{\ell^{\prime},p}_{\mathbb{H}}(\mathbb{H}^{d})$ to
$W^{\ell/2,p}_{\mathbb{H}}(\mathbb{H}^{d})$.

###### Proof.

Let $\varphi\in\mathcal{D}(\mathbb{H}^{d})$. The multiplication operator by
$\varphi$ denoted by $M_{\varphi}$ is a bounded linear operator from
$W_{\mathbb{H}}^{\ell^{\prime},p}(\mathbb{H}^{d})$ to
$W^{\ell^{\prime}/2,p}(\mathbb{R}^{2d+1})$ (see [19, Theorem 4.4.24, p. 240]).
Besides, the multiplication by a smooth compactly supported function is a
compact operator from $W^{\ell^{\prime}/2,p}(\mathbb{R}^{2d+1})$ to
$W^{\ell/2,p}(\mathbb{R}^{2d+1})$ and a bounded operator from
$W^{\ell/2,p}(\mathbb{R}^{2d+1})$ to
$W^{\ell/2,p}_{\mathbb{H}}(\mathbb{H}^{d})$ (see [19, Theorem 4.4.24, p.
240]). Consequently, $M_{\varphi}$ is a compact operator from
$W_{\mathbb{H}}^{\ell^{\prime},p}(\mathbb{H}^{d})$ to
$W_{\mathbb{H}}^{\ell/2,p}(\mathbb{H}^{d})$. ∎

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