# slow exponential growth representations of Sp(n, 1) at the edge of Cowling’s
strip

Pierre Julg  and  Shintaro Nishikawa  P.J.: Institut Denis Poisson, Université
d’Orléans, Collegium Sciences et Techniques Bâtiment de mathématiques, Rue de
Chartres B.P. 6759 F-45067 Orléans Cedex 2 - France  pierre.julg@univ-
orleans.fr S.N.: Mathematical Institute, WWU, Münster, Einsteinstr. 62, 48159
Münster, GERMANY<EMAIL_ADDRESS>
###### Abstract.

We obtain a slow exponential growth estimate for the spherical principal
series representation $\rho_{s}$ of Lie group $\mathrm{Sp}(n,1)$ at the edge
$(\mathrm{Re}(s)=1)$ of Cowling’s strip $(|\mathrm{Re}(s)|<1)$ on the Sobolev
space $\mathcal{H}^{\alpha}(G/P)$ when $\alpha$ is the critical value
$Q/2=2n+1$. As a corollary, we obtain a slow exponential growth estimate for
the homotopy $\rho_{s}$ ($s\in[0,1]$) of the spherical principal series which
is required for the first author’s program for proving the Baum–Connes
conjecture with coefficients for $\mathrm{Sp}(n,1)$.

S.N. was supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy EXC 2044-390685587,
Mathematics Münster: Dynamics-Geometry-Structure.

## Introduction

Let $G=\mathrm{SO}_{0}(n,1)$, resp. $\mathrm{SU}(n,1)$, resp.
$\mathrm{Sp}(n,1)$, resp. $\mathrm{F}_{4(-20)}$, a simple Lie group of real
rank one. Consider its spherical principal series representation $\rho_{s}(g)$
on $C^{\infty}(G/P)$ for $s\in\mathbb{C}$ where $P$ is a minimal parabolic
subgroup of $G$. Let us normalize the parameters $s$ so that $\rho_{it}(g)$ is
unitary for $t\in\mathbb{R}$ with respect to the canonical $L^{2}$-norm on
$G/P$ and so that it contains the trivial sub-representation at $s=1$. More
specifically, we have

$\rho_{s}(g)=\lambda_{g}^{Q/2(1-s)}\rho(g)$

on $C^{\infty}(G/P)$ where $\rho(g)$ is the left-translation by $g$,
$\lambda_{g}$ is a cocycle, and $Q=n-1,2n,4n+2,22$ respectively.

The picture of the unitary dual is rather different between the case of
$\mathrm{SO}_{0}(n,1)$ and $\mathrm{SU}(n,1)$, which have the Haagerup
property (or Gromov’s a-T-menability), and the case of $\mathrm{Sp}(n,1)$ and
$\mathrm{F}_{4(-20)}$ which have Kazhdan’s property T (see [6, Chapter 5]).
The complementary series (i.e. the $\rho_{s}$, $s$ real, which are unitary for
a suitable Hilbert scalar product) is defined for $-1<s<1$ in the first case,
whereas in the second case a gap appears between the trivial representation
and the complementary series, which ranges for $-{2n-1\over 2n+1}<s<{2n-1\over
2n+1}$ for $\mathrm{Sp}(n,1)$, $-{5\over 11}<s<{5\over 11}$ for
$\mathrm{F}_{4(-20)}$ (see [8]).

However, it was pointed out in the 1980’s by Michael Cowling that if one
replaces “unitary” by “uniformly bounded”, the difference disappears (see
[4]). Namely, the representation $\rho_{s}$ for $s$ in the strip
$-1<\mathrm{Re}(s)<1$ becomes a uniformly bounded representation of $G$ on a
Hilbert space, if we consider an appropriate Sobolev space. Here, a
representation is uniformly bounded in a sense that the operator norm
$||\rho_{s}(g)||$ of $\rho_{s}(g)$ is bounded by a constant $C>0$ which is
independent of $g$ in $G$.

Analysis on Heisenberg groups plays an important role as a nilpotent subgroup
$N$ of $G$ provides the so-called open picture of the principal series
representation $\rho_{s}$. For $-1<\mathrm{Re}(s)<1$, the representation
$\rho_{s}$ is equivalently represented on the Sobolev space
$\mathcal{H}^{\alpha}(N)$ on $N$ with respect to a sub-Laplacian on $N$. This
picture has an advantage that it uses the canonical norm for which
$\rho_{s}(g)$ is unitary for $g$ in $P$. It then suffices to show that a
single element $w$ in $G$ is bounded to see $\rho_{s}$ is uniformly bounded.
On the other hand, for $s$ outside the strip $-1<\mathrm{Re}(s)<1$, $w$ is not
bounded with respect to such a norm and hence the representation itself would
not be well-defined.

In the compact picture, Cowling’s result reads as follows: the representation
$\rho_{s}$ on the Sobolev space $\mathcal{H}^{\alpha}(G/P)$ for $s$ in the
strip $-1<\mathrm{Re}(s)<1$ is uniformly bounded for $\alpha=(Q/2)s$ (see
[1]). Here, the Sobolev space $\mathcal{H}^{\alpha}(G/P)$ is defined as the
completion of $C^{\infty}(G/P)$ with respect to the Hilbert space norm
$||(1+\Delta_{E})^{\alpha/2}\xi||_{L^{2}(G/P)}$ where $\Delta_{E}$ is a
K-invariant sub-Laplacian on $G/P$ for a maximal compact subgroup $K$ of $G$.
The compact picture has an advantage that the representation $\rho_{s}$ on
$\mathcal{H}^{\alpha}(G/P)$ is well-defined (bounded) for all $s$, in
particular for $s=1$.

In the first author’s program for proving the Baum–Connes conjecture with
coefficients for $\mathrm{Sp}(n,1)$ [7], it is crucial to consider the growth
of the operator norm
$||\rho_{s}(g)||_{\mathcal{H}^{\alpha}(G/P)\to\mathcal{H}^{\alpha}(G/P)}$ of
the representation $\rho_{s}(g)$ on $\mathcal{H}^{\alpha}(G/P)$ for
$\alpha=(Q/2)s$ and $0\leq s\leq 1$ as $s$ approaches to $1$ (see [7, Section
8.3]). In particular, we would like to show that the homotopy $\rho_{s}(g)$
($s\in[0,1]$) of representations has slow exponential growth in a sense that
for any $\epsilon>0$, there is $C=C(\epsilon)>0$ such that

$||\rho_{s}(g)||_{\mathcal{H}^{(Q/2)s}(G/P)\to\mathcal{H}^{(Q/2)s}(G/P)}\leq
Ce^{\epsilon l(g)}$

for all $s\in[0,1]$ and for all $g$ in $G$. Here, $l(g)$ is a K-bi-invariant
length function on $G$ defined as $l(g)=d_{G/K}(gK,K)$, i.e.
$l(ka_{t}k^{\prime})=|t|$ for any $k,k^{\prime}$ in $K$ and $t\in\mathbb{R}$.

With an application to this problem in mind, Astengo, Cowling and Di Blasio
obtained a similar type of estimates for $\rho_{s}(g)$ on
$\mathcal{H}^{\alpha}(G/P)$. They showed that [1, Theorem 5.1] for any
$\alpha\in(-Q/2,Q/2)$ fixed, there is $C>0$ such that

$||\rho_{s}(g)||_{\mathcal{H}^{\alpha}(G/P)\to\mathcal{H}^{\alpha}(G/P)}\leq
Ce^{((Q/2)\mathrm{Re}(s)-\alpha)l(g)}$

for all $s\in\mathbb{C}$ and for all $g$ in $G$.

Although the above estimate suggests that the Sobolev space
$\mathcal{H}^{Q/2}(G/P)$ should provide a Hilbert space for which
$\rho_{s}(g)$ has slow exponential growth when $\mathrm{Re}(s)=1$, the
estimate itself is not enough to conclude that the slow exponential growth of
the homotopy $\rho_{s}(g)$ ($s\in[0,1]$). Because of this, we shall study the
growth of the representation $\rho_{s}(g)$ on $\mathcal{H}^{\alpha}(G/P)$ when
$\alpha$ is the critical value $\alpha=Q/2$. The following is our main result.

######  Theorem.

(See Theorem 2.1) For any $\epsilon>0$, there is $C=C(\epsilon)>0$ such that
for all $s$ satisfying $\mathrm{Re}(s)=1$, we have the following upper-bound
for the operator norm
$||\rho_{s}(g)||_{\mathcal{H}^{Q/2}(G/P)\to\mathcal{H}^{Q/2}(G/P)}$ of
$\rho_{s}(g)$ on $\mathcal{H}^{Q/2}(G/P)$ for all $g$ in $G$:

$||\rho_{s}(g)||_{\mathcal{H}^{Q/2}(G/P)\to\mathcal{H}^{Q/2}(G/P)}\leq
Ce^{\epsilon l(g)}(1+|\mathrm{Im}(s)|)^{Q/2}.$

As a corollary, by a simple application of complex interpolation, we show the
desired slow exponential growth estimate for the homotopy $\rho_{s}(g)$
($s\in[0,1]$).

######  Corollary.

(See Corollary 2.2) For any $\epsilon>0$, there is $C=C(\epsilon)>0$ such that
for any $s\in[0,1]$, we have for all $g$ in $G$,

$||\rho_{s}(g)||_{\mathcal{H}^{(Q/2)s}(G/P)\to\mathcal{H}^{(Q/2)s}(G/P)}\leq
Ce^{\epsilon l(g)}.$

## Acknowledgements

The first author would like to thank Michael Cowling for many fruitful
discussions during the last 30 years. He is also grateful to Nigel Higson and
Vincent Lafforgue for pointing out the necessity to consider slow exponential
growth representations in the context of $K$-theory of group $C^{*}$-algebras
and Baum-Connes conjecture.

The second author would like to thank Nigel Higson for teaching him about
representation theory during the past years.

##  1\. Preliminaries

Let $\mathbb{F}=\mathbb{R},\mathbb{C}$ or $\mathbb{H}$ be the field of real
numbers, complex numbers or quaternions. For $z\in\mathbb{F}$, we define

$|z|^{2}=z^{\ast}z,\,\,\,\mathrm{Re}(z)=\frac{z+z^{\ast}}{2},\,\,\mathrm{Im}(z)=\frac{z-z^{\ast}}{2}.$

We also write $\mathrm{Im}(\mathbb{F})\subset\mathbb{F}$ to be the image of
$\mathrm{Im}(\,\,)$ on $\mathbb{F}$.

A sesquilinear form $q$ on a right vector space $\mathbb{F}^{n+1}$ over
$\mathbb{F}$ is given by

$q(z,w)=-\bar{z}_{0}w_{0}+\sum_{j=1}^{j=n}\bar{z}_{j}w_{j}$

for $z,w$ in $\mathbb{F}^{n+1}$.

Let $O(q)$ be the group of $(n+1)\times(n+1)$ matrices over $\mathbb{F}$ which
act on $\mathbb{F}^{n+1}$ from left and preserve the quadratic form $q$.

The Lie group $\mathrm{SO}_{0}(n,1)$ is the connected component of the
identity of $O(q)$ for $\mathbb{F}=\mathbb{R}$, $\mathrm{SU}(n,1)$ is
$O(q)\cap SL(n+1,\mathbb{C})$ for $\mathbb{F}=\mathbb{C}$ and
$\mathrm{Sp}(n,1)$ is $O(q)$ for $\mathbb{F}=\mathbb{H}$.

In this paper, we consider $G=\mathrm{Sp}(n,1)$ ($n\geq 2$) and thus
$\mathbb{F}=\mathbb{H}$ but all the results have obvious analogues for
$\mathrm{SO}_{0}(n,1)$ and $\mathrm{SU}(n,1)$. The closed subgroup $K$ of $G$
that preserves the canonical Euclidean metric on $\mathbb{F}^{n+1}$ is a
maximal compact subgroup of $G$.

The Lie group $G$ naturally acts on the projective space
$\mathbb{P}(\mathbb{F}^{n+1})$ over $\mathbb{F}$ and we have

$\displaystyle G\cdot[1,0,\cdots,0]^{T}$
$\displaystyle=\\{\,[z_{0},z_{1},\cdots,z_{n}]^{T}\in\mathbb{P}(\mathbb{F}^{n+1})\,\mid\,\,\,\sum_{1\leq
j\leq n}|z_{j}|^{2}<|z_{0}|^{2}\,\\}$
$\displaystyle=\\{\,[1,z_{1},\cdots,z_{n}]^{T}\in\mathbb{P}(\mathbb{F}^{n+1})\mid\,\,\,\sum_{1\leq
j\leq n}|z_{j}|^{2}<1\,\\}.$

The isotropy subgroup of $G$ at the point $[1,0,\cdots,0]^{T}$ is $K$. In this
way, $G/K$ can be viewed as the disk $\mathbb{D}^{4n}$ in $\mathbb{F}^{n}$.

The boundary of $G/K$ in $\mathbb{P}(\mathbb{F}^{n+1})$ is

$\displaystyle G\cdot[1,0,\cdots,0,1]^{T}$
$\displaystyle=\\{\,[z_{0},z_{1},\cdots,z_{n}]^{T}\in\mathbb{P}(\mathbb{F}^{n+1})\,\mid\,\,\,\sum_{1\leq
j\leq n}|z_{j}|^{2}=|z_{0}|^{2}\,\\}$
$\displaystyle=\\{\,[1,z_{1},\cdots,z_{n}]^{T}\in\mathbb{P}(\mathbb{F}^{n+1})\mid\,\,\,\sum_{1\leq
j\leq n}|z_{j}|^{2}=1\,\\}.$

The isotropy subgroup $P$ of $G$ at the point $[1,0,\cdots,0,1]^{T}$ is a
minimal parabolic subgroup of $G$. In this way, $G/P$ can be viewed as the
sphere $S^{4n-1}$ in $\mathbb{F}^{n}$.

Let $A$ be a closed subgroup of $G$ defined as

$A=\\{\,a_{t}\in G\mid t\in\mathbb{R}\,\\},\,\,\,a_{t}=\begin{bmatrix}\cosh
t&0&\sinh t\\\ 0&1&0\\\ \sinh t&0&\cosh
t\end{bmatrix}=U\begin{bmatrix}e^{-t}&0&0\\\ 0&1&0\\\
0&0&e^{t}\end{bmatrix}U^{-1},$

where each matrix has an $(n-1)\times(n-1)$ matrix in the middle entry and

$U=U^{*}=U^{-1}=\begin{bmatrix}-1/\sqrt{2}&0&1/\sqrt{2}\\\ 0&1&0\\\
1/\sqrt{2}&0&1/\sqrt{2}\end{bmatrix}.$

We have $KA^{+}K$ decomposition

$G=KA^{+}K$

where $A^{+}$ consists of $a_{t}$ for $t\geq 0$. Let $M$ be the centralizer of
$A$ in $K$. Consider the restricted root space decomposition of $\mathfrak{g}$
with respect $\mathfrak{a}$:

$\mathfrak{g}=\mathfrak{a}\oplus\mathfrak{m}\oplus\mathfrak{n}\oplus\overline{\mathfrak{n}}.$

We have

$\mathfrak{n}=\mathfrak{n}_{1}\oplus\mathfrak{n}_{2},$
$\mathfrak{n}_{1}=\\{\,U\begin{bmatrix}0&0&0\\\ -X/\sqrt{2}&0&0\\\
0&X^{\ast}/\sqrt{2}&0\end{bmatrix}U^{-1}=\begin{bmatrix}0&X^{\ast}/2&0\\\
X/2&0&-X/2\\\
0&X^{\ast}/2&0\end{bmatrix}\in\mathfrak{g}\mid\,X\in\mathbb{F}^{n-1}\\},$
$\mathfrak{n}_{2}=\\{\,U\begin{bmatrix}0&0&0\\\ 0&0&0\\\
-Z&0&0\end{bmatrix}U^{-1}=\begin{bmatrix}Z/2&0&-Z/2\\\ 0&0&0\\\
Z/2&0&-Z/2\end{bmatrix}\in\mathfrak{g}\mid\,Z\in\mathrm{Im}(\mathbb{F})\\},$

and

$\overline{\mathfrak{n}}=\overline{\mathfrak{n}}_{1}\oplus\overline{\mathfrak{n}}_{2},$
(1.1)
$\overline{\mathfrak{n}}_{1}=\\{\,U\begin{bmatrix}0&X^{\ast}/\sqrt{2}&0\\\
0&0&-X/\sqrt{2}\\\ 0&0&0\end{bmatrix}U^{-1}=\begin{bmatrix}0&-X^{\ast}/2&0\\\
-X/2&0&-X/2\\\
0&X^{\ast}/2&0\end{bmatrix}\in\mathfrak{g}\mid\,X\in\mathbb{F}^{n-1}\\},$
(1.2) $\overline{\mathfrak{n}}_{2}=\\{\,U\begin{bmatrix}0&0&-Z\\\ 0&0&0\\\
0&0&0\end{bmatrix}U^{-1}=\begin{bmatrix}Z/2&0&Z/2\\\ 0&0&0\\\
-Z/2&0&-Z/2\end{bmatrix}\in\mathfrak{g}\mid\,Z\in\mathrm{Im}(\mathbb{F})\\}.$

With respect to these expressions, we shall use the coordinates
$(X,Z)\in\mathbb{F}^{n-1}\oplus\mathrm{Im}(\mathbb{F})$ to express elements in
$\overline{\mathfrak{n}}$. In these coordinates, the Lie bracket on
$\overline{\mathfrak{n}}$ is

$[(X_{1},Z_{1}),(X_{2},Z_{2})]=(0,\mathrm{Im}(X_{1}^{\ast}X_{2})).$

The exponential map $\exp\colon\overline{\mathfrak{n}}\to\overline{N}\subset
G$ (as well as $\exp\colon{\mathfrak{n}}\to N$) is a diffeomorphism. For
$(X,Z)$ in $\overline{\mathfrak{n}}$, we have

$\mathrm{exp}(X,Z)=\begin{bmatrix}1+X^{\ast}X/8+Z/2&-X^{\ast}/2&X^{\ast}X/8+Z/2\\\
-X/2&1&-X/2\\\
-X^{\ast}X/8-Z/2&X^{\ast}/2&1-X^{\ast}X/8-Z/2\end{bmatrix}\in\overline{N}\subset
G.$

We shall identify $\overline{\mathfrak{n}}$ as
$\mathbb{F}^{n-1}\oplus\mathrm{Im}(\mathbb{F})$ and equip it with the standard
Euclidean metric. The group $\overline{N}$ is unimodular and we fix a Haar
measure given by
$d\overline{n}=2^{\mathrm{dim}(\overline{\mathfrak{n}}_{2})}dXdZ$ on
$\overline{N}\cong\overline{\mathfrak{n}}\cong
F^{n-1}\oplus\mathrm{Im}(\mathbb{F})$ where $dX$ and $dZ$ are the elements of
the Lebesgue measures on $\mathbb{F}^{n-1}$ and $\mathrm{Im}(\mathbb{F})$.

We have $KAN$ decomposition

$G=KAN,$

Langlands decomposition

$P=MAN,$

and Bruhat decomposition

$G=PwP\sqcup P$

where $w\in K$ is any representative of the non-trivial element in the Weyl
group $N_{K}(A)/Z_{K}(A)$.

Let $o=[1,0,\cdots,0,1]^{T}$ in $G/P\subset\mathbb{P}(\mathbb{F}^{n+1})$.
Recall that the minimal parabolic subgroup $P$ is the isotropy subgroup of $G$
at $o$. The Cayley transform

(1.3) $\mathcal{C}\colon\overline{N}\to G/P$

is defined by sending $n$ in $\overline{N}$ to $n\cdot o=nP$ in $G/P$. The
Cayley transform is a diffeomorphism from $\overline{N}$ onto an open dense
subset $G/P-\\{-o\\}$ where $-o=[1,0,\cdots,0,-1]^{T}=w\cdot o$.

Recall that $G/P$ is naturally viewed as the sphere
$S^{4n-1}\subset\mathbb{F}^{n}$ and the point $o$ corresponds to the point
$(0,\cdots,0,1)$ in the sphere $S^{4n-1}$.

With this identification of $G/P\subset\mathbb{P}(\mathbb{F}^{n+1})$ and
$S^{4n-1}\subset\mathbb{F}^{n}$, we have for
$(X,Z)\in\mathbb{F}^{n-1}\oplus\mathrm{Im}(\mathbb{F})=\overline{\mathfrak{n}}$,

$\displaystyle\mathcal{C}(\exp(X,Z)$ $\displaystyle)=\exp(X,Z)\cdot
o=\begin{pmatrix}-X\\\
1-X^{*}X/4-Z\end{pmatrix}\left(1+X^{*}X/4+Z\right)^{-1}$
$\displaystyle=\begin{pmatrix}-X(1+X^{*}X/4-Z)\\\
1-(X^{*}X)^{2}/16-|Z|^{2}-2Z\end{pmatrix}\left((1+X^{*}X/4)^{2}+|Z|^{2}\right)^{-1}$

in $S^{4n-1}\subset\mathbb{F}^{n}$. Here, the first row represents a vector in
$\mathbb{F}^{n-1}$ and the second row represents a vector in $\mathbb{F}$.
Setting

$||X||^{2}=X^{\ast}X,\,\,\,B(X,Z)=(1+||X||^{2}/4)^{2}+|Z|^{2},$

we have

(1.4) $\mathcal{C}(\exp(X,Z))=B(X,Z)^{-1}\begin{pmatrix}-X(1+||X||^{2}/4-Z)\\\
1-||X||^{4}/16-|Z|^{2}-2Z\end{pmatrix}.$

This is essentially the same formula as the one in [1]. In [1], it is defined
from $N$ to $G/P-\\{o\\}$.

The Riemannian symmetric space $G/K\subset\mathbb{P}(\mathbb{F}^{n+1})$ has a
canonical $G$-invariant metric for which the distance between
$eK=[1,0,\cdots,0]^{T}$ and $a_{t}K=[\cosh t,0,\cdots,0,\sinh
t]^{T}=[1,0,\cdots,0,\tanh t]^{T}$ is $|t|$ for $t\in\mathbb{R}$.

For $x,y$ in $G/K$, the Busemann cocycle $\gamma_{x,y}$ is a smooth function
on $G/P$ defined as

$\gamma_{x,y}(z)=\lim_{z^{\prime}\to
z}\left(d_{G/K}(z^{\prime},y)-d_{G/K}(z^{\prime},x)\right)=\log\left|\frac{q(y,z)q(x,x)^{1/2}}{q(x,z)q(y,y)^{1/2}}\right|$

for $z\in G/P$ (see [3, Proposition 3.1.1]).

Let $d\mu_{x}$ be the normalized $K_{x}$ invariant measure on $G/P$ for the
isotropy group $K_{x}$ at $x$ of the $G$-action on $G/K$. We have [3,
Proposition 3.1.2]

$d\mu_{y}=e^{-Q\gamma_{x,y}}d\mu_{x}$

for
$Q=\mathrm{dim}(\mathfrak{n}_{1})+2\mathrm{dim}(\mathfrak{n}_{2})=4(n-1)+2(3)=4n+2$.

We simply write $0$ for the fixed origin $eK$ of $G/K$. Let

(1.5) $\lambda_{g}=e^{-\gamma_{0,g0}}$

so that

$d\mu_{g0}=\lambda_{g}^{Q}d\mu_{0}.$

We have for $a_{t}\in A$ and for $z=[1,z_{1},\cdots,z_{n}]^{T}$ in $G/P$,

$\displaystyle\gamma_{0,a_{t}0}(z)$ $\displaystyle=\log\left|1-(\tanh
t)z_{n}\right|-(1/2)\log(1-(\tanh t)^{2})$ $\displaystyle=\log\left|\cosh
t-(\sinh t)z_{n}\right|$

and

(1.6) $\lambda_{a_{t}}=e^{-\gamma_{0,a_{t}0}}=\left|1-(\tanh
t)z_{n}\right|^{-1}\left(\cosh t\right)^{-1}=\left|\cosh t-(\sinh
t)z_{n}\right|^{-1}.$

We now briefly recall from [7], the definition of a $G$-equivariant sub-bundle
$E$ of the tangent-bundle $T(G/P)$ and its associated sub-Laplacian
$\Delta_{E}$. Denote by $P_{x}$, the isotropy subgroup of $G$ at $x$ in $G/P$.
The fiber of the cotangent bundle $T^{\ast}(G/P)$ at $x$ is naturally
identified as $(\mathfrak{g}/\mathfrak{p}_{x})^{\ast}$. The latter space is
naturally identified as the annihilator $\mathfrak{p}_{x}^{\perp}$ of
$\mathfrak{p}_{x}$ in $\mathfrak{g}$ with respect to the Killing form on
$\mathfrak{g}$. The space $\mathfrak{p}_{x}^{\perp}$ coincides with the
nilpotent radical $\mathfrak{n}_{x}$ of $\mathfrak{p}_{x}$ which is a 2-step
nilpotent Lie algebra with center
$\mathfrak{z}_{x}=[\mathfrak{n}_{x},\mathfrak{n}_{x}]$. A $G$-equivariant sub-
bundle $F$ of $T^{\ast}(G/P)$ is defined so that the fiber at $x$ is
$\mathfrak{z}_{x}$. A $G$-equivariant sub-bundle $E$ of the tangent-bundle
$T(G/P)$ is defined as the annihilator $F^{\perp}$ of $F$.

Let $d_{E}$ be the differential operator from $C^{\infty}(G/P)$ to the section
$\Gamma(E^{\ast})$ of $E^{\ast}$, defined as the composition of the de-Rham
differential operator and the restriction map from $\Gamma(T^{\ast}(G/P))$ to
$\Gamma(E^{\ast})$. The adjoint $d_{E}^{\ast}$ is defined with respect to the
standard $K$-invariant metric on $G/P$. The $K$-invariant sub-Laplacian on
$C^{\infty}(G/P)$ is defined by $\Delta_{E}=d_{E}^{\ast}d_{E}$. The sub-
Laplacian $\Delta_{E}$ is a self-adjoint, positive operator on the Hilbert
space $L^{2}(G/P,d\mu_{0})$. It has compact resolvent and one-dimensional
kernel consisting of constant functions on $G/P$.

For any $\alpha$ in $\mathbb{R}$, let $\mathcal{H}^{\alpha}(G/P)$ be the
Sobolev space defined by the completion of $C^{\infty}(G/P)$ by the norm

$||\xi||_{\mathcal{H}^{\alpha}(G/P)}=||(1+\Delta_{E})^{\alpha/2}\xi||_{L^{2}(G/P,d\mu_{0})}$

for $\xi$ in $C^{\infty}(G/P)$.

##  2\. Main result

As in [7, Section 8.1], for $0\leq\mathrm{Re}(s)\leq 1$, let

$\pi_{s}(g)=(1+\Delta_{E})^{(Q/4)s}\lambda_{g}^{(Q/2)(1-s)}\rho(g)(1+\Delta_{E})^{-(Q/4)s}$

defined on the Hilbert space $\mathcal{H}^{0}(G/P)=L^{2}(G/P,d\mu_{0})$. Here,
$\rho(g)$ for $g$ in $G$ denotes the left-translation action on the functions
on $G/P$ and the cocycle $\lambda_{g}$ is as in (1.5).

Note,

$\pi_{0}(g)=\lambda_{g}^{Q/2}\rho(g)$

is unitary on $L^{2}(G/P,d\mu_{0})$. This is because

$\rho(g)d\mu_{0}=(g^{-1})^{\ast}d\mu_{0}=d\mu_{g0}=\lambda_{g}^{Q}d\mu_{0}$

so we have

$\langle
f_{1},f_{2}\rangle_{\mathcal{H}^{0}}=\int_{G/P}\overline{f_{1}}f_{2}d\mu_{0}=\int_{G/P}\rho(g)(\overline{f_{1}}f_{2}d\mu_{0})$
$=\int_{G/P}\rho(g)(\overline{f_{1}}f_{2})\lambda_{g}^{Q}d\mu_{0}=\langle\pi_{0}(g)f_{1},\pi_{0}(g)f_{2}\rangle.$

Similarly, if $\mathrm{Re}(s)=0$, $\pi_{s}(g)$ is unitary on
$L^{2}(G/P,d\mu_{0})$.

The following is our main result.

######  2.1 Theorem.

For any $\epsilon>0$, there is $C=C(\epsilon)>0$ such that for all $s$
satisfying $\mathrm{Re}(s)=1$, we have the following upper-bound for the
operator norm $||\pi_{s}(g)||$ of $\pi_{s}(g)$ on $\mathcal{H}^{0}(G/P)$ for
all $g$ in $G$:

$||\pi_{s}(g)||\leq Ce^{\epsilon l(g)}(1+|\mathrm{Im}(s)|)^{Q/2}.$

Here, $l(g)$ is a K-bi-invariant length function on $G$ defined as
$l(g)=d_{G/K}(gK,K)$, i.e. $l(ka_{t}k^{\prime})=|t|$ for any $k,k^{\prime}$ in
$K$ and $t\in\mathbb{R}$.

######  2.2 Corollary.

For any $\epsilon>0$, there is $C=C(\epsilon)>0$ such that for any
$t\in[0,1]$, we have for all $g$ in $G$,

$||\pi_{t}(g)||\leq Ce^{\epsilon l(g)}.$

###### Proof.

For any $g$ in $G$ and for any $K$-finite functions $u$ and $v$ in
$L^{2}(G/P)$, we consider a holomorphic function

$s\mapsto\langle\pi_{s}(g)u,v\rangle$

for $s$ in the strip $0\leq\mathrm{Re}(s)\leq 1$. For each fixed $g$ and
$u,v$, it is not hard to see that this holomorphic function is bounded on the
strip. Now fix any positive constant $A>0$ and consider a holomorphic function

$s\mapsto\langle\pi_{s}(g)u,v\rangle e^{As^{2}}$

on the strip. For each $g$ and $u,v$ fixed, by the three lines theorem, we
have

$\sup_{\mathrm{Re}(s)=t}|\langle\pi_{s}(g)u,v\rangle
e^{As^{2}}|\leq\left(\sup_{\mathrm{Re}(s)=0}|\langle\pi_{s}(g)u,v\rangle
e^{As^{2}}|\right)^{1-t}\left(\sup_{\mathrm{Re}(s)=1}|\langle\pi_{s}(g)u,v\rangle
e^{As^{2}}|\right)^{t}$

for any $0\leq t\leq 1$. We have

$|\langle\pi_{t}(g)u,v\rangle|e^{At^{2}}\leq\sup_{\mathrm{Re}(s)=t}|\langle\pi_{s}(g)u,v\rangle
e^{As^{2}}|.$

For $\mathrm{Re}(s)=0$, $\pi_{s}(g)$ is unitary so we have

$\sup_{\mathrm{Re}(s)=0}|\langle\pi_{s}(g)u,v\rangle
e^{As^{2}}|\leq||u||||v||.$

For $\mathrm{Re}(s)=1$, by Theorem 2.1,

$\displaystyle\sup_{\mathrm{Re}(s)=1}|\langle\pi_{s}(g)u,v\rangle e^{As^{2}}|$
$\displaystyle\leq Ce^{\epsilon
l(g)}||u||||v||\sup_{b\in\mathbb{R}}\left|(1+|b|)^{Q/2}e^{A(1+ib)^{2}}\right|$
$\displaystyle\leq C^{\prime}e^{\epsilon l(g)}||u||||v||$

where

$C^{\prime}=C\sup_{b\in\mathbb{R}}\left|(1+|b|)^{Q/2}e^{A(1+ib)^{2}}\right|<+\infty.$

Combining all these, we get

$\sup_{0\leq t\leq 1}|\langle\pi_{t}(g)u,v\rangle|\leq C^{\prime}e^{\epsilon
l(g)}||u||||v||.$

The constant $C^{\prime}$ does not depend on $g$, $u$ and $v$ so we are done.
∎

##  3\. Some reduction

We begin our proof of Theorem 2.1. We will eventually reduce this problem to a
certain technical estimate of functions. First of all,

######  3.1 Proposition.

For $s=1$, we have

$||\pi_{1}(g)||\leq C(1+l(g))$

for $C>0$ independent of $g$ in $G$, i.e. $\pi_{1}(g)$ is of linear growth.

###### Proof.

Recall

$\pi_{1}(g)=(1+\Delta_{E})^{(Q/4)}\rho(g)(1+\Delta_{E})^{-(Q/4)}.$

The left-translation $\rho(g)$ defines a bounded representation of $G$ on the
Sobolev space $\mathcal{H}^{\alpha}(G/P)$. Our assertion is that

$||\rho(g)||_{\mathcal{H}^{Q/2}(G/P)\to\mathcal{H}^{Q/2}(G/P)}\leq C(1+l(g)).$

This follows from the following two lemmas. ∎

######  3.2 Lemma.

Define a new Hilbert space norm $||\,\,||^{\prime}$ on $C^{\infty}(G/P)$ by

$||\xi||^{\prime
2}=||\xi||^{2}_{L^{2}(G/P)}+||(1+\Delta_{E})^{Q/4-1/2}d_{E}\xi||^{2}_{\Gamma_{L^{2}}(E^{\ast})}.$

The norm $||\,\,||^{\prime}$ and the Sobolev norm
$||\,\,||_{\mathcal{H}^{Q/2}(G/P)}$ are equivalent. Here,
$\Gamma_{L^{2}}(E^{\ast})$ is the Hilbert space completion of
$\Gamma(E^{\ast})$ with respect to the $K$-invariant metric and a K-invariant
sub-Laplacian $\Delta_{E}$ on $\Gamma_{L^{2}}(E^{\ast})$ is defined as
$\Delta_{E}=\nabla_{E}^{\ast}\nabla_{E}$ where a differential operator
$\nabla_{E}\colon\Gamma(E^{\ast})\to\Gamma(E^{\ast}\otimes E^{\ast})$ is the
composition of a fixed K-invariant connection $\nabla$ on $E^{\ast}$ and the
restriction map from $T^{\ast}(G/P)$ to $E^{\ast}$.

###### Proof.

Both operators

$(1+\Delta_{E})^{-Q/4}(1+d_{E}^{\ast}\Delta_{E}^{Q/2-1}d_{E})^{1/2},\,\,\,(1+d_{E}^{\ast}\Delta_{E}^{Q/2-1}d_{E})^{-1/2}(1+\Delta_{E})^{Q/4}$

on $C^{\infty}(G/P)$ have weighted order zero in a sense of [7, Section 4.1],
and thus are bounded on $L^{2}(G/P,d\mu_{0})$. These two operators give an
equivalence between $\mathcal{H}^{Q/2}(G/P)$ and the completion of
$C^{\infty}(G/P)$ with respect to $||\,\,||^{\prime}$. ∎

######  3.3 Lemma.

The left-translation $\rho(g)$ defines a bounded representation on the
completion of $C^{\infty}(G/P)$ with respect to the new norm
$||\,\,||^{\prime}$ and it has linear growth.

###### Proof.

Denote by $W^{\prime}$, the completion of $C^{\infty}(G/P)$ with respect to
$||\,\,||^{\prime}$. Note that $W^{\prime}$ contains the trivial sub-
representation $\mathbb{C}1_{G/P}$ spanned by the constant functions. Our
claim follows from the fact that the induced representation on the quotient
space $W_{0}=W^{\prime}/\mathbb{C}1_{G/P}$ is uniformly bounded. This fact was
proved in [9, Corollary 2.4] by observing that $W_{0}$ is naturally viewed as
a sub-representation of the representation of $G$ on (the completion of)
$\Gamma(E^{\ast})$ with respect to the norm $||(1+\Delta_{E})^{Q/4-1/2}\xi||$
for $\xi\in\Gamma(E^{\ast})$. This representation of $G$ on $\Gamma(E^{\ast})$
is a principal series representation and it is uniformly bounded with respect
to the given norm (See [9, Theorem 2.3]). ∎

Let $s=1+ib$ for $b\in\mathbb{R}$. We have

$\pi_{s}(g)=(1+\Delta_{E})^{(Q/4)ib}(1+\Delta_{E})^{(Q/4)}\lambda_{g}^{-(Q/2)(ib)}\rho(g)(1+\Delta_{E})^{-(Q/4)}(1+\Delta_{E})^{-(Q/4)ib},$

so

$\displaystyle||\pi_{s}(g)||$
$\displaystyle=||(1+\Delta_{E})^{(Q/4)}\lambda_{g}^{-(Q/2)(ib)}\rho(g)(1+\Delta_{E})^{-(Q/4)}||$
$\displaystyle=||(1+\Delta_{E})^{(Q/4)}\lambda_{g}^{-(Q/2)(ib)}(1+\Delta_{E})^{-(Q/4)}\pi_{1}(g)||.$

In view of Proposition 3.1, using $G=KA^{+}K$, in order to prove Theorem 2.1,
we just need to show, for any $\epsilon>0$, the existence of $C=C(\epsilon)>0$
such that for $t\geq 0$ and for $b\in\mathbb{R}$,

$||(1+\Delta_{E})^{(Q/4)}\lambda_{a_{t}}^{-(Q/2)(ib)}(1+\Delta_{E})^{-(Q/4)}||\leq
Ce^{\epsilon t}(1+|b|)^{Q/2}.$

Let $\chi$ be a smooth function on $G/P$ which is $1$ near the point
$o=[1,0,\cdots,0,1]^{T}$ and vanishes outside a small neighborhood of $o$.

######  3.4 Lemma.

There is $C>0$ such that for $t\geq 0$ and for $b\in\mathbb{R}$,

$||(1+\Delta_{E})^{(Q/4)}(1-\chi)\lambda_{a_{t}}^{-(Q/2)(ib)}(1+\Delta_{E})^{-(Q/4)}||\leq
C(1+|b|)^{Q/2}.$

###### Proof.

Recall from (1.6)

$\lambda_{a_{t}}=\left|1-(\tanh t)z_{n}\right|^{-1}\left(\cosh t\right)^{-1}$

so

$\lambda_{a_{t}}^{-(Q/2)(ib)}=\left|1-(\tanh
t)z_{n}\right|^{(Q/2)ib}\left(\cosh t\right)^{Q/2(ib)}.$

Note that the second factor is a constant function on $G/P$ of modulus one and
that

$(1-\chi)\left|1-(\tanh t)z_{n}\right|^{(Q/2)ib}$

smoothly converges as $t\to+\infty$ to the smooth function

$(1-\chi)\left|1-z_{n}\right|^{(Q/2)ib}$

on $G/P$. The claim follows from these. ∎

Thus, we reduced our proof of Theorem 2.1 to the following.

######  3.5 Lemma.

Let $\chi$ be a smooth function on $G/P$ which is supported near the point $o$
($z_{n}=1$). Consider the multiplication operator

$\chi\lambda_{a_{t}}^{-(Q/2)(ib)}\colon\mathcal{H}^{Q/2}(G/P)\to\mathcal{H}^{Q/2}(G/P).$

For any $\epsilon>0$, there is $C=C(\epsilon)>0$ such that for $t\geq 0$ and
for $b\in\mathbb{R}$,

$||\chi\lambda_{a_{t}}^{-(Q/2)(ib)}||_{\mathcal{H}^{Q/2}(G/P)\to\mathcal{H}^{Q/2}(G/P)}\leq
Ce^{\epsilon t}(1+|b|)^{Q/2}.$

We will prove Lemma 3.5 using a local model $\overline{N}\cong N$ of $G/P$
around the point $o$ via the Cayley transform $\mathcal{C}$ (1.3).

Fix an orthonormal basis $\\{E_{j}\\}_{1\leq
j\leq\mathrm{dim}(\overline{\mathfrak{n}}_{1})}$ of
$\overline{\mathfrak{n}}_{1}$. We identify $\overline{\mathfrak{n}}$ as the
left-invariant vector field on $\overline{N}$. The sub-Laplacian
$\Delta_{\overline{\mathfrak{n}}}$ on $\overline{N}$ is defined by

$\Delta_{\overline{\mathfrak{n}}}=-\sum_{j=1}^{\mathrm{dim}(\overline{\mathfrak{n}}_{1})}E_{j}^{2}$

on the space $C_{c}^{\infty}(\overline{N})$ of compactly supported smooth
functions on $\overline{N}$. It defines an essentially self-adjoint, positive
operator on $L^{2}(\overline{N})$. For any $\alpha\in\mathbb{R}$, we define
the homogeneous Sobolev space $\dot{\mathcal{H}}^{\alpha}(\overline{N})$ as
the completion of $C_{c}^{\infty}(\overline{N})$ by the norm

$||\xi||_{\dot{\mathcal{H}}^{\alpha}(\overline{N})}=||\Delta_{\overline{\mathfrak{n}}}^{\alpha/2}\xi||_{L^{2}(\overline{N})}.$

For a technical purpose, we also define the non-homogenous Sobolev space
$\mathcal{H}^{\alpha}(\overline{N})$ as the completion of
$C_{c}^{\infty}(\overline{N})$ by the norm

$||\xi||_{\mathcal{H}^{\alpha}(\overline{N})}=||(1+\Delta_{\overline{\mathfrak{n}}})^{\alpha/2}\xi||_{L^{2}(\overline{N})}.$

Multiplication by any compactly supported smooth function on $\overline{N}$
defines a bounded operator on $\mathcal{H}^{\alpha}(\overline{N})$ for any
$\alpha$ [5, Corollary 4.15]. It is also bounded on the homogeneous Sobolev
space $\dot{\mathcal{H}}^{\alpha}(\overline{N}$) for $-Q/2<\alpha<Q/2$ but not
necessarily for $|\alpha|\geq Q/2$.

For any open subset $U$ of $\overline{N}$, we define $\mathcal{H}^{\alpha}(U)$
and $\dot{\mathcal{H}}^{\alpha}(U)$ as the closures of $C^{\infty}_{c}(U)$ in
$\mathcal{H}^{\alpha}(\overline{N})$ and
$\dot{\mathcal{H}}^{\alpha}(\overline{N})$ respectively. Similarly, for any
open subset $U_{G/P}$ of $G/P$, we define $\mathcal{H}^{\alpha}(U_{G/P})$ as
the closure of $C^{\infty}_{c}(U_{G/P})$ in $\mathcal{H}^{\alpha}(G/P)$.

We note two things. First, if $U\subset\overline{N}$ is relatively compact,
the inclusion $\mathcal{H}^{\alpha}(U)$ to $\dot{\mathcal{H}}^{\alpha}(U)$ is
an isomorphism for all $\alpha\geq 0$. This is because the sub-Laplacian
$\Delta_{\overline{\mathfrak{n}}}$ is locally, bounded away from zero.
Secondly, for any $\alpha$, locally but not globally, the Cayley transform
$\mathcal{C}\colon\overline{N}\to G/P$ induces an isomorphism on Sobolev
spaces $\mathcal{H}^{\alpha}$. That is, the composition by the Cayley
transform $\mathcal{C}$ induces an isomorphism from
$\mathcal{H}^{\alpha}(U_{G/P})$ to $\mathcal{H}^{\alpha}(U_{\overline{N}})$
for any relatively compact open subset $U_{\overline{N}}$ and for
$U_{G/P}=\mathcal{C}U_{\overline{N}}$. We only use this fact for $\alpha=Q/2$.

Thus, Lemma 3.5 follows from the following claim. Notice that we are using the
non-homogenous Sobolev space in this statement.

######  3.6 Claim.

Let $\chi$ be a smooth compactly supported function on $\overline{N}$. For any
$\epsilon>0$, there is $C=C(\epsilon)>0$ such that for $t\geq 0$ and for
$b\in\mathbb{R}$, the multiplication operator

$\chi(\lambda_{a_{t}}\circ\mathcal{C})^{-(Q/2)(ib)}\colon\mathcal{H}^{Q/2}(\overline{N})\to\mathcal{H}^{Q/2}(\overline{N})$

has norm bound

$||\chi(\lambda_{a_{t}}\circ\mathcal{C})^{-(Q/2)(ib)}||_{\mathcal{H}^{Q/2}(\overline{N})\to\mathcal{H}^{Q/2}(\overline{N})}\leq
Ce^{\epsilon t}(1+|b|)^{Q/2}.$

In order to prove this claim, we give an explicit expression of
$\lambda_{a_{t}}\circ\mathcal{C}$ on $\overline{N}$. From now on, when there
is no confusion, we identify $\overline{N}$ with $\overline{\mathfrak{n}}$ via
the exponential map and use the coordinates
$(X,Z)\in\mathbb{F}^{n-1}\oplus\mathrm{Im}(\mathbb{F})$ for
$\overline{\mathfrak{n}}$ to express the element $\exp(X,Z)$ in $\overline{N}$
as well.

######  3.7 Lemma.

(c.f. [1, Corollary 3.9]) For any $t\in\mathbb{R}$, for any
$(X,Z)\in\mathbb{F}^{n-1}\oplus\mathrm{Im}(\mathbb{F})$, we have

$(\lambda_{a_{t}}\circ\mathcal{C})(X,Z)=e^{-t}\frac{B(X,Z)^{1/2}}{B_{t}(X,Z)^{1/2}}$

where

$B(X,Z)=(1+||X||^{2}/4)^{2}+|Z|^{2},\,\,B_{t}(X,Z)=(e^{-2t}+||X||^{2}/4)^{2}+|Z|^{2}.$

###### Proof.

Recall that

$d\mu_{a_{t}0}=(a_{-t})^{\ast}d\mu_{0}=\lambda_{a_{t}}^{Q}d\mu_{0}.$

That is, $\lambda_{a_{t}}^{Q}$ is the Jacobian $J_{a_{-t}}$ of the map
$a_{-t}$ on $G/P$ (with respect to $d\mu_{0}$). Let
$\delta_{t}\colon\overline{N}\to\overline{N}$ be the map
$(X,Z)\to(e^{t}X,e^{2t}Z)$. We have

$\mathrm{Ad}_{a_{-t}}=\delta_{t}$

on $\overline{N}$. By the chain rule, we have

$\lambda_{a_{t}}^{Q}\circ\mathcal{C}=J_{a_{-t}}\circ\mathcal{C}=J_{\mathcal{C}}\circ\delta_{t}\cdot
J_{\delta_{t}}\cdot J_{\mathcal{C}}^{-1}$

on $\overline{N}$ where $J_{\mathcal{C}}$ and $J_{\delta_{t}}$ are the
Jacobian of the map $\mathcal{C}$ and $\delta_{t}$ respectively. We have

$J_{\delta_{t}}=e^{Qt}$

and

$J_{\mathcal{C}}=B(X,Z)^{-Q/2}$

(see the end of Section 2.3 of [1]). Thus,

$\lambda_{a_{t}}^{Q}\circ\mathcal{C}=J_{\mathcal{C}}\circ\delta_{t}\cdot
J_{\delta_{t}}\cdot
J_{\mathcal{C}}^{-1}=e^{Qt}\left(\frac{B(X,Z)}{B(e^{t}X,e^{2t}Z)}\right)^{Q/2}.$

We get

$\lambda_{a_{t}}\circ\mathcal{C}=e^{t}\frac{B(X,Z)^{1/2}}{B(e^{t}X,e^{2t}Z)^{1/2}}=e^{-t}\frac{B(X,Z)^{1/2}}{B_{t}(X,Z)^{1/2}}.$

∎

######  3.8 Remark.

Recall

$\lambda_{a_{t}}=e^{-\gamma_{0,a_{t}0}}=|\cosh t-\sinh t|^{-1}=|1-(\tanh
t)z_{n}|^{-1}(\cosh t)^{-1}.$

The link with Lemma 3.7 is the following. For
$z_{n}=z_{n}\circ\mathcal{C}(X,Z)$, using (1.4),

$\displaystyle z_{n}=$ $\displaystyle
B^{-1}(1-||X||^{4}/16-|Z|^{2}-2Z)=B^{-1}(-N^{2}+2N-|Z|^{2}-2Z)$
$\displaystyle=-1+2B^{-1}N-2B^{-1}Z$

where $N=1+\|X\|^{2}/4$ and $B=N^{2}+|Z|^{2}$. An easy calculation yields
(denote $r=\tanh t$)

$\lambda_{a_{t}}^{-2}\circ\mathcal{C}=(1-r^{2})^{-1}|1-rz_{n}|^{2}={1+r\over
1-r}(1-{4r\over 1+r}B^{-1}N+{4r^{2}\over(1+r)^{2}}B^{-1}).$

On the other hand

$B_{t}=(\|X\|^{2}/4+e^{-2t})^{2}+|Z|^{2}=(N-{2r\over
1+r})^{2}+|Z|^{2}=B-{4r\over 1+r}N+{4r^{2}\over(1+r)^{2}},$

so that $\lambda_{a_{t}}^{-2}\circ\mathcal{C}=e^{2t}B^{-1}B_{t}$ which
coincides with the above formula.

In the light of the formula in Lemma 3.7, we have

$(\lambda_{a_{t}}\circ\mathcal{C})^{-(Q/2)(ib)}=e^{Q/2(t)(ib)}B(X,Z)^{-Q/4(ib)}B_{t}(X,Z)^{(Q/4)ib}.$

Note that the first factor $e^{Q/2(t)(ib)}$ is a constant function on
$\overline{N}$ of modulus one. We reduced Claim 3.6, and hence Theorem 2.1, to
the following.

######  3.9 Lemma.

Let $\chi$ be a smooth compactly supported function on $\overline{N}$. For any
$\epsilon>0$, there is $C=C(\epsilon)>0$ such that for $t\geq 0$ and for
$b\in\mathbb{R}$, the multiplication operator

$\chi
B_{t}(X,Z)^{ib}B(X,Z)^{-ib}\colon\mathcal{H}^{Q/2}(\overline{N})\to\mathcal{H}^{Q/2}(\overline{N})$

has norm bound

$||\chi
B_{t}(X,Z)^{ib}B(X,Z)^{-ib}||_{\mathcal{H}^{Q/2}(\overline{N})\to\mathcal{H}^{Q/2}(\overline{N})}\leq
Ce^{\epsilon t}(1+|b|)^{Q/2}.$

From now on, we focus on proving the lemma 3.9. We have the following easy
estimates for $B_{t}(X,Z)$:

  1. (1)

$B_{t}(X,Z)\geq e^{-4t}$,

  2. (2)

$B_{t}(X,Z)\geq||X||^{4}/16+|Z|^{2}=\mathcal{N}^{4}$,

where

$\mathcal{N}=(||X||^{4}/16+|Z|^{2})^{1/4}$

is a homogeneous gauge on $\overline{N}$.

##  4\. Proof of Lemma 3.9

Let us follow some convention from [1]: for a positive integer
$k\in\mathbb{Z}$, $\mathfrak{D}^{k}$ denotes the set of all differential
operators of the form

$E_{j_{1}}E_{j_{2}}\cdots E_{j_{k}},$

where $1\leq
j_{1},j_{2},\cdots,j_{k}\leq\mathrm{dim}(\overline{\mathfrak{n}}_{1})$. The
following result from [1] will be the key. Note that the homogeneous Sobolev
space is used in this statement.

######  4.1 Lemma.

[1, Theorem 3.6] Let $0\leq\alpha\leq\beta<Q/2$. Let $m$ be a smooth function
on $\overline{N}-\\{0\\}$. Suppose that on $\overline{N}-\\{0\\}$,

$|D^{j}m|\leq C_{j}\mathcal{N}^{-d-j}\,\,\,\,$

holds for any $0\leq j\leq k=\left\lceil{\alpha}\right\rceil$,
$D^{j}\in\mathfrak{D}^{j}$ and $d=\beta-\alpha$. Then,

$||m||_{\dot{\mathcal{H}}^{\beta}(\overline{N})\to\dot{\mathcal{H}}^{\alpha}(\overline{N})}\leq
C(\alpha,\beta)(C_{0}+C_{1}+\cdots+C_{k})$

where the constant $C(\alpha,\beta)$ only depends on $\alpha$ and $\beta$.

###### Proof.

This is what is proven in [1, Theorem 3.6] and used in the proof of [1,
Corollary 3.9]. For the detail of its proof, see the proof of [2, Theorem 7].
∎

The following technical estimate for the derivatives of $B_{t}$ very roughly
means $B_{t}$ behaves like a homogeneous function of degree $4$.

######  4.2 Lemma.

Let $U$ be a relatively compact open subset of $\overline{N}$. For any
positive integer $s\geq 0$, there is $C=C(s)>0$ so that for any
$D^{s}\in\mathfrak{D}^{s}$ and for any $(X,Z)$ in $U$,

$|D^{s}(B_{t}^{ib})|(X,Z)\leq C\frac{1}{B_{t}(X,Z)^{s/4}}(1+|b|)^{s}$

for all $t\geq 0$ and $b\in\mathbb{R}$.

###### Proof.

The proof will be given in the next (last) section. ∎

Now we use Lemma 4.1 together with Lemma 4.2 to prove Lemma 3.9, that is we
show that for any smooth compactly supported function $\chi$ on $\overline{N}$
and for any $\epsilon>0$, the multiplication operator

$\chi
B_{t}^{ib}B^{-ib}\colon\mathcal{H}^{Q/2}(\overline{N})\to\mathcal{H}^{Q/2}(\overline{N})$

has norm bound

$||\chi
B_{t}^{ib}B^{-ib}||_{\mathcal{H}^{Q/2}(\overline{N})\to\mathcal{H}^{Q/2}(\overline{N})}\leq
Ce^{\epsilon t}(1+|b|)^{Q/2},$

for $C=C(\epsilon)>0$ independent of $t\geq 0$ and $b\in\mathbb{R}$.

Note that for $\xi$ in $\mathcal{H}^{Q/2}(\overline{N})$, the following two
norms

$||\xi||_{\mathcal{H}^{Q/2}(\overline{N})},\,\,\,||\xi||_{\mathcal{H}^{0}(\overline{N})}+\sum_{j=1}^{\mathrm{dim}(\overline{\mathfrak{n}}_{1})}||E_{j}\xi||_{\mathcal{H}^{Q/2-1}(\overline{N})}$

are equivalent. Using this, we see that it is enough to show that for any
compactly supported smooth function $\chi$ and for any $\epsilon>0$, there is
$C=C(\epsilon)>0$ so that for any $t\geq 0$ and $b\in\mathbb{R}$,

(4.1) $||\chi
B_{t}^{ib}B^{-ib}||_{\mathcal{H}^{Q/2-1}(\overline{N})\to\mathcal{H}^{Q/2-1}(\overline{N})}\leq
Ce^{\epsilon t}(1+|b|)^{Q/2},$

and

(4.2) $||\chi
E_{j}(B_{t}^{ib}B^{-ib})||_{\mathcal{H}^{Q/2}(\overline{N})\to\mathcal{H}^{Q/2-1}(\overline{N})}\leq
Ce^{\epsilon t}(1+|b|)^{Q/2}.$

By Lemma 4.2, on a ball $U$ which contains the support of $\chi$,

$|D^{s}(B_{t}^{ib})|\leq C_{s}B_{t}^{-s/4}(1+|b|)^{s}\leq
C_{s}(1+|b|)^{s}\mathcal{N}^{-s}.$

It follows that there is $C>0$ such that for $0\leq s\leq Q/2-1$,

$|D^{s}(\chi B_{t}^{ib}B^{-ib})|\leq C(1+|b|)^{s}\mathcal{N}^{-s}.$

By Lemma 4.1, it follows that for some $C>0$,

$||\chi
B_{t}^{ib}B^{-ib}||_{\dot{\mathcal{H}}^{Q/2-1}(\overline{N})\to\dot{\mathcal{H}}^{Q/2-1}(\overline{N})}\leq
C(1+|b|)^{Q/2-1}\leq Ce^{\epsilon t}(1+|b|)^{Q/2}$

for all $t\geq 0$ and $b\in\mathbb{R}$. Since $\chi$ is compactly supported,
we can replace the homogeneous Sobolev space $\dot{\mathcal{H}}^{\alpha}$ by
the non-homogeneous Sobolev space $\mathcal{H}^{\alpha}$ in this inequality.
This proves (4.1).

As for (4.2), first note that for any compactly supported smooth function
$\chi_{0}$,

$\chi_{0}\colon\mathcal{H}^{Q/2}(\overline{N})\to\mathcal{H}^{Q/2-\epsilon}(\overline{N})$

is bounded for any $\epsilon>0$. Taking $\chi_{0}$ such that
$\chi_{0}\chi=\chi$, it is enough to bound the norm

$\chi
E_{j}(B_{t}^{ib}B^{-ib})\colon\dot{\mathcal{H}}^{Q/2-\epsilon}(\overline{N})\to\dot{\mathcal{H}}^{Q/2-1}(\overline{N}).$

Again, we are using that $\chi$ is compactly supported to replace the
homogeneous Sobolev space $\dot{\mathcal{H}}^{\alpha}$ by the non-homogeneous
Sobolev space $\mathcal{H}^{\alpha}$.

Let $\beta=Q/2-\epsilon$, $\alpha=Q/2-1$ and $d=\beta-\alpha=1-\epsilon$. By
Lemma 4.2, we have on a ball $U$ which contains the support of $\chi$,

$\displaystyle|D^{s}(E_{j}(B_{t}^{ib}))|=|D^{s+1}B_{t}^{ib}|\leq
C_{s+1}B_{t}^{-(s+1)/4}(1+|b|)^{s+1}$ $\displaystyle\leq
C_{s+1}B_{t}^{-\epsilon/4}B_{t}^{-(s+1-\epsilon)/4}(1+|b|)^{s+1}\leq
C_{s+1}B_{t}^{-\epsilon/4}\mathcal{N}^{-d-s}(1+|b|)^{s+1}.$

Moreover, we have

$B_{t}^{-\epsilon/4}\leq e^{\epsilon t}$

for $t\geq 0$. It follows that there is $C>0$ such that for $0\leq s\leq
Q/2-1$,

$|D^{s}(\chi E_{j}(B_{t}^{ib}B^{-ib}))|\leq Ce^{\epsilon
t}\mathcal{N}^{-d-s}(1+|b|)^{s+1}.$

Hence, by Lemma 4.1, it follows that for some $C>0$,

$||\chi
E_{j}(B_{t}^{ib}B^{-ib})||_{\dot{\mathcal{H}}^{Q/2-\epsilon}(\overline{N})\to\dot{\mathcal{H}}^{Q/2-1}(\overline{N})}\leq
C(1+|b|)^{Q/2}e^{\epsilon t}.$

This gives (4.2). This proves Lemma 3.9, and hence Theorem 2.1, modulo the
technical estimate, Lemma 4.2.

##  5\. Technical Estimate

We give a proof of Lemma 4.2. We fix a relatively compact open set $U$ of
$\overline{N}$ and let $C_{0}>0$ so that

$B_{t}(X,Z)=(e^{-2t}+||X||^{2}/4)^{2}+|Z|^{2}\leq C_{0}$

for all $t\geq 0$.

We need a small lemma. Again, this roughly means $B_{t}$ behaves like a
homogeneous function of degree $4$.

######  5.1 Lemma.

On $\overline{N}$, we have

  1. (1)

$|D^{1}B_{t}|\leq C_{1}B_{t}^{3/4}$,

  2. (2)

$|D^{2}B_{t}|\leq C_{2}B_{t}^{2/4}$,

  3. (3)

$|D^{3}B_{t}|\leq C_{3}B_{t}^{1/4}$,

  4. (4)

$|D^{4}B_{t}|\leq C_{4}$

  5. (5)

$D^{5}B_{t}=0\,\,\,(j\geq 5)$,

for some constants $C_{j}\geq 0$ independent of $t\geq 0$ and of
$D^{j}\in\mathfrak{D}^{j}$.

###### Proof.

Let us write

$B_{t}=e^{-4t}+e^{-2t}||X||^{2}/2+P_{4}$

where $P_{4}=\mathcal{N}^{4}=||X||^{4}/16+|Z|^{2}$. We have

$\displaystyle D^{1}B_{t}$ $\displaystyle=e^{-2t}D^{1}(||X||^{2})/2+P_{3}$
$\displaystyle D^{2}B_{t}$ $\displaystyle=e^{-2t}D^{2}(||x||^{2})/2+P_{2}$
$\displaystyle D^{3}B_{t}$ $\displaystyle=P_{1}$ $\displaystyle D^{4}B_{t}$
$\displaystyle=P_{0}$

where $P_{d}$ is a homogenous polynomial of degree $d$. Note that for each
such $P_{d}$, we have

$|P_{d}|\leq C^{\prime}_{d}\mathcal{N}^{d}\leq C^{\prime}_{d}B_{t}^{d/4}$

for some $C^{\prime}_{d}>0$. We also have

$\displaystyle|e^{-2t}D^{1}(||X||^{2})/2|$ $\displaystyle\leq B_{t}^{3/4}$
$\displaystyle|e^{-2t}D^{2}(||x||^{2})/2|$ $\displaystyle\leq B_{t}^{2/4}.$

The first one follows from

$B_{t}^{3}\geq(e^{-4t}+||X||^{4}/16)^{3}\geq e^{-8t}||X||^{4}/16.$

The second one follows from

$B_{t}\geq e^{-4t}.$

The claim follows from these. ∎

###### Proof of Lemma 4.2.

For any $s>0$, we need to show that for some $C=C(s)>0$, on $U$,

$|D^{s}(B_{t}^{ib})|\leq C\frac{1}{(B_{t})^{s/4}}(1+|b|)^{s}.$

For $s=0$, this is trivial. For $s=1$, we have, on $U$,

$|D^{1}(B_{t}^{ib})|=|D^{1}e^{ib\log(B_{t})}|=\frac{|D^{1}(B_{t})|}{B_{t}}|b|\leq
C_{1}\frac{B_{t}^{3/4}}{B_{t}}(1+|b|)=C_{1}B_{t}^{-1/4}(1+|b|).$

The general case for $s\geq 1$ follows from that $D^{s}e^{ib\log(B_{t})}$ is a
finite sum of the product

$(ib)^{l}e^{ib\log(B_{t})}\prod_{1\leq j\leq l}(D^{k_{j}}\log(B_{t}))$

where $k_{1}+\cdots+k_{l}=s$ ($k_{j}\geq 1$) and
$D^{k_{j}}\in\mathfrak{D}^{k_{j}}$ and that on $U$,

$|D^{j}(\log(B_{t}))|\leq C^{\prime}_{j}(B_{t})^{-j/4}\,\,\,(j\geq 1).$

We can see the latter from that $D^{s}(\log(B_{t}))$ is a finite sum of the
product

$(B_{t})^{-(m_{1}+\cdots+m_{s})}\prod_{1\leq j\leq s}(D^{j}B_{t})^{m_{j}}$

where $m_{1}+2m_{2}+\cdots+sm_{s}=s$ and $D^{j}\in\mathfrak{D}^{j}$ and that

$\left|\frac{D^{j}B_{t}}{B_{t}}\right|\leq C_{j}(B_{t})^{-j/4},$

that is for $j\geq 1$,

$|D^{j}B_{t}|\leq C_{j}(B_{t})^{1-j/4}.$

The last one follows from Lemma 5.1. Note that in this argument, we have used
that $B_{t}\leq C_{0}$ on $U$ thus

$B_{t}^{-s_{1}}\leq C_{0}^{s_{2}-s_{1}}B_{t}^{-s_{2}}$

on $U$ for $0\leq s_{1}\leq s_{2}$.

∎

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