# Sharp endpoint $L_{p}$ estimates of quantum Schrödinger groups

Zhijie Fan  Zhijie Fan, Department of Mathematics, Wuhan University,
<EMAIL_ADDRESS>,  Guixiang Hong  Guixiang Hong, Department of
Mathematics, Wuhan University<EMAIL_ADDRESS>and  Liang Wang
Liang Wang, Department of Mathematics, Wuhan University<EMAIL_ADDRESS>
###### Abstract.

In this article, we establish sharp endpoint $L_{p}$ estimates of Schrödinger
groups on general measure spaces which may not be equipped with good metrics
but admit submarkovian semigroups satisfying purely algebraic assumptions. One
of the key ingredients of our proof is to introduce and investigate a new
noncommutative high-cancellation BMO space by constructing an abstract form of
P-metric codifying some sort of underlying metric and position. This provides
the first form of Schrödinger group theory on arbitrary von Neumann algebras
and can be applied to many models, including Schrödinger groups associated
with non-negative self-adjoint operators satisfying purely Gaussian upper
bounds on doubling metric spaces, standard Schrödinger groups on quantum
Euclidean spaces, matrix algebras and group von Neumann algebras with finite
dimensional cocycles.

###### Key words and phrases:

Sharp estimates, Schrödinger groups, Noncommutative $L_{p}$ spaces, P-metrics,
High-cancellation BMO spaces

###### 2010 Mathematics Subject Classification:

42B37, 35J10, 47F05

###### Contents

  1. 1 Introduction
    1. 1.1 Background and motivation
    2. 1.2 Main result
    3. 1.3 Applications
    4. 1.4 Notation and structure of the paper
  2. 2 P-metrics and non-commutative BMO spaces
    1. 2.1 P-metrics
    2. 2.2 Semigroup $BMO_{\mathcal{S}}$
    3. 2.3 Metric type $BMO_{\mathcal{Q},M}$
    4. 2.4 Operator-valued weights
    5. 2.5 Algebraic, analytic, topological and operator conditions
  3. 3 Proof of Theorem 1.1
    1. 3.1 $L_{\infty}\rightarrow BMO$ boundedness
    2. 3.2 $L_{p}$ boundedness
  4. 4 Applications
    1. 4.1 Operator-valued setting
    2. 4.2 Quantum Euclidean space
    3. 4.3 Matrix algebra
    4. 4.4 Group von Neumann algebra

##  1\. Introduction

###  1.1. Background and motivation

Schrödinger equation is a basic equation in quantum mechanic. Denote by
$\Delta$ the Laplacian operator on the Euclidean space $\mathbb{R}^{n}$ given
by $\Delta=\sum_{i=1}^{n}\partial_{x_{i}}^{2}$, then the solution of initial
value problem for Schrödinger equation

(1.3) $\displaystyle\left\\{\begin{array}[]{ll}i{\partial_{t}u(x,t)}-\Delta
u(x,t)=0,\quad x\in\mathbb{R}^{n},\ \ t>0,\\\\[5.0pt]
u(x,0)=f(x),\end{array}\right.$

with initial value $f\in L_{2}(\mathbb{R}^{n})$ can be represented as

$u(x,t)=e^{-it\Delta}f(x).$

It can be seen from Hörmander’s result [50] that the family of solution
operators $(e^{-it\Delta})_{t\in\mathbb{R}}$, also called Schrödinger group,
are bounded on $L^{p}(\mathbb{R}^{n})$ only when $p=2$. For general
$1<p<\infty$, based on the proceeding inspiring work [4, 61, 78], Miyachi [66,
67] showed that the operator $e^{-it\Delta}$ is bounded from the Sobolev space
$W_{p}^{2s}(\mathbb{R}^{n})$ to $L_{p}(\mathbb{R}^{n})$ for
$s\geq\sigma_{p}:=n|1/p-1/2|$ with the optimal bound $C(1+|t|)^{\sigma_{p}}$
(see also [32] due to Fefferman and Stein). Equivalently, the operator
$e^{-it\Delta}(I-\Delta)^{-s}$ is bounded on $L_{p}(\mathbb{R}^{n})$. This
result is said to be an endpoint $L_{p}$-estimate since such a mapping
property does not hold for $0<s<\sigma_{p}$. The proof of the result depends
heavily on the theory of Hardy spaces and the Fourier transform estimate of
Fourier multiplier $e^{it|\cdot|^{2}}$—the symbol of the operator
$e^{-it\Delta}$.

Motivated by further applications to partial differential equations and
geometric analysis, harmonic analysis associated to generalized semigroups, or
equivalently generalized Laplacians, such as Laplace-Beltrami operators,
divergence forms and Schrödinger operators etc., has been developing rapidly
in the past twenty years (see [1, 27, 28, 45] and the references therein). In
particular, over the last decade, the $L_{p}$-regularity problem for the
Schrödinger groups $e^{itL}$ associated to a large class of self-adjoint
operators $L$ has attracted extensive attention. Let $L$ be a non-negative
self-adjoint operator on $L_{2}(X)$, where $X$ is a metric measure space with
a distance function $d$, a measure $\mu$ and satisfying the doubling property:

(1.4) $\displaystyle\mu(B(x,\lambda r))\leq C\lambda^{n}\mu(B(x,r))$

for some $C,n>0$ independent of $\lambda\geq 1$, $r>0$ and $x\in X$. Here we
denote by $B(x,r)$ the ball centred at $x$ with radius $r$. The smallest $n$
so that the above inequality valid is called the dimension of $X$. Then by
spectral theory, the solution of Schrödinger equation (1.3) with $-\Delta$
replaced by $L$ and with initial value $f\in L_{2}(X)$ can be represented as

$u(x,t)=e^{itL}f(x).$

After several hard attempts [3, 5, 9, 17, 51, 52, 70], Chen–Duong–Li–Yan [12]
established the sharp endpoint $L_{p}$-regularity estimate of $e^{itL}$ for a
large class of $L$. Specifically, they used the Littlewood-Paley inequalities
and a variant of Fefferman-Stein’s sharp maximal function to show that if the
heat kernel $p_{t}(x,y)$ associated with $L$ satisfies $m$-order Gaussian
upper bound

(${\rm GE}_{m}$)
$|p_{t}(x,y)|\leq{C\over\mu(B(x,t^{1/m}))}\exp\left(-c\,{\left({d(x,y)^{m}\over
t}\right)^{1\over m-1}}\right)$

for some constants $m\geq 2$ and $C,c>0$ independent of $t>0$ and $x,y\in X$,
then for any $1<p<\infty$,

$\|e^{itL}(I+L)^{-s}f\|_{L_{p}(X)}\leq C(1+|t|)^{\sigma_{p}}\|f\|_{L_{p}(X)},\
{\rm for}\ {\rm any}\ s\geq\sigma_{p}.$

See also [6, 11, 12, 13, 30] for further progresses.

On the other hand, inspired by operator algebras, noncommutative martingale
inequalities and quantum information theory, there have appeared several
fundamental work on harmonic analysis associated to quantum semigroups
$(e^{-tL})_{t\geq 0}$ (see e.g. [33, 54, 58, 59, 63]). Moreover, this part of
noncommutative harmonic analysis plays an instrumental role in Fourier
analysis on group algebras (see e.g. [55, 56, 72]). However, regarding the
Schrödinger group $(e^{itL})_{t\in\mathbb{R}}$—an analytic extension of
quantum semigroup to imaginary time—on the truly noncommutative measure
spaces, there seems no result in the existing literature. The reason behind
might be that the key ingredients such as the Fourier transform/kernel
estimates like (${\rm GE}_{m}$), Hardy space theory and the sharp maximal
function techniques etc. are not available or do not admit perfect analogues
in the noncommutative setting.

The main task of this article is to investigate the mapping properties of
$e^{itL}$ for a fixed $t$ and thus to provide the $L_{p}$-regularity estimate
for solution of the following Schrödinger equation on general noncommutative
measure spaces:

(1.7) $\displaystyle\left\\{\begin{array}[]{ll}i{\partial_{t}u(t)}+Lu(t)=0,\
t>0,\\\\[5.0pt] u(0)=f\in L_{2}(\mathcal{M}),\end{array}\right.$

where $\mathcal{M}$ is semifinite von Neumann algebra with a normal semifinite
faithful trace $\tau$ and $L$ is a non-negative self-adjoint operator on
$L_{2}(\mathcal{M})$, the noncommutative $L_{2}$ space over $\mathcal{M}$. We
refer the reader to [74] for more terminologies from noncommutative
integration theory. It is worthwhile to mention that the $L_{p}$-regularity
estimate is one of the most important estimates for the regularity problem of
solution of Schrödinger equation (1.3), while a generalization to the
noncommutative setting can be regarded as a starting point for the research of
the subsequent related problems, including noncommutative local smoothing
estimate and noncommutative Schrödinger maximal inequality, both of which lie
in the central topics of modern harmonic analysis even when going back to the
commutative setting (see for example [23, 24, 34, 75]).

In order to achieve our goal, we first observe that even in the Euclidean
setting, the target operator $(I+L)^{-\sigma_{p}}e^{itL}$ is a strong singular
integral operator beyond Calderón-Zygmund framework (see for example [7, 10,
14]). Over the last twenty decades, lots of significant progress toward
noncommutative singular integral operator theory has been achieved [38, 48,
49, 53, 55, 57, 65, 71]. Among these results, we would like to highlight a
pioneering work due to Junge–Mei–Parcet–Xia [57], who first introduced a
noncommutative form of Calderón-Zygmund theory under algebraic framework.
Specifically, in the lack of good metrics, they established the
$L_{p}$-boundedness of algebraic Calderón-Zygmund operators on semi-finite von
Neumann algebra admitting a Markov semigroup which only satisfies purely
algebraic assumptions. This result covers many concrete examples in both
classical setting and noncommutative setting. Besides, it is also particularly
interesting for measure spaces with poor geometric information.

Although our target operator is a class of singular integral, it is a strong
type beyond Calderón-Zygmund framework, which means that Calderón-Zygmund
theory cannot be applied directly to this operator. Instead, the extra
oscillation information about $e^{itL}$ should be dug out and the singularity
of the target operator should be handled more delicately. Inspired by the
establishment of algebraic Calderón-Zygmund theory, we will introduce the
first form of Schrödinger group theory which is valid for general semi-finite
von Neumann algebras under purely algebraic framework.

###  1.2. Main result

A semigroup $\mathcal{S}=(S_{t})_{t\geq 0}$ over $\mathcal{M}$ is a family of
operators $S_{t}:\mathcal{M}\rightarrow\mathcal{M}$ such that
$S_{0}=id_{\mathcal{M}}$ and $S_{t}S_{s}=S_{t+s}$ for any $t,s\geq 0$. It is
said to be a bounded semigroup if it is uniformly bounded and satisfies the
following properties:

(i) each $S_{t}$ is a weak-$*$ continuous and completely positive map;

(ii) each $S_{t}$ is self-adjoint in the sense that

(1.8) $\displaystyle\tau(S_{t}(f)g)=\tau(fS_{t}(g)),\ \ {\rm for\ all}\
f,g\in\mathcal{M}\cap L_{1}(\mathcal{M});$

(iii) the map $t\rightarrow S_{t}$ is a pointwise weak-$*$ continuous map.

A bounded semigroup $\mathcal{S}$ is said to be submarkovian if each $S_{t}$
is a contractive map. Besides, a submarkovian semigroup $\mathcal{S}$ is said
to be Markovian if it satisfies an additional unital property, i.e.
$S_{t}(1)=1$. Throughout the article, unless we mention the contrary, we will
assume that the underlying noncommutative measure space $(\mathcal{M},\tau)$
comes equipped with a bounded semigroup $\mathcal{S}=(S_{t})_{t\geq 0}$. Note
that $\mathcal{S}$ admits an infinitesimal negative generator

$Af=\lim_{t\rightarrow 0}\frac{S_{t}(f)-f}{t}$

defined on $dom(A)=\cup_{1\leq p\leq\infty}dom_{p}(A)$, where $dom_{p}(A)$ is
given by

$\displaystyle dom_{p}(A)=\left\\{f\in L_{p}(\mathcal{M}):\lim_{t\rightarrow
0}\frac{S_{t}(f)-f}{t}\ {\rm converges}\ {\rm in}\
L_{p}(\mathcal{M})\right\\}.$

It can be verified easily that for any $s>0$ and $f\in L_{p}(\mathcal{M})$,
one has $\frac{1}{s}\int_{0}^{s}S_{t}(f)dt\in dom_{p}(A)$, which means that
$dom_{p}(A)$ is dense in $L_{p}(\mathcal{M})$ for $p<\infty$, and is weak-$*$
dense in $\mathcal{M}$ for $p=\infty$. We denote $A_{p}$ by the restriction of
$A$ on $dom_{p}(A)$. It follows from (1.8) that $L:=-A$ is a non-negative
self-adjoint operator acting on $L_{2}(\mathcal{M})$. By the spectral theorem,
for any bounded Borel function $F:[0,\infty)\rightarrow\mathbb{C}$, one can
define an $L_{2}(\mathcal{M})$-bounded operator $F(L)$ given by

$F(L)=\int_{0}^{\infty}F(\lambda)dE_{L}(\lambda),$

where $E_{L}(\lambda)$ is the projection-valued measure supported on the
spectrum of $L$. By this spectral decomposition, the solution of
noncommutative Schrödinger equation (1.7) can be represented as

$u(t)=e^{itL}f,\ {\rm for}\ {\rm any}\ t>0.$

In order to establish an algebraic Schrödinger group theory as general as
possible, many difficulties occur due to the non-commutativity and the lack of
geometric concepts. To the best of our knowledge, even in the Euclidean
setting, there are only two methods in the literatures to establish the
$L_{p}$ estimate of Schrödinger groups. The first one is to show the
$L_{p}$-boundedness directly, which relies heavily on the $L_{p}$-boundedness
of sharp maximal function [12]. Another one is to establish the
$H^{1}\rightarrow L_{1}$ boundedness of targeted operator
$e^{itL}(I+L)^{-n/2}$ for appropriate Hardy space first [11], which relies on
atom (or molecule) decomposition of Hardy space, and then use complex
interpolation with the trivial $L_{2}$ boundedness. However, in the general
non-commutative setting, these approaches are not valid. To overcome these
difficulties, our strategy is to establish the $L_{\infty}\rightarrow BMO$
boundedness of $e^{itL}(I+L)^{-n(1+iy)/2}$ for appropriate BMO space directly
without using its duality with the corresponding Hardy space, which is a
complete new proof even in the most classical setting. The implementation of
this strategy can be divided into three steps:

Step 1: Construct a ‘P-metric’ and show that the associated metric-type BMO
space can be embedded into semigroup BMO space.

We shall introduce a notion called P-metric, which codifies some sort of
underlying metric and position in $(\mathcal{M},\tau)$. This concept is a
modification of the Markov metric introduced in [57], but it contains certain
extra ’position’ information of the underlying metric in $(\mathcal{M},\tau)$,
so we call this a P-metric. It is determined by a family

$\mathcal{Q}=\\{(R_{j,t},\sigma_{j,t}):j\in\mathbb{N},t>0\\},$

where each $R_{j,t}:\mathcal{M}\rightarrow\mathcal{M}$ is a normal completely
positive unital map satisfying $R_{j,t}(f)=f$ for $f\in kerA_{\infty}$, the
fixed-point subspace of the semigroup $\mathcal{S}$, and $\sigma_{j,t}$ are
elements of the von Neumann algebra $\mathcal{M}$ such that the estimates
below hold for some $m\geq 2$:

$\displaystyle{\rm(i)}\ semigroup\ majorization:\
S_{t}(|\xi|^{2})\leq\sum_{j\geq
0}\sigma_{j,t}^{*}R_{j,\sqrt[m]{t}}(|\xi|^{2})\sigma_{j,t},\forall\ t>0\ {\rm
and}\ \xi\in\mathcal{M};$ $\displaystyle{\rm(ii)}\ average\ domination\
condition:\
\big{\|}R_{j,t}(|\xi|^{2})\big{\|}_{\mathcal{M}}\lesssim\big{\|}R_{t}(|\xi|^{2})\big{\|}_{\mathcal{M}},\forall
j\in\mathbb{N},t>0\ {\rm and}\ \xi\in\mathcal{M};$ $\displaystyle{\rm(iii)}\
metric\ integrability\ condition:\
k_{\mathcal{Q}}:=\sup\limits_{t>0}\Big{\|}\sum_{j\geq
0}|\sigma_{j,t}|^{2}\Big{\|}_{\mathcal{M}}^{1/2}<\infty.$

Here and in what follows, in order to simplify the notation, we set
$R_{t}(f):=R_{0,t}(f)$ for any $f\in\mathcal{M}$. By constructing concrete
P-metric in specific examples, we shall explain more in Section 2.1 and
Section 4 what motivates our definition.

Let

$\displaystyle\|f\|_{{BMO}_{\mathcal{S}}^{c}(\mathcal{M})}:=\sup_{t>0}\big{\|}S_{t}|(I-S_{t})f|^{2}\big{\|}_{\mathcal{M}}^{1/2}$

and
$\|f\|_{{BMO}_{\mathcal{S}}(\mathcal{M})}:=\max\big{\\{}\|f\|_{{BMO}_{\mathcal{S}}^{c}(\mathcal{M})},\|f^{*}\|_{{BMO}_{\mathcal{S}}^{c}(\mathcal{M})}\big{\\}}$.
Inspired by [54, 57, 64], the semigroup type BMO space
${BMO}_{\mathcal{S}}(\mathcal{M})$ is defined as the weak-$*$ closure of
$\mathcal{M}$ in certain direct sum of Hilbert modules determined by
$\mathcal{S}$ (see Section 2.2 for more details). $BMO_{\mathcal{S}}(M)$
interpolates with $L_{p}(\mathcal{M})$ provided $\mathcal{S}$ is a Markov
semigroup admitting a Markov dilation (see Section 3.2). Given a P-metric
associated to this semigroup and $M\in\mathbb{Z}_{+}$, we define metric-type
BMO (semi-)norm of $f\in\mathcal{M}$ as follows:

$\|f\|_{BMO_{\mathcal{Q}}(\mathcal{M})}=\max\big{\\{}\|f\|_{BMO_{\mathcal{Q}}^{c}(\mathcal{M})},\|f^{*}\|_{BMO_{\mathcal{Q}}^{c}(\mathcal{M})}\big{\\}},$
$\displaystyle\|f\|_{BMO_{\mathcal{Q}}^{c}(\mathcal{M})}:=\sup_{t>0}\big{\|}R_{\sqrt[m]{t}}|(I-S_{t})f|^{2}\big{\|}_{\mathcal{M}}^{1/2}.$

Then it is not difficult to see that
$BMO_{\mathcal{Q}}^{c}(\mathcal{M})\subset BMO_{\mathcal{S}}^{c}(\mathcal{M})$
(see Lemma 2.2 for more details).

Step 2: Provide suitable operator conditions on $L$ such that metric-type BMO
space coincides with high-cancellation metric-type BMO space.

In order to ensure all the definitions in our abstract framework are well-
defined, we will impose a priori assumption that for any function $F$ mapping
$[0,\infty)$ to $\mathbb{C}$ and having bounded derivatives at any order, the
spectral multiplier
$F(L):\mathcal{A}_{\mathcal{M}}\rightarrow\mathcal{A}_{\mathcal{M}}$ for some
weak-$*$ dense subalgebra $\mathcal{A}_{\mathcal{M}}\subset\mathcal{M}$ such
that $\mathcal{A}_{\mathcal{M}}$ is dense in $L_{p}(\mathcal{M})$ for all
$1\leq p<\infty$. One shall see in the body paragraph that this assumption
holds trivially for many non-commutative models and can be relaxed in the
semi-commutative setting that we shall consider.

Motivated by some classical ideas initially originated from [45, 46], we shall
introduce a non-commutative high-cancellation BMO space in this article, which
has not been studied in the previous literature in the field of non-
commutative analysis. Given a P-metric associated to this semigroup and
$M\in\mathbb{Z}_{+}$, we define $M$-order metric-type BMO (semi-)norm of
$f\in\mathcal{M}$ as follows:

$\|f\|_{BMO_{\mathcal{Q},M}(\mathcal{M})}=\max\big{\\{}\|f\|_{BMO_{\mathcal{Q},M}^{c}(\mathcal{M})},\|f^{*}\|_{BMO_{\mathcal{Q},M}^{c}(\mathcal{M})}\big{\\}},$
$\displaystyle\|f\|_{BMO_{\mathcal{Q},M}^{c}(\mathcal{M})}:=\sup_{t>0}\big{\|}R_{\sqrt[m]{t}}|(I-S_{t})^{M}f|^{2}\big{\|}_{\mathcal{M}}^{1/2}.$

We will show that under suitable algebraic, analytic and topological
conditions (explained more in Step 3), for any
$f\in\mathcal{A}_{\mathcal{M}}$,

$\|f\|_{BMO_{\mathcal{Q},1}(\mathcal{M})}\simeq\|f\|_{BMO_{\mathcal{Q},M}(\mathcal{M})}.$

This self-improved property plays a crucial role in establishing spectral
multiplier theorem associated with operators of various types (e.g. Hörmander-
type spectral multipliers [26], Schrödinger groups multipliers [12]) since it
endows BMO function a higher cancellation property in the low frequency.

Step 3: Provide suitable operator conditions on $L$ to establish the
$L_{\infty}\rightarrow BMO$ boundedness for high-cancellation metric-type BMO
space.

Inspired by the idea of introducing Markov metric in [57], the commutative
idea behind the notion of P-metric is also to find pointwise majorants of the
heat kernels of semigroups $\mathcal{S}$, such that $S_{t}$ can be dominated
by a sum of averaging operators $R_{j,\sqrt[m]{t}}$ over a distinguished
family of algebraic balls ‘with radius $2^{j}\sqrt[m]{t}$’. In the
noncommutative setting, the maps $R_{t}$ (resp. $R_{j,t}$) must be averages
over certain projections $q_{t}$ (resp. $q_{j,t}$). It is too technical to
explain this concept in full generality at this point of our article. However,
the authors may first care about a non-trivial model example given by

$R_{t}f=\Big{(}(id\otimes\tau)(q_{t})\Big{)}^{-\frac{1}{2}}(id\otimes\tau)\big{(}q_{t}(1\otimes
f)q_{t}\big{)}\Big{(}(id\otimes\tau)(q_{t})\Big{)}^{-\frac{1}{2}}$

for certain increasing family of projections ${\bf q}:=\\{q_{t}\\}_{t\geq
0}\in\mathcal{M}\bar{\otimes}\mathcal{M}$ determined later on.

As it happens in classical Schrödinger group theory and algebraic Calderón-
Zygmund theory [57], one need to impose some additional but appropriate
conditions (which will be split into algebraic, analytic and topological
conditions) on the P-metric to connect closely with the underlying
(noncommutative) measure. We would like to just mention at this stage that the
imposed algebraic conditions are very similar to that imposed in the algebraic
Calderón-Zygmund theory, which are inherent to noncommutativity and hold
trivially in the commutative settings. The analytic one provides a strong
doubling condition, which is equivalent to the doubling condition (1.4) in the
setting of metric measure space (see Section 4.1) but is stronger than the
doubling condition in the general framework since it can reflect extra
covering property. For the model example mentioned in the previous paragraph,
the strong doubling condition states that there exist constants $C,n>0$,
$D\geq 0$, such that for any $r_{1}\geq r_{2}\geq r_{3}$ and
$f\in\mathcal{M}$,

$\displaystyle\Bigg{\|}(id\otimes\tau)\bigg{(}\bigg{|}(1\otimes
f)q_{r_{1}}(id\otimes\tau)(q_{r_{3}})^{-\frac{1}{2}}\bigg{|}^{2}\bigg{)}\Bigg{\|}_{\mathcal{M}}\leq
C\left(\frac{r_{1}}{r_{2}}\right)^{D}\left(\frac{r_{1}}{r_{3}}\right)^{n}\Bigg{\|}(id\otimes\tau)\bigg{(}\bigg{|}(1\otimes
f)q_{r_{2}}(id\otimes\tau)(q_{r_{2}})^{-\frac{1}{2}}\bigg{|}^{2}\bigg{)}\Bigg{\|}_{\mathcal{M}}.$

In particular, with the choices $f=1$ and $r_{1}=r_{2}$, this implies the
doubling condition:

$\left\|\Big{(}(id\otimes\tau)(q_{r_{3}})\Big{)}^{-\frac{1}{2}}\Big{(}(id\otimes\tau)(q_{r_{1}})\Big{)}\Big{(}(id\otimes\tau)(q_{r_{3}})\Big{)}^{-\frac{1}{2}}\right\|_{\mathcal{M}}\leq
C\left(\frac{r_{1}}{r_{3}}\right)^{n}.$

Instead, the smallest constant $n=n_{\bf q}$ such that the above inequality
valid is called the P-metric dimension of $\mathcal{M}$ which, together with
an appropriate choice of $q$, goes back to the homogeneous dimension of $X$
when $\mathcal{M}=L_{\infty}(X)$. The topological conditions provide certain
natural continuity of projections and amplification map.

Next we impose operator conditions on $L$. In the model example, our operator
condition is just an $L_{2}$-Gaussian estimate, which can be regarded as an
‘integral-type’ replacement of Gaussian upper bound (${\rm GE}_{m}$) and is
stated as follows:

There exist constants $C,c>0$ and $m\geq 2$ such that for any $t>0$, $k\geq
1$, $0<r_{1}\leq r_{2}$ and $f\in\mathcal{M}\bar{\otimes}\mathcal{M}$,

$\displaystyle\left\|(id\otimes\tau)\Big{|}(id\otimes
S_{t})(fb_{k,r_{2}})q_{r_{1}}\Big{(}(id\otimes\tau)(q_{r_{1}})\Big{)}^{-\frac{1}{2}}\Big{|}^{2}\right\|_{\mathcal{M}}^{\frac{1}{2}}$
(1.9) $\displaystyle\leq
CE_{k}\left(-c\left(\frac{r_{2}}{t^{1/m}}\right)^{\frac{m}{m-1}}\right)\Big{\|}(id\otimes\tau)\Big{|}fb_{k,r_{2}}\Big{(}(id\otimes\tau)(q_{\max\\{r_{1},t^{1/m}\\}})\Big{)}^{-\frac{1}{2}}\Big{|}^{2}\Big{\|}_{\mathcal{M}}^{\frac{1}{2}},$

where

$b_{k,r}=\left\\{\begin{array}[]{ll}q_{2r},&{\rm if}\ k=1,\\\
q_{2^{k}r}-q_{2^{k-1}r},&{\rm if}\ k\geq 2,\end{array}\right.$
$E_{k}(x)=\left\\{\begin{array}[]{ll}1,&{\rm if}\ k=1,\\\
\exp(2^{\frac{mk}{m-1}}x),&{\rm if}\ k\geq 2.\end{array}\right.$

In condition (2), $b_{k,r}$ and $q_{r}$ play the roles of indicators with
respect to the algebraic annuli with radius $2^{k}r$ and algebraic balls with
radius $r$, respectively. In the Euclidean setting, this condition is a simple
consequence of (${\rm GE}_{m}$) (see Section 4.1 for more details). In the
above text, we just formulate our conditions for special amplification algebra
model $\mathcal{M}\bar{\otimes}\mathcal{M}$, but our general conditions will
include many more amplification algebras other than this model, which will be
specific in Section 2.5. With these more general conditions, our main result
can be formulated as follows.

######  Theorem 1.1.

Let $(\mathcal{M},\tau)$ be a semifinite von Neumann algebra equipped with a
bounded semigroup $\mathcal{S}=(e^{-tL})_{t\geq 0}$ with associated P-metric
$\mathcal{Q}$ fulfilling our algebraic, analytic and topological assumptions.
Then there exists a constant $C=C(n)>0$ such that for any
$f\in\mathcal{A}_{\mathcal{M}}$,

$\displaystyle\left\|e^{itL}(I+L)^{-s}f\right\|_{{BMO}_{\mathcal{S}}(\mathcal{M})}\leq
C(1+|t|)^{n/2}\|f\|_{\mathcal{M}},\ {\rm for}\ s\geq\frac{n}{2}.$

If in addition $\mathcal{S}$ is a Markovian semigroup admitting a Markov
dilation, then for any $1<p<\infty$,

$\displaystyle\left\|e^{itL}(I+L)^{-s}f\right\|_{L_{p}(\mathcal{M})}\leq
C(1+|t|)^{\sigma_{p}}\|f\|_{L_{p}(\mathcal{M})},\ {\rm for}\
s\geq\sigma_{p}=n\left|\frac{1}{2}-\frac{1}{p}\right|.$

######  Remark 1.2.

Theorem 1.1 is sharp in the sense that under the imposed conditions, the
dependency on $t$ of the upper bound and the minimal regularity of $s$ cannot
be improved, which can be seen by the classical example $L=-\Delta$ mentioned
at the beginning of the introduction. Note that one cannot expect to show the
sharpness of this estimate for all von Neumann algebra $\mathcal{M}$ and all
generator $L$ under our framework. A typical counterexample appears when
$\mathcal{M}$ and $L$ chosen to be $L_{\infty}$-space over Heisenberg group
and its associated sub-Laplacian operator, respectively. In this case, the
threshold dimension should be the topological dimension instead of the
homogeneous one [14].

###  1.3. Applications

By verifying all the algebraic, analytic, topological and operator conditions
of the main theorem (with some minor modification), we shall see how our main
result applies to a wide variety of concrete von Neumann algebras with
specific P-metrics. We list these models in the following.

Example A. Doubling metric spaces

In this commutative model $\mathcal{M}=L_{\infty}(X)$, we shall see that our
algebraic theory recover the result obtained by Chen–Duong–Li–Yan (see more
comment on Remark 4.4). In other words, we provide a new proof of
$L_{p}$-boundedness for $(I+L)^{-\sigma_{p}}e^{itL}$ via
$L_{\infty}\rightarrow BMO_{L}$ boundedness together with a standard complex
interpolation argument (see for example [11, Section 4]) under the assumption
that $L$ is a non-negative self-adjoint operator on $L_{2}(X)$ satisfying
Gaussian upper bound (${\rm GE}_{m}$).

Example B. Operator-valued setting

As a generalization of matrix-valued harmonic analysis, operator-valued (semi-
commutative) harmonic analysis has became an active area of research (see for
example [47, 49, 60, 63, 71] and the references therein). The operator-valued
model is perhaps the easiest non-commutative model, but it usually require
revolutionary arguments to do analysis due to noncommutativity. To illustrate
our application on this model, we let $X$ be a doubling metric space with a
dimension $n$, a distance function $d$ and a nonnegative Borel doubling
measure $\mu$. Besides, let $\mathcal{M}$ be a semifinite von Neumann algebra
equipped with a normal semifinite faithful trace $\tau$ and
$\mathcal{N}=L_{\infty}(X)\bar{\otimes}\mathcal{M}$ be the von Neumann algebra
tensor product equipped with tensor trace $\int\otimes\tau$. Moreover, let $L$
be a non-negative self-adjoint operator on $L_{2}(X)$. Then under the
assumption that the heat kernel associated with $L$ is non-negative, satisfies
Gaussian upper bound (${\rm GE}_{m}$) and induces a Markovian semigroup
admitting a Markov dilation, our algebraic theory can be applied to establish
the $L_{p}$-boundedness of $(id_{\mathcal{N}}+L\otimes
id_{\mathcal{M}})^{-s}e^{it(L\otimes id_{\mathcal{M}})}$ for $s\geq
n\big{|}{1\over 2}-{1\over p}\big{|}$ (see Theorem 4.3 for more details and
Remark 4.5 for more comments). There are many concrete generators fulfilling
all of our assumptions, such as Neumann-Laplacian operator $\Delta_{N}$ (resp.
$\Delta_{{N_{\pm}}}$) on $\mathbb{R}^{n}$ (resp. $\mathbb{R}^{n}_{\pm}$) [21],
Dunkl operators $\Delta_{{\rm Dunkl}}$ on $\mathbb{R}^{N}$ equipped with Dunkl
measure [29], and Laplace-Beltrami operator $\Delta_{g}$ on $n$-dimensional
complete Riemannian manifold with non-negative Ricci curvature [19, Section
5].

Example C. Quantum Euclidean spaces

Quantum Euclidean spaces (also called Moyal spaces) are one of the archetypal
algebras of noncommutative geometry, which were first introduced by many
authors, such as Groenewold [40] and Moyal [68], for the study of quantum
mechanics in phase space. Besides, being a kind of geometrical spaces with
noncommutating spatial coordinates, these spaces have appeared frequently in
several branches of mathematical physics [22], including string theory [77]
and noncommutative field theory [69].

Recently, many breakthroughs have been obtained in several branches of
harmonic analysis over quantum Euclidean space, such as singular integral
theory [38], function space theory [38, 82] and quantum differentiability
[62]. To illustrate our application on this space, we let
$\mathcal{R}_{\Theta}$ be a quantum Euclidean space associated with an anti-
symmetric real-valued $n\times n$ matrix $\Theta$. Denote $\Delta_{\Theta}$ be
the natural quantum Laplacian operator over it (more details are referred to
Section 4.2). Then our algebraic theory can be applied to establish the
$L_{p}(\mathcal{R}_{\Theta})$-boundedness of
$(id_{\mathcal{R}_{\Theta}}-\Delta_{\Theta})^{-s}e^{-it\Delta_{\Theta}}$ for
$s\geq n\big{|}{1\over 2}-{1\over p}\big{|}$ (see Theorem 4.6).

Example D. Matrix algebras

Denote $L_{B(\ell_{2})}$ be the non-negative infinitesimal generator induced
from a Markov semigroup of convolution type on $B(\ell_{2})$ defined by

$S_{t}\Big{(}\sum_{m,k}a_{m,k}e_{m,k}\Big{)}=\sum_{m,k}e^{-t|m-k|^{2}}a_{m,k}e_{m,k},$

where $e_{m,k}$ is the matrix unit for $B(\ell_{2})$. Then our algebraic
theory can be applied to establish the $S_{p}$-boundedness of
$(id_{B(\ell_{2})}+L_{B(\ell_{2})})^{-1/2}e^{itL_{B(\ell_{2})}}$ for
$s\geq\big{|}{1\over 2}-{1\over p}\big{|}$ (see Theorem 4.7), where $S_{p}$
denotes the Schatten-$p$ class on the Hilbert space $\ell_{2}$.

Example E. Group von Neumann algebras

The interest of Fourier multipliers over group von Neumann algebras dates back
to a pioneering work by Haarerup on free groups [41]. Fourier multiplier is an
effective tool to reveal the geometry of these algebras via approximation
properties [16, 20, 41, 44]. It is worthwhile to note that there have been
fruitful results about Fourier multiplier $L_{p}$-theory over group von
Neumann algebras in the past decade (see for example [8, 15, 37, 55, 56]). In
particular, by investigating the $L_{\infty}\rightarrow BMO$ endpoint
inequality, Junge–Mei–Parcet [55] established $L_{p}$ Hörmander-Mihlin
multiplier theorem with optimal smoothness condition on discrete group von
Neumann algebra equipped with finite-dimensional cocycles. To illustrate our
application on these algebras, we let $\mathcal{L}(G)$ be the group von
Neumann algebra of a discrete group $G$ equipped with $n$-dimensional cocycle.
Then our algebraic theory can be applied to obtain the
$L_{p}(\mathcal{L}(G))$-boundedness of Fourier multiplier operator with
multiplier function $e^{it\psi(g)}(1+\psi(g))^{-s}$ for $s\geq n\big{|}{1\over
2}-{1\over p}\big{|}$ (see Theorem 4.11), where
$\psi:G\rightarrow\mathbb{R}_{+}$ is a length function on $G$. This provides
an alternative proof for a specific radial Fourier multiplier of [55, Theorem
D] (see also the first paragraph in the proof of Theorem 4.11).

###  1.4. Notation and structure of the paper

We use $A\lesssim B$ to denote the statement that $A\leq CB$ for some constant
$C>0$, and $A\simeq B$ to denote the statement that $A\lesssim B$ and
$B\lesssim A$.

This paper is organized as follows. In Section 2, we first introduce the
concept of P-metric and then illustrate how metrics in homogeneous space fit
in. Secondly, we introduce non-commutative $M$-order semigroup BMO space
together with the associated metric-type BMO space, and then show an embedding
theorem between them. Thirdly, we recall some basic concepts about operator-
valued weight. Finally, we formulate the algebraic conditions, analytic
conditions and operator conditions in the full generality, and then build the
bridge between low-order metric BMO space and the higher one under these
assumptions. In Section 3, we provide the proof of Theorem 1.1. In Section 4.
we provide several applications of Theorem 1.1 by verifying the imposed
conditions in different settings.

##  2\. P-metrics and non-commutative BMO spaces

###  2.1. P-metrics

This subsection will provide more explanation about the P-metric introduced in
the introduction. One shall see below that the notion P-metric is easily
understood for commutative model case. We illustrate this in a more general
von-Neumann algebra $L_{\infty}(X)$ by constructing a P-metric on
$L_{\infty}(X)$. To this end, let $\xi$ be a bounded complex-valued function
on $X$. Then it satisfies the following semigroup majorization

$\displaystyle\int_{X}p_{t}(x,y)|\xi(y)|^{2}d\mu(y)$ $\displaystyle\leq
C\sum_{j\geq
0}\frac{\mu(B(x,2^{j}t^{1/m}))}{\mu(B(x,t^{1/m}))}\exp\big{(}-c2^{\frac{jm}{m-1}}\big{)}\fint_{B(x,2^{j}t^{1/m})}|\xi(y)|^{2}d\mu(y)$

for some $C,c>0$ independent of $x\in X$ and $t>0$. Here we used the fact that
$\exp\big{(}-c2^{\frac{jm}{m-1}}\big{)}\simeq 1$ when $j=1,2$. This means that
$R_{j,t}\xi(x)$ is the average of $\xi\in L_{\infty}(X)$ over the ball
$B(x,2^{j}t)$. Next, note that $B(x,2^{j}t^{1/m})$ can be covered by at most
$c^{jn}$ balls with radius $t^{1/m}$ and with finite overlap. For simplicity,
we denote these balls by $B_{j,1}(x,t^{1/m}),\cdots,B_{j,d_{j,n}}(x,t^{1/m})$,
where $d_{j,n}\lesssim c^{jn}$. Such a cover is not unique, but we fix a kind
of cover. Then for any $j\geq 0$ and $t>0$, the following average domination
inequality holds:

$\displaystyle\left\|\fint_{B(x,2^{j}t^{1/m})}|\xi(y)|^{2}d\mu(y)\right\|_{L_{\infty}(X)}$
$\displaystyle\leq\left\|\sum_{\ell=1}^{d_{j,n}}\frac{\mu(B_{j,\ell}(x,t^{1/m}))}{\mu(B(x,2^{j}t^{1/m}))}\fint_{B_{j,\ell}(x,t^{1/m})}|\xi(y)|^{2}d\mu(y)\right\|_{L_{\infty}(X)}$
$\displaystyle\leq\left\|\sum_{\ell=1}^{d_{j,n}}\frac{\mu(B_{j,\ell}(x,t^{1/m}))}{\mu(B(x,2^{j}t^{1/m}))}\right\|_{L_{\infty}(X)}\left\|\fint_{B(x,t^{1/m})}|\xi(y)|^{2}d\mu(y)\right\|_{L_{\infty}(X)}$
$\displaystyle\leq
C\left\|\fint_{B(x,t^{1/m})}|\xi(y)|^{2}d\mu(y)\right\|_{L_{\infty}(X)}.$

If we choose

$|\sigma_{j,t}(x)|^{2}=C\frac{\mu(B(x,2^{j}t^{1/m}))}{\mu(B(x,t^{1/m}))}\exp\big{(}-c2^{\frac{jm}{m-1}}),$

then by the doubling condition (1.4),

$\displaystyle k_{\mathcal{Q}}=C\left\|\sum_{j\geq
0}\frac{\mu(B(x,2^{j}t^{1/m}))}{\mu(B(x,t^{1/m}))}\exp\big{(}-c2^{\frac{jm}{m-1}}\big{)}\right\|_{L_{\infty}(X)}^{1/2}\leq
C\left(\sum_{j\geq
0}2^{jn}\exp\big{(}-c2^{\frac{jm}{m-1}}\big{)}\right)^{1/2}<\infty,$

which verifies the conditions (i)-(iii) and then ends the construction of
P-metric for $L_{\infty}(X)$.

###  2.2. Semigroup $BMO_{\mathcal{S}}$

Given $M\in\mathbb{Z}_{+}$ and a noncommutative measure space
$(\mathcal{M},\tau)$ together with a B-submarkovian semigroup
$\mathcal{S}=(S_{t})_{t\geq 0}$ acting on $(\mathcal{M},\tau)$, we define the
semigroup $BMO_{\mathcal{S},M}(\mathcal{M})$ semi-norm for $f\in\mathcal{M}$
as

$\displaystyle\|f\|_{{BMO}_{\mathcal{S},M}(\mathcal{M})}:=\max\big{\\{}\|f\|_{{BMO}_{\mathcal{S},M}^{r}(\mathcal{M})},\|f\|_{{BMO}_{\mathcal{S},M}^{c}(\mathcal{M})}\big{\\}},$

where the column BMO semi-norm is given by

$\displaystyle\|f\|_{{BMO}_{\mathcal{S},M}^{c}(\mathcal{M})}:=\sup_{t>0}\big{\|}S_{t}|(I-S_{t})^{M}f|^{2}\big{\|}_{\mathcal{M}}^{1/2}$

and the row BMO semi-norm is given by
$\|f\|_{{BMO}_{\mathcal{S},M}^{r}(\mathcal{M})}:=\|f^{*}\|_{{BMO}_{\mathcal{S},M}^{c}(\mathcal{M})}.$
For simplicity, we denote
${BMO}_{\mathcal{S}}^{\dagger}(\mathcal{M}):={BMO}_{\mathcal{S},1}^{\dagger}(\mathcal{M})$
and ${BMO}_{\mathcal{S}}(\mathcal{M}):={BMO}_{\mathcal{S},1}(\mathcal{M})$,
where $\dagger\in\\{c,r\\}$. Here we say that
$\|\cdot\|_{BMO_{\mathcal{S},M}(\mathcal{M})}$ and
$\|\cdot\|_{{BMO}_{\mathcal{S},M}^{\dagger}(\mathcal{M})}$ are semi-norms on
$\mathcal{M}$ since one can establish the following lemma:

######  Lemma 2.1.

Let $\mathcal{M},\mathcal{N}$ be two von Neumann algebras, and $T$ be a
positive linear operator from $\mathcal{N}$ to $\mathcal{M}$. Then
$\|\cdot\|_{Z}$ is a semi-norm on $\mathcal{N}$, where
$\|x\|_{Z}:=\|T(x^{*}x)\|_{\mathcal{M}}^{1/2}$.

###### Proof.

Let $\phi$ be a state on $\mathcal{M}$. For any $x,y\in\mathcal{N}$, we define
$\tilde{\phi}(x,y):=\phi(T(x^{*}y))$, then one could easily deduce the
following Cauchy-Schwarz inequality:

(2.1) $|\tilde{\phi}(x,y)|^{2}\leq\tilde{\phi}(x,x)\tilde{\phi}(y,y),$

which implies that
$\tilde{\phi}(x+y,x+y)^{1/2}\leq\tilde{\phi}(x,x)^{1/2}+\tilde{\phi}(y,y)^{1/2}.$
Therefore,

$\displaystyle\|x+y\|_{Z}$ $\displaystyle=\sup_{\phi\text{ is a
state}}\phi(T((x+y)^{*}(x+y)))^{1/2}\leq\|x\|_{Z}+\|y\|_{Z},$

which implies the desired result. ∎

After quotienting out the null space of the semi-norm
$\|\cdot\|_{{BMO}_{\mathcal{S},M}^{c}(\mathcal{M})}$ on $\mathcal{M}$, we get
a normed vector space which is denoted by
$\Big{(}\mathcal{M},\|\cdot\|_{{BMO}_{\mathcal{S},M}^{c}(\mathcal{M})}\Big{)}$.
Now we need to complete this space with respect to some suitable topology as
in [36, 54, 57] via using the concept of Hilbert module (see [73]).

Given a complete positive map $S_{t}$ from $\mathcal{M}$ to $\mathcal{M}$, one
can construct the Hilbert module $\mathcal{M}\bar{\otimes}_{S_{t}}\mathcal{M}$
as follows: let $\langle\cdot,\cdot\rangle_{S_{t}}$ be the
$\mathcal{M}$-valued inner product on $\mathcal{M}\otimes\mathcal{M}$ which is
defined by

$\displaystyle\Big{\langle}\sum_{i}a_{i}\otimes b_{i},\sum_{j}c_{j}\otimes
d_{j}\Big{\rangle}_{S_{t}}:=\sum_{i,j}b_{i}^{*}S_{t}(a_{i}^{*}c_{j})d_{j}$

for $a_{i},b_{i},c_{j},d_{j}\in\mathcal{M}$. Then we consider the semi-norm
$\|\cdot\|_{\mathcal{M}\bar{\otimes}_{S_{t}}\mathcal{M}}$ on
$\mathcal{M}\otimes\mathcal{M}$ determined by the $\mathcal{M}$-valued inner
product

$\displaystyle\|x\|_{\mathcal{M}\bar{\otimes}_{S_{t}}\mathcal{M}}:=\|\langle
x,x\rangle_{S_{t}}^{1/2}\|_{\mathcal{M}}.$

After quotienting out the null space of the semi-norm, we obtain a normed
vector space denoted by
$\Big{(}\mathcal{M}\otimes\mathcal{M},\|\cdot\|_{\mathcal{M}\bar{\otimes}_{S_{t}}\mathcal{M}}\Big{)}$,
and then $\mathcal{M}\bar{\otimes}_{S_{t}}\mathcal{M}$ denotes the completion
of
$\Big{(}\mathcal{M}\otimes\mathcal{M},\|\cdot\|_{\mathcal{M}\bar{\otimes}_{S_{t}}\mathcal{M}}\Big{)}$
with respect to the strong operator topology, i.e. the coarsest topology
making the maps on $\mathcal{M}\otimes\mathcal{M}$

$x\rightarrow\tau(a\langle x,x\rangle_{S_{t}})$

continuous for all $a\in L_{1}(\mathcal{M})$.

Let
$u:\Big{(}\mathcal{M},\|\cdot\|_{{BMO}_{\mathcal{S},M}^{c}(\mathcal{M})}\Big{)}\rightarrow\bigoplus_{t>0}\mathcal{M}\bar{\otimes}_{S_{t}}\mathcal{M}$
be the isometric map given by

$u(x):=\left((I-S_{t})^{M}x)\otimes 1\right)_{t>0}.$

Then the column BMO space ${BMO}_{\mathcal{S},M}^{c}(\mathcal{M})$ is defined
as the weak-$*$ closure of $u(\mathcal{M})$ in
$\bigoplus_{t>0}\mathcal{M}\bar{\otimes}_{S_{t}}\mathcal{M}$, see [57] for
more details. ${BMO}_{\mathcal{S},M}^{r}(\mathcal{M})$ is defined as the space
consisting of the elements whose adjoint belongs to
${BMO}_{\mathcal{S},M}^{c}(\mathcal{M})$. And
${BMO}_{\mathcal{S},M}(\mathcal{M})$ is defined to be the intersection of the
row and column BMO spaces.

###  2.3. Metric type $BMO_{\mathcal{Q},M}$

Inspired by [57], we introduce a suitable metric type BMO space for von
Neumann algebra and relate it with the semigroup type BMO spaces defined in
Section 2.2. Given a bounded semigroup $\mathcal{S}=(S_{t})_{t\geq 0}$ acting
on $(\mathcal{M},\tau)$, consider a P-metric
$\mathcal{Q}=\\{(R_{j,t},\sigma_{j,t}):j\in\mathbb{N},t>0\\}$ given in Section
2.1. For any $M\in\mathbb{Z}_{+}$, we define the semigroup
$BMO_{\mathcal{Q},M}(\mathcal{M})$ semi-norm for $f\in\mathcal{M}$ as

$\|f\|_{BMO_{\mathcal{Q},M}(\mathcal{M})}=\max\big{\\{}\|f\|_{BMO_{\mathcal{Q},M}^{c}(\mathcal{M})},\|f\|_{BMO_{\mathcal{Q},M}^{r}(\mathcal{M})}\big{\\}},$

where the column BMO semi-norm is given by

$\displaystyle\|f\|_{BMO_{\mathcal{Q},M}^{c}(\mathcal{M})}:=\sup_{t>0}\big{\|}R_{\sqrt[m]{t}}|(I-S_{t})^{M}f|^{2}\big{\|}_{\mathcal{M}}^{1/2}$

and the row BMO semi-norm is given by
$\|f\|_{{BMO}_{\mathcal{Q},M}^{r}(\mathcal{M})}:=\|f^{*}\|_{{BMO}_{\mathcal{Q},M}^{c}(\mathcal{M})}$.
For simplicity, we denote
${BMO}_{\mathcal{Q}}^{\dagger}(\mathcal{M}):={BMO}_{\mathcal{Q},1}^{\dagger}(\mathcal{M})$
and ${BMO}_{\mathcal{Q}}(\mathcal{M}):={BMO}_{\mathcal{Q},1}(\mathcal{M})$,
where $\dagger\in\\{c,r\\}$.

With the P-metric, the metric type BMO space for classical model
$L_{\infty}(X)$ equipped with the semigroup $\\{S_{t}\\}_{t\geq 0}$, where $L$
is a non-negative self-adjoint operator and its associated heat kernel
satisfies $({\rm GE}_{m})$, is determined by

$\displaystyle\|f\|_{BMO_{\mathcal{Q}}}:=\sup_{t\geq 0}\left(\sup_{y\in
X}\fint_{B(y,\sqrt[m]{t})}|(I-S_{t})f(x)|^{2}dx\right)^{1/2},$

which induces the $BMO_{L}(X)$ space considered in [27, 28, 45].

Next, recalling that the value of the constant $k_{\mathcal{Q}}$ in the
definition of P-metric, we show that for any $M\in\mathbb{Z}_{+}$,
$BMO_{\mathcal{Q},M}$ embeds in $BMO_{\mathcal{S},M}$.

######  Lemma 2.2.

Let $(\mathcal{M},\tau)$ be a noncommutative measure space equipped with a
bounded semigroup $\mathcal{S}=(S_{t})_{t\geq 0}$ and a P-metric
$\mathcal{Q}=\\{(R_{j,t},\sigma_{j,t}):j\in\mathbb{N},t>0\\}$. Then for any
$M\in\mathbb{Z}_{+}$ and $f\in\mathcal{M}$,

$\displaystyle\|f\|_{{BMO}_{\mathcal{S},M}^{c}}\lesssim
k_{\mathcal{Q}}\|f\|_{{BMO}_{\mathcal{Q},M}^{c}}.$

###### Proof.

By the definition of P-metric defined in section 2.1, one has

$\displaystyle\|f\|_{{BMO}_{\mathcal{S},M}^{c}}$
$\displaystyle=\sup_{t>0}\big{\|}S_{t}|(I-S_{t})^{M}f|^{2}\big{\|}_{\mathcal{M}}^{1/2}$
$\displaystyle\leq\sup_{t>0}\Big{\|}\sum_{j\geq
0}\sigma_{j,t}^{*}R_{j,\sqrt[m]{t}}(|(I-S_{t})^{M}f|^{2})\sigma_{j,t}\Big{\|}_{\mathcal{M}}^{1/2}$
$\displaystyle\leq\sup_{t>0}\Big{\|}\sum_{j\geq
0}|\sigma_{j,t}|^{2}\big{\|}R_{j,\sqrt[m]{t}}(|(I-S_{t})^{M}f|^{2})\big{\|}_{\mathcal{M}}\Big{\|}_{\mathcal{M}}^{1/2}$
$\displaystyle\leq\sup_{t>0}\Big{\|}\sum_{j\geq
0}|\sigma_{j,t}|^{2}\Big{\|}_{\mathcal{M}}^{1/2}\sup_{t>0}\Big{\|}R_{\sqrt[m]{t}}(|(I-S_{t})^{M}f|^{2})\Big{\|}_{\mathcal{M}}^{1/2}$
$\displaystyle\lesssim k_{\mathcal{Q}}\|f\|_{{BMO}_{\mathcal{Q},M}^{c}}.$

This ends the proof of Lemma 2.2. ∎

###  2.4. Operator-valued weights

In this subsection we collect the definition and basic properties of operator-
valued weights from [42, 43] (see also [57] for the summary).

Let $\mathcal{M}$ be a von Neumann algebra, then the extended positive part
$\widehat{\mathcal{M}}_{+}$ is the set consisting of all lower semi-continuous
maps $m:\mathcal{M}_{*,+}\rightarrow[0,\infty]$ satisfying the linear law: for
$a,b\geq 0$ and $f,g\in\mathcal{M}_{*,+}$, one has

$m(af+bg)=am(f)+bm(g).$

The extended positive part is closed under addition, scalar multiplication,
increasing limits and conjugation map defined by $x\mapsto a^{*}xa$, for any
$a\in\mathcal{M}$. It is direct that $\mathcal{M}_{+}$ lies in
$\widehat{\mathcal{M}}_{+}$. Recall also that when $\mathcal{M}$ is abelian,
then indeed one has $\mathcal{M}\simeq L_{\infty}(\Omega,\mu)$ for some
measure space $(\Omega,\mu)$. Besides, in this case, the extended positive
part corresponds to the set of $\mu$-measurable functions on $\Omega$ (module
sets of zero measure) with values in $[0,\infty]$.

Operator-valued weights appears as ”unbounded conditional expectations”. The
simplest nontrivial example about this concept is probably the partial trace
$E_{\mathcal{M}}={\rm tr}_{\mathcal{A}}\otimes id_{\mathcal{M}}$, where
$\mathcal{A}$ is a semifinite non-finite von Neumann algebra. Generally, an
operator-valued weight is a kind of generalized conditionally expectation from
a von Neumann algebra $\mathcal{N}$ to a von Neumann subalgebra $\mathcal{M}$.
Specifically, it is a linear map from $\mathcal{N}_{+}$ to
$\widehat{\mathcal{M}_{+}}$ satisfying

(2.2) $\displaystyle E_{\mathcal{M}}(a^{*}fa)=a^{*}E_{\mathcal{M}}(f)a$

for all $a\in\mathcal{M}$. $E_{\mathcal{M}}$ is called normal when it
satisfies
$\sup\limits_{\alpha}E_{\mathcal{M}}(f_{\alpha})=E_{\mathcal{M}}(\sup\limits_{\alpha}f_{\alpha})$
for bounded increasing nets $(f_{\alpha})$ in $\mathcal{N}_{+}$.

Note that when $\mathcal{M}=\mathbb{C}$, the operator-valued weight
$E_{\mathcal{M}}$ becomes an ordinary weight on $\mathcal{N}$. In analogy with
ordinary weights, we consider

$\displaystyle
L_{\infty}^{c}(\mathcal{N};E_{\mathcal{M}})=\Big{\\{}f\in\mathcal{N}:\|E_{\mathcal{M}}(f^{*}f)\|_{\mathcal{M}}<\infty\Big{\\}}$

endowed with the natural (semi-)norm

$\displaystyle\|f\|_{L_{\infty}^{c}(\mathcal{N};E_{\mathcal{M}})}:=\|E_{\mathcal{M}}(f^{*}f)\|_{\mathcal{M}}^{1/2}.$

Note that in the case of $E_{\mathcal{M}}=tr_{\mathcal{A}}\otimes
id_{\mathcal{M}}$ with $\mathcal{N}=\mathcal{A}\bar{\otimes}\mathcal{M}$,
$L_{\infty}^{c}(\mathcal{N};E_{\mathcal{M}})$ becomes the Hilbert space valued
noncommutative $L_{\infty}$ spaces
$L_{2}^{c}(\mathcal{A})\bar{\otimes}\mathcal{M}$ (see [57]), which is defined
in [58].

By identifying $\mathcal{M}$ as a von Neumann subalgebra of
$B(L_{2}(\mathcal{M}))$ and then applying inequality (2.1), one has the
following equivalent (semi-)norms

(2.3)
$\displaystyle\|f\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}=\sup\limits_{\|b\|_{L_{2}(\mathcal{M})}=1}\tau(b^{*}E_{\rho}(|f|^{2})b)^{\frac{1}{2}},$

which is an effective tool to deal with the strong operator topology defined
in the next subsection.

Let $\mathcal{N}_{E_{\mathcal{M}}}$ be the linear span of $f_{1}^{*}f_{2}$
with $f_{1}$, $f_{2}\in L_{\infty}^{c}(\mathcal{N};E_{\mathcal{M}})$. Then one
has the following properties:

(i)
$\mathcal{N}_{E_{\mathcal{M}}}$=span$\\{f\in\mathcal{N}_{+}:\|E_{\mathcal{M}}f\|<\infty\\}$;

(ii) $L_{\infty}^{c}(\mathcal{N};E_{\mathcal{M}})$ and
$\mathcal{N}_{E_{\mathcal{M}}}$ are two-sided modules over $\mathcal{M}$;

(iii) $E_{\mathcal{M}}$ has a unique linear extension
$E_{\mathcal{M}}:\mathcal{N}_{E_{\mathcal{M}}}\rightarrow\mathcal{M}$
satisfying the bimodularity. That is, for any
$f\in\mathcal{N}_{E_{\mathcal{M}}}$ and $a,b\in\mathcal{M}$,

$E_{\mathcal{M}}(afb)=aE_{\mathcal{M}}(f)b.$

###  2.5. Algebraic, analytic, topological and operator conditions

In what follows, we shall assume that the completely positive unital map
$R_{r}$ from the P-metric $\mathcal{Q}$ defined in Section 2.1 is of the
following form:

$\displaystyle\mathcal{M}\stackrel{{\scriptstyle\rho_{j}}}{{\longrightarrow}}\mathcal{N}_{\rho}\stackrel{{\scriptstyle
E_{\rho}}}{{\longrightarrow}}\rho_{1}(\mathcal{M})\simeq\mathcal{M}$ (2.4)
$\displaystyle R_{r}f$
$\displaystyle=E_{\rho}(q_{r})^{-\frac{1}{2}}E_{\rho}(q_{r}\rho_{2}(f)q_{r})E_{\rho}(q_{r})^{-\frac{1}{2}},$

where $\rho_{1},\rho_{2}:\mathcal{M}\rightarrow\mathcal{N}_{\rho}$ are unital
injective $*$-homomorphisms into certain von Neumann algebra
$\mathcal{N}_{\rho}$ and the map
$E_{\rho}:\mathcal{N}_{\rho}\rightarrow\rho_{1}(\mathcal{M})$ is a normal
operator-valued weight defined in Section 2.4. The elements $q_{r}$ are
increasing projections with respect to $r$ in $\mathcal{N}_{\rho}$ and satisfy
some properties determined later on. In particular, we shall assume that
$q_{r}$ and $q_{r}\mathcal{N}_{\rho}q_{r}$ belong to the domain of $E_{\rho}$
so that our formula for $R_{r}f$ make sense. Our model provides a quite
general form of P-metric which includes the P-metric for the heat semigroup
considered before. Indeed, in the model case one could take
$\mathcal{N}_{\rho}=L_{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n})$ with
$\rho_{1}(f)(x,y)=f(x)$ and $\rho_{2}(f)(x,y)=f(y)$. Let $E_{\rho}$ be the
integral in $\mathbb{R}^{n}$ with respect to the variable $y$ and set
$q_{r}=\chi_{B(x,r)}(y).$ Then it is direct to check that we recover from
(2.5) the $R_{r}$’s for the heat semigroup. Note that the $q_{r}$ reproduce in
this case all the Euclidean balls in $\mathbb{R}^{n}$.

Now we establish some preliminary algebraic, analytic and topological
conditions on the P-metric. Let $C_{b}^{\infty}$ be the space of bounded
functions mapping $[0,\infty)$ to $\mathbb{C}$ and having bounded derivatives
at any order, we impose a priori assumption that for any $F\in
C_{b}^{\infty}$, the spectral multiplier
$F(L):\mathcal{A}_{\mathcal{M}}\rightarrow\mathcal{A}_{\mathcal{M}}$ for some
weak-$*$ dense subalgebra $\mathcal{A}_{\mathcal{M}}\subset\mathcal{M}$ such
that $\mathcal{A}_{\mathcal{M}}$ is dense in $L_{p}(\mathcal{M})$ for any
$1<p<\infty$. The above assumption is not a crucial condition as we mentioned
in the introduction. This can be easily seen for the classical case
$\mathcal{M}=L_{\infty}(\mathbb{R}^{n})$ with $L=-\Delta$ and
$\mathcal{A}_{\mathcal{M}}=\mathcal{S}(\mathbb{R}^{n})$. For simplicity, in
what follows we denote
$V_{\rho}=L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})+\rho_{2}(\mathcal{A}_{\mathcal{M}})$
and $\mathcal{L}(X)$ be the space consisting of all linear operators over a
linear space $X$. Assume that for any $F\in C_{b}^{\infty}$, there exists a
homomorphism $\pi$: $\\{F(L)\\}_{F\in C_{b}^{\infty}}\subset
B(L_{2}(\mathcal{M}))\rightarrow\mathcal{L}(V_{\rho})$ such that the following
intertwining equality holds: for any $F\in C_{b}^{\infty}$,

(2.5) $\pi(F(L))\circ\rho_{2}=\rho_{2}\circ F(L)\ {\rm on}\
\mathcal{A}_{\mathcal{M}}.$

Algebraic conditions

(ALi) $\mathcal{Q}$-monotonicity of $E_{\rho}$

$\displaystyle E_{\rho}(b_{k,r}|\xi|^{2}b_{k,r})\leq E_{\rho}(|\xi|^{2})$

for any $k\geq 1$ and $\xi\in\mathcal{N}_{\rho}$, where

$b_{k,r}=\left\\{\begin{array}[]{ll}q_{2r},&{\rm if}\ k=1,\\\
q_{2^{k}r}-q_{2^{k-1}r},&{\rm if}\ k\geq 2,\end{array}\right.$

every projection $q_{r}$ determined by identity in $\eqref{hhhh}$.

(ALii) Right modularity of $\pi(F(L))$

$\displaystyle\pi(F(L))(\eta
E_{\rho}(q_{r})^{-\frac{1}{2}})=\left(\pi(F(L))(\eta)\right)E_{\rho}(q_{r})^{-\frac{1}{2}}$

for all $\eta\in L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})$, $F\in
C_{b}^{\infty}$ and $r>0$.

######  Remark 2.3.

We will see below that all the algebraic conditions hold trivially in the
classical theory, where the first one essentially says that the integration of
a positive function over a “ball” or an “annulus” is always smaller than over
the whole space, while the second condition says that one could take out
$|B(x,r)|^{-1/2}$ from the $y$-dependent integral defining $F(L)$. Besides,
the first condition also suggests that certain amount of commutativity for the
projections is needed. In particular, we will frequently use the following
inequality: for any $a\in\mathcal{N}_{\rho}$, $b\in\mathcal{M}$ and $r>0$,

(2.6)
$\displaystyle\left\|aq_{r}b\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}\leq\left\|ab\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}.$

The above inequality can be deduced, by the property of operator-valued weight
(2.2) together with condition (ALi), as follows:

$E_{\rho}(b^{*}q_{r}|a|^{2}q_{r}b)=b^{*}E_{\rho}(q_{r}|a|^{2}q_{r})b\leq
b^{*}E_{\rho}(|a|^{2})b=E_{\rho}(b^{*}|a|^{2}b).$

Topological conditions

######  Definition 2.4.

Let $\\{f_{k}\\}_{k}\subset L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})$ and
$f\in L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})$. We say that $f_{k}$
converges to $f$ in the strong operator topology if for any $b\in
L_{2}(\mathcal{M})$, we have

$\displaystyle\lim_{k\rightarrow\infty}\tau\Big{(}b^{*}E_{\rho}((f_{k}-f)^{*}(f_{k}-f))b\Big{)}=0.$

(Ti) Continuity of $q_{r}$

$q_{s}$ converges to $1$ in the sense that for any $r>0$,
$F\in\mathcal{S}(\mathbb{R})$ and $f\in V_{\rho}$,

$\pi(F(L))(fq_{s})q_{r}E_{\rho}(q_{r})^{-\frac{1}{2}}\rightarrow\pi(F(L))(f)q_{r}E_{\rho}(q_{r})^{-\frac{1}{2}}\
{\rm in}\ {\rm the}\ {\rm strong}\ {\rm operator}\ {\rm topology}.$

(Tii) Continuity of $\pi$

If $\mathcal{S}(\mathbb{R})\ni F_{k}\rightarrow F\in\mathcal{S}(\mathbb{R})$
pointwise and $\sup\limits_{k}\|F_{k}\|_{\infty}<+\infty$, then for any $r>0$
and $f\in V_{\rho}$,

$\pi(F_{k}(L))(f)q_{r}E_{\rho}(q_{r})^{-\frac{1}{2}}\rightarrow\pi(F(L))(f)q_{r}E_{\rho}(q_{r})^{-\frac{1}{2}}\
{\rm in}\ {\rm the}\ {\rm strong}\ {\rm operator}\ {\rm topology}.$

Operator conditions

(Oi) Boundedness condition

$\|\pi(F(L))\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})\rightarrow
L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}\leq\|F\|_{L_{\infty}(\mathbb{R})}$

for all $F\in C_{b}^{\infty}$.

(Oii) $L_{2}$-Gaussian estimate

There exist constants $C,c>0$ and $m\geq 2$ such that for any $t>0$, $k\geq
1$, $r_{2}\geq r_{1}$ and $f\in\mathcal{N}_{\rho}$,

$\displaystyle\Big{\|}\left(\pi(e^{-tL})(fb_{k,r_{2}})\right)q_{r_{1}}E_{\rho}(q_{r_{1}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
(2.7) $\displaystyle\leq
CE_{k}\left(-c\left(\frac{r_{2}}{t^{1/m}}\right)^{\frac{m}{m-1}}\right)\Big{\|}fb_{k,r_{2}}E_{\rho}(q_{\max\\{r_{1},t^{1/m}\\}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})},$

where

$E_{k}(x)=\left\\{\begin{array}[]{ll}1,&{\rm if}\ k=1,\\\
\exp(2^{\frac{mk}{m-1}}x),&{\rm if}\ k\geq 2.\end{array}\right.$

######  Remark 2.5.

We shall see in Section 4 that in the lack of good metrics on general von
Neumann algebra, the $L_{2}$-Gaussian condition is a suitable replacement of
pointwise Gaussian estimate (${\rm GE}_{m}$).

Analytic condition

(ANi) Strong doubling condition

There exist constants $C,n>0$, $D\geq 0$, such that for any $r_{1}\geq
r_{2}\geq r_{3}$ and $f\in\mathcal{A}_{\mathcal{M}}$,

$\displaystyle\Bigg{\|}E_{\rho}\bigg{(}\bigg{|}\rho_{2}(f)q_{r_{1}}E_{\rho}(q_{r_{3}})^{-\frac{1}{2}}\bigg{|}^{2}\bigg{)}\Bigg{\|}_{\mathcal{M}}\leq
C\left(\frac{r_{1}}{r_{2}}\right)^{D}\left(\frac{r_{1}}{r_{3}}\right)^{n}\Bigg{\|}E_{\rho}\bigg{(}\bigg{|}\rho_{2}(f)q_{r_{2}}E_{\rho}(q_{r_{2}})^{-\frac{1}{2}}\bigg{|}^{2}\bigg{)}\Bigg{\|}_{\mathcal{M}}.$

######  Remark 2.6.

Due to the lack of concepts of pointwise and position in general metric space,
the procedure of covering cannot be proceeded. Instead, in this condition, we
apply an energy inequality to suggest certain kind of ’covering phenomenon’
over the underlying space. More details could be found in the applications
given in the last section. Besides, we would like to mention that by choosing
the parameters $f=1$ and $r_{1}=r_{2}$, this condition implies the following
doubling inequality: for any $r_{1}\geq r_{2}$, there are constants $C,n>0$
such that

(2.8) $\displaystyle
E_{\rho}(q_{r_{2}})^{-\frac{1}{2}}E_{\rho}(q_{r_{1}})E_{\rho}(q_{r_{2}})^{-\frac{1}{2}}\leq
C\left(\frac{r_{1}}{r_{2}}\right)^{n}.$

The smallest constant $n=n_{\bf q}$ such that the above inequality valid is
called the P-metric dimension of $\mathcal{M}$.

$\bullet$ Note: Throughout Sections 2.5 and 3.1, unless we mention the
contrary, we assume all the conditions stated as above.

######  Lemma 2.7.

There exists a constant $C>0$ such that for any $t>0$, $r>0$ and $f\in
L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})$,

$\displaystyle\Big{\|}\pi(e^{-tL})(f)q_{r}E_{\rho}(q_{r})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}\leq
C\Big{\|}fE_{\rho}(q_{\max\\{r,t^{1/m}\\}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}.$

###### Proof.

By equality (2.3),

(2.9) $\displaystyle{\rm LHS}$
$\displaystyle=\sup\limits_{\|b\|_{L_{2}(\mathcal{M})}=1}\Big{|}\tau\Big{(}b^{*}E_{\rho}\big{(}|(\pi(e^{-tL})(f)q_{r}E_{\rho}(q_{r}\big{)}^{-\frac{1}{2}})|^{2}\big{)}b\Big{)}\Big{|}^{1/2}.$

Applying Lemma 2.1, we see that for any $N>0$,

$\displaystyle\Big{|}\tau\Big{(}b^{*}E_{\rho}\big{(}|(\pi(e^{-tL})(f)q_{r}E_{\rho}(q_{r}\big{)}^{-\frac{1}{2}})|^{2}\big{)}b\Big{)}\Big{|}^{1/2}$
$\displaystyle\leq\Big{|}\tau\Big{(}b^{*}E_{\rho}\big{(}|(\pi(e^{-tL})(f(1-q_{2^{N+1}\max\\{r,t^{1/m}\\}}))q_{r}E_{\rho}(q_{r}\big{)}^{-\frac{1}{2}})|^{2}\big{)}b\Big{)}\Big{|}^{1/2}$
(2.10)
$\displaystyle+\sum_{k=1}^{N}\Big{|}\tau\Big{(}b^{*}E_{\rho}\big{(}|(\pi(e^{-tL})(fb_{k,\max\\{r,t^{1/m}\\}})q_{r}E_{\rho}(q_{r}\big{)}^{-\frac{1}{2}})|^{2}\big{)}b\Big{)}\Big{|}^{1/2}.$

By inequalities (2.9) and (2.5) together with the topological condition (Ti),
one can easily conclude that

(2.11)
$\displaystyle\Big{\|}\pi(e^{-tL})(f)q_{r}E_{\rho}(q_{r})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}\leq\sum_{k\geq
1}\Big{\|}\left(\pi(e^{-tL})(fb_{k,\max\\{r,t^{1/m}\\}})\right)q_{r}E_{\rho}(q_{r})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}.$

Applying the operator condition (Oii) and the algebraic condition (ALi), one
see that

$\displaystyle{\rm RHS}\ {\rm of}\ \eqref{eeee}$ $\displaystyle\leq
C\sum_{k\geq
1}E_{k}(-c)\Big{\|}fb_{k,\max\\{r,t^{1/m}\\}}E_{\rho}(q_{\max\\{r,t^{1/m}\\}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
$\displaystyle=C\sum_{k\geq
1}E_{k}(-c)\Big{\|}E_{\rho}(q_{\max\\{r,t^{1/m}\\}})^{-\frac{1}{2}}E_{\rho}(b_{k,\max\\{r,t^{1/m}\\}}|f|^{2}b_{k,\max\\{r,t^{1/m}\\}})E_{\rho}(q_{\max\\{r,t^{1/m}\\}})^{-\frac{1}{2}}\Big{\|}_{\mathcal{M}}^{\frac{1}{2}}$
$\displaystyle\leq C\sum_{k\geq
1}E_{k}(-c)\Big{\|}E_{\rho}(q_{\max\\{r,t^{1/m}\\}})^{-\frac{1}{2}}E_{\rho}(|f|^{2})E_{\rho}(q_{\max\\{r,t^{1/m}\\}})^{-\frac{1}{2}}\Big{\|}_{\mathcal{M}}^{\frac{1}{2}}$
$\displaystyle\leq
C\Big{\|}fE_{\rho}(q_{\max\\{r,t^{1/m}\\}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}.$

This ends the proof of Lemma 2.7. ∎

We need the following version of Phragmen-Lindelöf Theorem (see [18, Lemma
9]).

######  Lemma 2.8.

Assume that $F$ is an analytic function in
$\mathbb{C}_{+}:=\\{z\in\mathbb{C}:{\rm Rez}>0\\}$ satisfying

$\displaystyle|F(z)|\leq a_{1}({\rm Re}z)^{-b_{1}},$ $\displaystyle|F(t)|\leq
a_{1}t^{-b_{1}}{\rm exp}(-a_{2}t^{-b_{2}}),$

for some $a_{1}$, $a_{2}>0$, $b_{1}\geq 0$, $b_{2}\in(0,1]$ independent of
$z\in\mathbb{C}_{+}$ and $t>0$. Then for any $z\in\mathbb{C}_{+}$,

$\displaystyle|F(z)|\leq a_{1}2^{b_{1}}({\rm Re}z)^{-b_{1}}{\rm
exp}\bigg{(}-\frac{a_{2}b_{2}}{2}|z|^{-b_{2}-1}{\rm Re}z\bigg{)}.$

We now apply Phragmen-Lindelöf Theorem to pass $L_{2}$-Gaussian estimate from
$\mathbb{R}$ to $\mathbb{C}_{+}$.

######  Lemma 2.9.

There is a constant $C>0$ such that for any $z\in\mathbb{C}_{+}$, $k\geq 1$,
$0<r_{1}\leq r_{2}$ and $f\in\mathcal{N}_{\rho}$,

$\displaystyle\Big{\|}\left(\pi(e^{-zL})(fb_{k,r_{2}})\right)q_{r_{1}}E_{\rho}(q_{r_{1}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
(2.12) $\displaystyle\leq
CE_{k}\left(-c\left(\frac{r_{2}}{|z|^{1/m}}\right)^{\frac{m}{m-1}}\cos\theta\right)\Big{\|}fb_{k,r_{2}}E_{\rho}(q_{\max\\{r_{1},({\rm
Re}z)^{1/m}\\}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})},$

where $\theta=\arccos\frac{{\rm Re}z}{|z|}$.

###### Proof.

We first consider the proof for the case $k=1$. To this end, noting that the
map $\pi$ is a homomorphism, we apply Lemma 2.7 and the right modularity
condition (ALii) together with the boundedness condition (Oi) to conclude that
for any $z\in\mathbb{C}_{+}$, $r_{1}\geq 0$ and $f\in
L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})$,

$\displaystyle\Big{\|}\left(\pi(e^{-zL})(f)\right)q_{r_{1}}E_{\rho}(q_{r_{1}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
$\displaystyle=\Big{\|}\left(\pi(e^{-\frac{{\rm
Re}z}{2}L})\pi(e^{-(z-\frac{{\rm
Re}z}{2})L})(f)\right)q_{r_{1}}E_{\rho}(q_{r_{1}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
$\displaystyle\leq C\Big{\|}\left(\pi(e^{-(z-\frac{{\rm
Re}z}{2})L})(f)\right)E_{\rho}(q_{\max\\{r_{1},(\frac{{\rm
Re}z}{2})^{1/m}\\}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
$\displaystyle=C\Big{\|}\pi(e^{-(z-\frac{{\rm
Re}z}{2})L})(fE_{\rho}(q_{\max\\{r_{1},(\frac{{\rm
Re}z}{2})^{1/m}\\}})^{-\frac{1}{2}})\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
$\displaystyle\leq C\Big{\|}fE_{\rho}(q_{\max\\{r_{1},(\frac{{\rm
Re}z}{2})^{1/m}\\}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
(2.13) $\displaystyle\leq C\Big{\|}fE_{\rho}(q_{\max\\{r_{1},({\rm
Re}z)^{1/m}\\}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})},$

where in the last inequality we applied the definition of
$L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})$ and the doubling inequality
(2.8) to get rid of the harmless multiple $\frac{1}{2}$ from $(\frac{{\rm
Re}z}{2})^{1/m}$, which implies the desired estimate for $k=1$.

To obtain the desired estimate for the case $k\geq 2$, we let $\phi$ be a
state of $\mathcal{M}$ and $\tilde{\phi}$ be a linear functional from
$L_{\infty}^{c}\times L_{\infty}^{c}$ to $\mathbb{C}$ given by
$\tilde{\phi}(x,y)=\phi(E_{\rho}(x^{*}y))$. Let $y\in
L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})$ such that $\tilde{\phi}(y,y)=1$.
For any $R>0$, we define an analytic function over right-half plane
$\mathbb{C}_{+}$ by

(2.14) $\displaystyle
F(z):=e^{-Rz}\tilde{\phi}\left(\left(\pi(e^{-zL})(fb_{k,r_{2}})\right)q_{r_{1}}E_{\rho}(q_{r_{1}})^{-\frac{1}{2}},y\right).$

By inequalities (2.1) and (2.5), we see that for any $z\in\mathbb{C}_{+}$,

$\displaystyle|F(z)|$ $\displaystyle\leq e^{-R{\rm
Re}z}\tilde{\phi}\left(\left(\pi(e^{-zL})(fb_{k,r_{2}})\right)q_{r_{1}}E_{\rho}(q_{r_{1}})^{-\frac{1}{2}},\left(\pi(e^{-zL})(fb_{k,r_{2}})\right)q_{r_{1}}E_{\rho}(q_{r_{1}})^{-\frac{1}{2}}\right)^{1/2}$
$\displaystyle\leq e^{-R{\rm
Re}z}\Big{\|}\left(\pi(e^{-zL})(fb_{k,r_{2}})\right)q_{r_{1}}E_{\rho}(q_{r_{1}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
$\displaystyle\leq Ce^{-R{\rm
Re}z}\Big{\|}fb_{k,r_{2}}E_{\rho}(q_{\max\\{r_{1},({\rm
Re}z)^{1/m}\\}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})},$

which, together with the definition of
$L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})$ and the doubling inequality
(2.8), implies the following pair of inequalities

(2.15) $\displaystyle|F(z)|\leq
C\Big{\|}fb_{k,r_{2}}E_{\rho}(q_{r_{1}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$

and

(2.16) $\displaystyle|F(z)|\leq Ce^{-R{\rm
Re}z}\Big{\|}fb_{k,r_{2}}E_{\rho}(q_{({\rm
Re}z)^{1/m}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}.$

To continue, by (2.16) and the doubling inequality (2.8), one has

$\displaystyle|F(z)|$ $\displaystyle\leq Ce^{-R{\rm
Re}z}\Big{\|}E_{\rho}(q_{({\rm
Re}z)^{1/m}})^{-\frac{1}{2}}E_{\rho}(q_{R^{-1/m}})E_{\rho}(q_{({\rm
Re}z)^{1/m}})^{-\frac{1}{2}}\Big{\|}_{\mathcal{M}}^{\frac{1}{2}}\Big{\|}fb_{k,r_{2}}E_{\rho}(q_{R^{-1/m}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
$\displaystyle\leq Ce^{-R{\rm Re}z}\bigg{(}1+\frac{1}{(R{\rm
Re}z)^{1/m}}\bigg{)}^{\frac{n}{2}}\Big{\|}fb_{k,r_{2}}E_{\rho}(q_{R^{-1/m}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
(2.17) $\displaystyle\leq C(R{\rm
Re}z)^{-\frac{n}{2m}}\Big{\|}fb_{k,r_{2}}E_{\rho}(q_{R^{-1/m}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}.$

On the other hand, for any $s>0$, by inequality (2.1) and $L_{2}$-Gaussian
estimate (Oii), we have

(2.18) $\displaystyle|F(s)|$ $\displaystyle\leq
Ce^{-Rs}\exp\left(-c\left(\frac{2^{k}r_{2}}{s^{1/m}}\right)^{\frac{m}{m-1}}\right)\Big{\|}fb_{k,r_{2}}E_{\rho}(q_{\max\\{r_{1},s^{1/m}\\}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})},$

which, together with the definition of
$L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})$ and the doubling inequality
(2.8), implies the following pair of inequalities

(2.19) $\displaystyle|F(s)|\leq
C\exp\left(-c\left(\frac{2^{k}r_{2}}{s^{1/m}}\right)^{\frac{m}{m-1}}\right)\Big{\|}fb_{k,r_{2}}E_{\rho}(q_{r_{1}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$

and

(2.20) $\displaystyle|F(s)|\leq
Ce^{-Rs}\exp\left(-c\left(\frac{2^{k}r_{2}}{s^{1/m}}\right)^{\frac{m}{m-1}}\right)\Big{\|}fb_{k,r_{2}}E_{\rho}(q_{s^{1/m}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}.$

To continue, by (2.20) and the doubling inequality (2.8), one has

(2.21) $\displaystyle|F(s)|$ $\displaystyle\leq
C(Rs)^{-\frac{n}{2m}}\exp\left(-c\left(\frac{2^{k}r_{2}}{s^{1/m}}\right)^{\frac{m}{m-1}}\right)\Big{\|}fb_{k,r_{2}}E_{\rho}(q_{R^{-1/m}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}.$

Applying Phragmén-Lindelöf Theorem 2.8 to inequalities (2.15) and (2.19), we
conclude that

$\displaystyle|F(z)|\leq
C\exp\left(-c\left(\frac{2^{k}r_{2}}{|z|^{1/m}}\right)^{\frac{m}{m-1}}\cos\theta\right)\Big{\|}fb_{k,r_{2}}E_{\rho}(q_{r_{1}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}.$

Since $F$ depends on $R$ but the constant $C$ is independent of $R$ and $z$,
we choose $R=({\rm Re}z)^{-1}$ to get that

$\displaystyle\big{|}\tilde{\phi}\left(\left(\pi(e^{-zL})(fb_{k,r_{2}})\right)q_{r_{1}}E_{\rho}(q_{r_{1}})^{-\frac{1}{2}},y\right)\big{|}$
(2.22) $\displaystyle\leq
C\exp\left(-c\left(\frac{2^{k}r_{2}}{|z|^{1/m}}\right)^{\frac{m}{m-1}}\cos\theta\right)\Big{\|}fb_{k,r_{2}}E_{\rho}(q_{r_{1}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}.$

Applying Phragmén-Lindelöf Theorem 2.8 again to inequalities (2.5) and (2.21),
we deduce that

(2.23) $\displaystyle|F(z)|\leq CR^{-\frac{n}{2m}}({\rm
Re}z)^{-\frac{n}{2m}}\exp\left(-c\left(\frac{2^{k}r_{2}}{|z|^{1/m}}\right)^{\frac{m}{m-1}}\cos\theta\right)\Big{\|}fb_{k,r_{2}}E_{\rho}(q_{R^{-1/m}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}.$

Choosing $R=({\rm Re}z)^{-1}$ again, we get that

$\displaystyle\big{|}\tilde{\phi}\left(\left(\pi(e^{-zL})(fb_{k,r_{2}})\right)q_{r_{1}}E_{\rho}(q_{r_{1}})^{-\frac{1}{2}},y\right)\big{|}$
(2.24) $\displaystyle\leq
C\exp\left(-c\left(\frac{2^{k}r_{2}}{|z|^{1/m}}\right)^{\frac{m}{m-1}}\cos\theta\right)\Big{\|}fb_{k,r_{2}}E_{\rho}(q_{({\rm
Re}z)^{1/m}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}.$

To continue, recall from inequality (2.1) that for $x\in
L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})$,

$\displaystyle\|x\|_{L_{\infty}^{c}(\mathcal{N_{\rho}},E_{\rho})}=\|E_{\rho}(x^{*}x)\|_{\mathcal{M}}^{1/2}$
$\displaystyle=\sup_{\phi\text{ is a state}}\phi(E_{\rho}(x^{*}x))^{1/2}$
$\displaystyle=\sup_{\phi\text{ is a state}}\tilde{\phi}(x,x)^{1/2}$
$\displaystyle=\sup_{\phi\text{ is a
state}}\sup_{\tilde{\phi}(y,y)=1}|\tilde{\phi}(x,y)|.$

Substituting $x$ with
$\left(\pi(e^{-zL})(fb_{k,r_{2}})\right)q_{r_{1}}E_{\rho}(q_{r_{1}})^{-\frac{1}{2}}$
into the above inequality and then using inequalities (2.5) and (2.5), we
deduce that

$\displaystyle\Big{\|}\left(\pi(e^{-zL})(fb_{k,r_{2}})\right)q_{r_{1}}E_{\rho}(q_{r_{1}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
(2.25) $\displaystyle\leq
C\exp\left(-c\left(\frac{2^{k}r_{2}}{|z|^{1/m}}\right)^{\frac{m}{m-1}}\cos\theta\right)\Big{\|}fb_{k,r_{2}}E_{\rho}(q_{r_{1}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$

and that

$\displaystyle\Big{\|}\left(\pi(e^{-zL})(fb_{k,r_{2}})\right)q_{r_{1}}E_{\rho}(q_{r_{1}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
(2.26) $\displaystyle\leq
C\exp\left(-c\left(\frac{2^{k}r_{2}}{|z|^{1/m}}\right)^{\frac{m}{m-1}}\cos\theta\right)\Big{\|}fb_{k,r_{2}}E_{\rho}(q_{({\rm
Re}z)^{1/m}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})},$

which implies the estimate (2.9). This ends the proof of Lemma 2.9. ∎

Next we define a Besov type norm of $f$ by

$\|F\|_{B^{N}}:=\int_{-\infty}^{\infty}|\hat{F}(\tau)|(1+|\tau|)^{N}d\tau,$

where $\hat{F}$ denotes the Fourier transform of $f$. It can be verified
easily that

$\|FG\|_{B^{N}}\leq\|F\|_{B^{N}}\|G\|_{B^{N}}.$

Besides, for any $R>0$, we denote the $R$-dilation of a given function $F$ by
$\delta_{R}F(x):=F(Rx)$.

######  Lemma 2.10.

There is a constant $C>0$ such that for any $R>0$, $k\geq 2$, $0<r_{1}\leq
r_{2}$ and $f\in\mathcal{N}_{\rho}$,

$\displaystyle\Big{\|}\left(\pi(F(L))(fq_{2r_{2}})\right)q_{r_{1}}E_{\rho}(q_{r_{1}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
(2.27) $\displaystyle\leq
C\|\delta_{R}F\|_{B^{0}}\Big{\|}fq_{2r_{2}}E_{\rho}(q_{\max\\{r_{1},R^{-1/m}\\}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$

and

$\displaystyle\Big{\|}\left(\pi(F(L))(fb_{k,r_{2}})\right)q_{r_{1}}E_{\rho}(q_{r_{1}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
(2.28) $\displaystyle\leq
C(\sqrt[m]{R}2^{k}r_{2})^{-N}\|\delta_{R}F\|_{B^{N}}\Big{\|}fb_{k,r_{2}}E_{\rho}(q_{\max\\{r_{1},R^{-1/m}\\}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$

for all smoothing functions $F$ such that supp$F\subset[-R,R]$.

###### Proof.

Let $G(\lambda)=F(R\lambda)e^{\lambda}$. We apply the Fourier inversion
formula

$F(L)=G(L/R)e^{-L/R}=\frac{1}{2\pi}\int_{\mathbb{R}}e^{(i\tau-1)R^{-1}L}\hat{G}(\tau)d\tau$

and the linearity of the amplification map $\pi$ together with the topological
condition (Tii) to get that

(2.29)
$\displaystyle\pi(F(L))(f)q_{r_{1}}E_{\rho}(q_{r_{1}})^{-\frac{1}{2}}=\frac{1}{2\pi}\int_{\mathbb{R}}\pi(e^{(i\tau-1)R^{-1}L})(f)q_{r_{1}}E_{\rho}(q_{r_{1}})^{-\frac{1}{2}}\hat{G}(\tau)d\tau,$

where the improper integral converges in the strong operator topology.

To continue, we apply Lemma 2.9 with the choice
$z=(1-i\tau)R^{-1}\in\mathbb{C}_{+}$ to deduce that there exist constants
$C,c>0$ such that for any $\tau\in\mathbb{R}$, $k\geq 1$, $R>0$, $0<r_{1}\leq
r_{2}$ and $f\in\mathcal{N}_{\rho}$,

$\displaystyle\Big{\|}\left(\pi(e^{(i\tau-1)R^{-1}L})(fb_{k,r_{2}})\right)q_{r_{1}}E_{\rho}(q_{r_{1}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
(2.30) $\displaystyle\leq
CE_{k}\left(-c\left(\frac{\sqrt[m]{R}r_{2}}{\sqrt{1+\tau^{2}}}\right)^{\frac{m}{m-1}}\right)\Big{\|}fb_{k,r_{2}}E_{\rho}(q_{\max\\{r_{1},R^{-1/m}\\}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}.$

This, in combination with equalities (2.3) and (2.29), yields

$\displaystyle\Big{\|}\left(\pi(F(L))(fb_{k,r_{2}})\right)q_{r_{1}}E_{\rho}(q_{r_{1}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
(2.31) $\displaystyle\leq
C\int_{\mathbb{R}}\Big{\|}\left(\pi(e^{(i\tau-1)R^{-1}L})(fb_{k,r_{2}})\right)q_{r_{1}}E_{\rho}(q_{r_{1}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}|\hat{G}(\tau)|d\tau$
$\displaystyle\leq
C\int_{\mathbb{R}}E_{k}\left(-c\left(\frac{\sqrt[m]{R}r_{2}}{\sqrt{1+\tau^{2}}}\right)^{\frac{m}{m-1}}\right)|\hat{G}(\tau)|d\tau\Big{\|}fb_{k,r_{2}}E_{\rho}(q_{\max\\{r_{1},R^{-1/m}\\}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}.$

Note that the last term is dominated by

$C\|G\|_{B^{0}}\Big{\|}fq_{2r_{2}}E_{\rho}(q_{\max\\{r_{1},R^{-1/m}\\}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$

if $k=1$ and

$C(\sqrt[m]{R}2^{k}r_{2})^{-N}\|G\|_{B^{N}}\Big{\|}fb_{k,r_{2}}E_{\rho}(q_{\max\\{r_{1},R^{-1/m}\\}})^{-\frac{1}{2}}\Big{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$

if $k\geq 2$. To continue, note that supp$F\subset[-R,R]$ and so
supp$F(R\cdot)\subset[-1,1]$. Hence taking a function $\psi\in C_{c}^{\infty}$
such that supp$\psi\subset[-2,2]$ and $\psi(\lambda)=1$ for
$\lambda\in[-1,1]$, we have

$\displaystyle
G(\lambda)=F(R\lambda)e^{\lambda}=F(R\lambda)\psi(\lambda)e^{\lambda},$

which implies that

(2.32) $\displaystyle\|G\|_{B^{N}}\leq
C\|\delta_{R}F\|_{B^{N}}\|\psi(\lambda)e^{\lambda}\|_{B^{N}}\leq
C\|\delta_{R}F\|_{B^{N}}.$

This finishes the proof of Lemma 2.10. ∎

######  Proposition 2.11.

For any $M\in\mathbb{Z}_{+}$ and for all $f\in\mathcal{A}_{\mathcal{M}}$,

(2.33)
$\displaystyle\|f\|_{{BMO}_{\mathcal{Q},1}^{c}}\simeq\|f\|_{{BMO}_{\mathcal{Q},M}^{c}},$

where the implicit constant depends on $M$.

###### Proof.

We first show that for any $f\in\mathcal{A}_{\mathcal{M}}$,

(2.34)
$\displaystyle\|f\|_{{BMO}_{\mathcal{Q},2}^{c}}\lesssim\|f\|_{{BMO}_{\mathcal{Q},1}^{c}}.$

To show this, we apply Kadison-Schwarz’s inequality and Lemma 2.2 to deduce
that

$\displaystyle{\rm LHS}$
$\displaystyle\leq\sup\limits_{t>0}\big{\|}R_{\sqrt[m]{t}}|(I-e^{-tL})f|^{2}\big{\|}_{\mathcal{M}}^{1/2}+\sup\limits_{t>0}\big{\|}R_{\sqrt[m]{t}}|e^{-tL}(I-e^{-tL})f|^{2}\big{\|}_{\mathcal{M}}^{1/2}$
$\displaystyle\leq\sup\limits_{t>0}\big{\|}R_{\sqrt[m]{t}}|(I-e^{-tL})f|^{2}\big{\|}_{\mathcal{M}}^{1/2}+\sup\limits_{t>0}\big{\|}e^{-tL}(I-e^{-tL})f\big{\|}_{\mathcal{M}}$
$\displaystyle\lesssim\sup\limits_{t>0}\big{\|}R_{\sqrt[m]{t}}|(I-e^{-tL})f|^{2}\big{\|}_{\mathcal{M}}^{1/2}+\sup\limits_{t>0}\big{\|}e^{-tL}|(I-e^{-tL})f|^{2}\big{\|}_{\mathcal{M}}^{1/2}$
$\displaystyle\lesssim\sup\limits_{t>0}\big{\|}R_{\sqrt[m]{t}}|(I-e^{-tL})f|^{2}\big{\|}_{\mathcal{M}}^{1/2}.$

This ends the proof of inequality (2.34). By induction on $M\geq 1$, one has

$\displaystyle\|f\|_{{BMO}_{\mathcal{Q},M}^{c}}\lesssim\|f\|_{{BMO}_{\mathcal{Q},1}^{c}},$

where the implicit constant depends on $M$.

It remains to show the converse direction of the above inequality. To begin
with, for any $M\geq 2$, we observe that

$\displaystyle I$ $\displaystyle=(I-e^{-tL})^{M-1}+(I-(I-e^{-tL})^{M-1})$
$\displaystyle=(I-e^{-tL})^{M-1}+\sum_{k=1}^{M-1}(-1)^{k}\frac{(M-1)!}{k!(M-1-k)!}e^{-ktL}.$

Then

$\displaystyle\left\|R_{\sqrt[m]{t}}|(I-e^{-tL})f|^{2}\right\|_{\mathcal{M}}^{1/2}$
$\displaystyle\leq\left\|R_{\sqrt[m]{t}}|(I-e^{-tL})^{M}f|^{2}\right\|_{\mathcal{M}}^{1/2}$
$\displaystyle+\sum_{k=1}^{M-1}\frac{(M-1)!}{k!(M-1-k)!}\left\|R_{\sqrt[m]{t}}|e^{-ktL}(I-e^{-tL})f|^{2}\right\|_{\mathcal{M}}^{1/2}.$

This means that it suffices to show that for any $1\leq k\leq M-1$,

(2.35)
$\displaystyle\sup\limits_{t>0}\left\|R_{\sqrt[m]{t}}|e^{-ktL}(I-e^{-tL})f|^{2}\right\|_{\mathcal{M}}^{1/2}\lesssim\sup\limits_{t>0}\left\|R_{\sqrt[m]{t}}|(I-e^{-tL})^{M}f|^{2}\right\|_{\mathcal{M}}^{1/2}.$

To show this, we first choose $\Phi$ be a smooth cut-off function supported in
$[1/2,2]$ such that

$\displaystyle\int_{0}^{\infty}\Phi(s\lambda)(I-e^{-s\lambda})^{M}\frac{ds}{s}=1,\
{\rm for}\ \lambda>0.$

This, together with the topological condition (Tii), implies that

$\displaystyle\pi(e^{-ktL}(I-e^{-tL}))(\rho_{2}(f))q_{\sqrt[m]{t}}E_{\rho}(q_{\sqrt[m]{t}})^{-\frac{1}{2}}$
(2.36)
$\displaystyle=\int_{0}^{\infty}\pi\Big{(}e^{-ktL}(I-e^{-tL})\Phi(sL)(I-e^{-sL})^{M}\Big{)}(\rho_{2}(f))q_{\sqrt[m]{t}}E_{\rho}(q_{\sqrt[m]{t}})^{-\frac{1}{2}}\frac{ds}{s},$

where the improper integral converges in the strong operator topology. This,
together with the intertwining equality (2.5) and formula (2.3), implies that

$\displaystyle\
\left\|R_{\sqrt[m]{t}}|e^{-ktL}(I-e^{-tL})f|^{2}\right\|_{\mathcal{M}}^{1/2}$
$\displaystyle=\left\|\pi(e^{-ktL}(I-e^{-tL}))(\rho_{2}(f))q_{\sqrt[m]{t}}E_{\rho}(q_{\sqrt[m]{t}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
(2.37)
$\displaystyle\leq\int_{0}^{\infty}\left\|\pi\left(e^{-ktL}(I-e^{-tL})\Phi(sL)(I-e^{-sL})^{M}\right)(\rho_{2}(f))q_{\sqrt[m]{t}}E_{\rho}(q_{\sqrt[m]{t}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}\frac{ds}{s}.$

To continue, let $\psi\in C_{c}^{\infty}(\mathbb{R})$ supported in
$\psi\in[1/8,4]$ and $\psi(\lambda)=1$ for $\lambda\in[1/2,2]$. For
simplicity, we denote

$F_{k,t,s}(\lambda):=e^{-kt\lambda}(I-e^{-t\lambda})\Phi(s\lambda),$

which satisfies for any $N\geq 0$, there is a constant $C=C_{N}>0$ such that

$\displaystyle\|\delta_{s^{-1}}F_{k,t,s}\|_{B^{N}}$
$\displaystyle\leq\left\|e^{-\frac{kt\lambda}{s}}(1-e^{-\frac{t\lambda}{s}})\Phi(\lambda)\right\|_{C^{N+2}}$
$\displaystyle\leq\left\|e^{-\frac{kt\lambda}{s}}\Phi(\lambda)\right\|_{C^{N+2}}\left\|(1-e^{-\frac{t\lambda}{s}})\psi(\lambda)\right\|_{C^{N+2}}$
(2.38) $\displaystyle\leq
C_{N}\min\left\\{\Big{(}\frac{t}{s}\Big{)}^{-N},\Big{(}\frac{t}{s}\Big{)}\right\\},$

where $C^{N}$ is the space consisting of all $N$-order differentiable
functions, endowed with the norm:

$\|f\|_{C^{N}}:=\sum_{k=1}^{N}\left\|\frac{d^{k}}{dx^{k}}f\right\|_{L_{\infty}}.$

By inequality (2.5),

$\displaystyle\left\|R_{\sqrt[m]{t}}|e^{-ktL}(I-e^{-tL})f|^{2}\right\|_{\mathcal{M}}^{1/2}$
$\displaystyle\leq\int_{0}^{t}\left\|\left(\pi(F_{k,t,s}(L))\rho_{2}((I-e^{-sL})^{M}f)\right)q_{\sqrt[m]{t}}E_{\rho}(q_{\sqrt[m]{t}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}\frac{ds}{s}$
$\displaystyle+\int_{t}^{\infty}\left\|\left(\pi(F_{k,t,s}(L))\rho_{2}((I-e^{-sL})^{M}f)\right)q_{\sqrt[m]{t}}E_{\rho}(q_{\sqrt[m]{t}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}\frac{ds}{s}$
$\displaystyle=:{\rm I}+{\rm II}.$

To estimate term ${\rm I}$, we follow the estimate of (2.11) and then apply
Lemma 2.10 to conclude that

$\displaystyle{\rm I}$ $\displaystyle\leq\int_{0}^{t}\sum_{j\geq
1}\left\|\left(\pi(F_{k,t,s}(L))(\rho_{2}((I-e^{-sL})^{M}f)b_{j,\sqrt[m]{t}})\right)q_{\sqrt[m]{t}}E_{\rho}(q_{\sqrt[m]{t}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}\frac{ds}{s}$
$\displaystyle\leq C\int_{0}^{t}\sum_{j\geq
1}2^{-jN}\|\delta_{s^{-1}}F_{k,t,s}\|_{B^{N}}\left\|\rho_{2}((I-e^{-sL})^{M}f)b_{j,\sqrt[m]{t}}E_{\rho}(q_{\sqrt[m]{t}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}\frac{ds}{s}$
$\displaystyle\leq C\int_{0}^{t}\sum_{j\geq
1}2^{-jN}\Big{(}\frac{t}{s}\Big{)}^{-N}\left\|\rho_{2}((I-e^{-sL})^{M}f)b_{j,\sqrt[m]{t}}E_{\rho}(q_{\sqrt[m]{t}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}\frac{ds}{s}.$

This, together with the strong doubling condition (ANi), implies that

$\displaystyle{\rm I}$ $\displaystyle\leq C\int_{0}^{t}\sum_{j\geq
1}2^{-jN}\Big{(}\frac{t}{s}\Big{)}^{-N}\Big{(}\frac{2^{j}\sqrt[m]{t}}{\sqrt[m]{s}}\Big{)}^{\frac{D}{2}}2^{\frac{jn}{2}}\left\|\rho_{2}((I-e^{-sL})^{M}f)q_{\sqrt[m]{s}}E_{\rho}(q_{\sqrt[m]{s}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}\frac{ds}{s}$
$\displaystyle\leq
C\int_{0}^{t}\Big{(}\frac{s}{t}\Big{)}^{N-\frac{D}{2m}}\left\|\rho_{2}((I-e^{-sL})^{M}f)q_{\sqrt[m]{s}}E_{\rho}(q_{\sqrt[m]{s}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}\frac{ds}{s}$
$\displaystyle\leq
C\sup\limits_{s>0}\left\|R_{\sqrt[m]{s}}|(I-e^{-sL})^{M}f|^{2}\right\|_{\mathcal{M}}^{1/2},$

provided we choose $N$ be a sufficiently large constant.

To estimate term ${\rm II}$, we follow the estimate of (2.11) and then apply
Lemma 2.10 to get that

$\displaystyle{\rm II}$ $\displaystyle\leq\int_{t}^{\infty}\sum_{j\geq
1}\left\|\left(\pi(F_{k,t,s}(L))(\rho_{2}((I-e^{-sL})^{M}f)b_{j,\sqrt[m]{s}})\right)q_{\sqrt[m]{t}}E_{\rho}(q_{\sqrt[m]{t}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}\frac{ds}{s}$
$\displaystyle\leq C\int_{t}^{\infty}\sum_{j\geq
1}2^{-jN}\|\delta_{s^{-1}}F_{k,t,s}\|_{B^{N}}\left\|\rho_{2}((I-e^{-sL})^{M}f)b_{j,\sqrt[m]{s}}E_{\rho}(q_{\sqrt[m]{s}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}\frac{ds}{s}$
$\displaystyle\leq C\int_{t}^{\infty}\sum_{j\geq
1}2^{-jN}\left(\frac{t}{s}\right)\left\|\rho_{2}((I-e^{-sL})^{M}f)b_{j,\sqrt[m]{s}}E_{\rho}(q_{\sqrt[m]{s}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}\frac{ds}{s}.$

This, together with the strong doubling condition (ANi), implies that

$\displaystyle{\rm II}$ $\displaystyle\leq C\int_{t}^{\infty}\sum_{j\geq
1}2^{-j(N-\frac{n}{2}-\frac{D}{2})}\left(\frac{t}{s}\right)\left\|\rho_{2}((I-e^{-sL})^{M}f)q_{\sqrt[m]{s}}E_{\rho}(q_{\sqrt[m]{s}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}\frac{ds}{s}$
$\displaystyle\leq
C\int_{t}^{\infty}\left(\frac{t}{s}\right)\left\|\rho_{2}((I-e^{-sL})^{M}f)q_{\sqrt[m]{s}}E_{\rho}(q_{\sqrt[m]{s}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}\frac{ds}{s}$
$\displaystyle\leq
C\sup\limits_{s>0}\left\|R_{\sqrt[m]{s}}|(I-e^{-sL})^{M}f|^{2}\right\|_{\mathcal{M}}^{1/2},$

provided we choose $N$ be a sufficiently large constant.

Combining the estimates of terms I and II, we complete the proof of
Proposition 2.11. ∎

##  3\. Proof of Theorem 1.1

###  3.1. $L_{\infty}\rightarrow BMO$ boundedness

In this subsection, we will provide a proof for the first statement of Theorem
1.1. For future use in the complex interpolation, we will show a stronger
argument under the same condition. That is, we will show that there is a
constant $C>0$ independent of $t>0$ and $y\in\mathbb{R}$ such that

(3.1)
$\displaystyle\left\|e^{itL}(I+L)^{-s(1+iy)}f\right\|_{{BMO}_{\mathcal{S}}(\mathcal{M})}\leq
C(1+|y|)^{s+3}(1+|t|)^{n/2}\|f\|_{\mathcal{M}},\ {\rm for}\ s\geq n/2.$

Then the first statement of Theorem 1.1 follows directly by choosing $y=0$.

To begin with, we will apply a simple argument to reduce the proof to the
column BMO estimate. To this end, we assume that
$e^{itL}(I+L)^{-s(1+iy)}:\mathcal{A}_{\mathcal{M}}\rightarrow
BMO_{\mathcal{S}}^{c}(\mathcal{M})$ is bounded by
$C(1+|y|)^{s+1}(1+|t|)^{n/2}$ with a constant $C>0$ independent of
$t\in\mathbb{R}$. Note that

$(e^{itL}(I+L)^{-s(1+iy)}f)^{*}=e^{-itL}(I+L)^{-s(1-iy)}f^{*},$

then

$\displaystyle\left\|e^{itL}(I+L)^{-s(1+iy)}f\right\|_{{BMO}_{\mathcal{S}}^{r}(\mathcal{M})}$
$\displaystyle=\left\|e^{-itL}(I+L)^{-s(1-iy)}f^{*}\right\|_{{BMO}_{\mathcal{S}}^{c}(\mathcal{M})}$
$\displaystyle\leq C(1+|y|)^{s+1}(1+|t|)^{n/2}\|f^{*}\|_{\mathcal{M}}$
$\displaystyle\leq C(1+|y|)^{s+1}(1+|t|)^{n/2}\|f\|_{\mathcal{M}}.$

This, together with Lemma 2.11, implies that it suffices to establish the
column BMO estimate of $e^{itL}(I+L)^{-s(1+iy)}$ by showing that for a given
large positive constant $M$, there is a constant $C>0$ independent of $t$ such
that for any $f\in\mathcal{A}_{\mathcal{M}}$,

(3.2) $\displaystyle\|F(L)f\|_{{BMO}_{\mathcal{Q},M}^{c}(\mathcal{M})}\leq
C(1+|y|)^{s+1}(1+|t|)^{n/2}\|f\|_{\mathcal{M}},$

where we denote $F(u)=e^{itu}(1+u)^{-s(1+iy)}$ for simplicity.

To show this, recalling that $\rho_{2}$ is a $*$-homomorphism from
$\mathcal{M}$ to $\mathcal{N}_{\rho}$, we apply equality (2.5) and the
property of operator-valued weight to get that

$\displaystyle\left\|R_{\sqrt[m]{\lambda}}|(I-e^{-\lambda
L})^{M}F(L)f|^{2}\right\|_{\mathcal{M}}^{1/2}$
$\displaystyle=\left\|\left(\pi((I-e^{-\lambda
L})^{M}F(L))(\rho_{2}(f))\right)q_{\sqrt[m]{\lambda}}E_{\rho}(q_{\sqrt[m]{\lambda}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
$\displaystyle\leq\left\|\left(\pi((I-e^{-\lambda
L})^{M}F(L))(\rho_{2}(f)q_{4(1+|t|)\sqrt[m]{\lambda}})\right)q_{\sqrt[m]{\lambda}}E_{\rho}(q_{\sqrt[m]{\lambda}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
$\displaystyle+\left\|\left(\pi((I-e^{-\lambda
L})^{M}F(L))(\rho_{2}(f)(1-q_{4(1+|t|)\sqrt[m]{\lambda}}))\right)q_{\sqrt[m]{\lambda}}E_{\rho}(q_{\sqrt[m]{\lambda}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
$\displaystyle=:{\rm I}+{\rm II}.$

To estimate term I, we apply the inequality (2.6) and condition (ALii) to get
that

$\displaystyle{\rm I}$ $\displaystyle=\left\|\left(\pi((I-e^{-\lambda
L})^{M}F(L))(\rho_{2}(f)q_{4(1+|t|)\sqrt[m]{\lambda}})\right)q_{\sqrt[m]{\lambda}}E_{\rho}(q_{\sqrt[m]{\lambda}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
$\displaystyle\leq\left\|\left(\pi((I-e^{-\lambda
L})^{M}F(L))(\rho_{2}(f)q_{4(1+|t|)\sqrt[m]{\lambda}})\right)E_{\rho}(q_{\sqrt[m]{\lambda}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
$\displaystyle=\left\|\pi((I-e^{-\lambda
L})^{M}F(L))(\rho_{2}(f)q_{4(1+|t|)\sqrt[m]{\lambda}}E_{\rho}(q_{\sqrt[m]{\lambda}})^{-\frac{1}{2}})\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}.$

Next we apply conditions (Oi) and (ANi) to get that

$\displaystyle{\rm I}$
$\displaystyle\leq\left\|\rho_{2}(f)q_{4(1+|t|)\sqrt[m]{\lambda}}E_{\rho}(q_{\sqrt[m]{\lambda}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
$\displaystyle\leq\left\|E_{\rho}(q_{\sqrt[m]{\lambda}})^{-\frac{1}{2}}E_{\rho}(q_{4(1+|t|)\sqrt[m]{\lambda}})E_{\rho}(q_{\sqrt[m]{\lambda}})^{-\frac{1}{2}}\right\|_{\mathcal{M}}^{1/2}\|f\|_{\mathcal{M}}$
$\displaystyle\leq C(1+|t|)^{n/2}\|f\|_{\mathcal{M}}.$

To estimate term II, we let $\phi$ be a non-negative $C_{c}^{\infty}$ function
on $\mathbb{R}$ such that supp$\phi\subset(1/4,1)$ and

$\sum_{\ell\in\mathbb{Z}}\phi(2^{-\ell}u)=1,\ \ {\rm for}\ {\rm any}\ u>0.$

For simplicity we set $\phi_{\ell}(u)=\phi(2^{-\ell}u)$. Pick
$\nu_{0}\in\mathbb{Z}_{+}$ such that

$\left\\{\begin{array}[]{ll}4\leq
2^{\nu_{0}-\ell(m-1)/m}\sqrt[m]{\lambda}<8,&{\rm if}\
2^{-\ell(m-1)/m}\sqrt[m]{\lambda}<1,\\\ \nu_{0}=2,&{\rm if}\
2^{-\ell(m-1)/m}\sqrt[m]{\lambda}\geq 1.\end{array}\right.$

Observe that if $2^{\ell(m-1)/m}<\sqrt[m]{\lambda}$, then one has

$q_{2^{\nu_{0}}(1+|t|)\sqrt[m]{\lambda}}=q_{4(1+|t|)\sqrt[m]{\lambda}}q_{2^{\nu_{0}}(1+|t|)\sqrt[m]{\lambda}}.$

This, together with a similar procedure as estimating (2.5) and (2.11) to deal
with the convergence of the infinity sum, implies that

(3.3) $\displaystyle{\rm II}$
$\displaystyle\leq\sum_{\ell\in\mathbb{Z}}\left\|\left(\pi((I-e^{-\lambda
L})^{M}F(L)\phi_{\ell}(L))(\rho_{2}(f)(1-q_{4(1+|t|)\sqrt[m]{\lambda}}))\right)q_{\sqrt[m]{\lambda}}E_{\rho}(q_{\sqrt[m]{\lambda}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
$\displaystyle\leq\sum_{\ell:2^{\ell(m-1)/m}>\sqrt[m]{\lambda}}\bigg{\|}\left(\pi((I-e^{-\lambda
L})^{M}F(L)\phi_{\ell}(L))(\rho_{2}(f)((1-q_{4(1+|t|)\sqrt[m]{\lambda}})q_{2^{\nu_{0}}(1+|t|)\sqrt[m]{\lambda}}))\right)\times$
$\displaystyle\hskip 293.06346pt\times
q_{\sqrt[m]{\lambda}}E_{\rho}(q_{\sqrt[m]{\lambda}})^{-\frac{1}{2}}\bigg{\|}_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
$\displaystyle+\sum_{\ell\in\mathbb{Z}}\sum_{\nu\geq\nu_{0}}\left\|\left(\pi((I-e^{-\lambda
L})^{M}F(L)\phi_{\ell}(L))(\rho_{2}(f)b_{\nu,(1+|t|)\sqrt[m]{\lambda}})\right)q_{\sqrt[m]{\lambda}}E_{\rho}(q_{\sqrt[m]{\lambda}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
$\displaystyle=:{\rm(a)}+{\rm(b)}.$

For term (a), we observe that the function $F_{\ell}(u):=(1-e^{-\lambda
u})^{M}F(u)\phi_{\ell}(u)$ satisfies

$\|F_{\ell}\|_{\infty}\leq C\min\\{1,(2^{\ell}\lambda)^{M}\\}\min\\{1,2^{-\ell
s}\\}\leq C\min\\{1,(2^{\ell}\lambda)^{M}\\}2^{-\ell n/2}.$

This, in combination with inequality (2.6) and the conditions (ALii) and (Oi),
yields

$\displaystyle{\rm(a)}$
$\displaystyle\leq\sum_{\ell:2^{\ell(m-1)/m}>\sqrt[m]{\lambda}}\left\|\left(\pi((I-e^{-\lambda
L})^{M}F(L)\phi_{\ell}(L))(\rho_{2}(f)((1-q_{4(1+|t|)\sqrt[m]{\lambda}})q_{2^{\nu_{0}}(1+|t|)\sqrt[m]{\lambda}}))\right)E_{\rho}(q_{\sqrt[m]{\lambda}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
$\displaystyle=\sum_{\ell:2^{\ell(m-1)/m}>\sqrt[m]{\lambda}}\left\|\pi((I-e^{-\lambda
L})^{M}F(L)\phi_{\ell}(L))(\rho_{2}(f)((1-q_{4(1+|t|)\sqrt[m]{\lambda}})q_{2^{\nu_{0}}(1+|t|)\sqrt[m]{\lambda}})E_{\rho}(q_{\sqrt[m]{\lambda}})^{-\frac{1}{2}})\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
$\displaystyle\leq
C\sum_{\ell:2^{\ell(m-1)/m}>\sqrt[m]{\lambda}}\min\\{1,(2^{\ell}\lambda)^{M}\\}2^{-\ell
n/2}\left\|\rho_{2}(f)((1-q_{4(1+|t|)\sqrt[m]{\lambda}})q_{2^{\nu_{0}}(1+|t|)\sqrt[m]{\lambda}})E_{\rho}(q_{\sqrt[m]{\lambda}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
(3.4) $\displaystyle\leq
C\sum_{\ell:2^{\ell(m-1)/m}>\sqrt[m]{\lambda}}\min\\{1,(2^{\ell}\lambda)^{M}\\}2^{-\ell
n/2}\left\|E_{\rho}(q_{\sqrt[m]{\lambda}})^{-\frac{1}{2}}E_{\rho}((1-q_{4(1+|t|)\sqrt[m]{\lambda}})q_{2^{\nu_{0}}(1+|t|)\sqrt[m]{\lambda}})E_{\rho}(q_{\sqrt[m]{\lambda}})^{-\frac{1}{2}}\right\|_{\mathcal{M}}^{1/2}\|f\|_{\mathcal{M}}.$

To continue, by doubling inequality (2.8) and the definition of $\nu_{0}$, we
see that if $2^{\ell(m-1)/m}>\sqrt[m]{\lambda}$, then

$\displaystyle\left\|E_{\rho}(q_{\sqrt[m]{\lambda}})^{-\frac{1}{2}}E_{\rho}((1-q_{4(1+|t|)\sqrt[m]{\lambda}})q_{2^{\nu_{0}}(1+|t|)\sqrt[m]{\lambda}})E_{\rho}(q_{\sqrt[m]{\lambda}})^{-\frac{1}{2}}\right\|_{\mathcal{M}}^{1/2}$
$\displaystyle\leq
C\left\|E_{\rho}(q_{\sqrt[m]{\lambda}})^{-\frac{1}{2}}E_{\rho}(q_{2^{\nu_{0}}(1+|t|)\sqrt[m]{\lambda}})E_{\rho}(q_{\sqrt[m]{\lambda}})^{-\frac{1}{2}}\right\|_{\mathcal{M}}^{1/2}$
$\displaystyle\leq C(1+|t|)^{n/2}2^{\frac{\nu_{0}n}{2}}$ $\displaystyle\leq
C(1+|t|)^{n/2}(2^{\ell(m-1)/m}(\sqrt[m]{\lambda})^{-1})^{n/2}.$

Substituting the above inequality into (3.1), we conclude that

$\displaystyle{\rm(a)}$ $\displaystyle\leq
C(1+|t|)^{n/2}\sum_{\ell\in\mathbb{Z}}\min\\{1,(2^{\ell}\lambda)^{M}\\}(2^{\ell}\lambda)^{-\frac{n}{2m}}\|f\|_{\mathcal{M}}.$
$\displaystyle\leq C(1+|t|)^{n/2}\|f\|_{\mathcal{M}}.$

For term (b) we applying Lemma 2.10 to conclude that

(3.5) $\displaystyle{\rm(b)}$
$\displaystyle\leq\sum_{\ell\in\mathbb{Z}}\sum_{\nu\geq\nu_{0}}(2^{\ell/m}2^{\nu}(1+|t|)\sqrt[m]{\lambda})^{-N}\|\delta_{2^{\ell}}(F_{\ell})\|_{B^{N}}\left\|\rho_{2}(f)b_{\nu,(1+|t|)\sqrt[m]{\lambda}}E_{\rho}(q_{\sqrt[m]{\lambda}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}.$

To continue, we claim that

(3.6) $\displaystyle\|\delta_{2^{\ell}}(F_{\ell})\|_{B^{N}}\leq
C\max\\{1,2^{(N-n/2)\ell}\\}(1+|y|)^{s+1}(1+|t|)^{N}\min\\{1,(2^{\ell}\lambda)^{M}\\}.$

To show this, we let $\psi\in C_{c}^{\infty}(\mathbb{R})$ supported in
$\psi\in[1/8,2]$ and $\psi(u)=1$ for $u\in[1/4,1]$. We have that for any
$\ell\in\mathbb{Z}$,

$\displaystyle\|\delta_{2^{\ell}}(F_{\ell})\|_{B^{N}}$
$\displaystyle=\|\phi(\cdot)(1-e^{2^{\ell}\lambda\cdot})^{M}F(2^{\ell}\cdot)\|_{B^{N}}$
$\displaystyle\leq\|\psi(\cdot)(1-e^{2^{\ell}\lambda\cdot})^{M}(1+2^{\ell}\cdot)^{-isy}\|_{B^{N}}\|\phi(\cdot)G(2^{\ell}\cdot)\|_{B^{N}}$
$\displaystyle\leq\|\psi(\cdot)(1-e^{2^{\ell}\lambda\cdot})^{M}(1+2^{\ell}\cdot)^{-isy}\|_{C^{N+2}}\|\phi(\cdot)G(2^{\ell}\cdot)\|_{B^{N}}$
$\displaystyle\leq
C(1+|y|)^{N+2}\min\\{1,(2^{\ell}\lambda)^{M}\\}\|\phi(\cdot)G(2^{\ell}\cdot)\|_{B^{N}},$

where we denote $G(u):=(1+u)^{-s}e^{itu}$.

Let us estimate $\|\phi(\cdot)G(2^{\ell}\cdot)\|_{B^{N}}$. It follows from the
Fourier transform $\mathcal{F}(\phi(\cdot)G(2^{\ell}\cdot))$ of
$\phi(\cdot)G(2^{\ell}\cdot)$ that

$\displaystyle\mathcal{F}(\phi(\cdot)G(2^{\ell}\cdot))(\tau)=\int_{\mathbb{R}}\phi(u)\frac{e^{i(2^{\ell}t-\tau)u}}{(1+2^{\ell}u)^{s}}du.$

Using integration by parts, we get that for any $a\in\mathbb{N}$,

$\displaystyle|\mathcal{F}(\phi(\cdot)G(2^{\ell}\cdot))(\tau)|$
$\displaystyle\leq C_{N}\min\\{1,2^{-\ell s}\\}(1+|2^{\ell}t-\tau|)^{-a}$
$\displaystyle\leq C_{N}\min\\{1,2^{-\ell n/2}\\}(1+|2^{\ell}t-\tau|)^{-a},$

which means that

$\displaystyle\|\phi(u)G(2^{\ell}u)\|_{B^{N}}$ $\displaystyle\leq
C\min\\{1,2^{-\ell
n/2}\\}\int_{\mathbb{R}}(1+|2^{\ell}t-\tau|)^{-s}(1+|2^{\ell}t-\tau|)^{N}d\tau$
$\displaystyle\leq C\max\\{1,2^{(N-n/2)\ell}\\}(1+|t|)^{N}.$

Combining these estimates together, we end the proof of (3.6).

Now, choosing $N=s+\frac{1}{2}$ such that $N>n/2$, we substitute (3.6) into
(3.5) and then apply the doubling inequality (2.8) to see that

$\displaystyle{\rm(b)}$ $\displaystyle\leq
C(1+|y|)^{s+3}\sum_{\ell\in\mathbb{Z}}\sum_{\nu\geq\nu_{0}}(2^{\ell/m}2^{\nu}\sqrt[m]{\lambda})^{-N}\max\\{1,2^{(N-s)\ell}\\}\min\\{1,(2^{\ell}\lambda)^{M}\\}$
$\displaystyle\hskip
170.71652pt\times\left\|\rho_{2}(f)b_{\nu,(1+|t|)\sqrt[m]{\lambda}}E_{\rho}(q_{\sqrt[m]{\lambda}})^{-\frac{1}{2}}\right\|_{L_{\infty}^{c}(\mathcal{N}_{\rho};E_{\rho})}$
$\displaystyle\leq
C(1+|y|)^{s+3}\sum_{\ell\in\mathbb{Z}}\sum_{\nu\geq\nu_{0}}(2^{\ell/m}2^{\nu}\sqrt[m]{\lambda})^{-N}\max\\{1,2^{(N-s)\ell}\\}\min\\{1,(2^{\ell}\lambda)^{M}\\}$
$\displaystyle\hskip
170.71652pt\times\left\|E_{\rho}(q_{\sqrt[m]{\lambda}})^{-\frac{1}{2}}E_{\rho}(q_{2^{\nu}(1+|t|)\sqrt[m]{\lambda}})E_{\rho}(q_{\sqrt[m]{\lambda}})^{-\frac{1}{2}}\right\|_{\mathcal{M}}^{1/2}\|f\|_{\mathcal{M}}$
(3.7) $\displaystyle\leq
C(1+|y|)^{s+3}(1+|t|)^{n/2}\sum_{\ell\in\mathbb{Z}}\sum_{\nu\geq\nu_{0}}2^{-\nu(N-n/2)}(2^{\ell/m}\sqrt[m]{\lambda})^{-N}\max\\{1,2^{(N-s)\ell}\\}\min\\{1,(2^{\ell}\lambda)^{M}\\}\|f\|_{\mathcal{M}}.$

From the definition of $\nu_{0}\geq 0$, we deduce that

$\displaystyle{\rm(b)}$ $\displaystyle\leq
C(1+|y|)^{s+3}(1+|t|)^{n/2}\sum_{\ell>0}(2^{-\ell(m-1)/m}\sqrt[m]{\lambda})^{N-n/2}(2^{\ell/m}\sqrt[m]{\lambda})^{-N}2^{(N-s)\ell}\min\\{1,(2^{\ell}\lambda)^{M}\\}\|f\|_{\mathcal{M}}$
$\displaystyle+C(1+|y|)^{s+2}(1+|t|)^{n/2}\sum_{\ell\leq
0}(2^{\ell}\lambda)^{-\frac{N}{m}}\min\\{1,(2^{\ell}\lambda)^{M}\\}\|f\|_{\mathcal{M}}$
$\displaystyle\leq
C(1+|y|)^{s+3}(1+|t|)^{n/2}\sum_{\ell>0}(2^{\ell}\lambda)^{-\frac{n}{2m}}\min\\{1,(2^{\ell}\lambda)^{M}\\}\|f\|_{\mathcal{M}}$
$\displaystyle+C(1+|y|)^{s+3}(1+|t|)^{n/2}\sum_{\ell\leq
0}(2^{\ell}\lambda)^{-\frac{N}{m}}\min\\{1,(2^{\ell}\lambda)^{M}\\}\|f\|_{\mathcal{M}}$
(3.8) $\displaystyle\leq C(1+|y|)^{s+3}(1+|t|)^{n/2}\|f\|_{\mathcal{M}}.$

Combining the estimates for (a) and (b), we deduce that

${\rm II}\leq C(1+|y|)^{s+3}(1+|t|)^{n/2}\|f\|_{\mathcal{M}}.$

Therefore, the proof of the first statement of Theorem 1.1 is completed.

###  3.2. $L_{p}$ boundedness

Before providing the proof for the remaining part of our main theorem, we
first recall the complex interpolation from [2, Section 4]. An interpolation
pair $(X_{0},X_{1})$ are two Banach spaces that can be continuously embedded
into a topological vector space $X$. Consider the sum space $X_{0}+X_{1}$ and
intersection space $X_{0}\cap X_{1}$, which are Banach spaces equipped with
the norm

$\displaystyle\|x\|_{X_{0}+X_{1}}:=\inf\\{\|x_{0}\|_{X_{0}}+\|x_{1}\|_{X_{1}}:x=x_{0}+x_{1},x_{0}\in
X_{1},x_{1}\in X_{1}\\}$ $\displaystyle\|x\|_{X_{0}\cap
X_{1}}:=\max\\{\|x\|_{X_{0}},\|x\|_{X_{1}}\\}.$

Let $S=\\{z\in\mathbb{C}:0<\operatorname{Re}z<1\\}$ denote the open strip in
the complex plane and $\bar{S}$ be its closure
$\bar{S}=\\{z\in\mathbb{C}:0\leq\operatorname{Re}z\leq 1\\}.$ Let
$\mathcal{F}(X_{0},X_{1})$ denote the family of all functions $f$ with values
in $X_{0}+X_{1}$ and satisfying:

  1. (1)

$f$ is bounded and continuous on $\bar{S}$ and analytic on $S$;

  2. (2)

$f(it)\in X_{0}$ and $f(1+it)\in X_{1}$ for all $t\in\mathbb{R}$;

  3. (3)

$\|f(it)\|_{X_{0}},\|f(1+it)\|_{X_{1}}$ tend to $0$ as $|t|\rightarrow\infty$.

Then $\mathcal{F}(X_{0},X_{1})$ is a Banach space equipped with the norm given
by

$\|f\|_{\mathcal{F}(X_{0},X_{1})}:=\max\Big{\\{}\sup_{t\in\mathbb{R}}\|f(it)\|_{X_{0}},\sup_{t\in\mathbb{R}}\|f(1+it)\|_{X_{1}}\Big{\\}}.$

For $0\leq\theta\leq 1$, we define a subspace $[X_{0},X_{1}]_{\theta}$ of
$X_{0}+X_{1}$, which consists of all $x\in X_{0}+X_{1}$ such that
$f(\theta)=x$ for some $f\in\mathcal{F}(X_{0},X_{1})$, and
$[X_{0},X_{1}]_{\theta}$ is a Banach space with the norm

$\|x\|_{[X_{0},X_{1}]_{\theta}}=\inf\\{\|f\|_{\mathcal{F}}:f(\theta)=x,f\in\mathcal{F}(X_{0},X_{1})\\}.$

Let $\mathcal{D}(X_{0},X_{1})$ be a subspace of $\mathcal{F}(X_{0},X_{1})$,
which consists of the functions of the form
$f(z)=\sum_{\ell=1}^{m}g_{\ell}(z)x_{\ell}$ with each $x_{\ell}\in X_{0}\cap
X_{1}$ and with each $g_{\ell}(z)$ being a scalar function such that it is
continuous on $\bar{S}$, analytic on $S$ and vanishes at infinity. We will use
the following lemma which was established in [79]:

######  Lemma 3.1.

For any $0<\theta<1$, $X_{0}\cap X_{1}$ is dense in $[X_{0},X_{1}]_{\theta}$.
Besides, for any $x\in X_{0}\cap X_{1}$,

$\|x\|_{[X_{0},X_{1}]_{\theta}}=\inf\\{\|f\|_{\mathcal{F}(X_{0},X_{1})}:f(\theta)=x,f\in\mathcal{D}(X_{0},X_{1})\\}.$

Throughout the sequel in this subsection, we assume that the semigroup
$(S_{t})_{t\geq 0}$ is a Markov semigroup admitting a Markov dilation (see
[54, 57] for the specific definition). We shall see that these assumptions are
only used to guarantee the $L_{p}$–interpolation theory available (see Lemma
3.2), but it is still a challenging problem to minimize the assumptions on
$(S_{t})_{t\geq 0}$ such that the $L_{p}$–interpolation theory works.

Note that for $1<p\leq\infty$,

$\ker(A_{p}):=\\{f\in dom_{p}(A),Af=0\\}=\\{f\in
L_{p}(\mathcal{M}),S_{t}f=f,\forall t>0\\}.$

Besides, under the Markovian assumption of $(S_{t})_{t\geq 0}$, it is easy to
show that $(S_{t})_{t\geq 0}$ is trace preserving and that the null space of
${BMO}_{\mathcal{S}}^{\dagger}(\mathcal{M})$ is the fixed point space $\ker
A_{\infty}$. Now let $L_{p}^{0}(\mathcal{M})$ be the complemented subspace of
$L_{p}(\mathcal{M})$ given by

$L_{p}^{0}(\mathcal{M}):=L_{p}(\mathcal{M})/\ker(A_{p})=\\{f\in
L_{p}(\mathcal{M}),\lim_{t\rightarrow\infty}S_{t}f=0\\}.$

Here the limit is taken with respect to the $L_{p}(\mathcal{M})$–norm for
$1<p<\infty$ and is taken with respect to the weak-$*$ topology for
$p=\infty$. Recall from [54, 59] that there is a canonical decomposition on
$L_{p}(\mathcal{M})$ for $1<p<\infty$:

(3.9) $L_{p}(\mathcal{M})=\ker(A_{p})\oplus L_{p}^{0}(\mathcal{M}).$

For any $1<p<\infty$, let $P_{p}$ be the contractive positive projection from
$L_{p}(\mathcal{M})$ onto $\ker(A_{p})$. By mean ergodic theorem, for any
$x\in L_{p}(\mathcal{M})$, $P_{p}(x)$ can be written as the $L_{p}$–limit of
ergodic mean given by

$M_{s}(x):=\frac{1}{s}\int_{0}^{s}S_{t}(x)dt.$

Note that $P_{p}$ and $P_{q}$ coincide on $\ker(A_{p})\cap\ker(A_{q})$ for two
disparate $p$, $q$, which allows us to denote the $P_{p}$’s by $P$.

The following interpolation result was established in [54].

######  Lemma 3.2.

Assume that $(S_{t})_{t\geq 0}$ is a Markov semigroup admitting a Markov
dilation, for any $2<p<\infty$, we have

$[L_{2}^{0}(\mathcal{M}),{BMO}_{\mathcal{S}}(\mathcal{M})]_{1-2/p}=L_{p}^{0}(\mathcal{M}).$

Now we provide a proof for the second statement of our main theorem.

###### The proof of $L_{p}$ boundedness:.

By duality and density argument, it suffices to show that when $2<p<\infty$,
we have

$\displaystyle\left\|e^{itL}(I+L)^{-s(1-2/p)}f\right\|_{L_{p}(\mathcal{M})}\leq
C(1+|t|)^{\sigma_{p}}\|f\|_{L_{p}(\mathcal{M})},\ {\rm for}\ s\geq\frac{n}{2}$

and all $f\in\mathcal{A}_{\mathcal{M}}$. Note that if $x\in\ker{A_{p}}$, we
have $e^{itL}(I+L)^{-s(1-2/p)}x=x$. Let $P$ be the projection from
$L_{p}(\mathcal{M})$ to $\ker{A_{p}}$, then for any
$f\in\mathcal{A}_{\mathcal{M}}$, we have

$P(e^{itL}(I+L)^{-s(1-2/p)}f)=e^{itL}(I+L)^{-s(1-2/p)}P(f)=P(f).$

Hence,

$\displaystyle\left\|e^{itL}(I+L)^{-s(1-2/p)}f\right\|_{L_{p}(\mathcal{M})}$
$\displaystyle\leq\left\|P(e^{itL}(I+L)^{-s(1-2/p)}f)\right\|_{L_{p}(\mathcal{M})}+\left\|(I-P)e^{itL}(I+L)^{-s(1-2/p)}(f)\right\|_{L_{p}(\mathcal{M})}$
$\displaystyle=\|P(f)\|_{L_{p}(\mathcal{M})}+\left\|(I-P)e^{itL}(I+L)^{-s(1-2/p)}(f)\right\|_{L_{p}^{0}(\mathcal{M})}$
$\displaystyle\leq\|f\|_{L_{p}(\mathcal{M})}+\left\|(I-P)e^{itL}(I+L)^{-s(1-2/p)}(f)\right\|_{L_{p}^{0}(\mathcal{M})}.$

Then it suffices to show that

(3.10)
$\left\|(I-P)e^{itL}(I+L)^{-s(1-2/p)}(f)\right\|_{L_{p}^{0}(\mathcal{M})}\leq
C(1+|t|)^{\sigma_{p}}\|f\|_{L_{p}(\mathcal{M})},\ {\rm for}\
s\geq\frac{n}{2}.$

We will use a complex interpolation argument to show the estimate (3.10). To
this end, suppose that $f\in\mathcal{A}_{\mathcal{M}}$ and
$\|f\|_{L_{p}(\mathcal{M})}=1$, then by Lemma 3.1, for any $\epsilon>0$, there
exists an
$F\in\mathcal{D}(L_{2}(\mathcal{M}),\overline{\mathcal{A}_{\mathcal{M}}})$,
such that $F(1-2/p)=f$ and

$\|F\|_{\mathcal{F}(L_{2}(\mathcal{M}),\overline{\mathcal{A}_{\mathcal{M}}})}\leq\|f\|_{[L_{2}(\mathcal{M}),\overline{\mathcal{A}_{\mathcal{M}}}]_{1-2/p}}+\epsilon\leq\|f\|_{L_{p}(\mathcal{M})}+\epsilon=1+\epsilon,$

where we denote the closure of $\mathcal{A}_{\mathcal{M}}$ under $L_{\infty}$
norm by $\overline{\mathcal{A}_{\mathcal{M}}}$, and where in the second
inequality we used the embedding

$L_{p}(\mathcal{M})\cap\mathcal{A}_{\mathcal{M}}\subset[L_{2}(\mathcal{M}),\overline{\mathcal{A}_{\mathcal{M}}}]_{1-2/p}.$

To continue, we define

$T(z):=e^{z^{2}}(1+|t|)^{-nz/2}(I-P)e^{itL}(I+L)^{-sz}(F(z)).$

Recall that for any $x\in L_{\infty}(\mathcal{M})$ with $Px\in\ker
A_{\infty}$, one has $\|Px\|_{{BMO}_{\mathcal{S}}(\mathcal{M})}=0$. Therefore,
by inequality (3.1), when $z=1+iy,y\in\mathbb{R}$, we have

$\displaystyle\left\|T(1+iy)\right\|_{{BMO}_{\mathcal{S}}(\mathcal{M})}$
$\displaystyle\leq
e^{1-y^{2}}(1+|t|)^{-n/2}\left\|e^{itL}(I+L)^{-s(1+iy)}(F(1+iy))\right\|_{{BMO}_{\mathcal{S}}(\mathcal{M})}$
$\displaystyle\leq C\|F(1+iy)\|_{L_{\infty}(\mathcal{M})}.$

Besides, when $z=iy,y\in\mathbb{R}$, we have

$\displaystyle\|T(iy)\|_{L_{2}^{0}(\mathcal{M})}$
$\displaystyle=\left\|e^{-y^{2}}(1+|t|)^{-iny/2}(I-P)e^{itL}(I+L)^{-iys}(F(iy))\right\|_{L_{2}^{0}(\mathcal{M})}$
$\displaystyle\leq
C\left\|(I-P)e^{itL}(I+L)^{-iys}(F(iy))\right\|_{L_{2}^{0}(\mathcal{M})}$
$\displaystyle\leq
C\left\|e^{itL}(I+L)^{-iys}(F(iy))\right\|_{L_{2}(\mathcal{M})}$
$\displaystyle\leq C\|F(iy)\|_{L_{2}(\mathcal{M})}.$

Hence,
$T\in\mathcal{F}(L_{2}^{0}(\mathcal{M}),{{BMO}_{\mathcal{S}}(\mathcal{M})})$
and

(3.11)
$\|T\|_{\mathcal{F}(L_{2}^{0}(\mathcal{M}),{{BMO}_{\mathcal{S}}(\mathcal{M})})}\leq
C\|F\|_{\mathcal{F}(L_{2}(\mathcal{M}),\overline{\mathcal{A}_{\mathcal{M}}})}\leq
C(1+\varepsilon).$

Note that

$(I-P)e^{itL}(I+L)^{-s(1-2/p)}(F(1-2/p))=e^{-(1-2/p)^{2}}(1+|t|)^{\sigma_{p}}T(1-2/p).$

This, in combination with Lemma 3.2, implies

$\displaystyle\left\|(I-P)e^{itL}(I+L)^{-s(1-2/p)}(f)\right\|_{L_{p}^{0}(\mathcal{M})}$
$\displaystyle=\left\|(I-P)e^{itL}(I+L)^{-s(1-2/p)}(f)\right\|_{[L_{2}^{0}(\mathcal{M}),{BMO}_{\mathcal{S}}(\mathcal{M})]_{1-2/p}}$
$\displaystyle\leq
e^{-(1-2/p)^{2}}(1+|t|)^{\sigma_{p}}\|T\|_{\mathcal{F}(L_{2}^{0}(\mathcal{M}),{BMO}_{\mathcal{S}}(\mathcal{M}))}$
$\displaystyle\leq C(1+|t|)^{\sigma_{p}}(1+\epsilon).$

Letting $\epsilon\rightarrow 0$, we get (3.10). Therefore, the proof of the
second statement of our main theorem is completed. ∎

##  4\. Applications

###  4.1. Operator-valued setting

In this subsection we apply our algebraic approach to investigate Schrödinger
groups in the operator-valued setting, which opens a door to study Schrödinger
groups over quantum Euclidean space, matrix algebra and group von Neumann
algebra.

Let $(X,d,\mu)$ be a doubling metric space such that the doubling condition
(1.4) holds. It follows that there exist $C>0$ and $0\leq D\leq n$ such that

(4.1) $\displaystyle\mu(B(y,r))\leq
C\left(1+\frac{d(x,y)}{r}\right)^{D}\mu(B(x,r))$

independent of $x,y\in X$ and $r>0$. Indeed, equality (4.1) holds trivially in
Euclidean setting with $D=0$ and holds directly in the abstract setting with
$D=n$ due to the triangle inequality for the metric $d$ in combination with
the doubling condition (1.4).

Now suppose that $L$ is a non-negative self-adjoint operator on $L_{2}(X)$ and
its associated heat kernel satisfies ${\rm(GE_{m})}$. The estimate (${\rm
GE}_{m}$) holds for many (sub-)elliptic differential operators of order $m$
(see for example [12, 19, 25, 39, 70, 81] and the references therein).

Let $\mathcal{M}$ be a semifinite von Neumann algebra equipped with a normal
semifinite faithful trace $\tau$. Then $\hat{\mathcal{S}}=(e^{-tL}\otimes
id_{\mathcal{M}})$ is called a semicommutative heat semigroup. Denote by
$\hat{L}=L\otimes id_{\mathcal{M}}$ the generator of $\hat{S}$. Next, consider
the algebra of essentially bounded operator-valued functions
$f:X\rightarrow\mathcal{M}$ equipped with the trace

${\rm Trace}(f)=\int_{X}\tau(f(x))d\mu(x).$

Observe that its weak-$*$ closure
$\mathcal{N}=L_{\infty}(X)\bar{\otimes}\mathcal{M}$ is a von Neumann algebra.
Now we let $\mathcal{Q}=\\{(R_{j,t},\sigma_{j,t}):j\in\mathbb{N},t>0\\}$ be a
P-metric associated to the original bounded semigroup
$\mathcal{S}=\\{e^{-tL}\\}_{t\geq 0}$ on $L_{\infty}(X)$, which is constructed
explicitly in Section 2.1. We assume in this subsection that $p_{t}(x,y)\geq
0$ for any $x,y\in X$ to ensure the positivity assumption of the semigroup
$\mathcal{S}$, then $\mathcal{Q}$ satisfies an operator-valued generalization
of semigroup majorization and average domination condition. Therefore,
$\mathcal{Q}$ extends to a P-metric $\mathcal{Q}_{\mathcal{N}}$ in
$\mathcal{N}$ by tensorizing with $id_{\mathcal{M}}$ and ${\bf
1}_{\mathcal{M}}$, which can be expressed explicitly as

$\mathcal{Q}_{\mathcal{N}}=\\{(R_{j,t}\otimes
id_{\mathcal{M}},\sigma_{j,t}\otimes{\bf
1}_{\mathcal{M}}):j\in\mathbb{N},t>0\\}.$

Before providing our application, we first establish two auxiliary lemmas. For
simplicity, we set $f_{x}(y):=f(x,y)$ for any $x,y\in X$.

######  Lemma 4.1.

Let $T$ be a linear bounded operator on $L_{2}(X)$, then for any $f\in
L_{2}^{c}(X)\bar{\otimes}\mathcal{N}$,

(4.2) $\displaystyle\left\|\left(\int_{X}|(T\otimes
id_{\mathcal{M}})(f_{x})(y)|^{2}d\mu(y)\right)^{1/2}\right\|_{\mathcal{N}}\leq\|T\|_{L_{2}(X)\rightarrow
L_{2}(X)}\left\|\left(\int_{X}|f(x,y)|^{2}d\mu(y)\right)^{1/2}\right\|_{\mathcal{N}}.$

###### Proof.

The lemma may be well-known to experts, but we are unable to locate a precise
reference. For completeness, we provide a proof here.

To begin with, using the embedding $\mathcal{N}\subset
B(\ell_{2}(\mathcal{N}))$, we obtain

$\displaystyle\left\|\left(\int_{X}|(T\otimes
id_{\mathcal{M}})(f_{x})(y)|^{2}d\mu(y)\right)^{1/2}\right\|_{\mathcal{N}}$
$\displaystyle=\sup\limits_{\|h\|_{L_{2}(\mathcal{N})}\leq
1}\left(\int_{X}\int_{X}\tau\left(|((T\otimes
id_{\mathcal{M}})(f_{x})(y))h(x)|^{2}\right)d\mu(y)d\mu(x)\right)^{\frac{1}{2}}$
$\displaystyle=\sup\limits_{\|h\|_{L_{2}(\mathcal{N})}\leq
1}\left(\int_{X}\int_{X}\tau\left(|(T\otimes
id_{\mathcal{M}})(f_{x}h(x))(y))|^{2}\right)d\mu(y)d\mu(x)\right)^{\frac{1}{2}}$
(4.3) $\displaystyle\leq\|T\otimes
id_{\mathcal{M}}\|_{L_{2}(\mathcal{N})\rightarrow
L_{2}(\mathcal{N})}\sup\limits_{\|h\|_{L_{2}(\mathcal{N})}\leq
1}\left(\int_{X}\int_{X}\tau\left(|f(x,y)h(x)|^{2}\right)d\mu(y)d\mu(x)\right)^{\frac{1}{2}}.$

To continue, recall the fact that every bounded linear operator $T$ on a
scalar-valued $L^{2}$-space can be extended to a bounded operator on an
Hilbert space valued $L^{2}$-space, i.e.

(4.4) $\displaystyle\|T\otimes
id_{\mathcal{M}}\|_{L_{2}(\mathcal{N})\rightarrow
L_{2}(\mathcal{N})}\leq\|T\|_{L_{2}(X)\rightarrow L_{2}(X)}.$

Combining the inequalities (4.1) and (4.4), we finish the proof of Lemma 4.1.
∎

######  Lemma 4.2.

Let $T\in B(L_{2}(X))$ and $\\{T_{k}\\}_{k}\subset B(L_{2}(X))$ such that
$\sup\limits_{k}\|T_{k}\|_{B(L_{2}(X))}<+\infty$ and $T_{k}\rightarrow T$ in
the strong operator topology of $B(L_{2}(X))$. Then for any $f\in
L_{2}^{c}(X)\bar{\otimes}\mathcal{N}$ and $x\in X$,

  1. (1)

$\int_{X}|(T\otimes
id_{\mathcal{M}})(f_{x}\chi_{B(x,s)^{c}})(y)|^{2}d\mu(y)\rightarrow 0,\ {\rm
as}\ s\rightarrow\infty$,

  2. (2)

$\int_{X}|((T_{k}-T)\otimes id_{\mathcal{M}})(f_{x})(y)|^{2}d\mu(y)\rightarrow
0,\ {\rm as}\ k\rightarrow\infty,$

in the strong operator topology of $B(L_{2}(\mathcal{M}))$.

###### Proof.

To show (1), from the definition of strong operator topology of
$B(L_{2}(\mathcal{M}))$, it suffices to verify that for any $b\in
L_{2}(\mathcal{M})$,

$\displaystyle\lim_{s\rightarrow\infty}\tau\left(\int_{X}|(T\otimes
id_{\mathcal{M}})(f_{x}\chi_{B(x,s)^{c}}b)(y)|^{2}d\mu(y)\right)=0.$

However, this can be verified by domination convergence theorem together with
inequality (4.4).

To show (2), we first note that the conditions imply that $T_{k}\otimes
id_{\mathcal{M}}$ converges to $T\otimes id_{\mathcal{M}}$ in the strong
operator topology of $B(L_{2}(\mathcal{N}))$. Therefore, for any $b\in
L_{2}(\mathcal{M})$,

$\displaystyle\lim_{k\rightarrow\infty}\tau\left(\int_{X}|((T_{k}-T)\otimes
id_{\mathcal{M}})(f_{x}b)(y)|^{2}d\mu(y)\right)=0.$

This shows (2) and finishes the proof of Lemma 4.2. ∎

Then our main theorem applied to the operator-valued setting can be formulated
as follows.

######  Theorem 4.3.

Assume that $(X,d,\mu)$ is an $n$-dimensional doubling metric space. Suppose
that $L$ is a non-negative self-adjoint operator on $L_{2}(X)$ and its
associated heat kernel $p_{t}(x,y)$ satisfies the Gaussian upper bound (${\rm
GE}_{m}$) and $p_{t}(x,y)\geq 0$ for any $x,y\in X$. Then there exists a
constant $C=C(n,m)>0$ independent of $t$ such that for any $f\in
L_{\infty}(\mathcal{N})\cap\big{(}L_{2}^{c}(X)\bar{\otimes}L_{\infty}(\mathcal{M})\big{)},$

(4.5)
$\displaystyle\left\|(id_{\mathcal{N}}+\hat{L})^{-s}e^{it\hat{L}}f\right\|_{BMO_{\hat{\mathcal{S}}}(\mathcal{N})}\leq
C(1+|t|)^{n/2}\|f\|_{\mathcal{N}},\ {\rm for}\ s\geq\frac{n}{2}.$

Furthermore, if in additionally $\mathcal{S}=\\{e^{-tL}\\}$ admits a Markov
dilation, then for any $1<p<\infty$, there exists a constant $C=C(n,m,p)>0$
independent of $t$ such that

(4.6)
$\displaystyle\left\|(id_{\mathcal{N}}+\hat{L})^{-s}e^{it\hat{L}}f\right\|_{L_{p}(\mathcal{N})}\leq
C(1+|t|)^{\sigma_{p}}\|f\|_{L_{p}(\mathcal{N})},\ {\rm for}\
s\geq\sigma_{p}=n\Big{|}{1\over 2}-{1\over p}\Big{|}.$

###### Proof.

We shall see that all the conditions imposed on Theorem 1.1 hold for the
special case $X=\mathbb{R}^{n}$ and $L=-\Delta$. Then making some minor
modifications with higher-level viewpoints toward parts of these conditions,
we could pass the result for this special model to a more general one as we
formulate in Theorem 4.3.

To begin with, we note that the assumption $p_{t}(x,y)\geq 0$ for any $x,y\in
X$ implies that $e^{-t\hat{L}}$ is a positive map. Next, consider two maps
$\omega_{1},\omega_{2}$ defined by $\omega_{1}(f)(x,y)=f(x)$ and
$\omega_{2}(f)(x,y)=f(y)$, for any $x,y\in X$ and any Borel measurable
function $f$. Define $\rho_{j}=\omega_{j}\otimes id_{\mathcal{M}}$, $j=1,2$.
Then
$\rho_{1},\rho_{2}:\mathcal{N}\rightarrow\mathcal{N}_{\rho}=L_{\infty}(X\times
X)\bar{\otimes}\mathcal{M}$ are injective $*$-homomorphisms. Besides,
$\rho_{1},\rho_{2}$ are maps from
$L_{2}^{c}(X)\bar{\otimes}L_{\infty}(\mathcal{M})$ to
$L_{2}^{c}(X)\bar{\otimes}L_{\infty}(\mathcal{N})$.

Now for any bounded Borel measurable function $F$ on $\mathbb{R}$, define
amplified spectral multiplier $\pi(F(\hat{L}))$ by
$\pi(F(\hat{L})):=id_{L_{\infty}(X)}\otimes F(\hat{L})$, then it is direct to
see that
$\pi(F(\hat{L})):L_{2}^{c}(X)\bar{\otimes}L_{\infty}(\mathcal{N})\rightarrow
L_{2}^{c}(X)\bar{\otimes}L_{\infty}(\mathcal{N})$ satisfies
$\pi(F(\hat{L}))\circ\rho_{2}=\rho_{2}\circ F(\hat{L})$ on
$\mathcal{A}_{\mathcal{N}}=L_{\infty}(\mathcal{N})\cap\big{(}L_{2}^{c}(X)\bar{\otimes}L_{\infty}(\mathcal{M})\big{)}$.
Next, choose $E_{\rho}=id_{L_{\infty}(X)}\otimes\int_{X}\otimes
id_{\mathcal{M}}$ and $q_{r}$ be the projection $\chi_{B(x,r)}\otimes{\bf
1}_{\mathcal{M}}$. The priori assumption trivially holds for the most
classical model $L=-\Delta$ and $X=\mathbb{R}^{n}$ with
$\mathcal{A}_{\mathcal{N}}$ chosen to be a smaller subalgebra
$\mathcal{S}(\mathbb{R}^{n})\otimes\mathcal{S}_{\mathcal{M}}\subset
L_{\infty}(\mathcal{N})\cap\big{(}L_{2}^{c}(\mathbb{R}^{n})\bar{\otimes}L_{\infty}(\mathcal{M})\big{)}$.
Here $\mathcal{S}_{\mathcal{M}}$ is the linear span of all positive elements
with finite support in $\mathcal{M}$ (See [74]). For the general setting
formulated in our theorem, we observe that the priori assumption can be
relaxed to the following more flexible condition: for any bounded Borel
measurable function $F:[0,\infty)\rightarrow\mathbb{C}$, the spectral
multiplier $F(\hat{L})$ is bounded on
$L_{2}^{c}(X)\bar{\otimes}L_{\infty}(\mathcal{M})$. This condition is a direct
consequence of Lemma 4.1 with $f$ being chosen to be independent of
$x-$variable. Under this condition, $f\mapsto\rho_{2}(F(\hat{L})f)$ and
$f\mapsto R_{r}(F(\hat{L})f)$ are well-defined operators in
$L_{2}^{c}(X)\bar{\otimes}L_{\infty}(\mathcal{M})$ and therefore, the proof of
Theorem 1.1 follows exactly as it was shown there.

The algebraic conditions (ALi) and (ALii) are direct consequences of
commutativity. Now we verify the analytic condition (ANi), which can be stated
precisely in this setting as

(4.7)
$\displaystyle\left\|\frac{1}{\mu(B(x,r_{3}))}\int_{B(x,r_{1})}|f(y)|^{2}d\mu(y)\right\|_{\mathcal{N}}\leq
C\left(\frac{r_{1}}{r_{2}}\right)^{D}\left(\frac{r_{1}}{r_{3}}\right)^{n}\left\|\frac{1}{\mu(B(x,r_{2}))}\int_{B(x,r_{2})}|f(y)|^{2}d\mu(y)\right\|_{\mathcal{N}}$

for any $r_{1}\geq r_{2}\geq r_{3}$ and $f\in\mathcal{N}$. To verify this, we
note that $B(x,r_{1})$ can be covered by at most
$c\big{(}\frac{r_{1}}{r_{2}}\big{)}^{n}$ balls with radius $r_{2}$. We denote
these balls by $B(x_{1},r_{2})$, $\cdots$, $B(x_{d_{n}},r_{2})$, where
$d_{n}\lesssim\big{(}\frac{r_{1}}{r_{2}}\big{)}^{n}$. By inequality (4.1), for
any $1\leq\ell\leq d_{n}$,

$\displaystyle\mu(B(x_{\ell},r_{2}))\leq
C\mu(B(x,r_{2}))\left(1+\frac{d(x_{\ell},x)}{r_{2}}\right)^{D}\leq
C\mu(B(x,r_{2}))\left(\frac{r_{1}}{r_{2}}\right)^{D}.$

Combining these facts together yields inequality (4.7).

To verify the operator conditions, we first observe that the boundedness
condition (Oi) is equivalent to

(4.8)
$\displaystyle\left\|\left(\int_{X}|F(\hat{L})f_{x}(y)|^{2}d\mu(y)\right)^{1/2}\right\|_{\mathcal{N}}$
$\displaystyle\leq\|F\|_{\infty}\left\|\left(\int_{X}|f(x,y)|^{2}d\mu(y)\right)^{1/2}\right\|_{\mathcal{N}},$

but this is an easy consequence of spectral theorem (see for example [83])
together with inequality (4.2).

The boundedness condition (Oii) is equivalent to for any $t>0$, $k\geq 1$,
$0<r_{1}\leq r_{2}$ and $f\in
L_{\infty}(X)\bar{\otimes}L_{\infty}(\mathcal{N})$,

$\displaystyle\left\|\left(\fint_{B(x,r_{1})}|e^{-t\hat{L}}(\chi_{U_{k}(B(x,r_{2}))}f_{x})(y)|^{2}d\mu(y)\right)^{\frac{1}{2}}\right\|_{\mathcal{N}}$
(4.9) $\displaystyle\leq
CE_{k}\left(-c\left(\frac{r_{2}}{t^{1/m}}\right)^{\frac{m}{m-1}}\right)\left\|\left(\frac{1}{\mu(B(x,\max\\{r_{1},t^{1/m}\\}))}\int_{U_{k}(B(x,r_{2}))}|f(x,y)|^{2}d\mu(y)\right)^{\frac{1}{2}}\right\|_{\mathcal{N}}$

for some constants $C,c>0$ and $m\geq 2$, where we denote $U_{1}(B):=2B$ and
$U_{k}(B):=2^{k}B\backslash 2^{k-1}B$ for $k\geq 2$.

Case 1: if $r_{1}\geq t^{1/m}$, then the estimate (4.1) in the case of $k=1$
is a direct consequence of spectral theorem (see for example [83]). For the
case $k\geq 2$, we apply the assumption imposed on $L$ to get that

$\displaystyle\left(\fint_{B(x,r_{1})}|e^{-t\hat{L}}(\chi_{U_{k}(B(x,r_{2}))}f_{x})(y)|^{2}d\mu(y)\right)^{\frac{1}{2}}$
$\displaystyle=\left(\fint_{B(x,r_{1})}\left|\int_{U_{k}(B(x,r_{2}))}p_{t}(y,w)f(x,w)d\mu(w)\right|^{2}d\mu(y)\right)^{\frac{1}{2}}$
$\displaystyle\leq\left(\fint_{B(x,r_{1})}\int_{U_{k}(B(x,r_{2}))}|p_{t}(y,w)|^{2}d\mu(w)d\mu(y)\right)^{\frac{1}{2}}\left(\int_{U_{k}(B(x,r_{2}))}|f(x,w)|^{2}d\mu(w)\right)^{\frac{1}{2}}$
(4.10) $\displaystyle\leq
C\left(\fint_{B(x,r_{1})}\int_{U_{k}(B(x,r_{2}))}\left|\frac{1}{\mu(B(y,t^{1/m}))}\exp\left(-c\left(\frac{2^{k}r_{2}}{t^{1/m}}\right)^{\frac{m}{m-1}}\right)\right|^{2}d\mu(w)d\mu(y)\right)^{\frac{1}{2}}\left(\int_{U_{k}(B(x,r_{2}))}|f(x,w)|^{2}d\mu(w)\right)^{\frac{1}{2}}.$

To continue, by inequality (4.1),

(4.11)
$\displaystyle\frac{1}{\mu(B(y,t^{1/m}))}\leq\frac{C}{\mu(B(x,t^{1/m}))}\bigg{(}1+\frac{d(x,y)}{t^{1/m}}\bigg{)}^{D}\leq\frac{C}{\mu(B(x,t^{1/m}))}\bigg{(}1+\frac{2^{k}r_{2}}{t^{1/m}}\bigg{)}^{D}.$

Note that the non-doubling factor
$\big{(}1+\frac{2^{k}r_{2}}{t^{1/m}}\big{)}^{D}$ can be absorbed in the
exponential term of the right-hand side of (4.1). Therefore, one has

$\displaystyle\left(\fint_{B(x,r_{1})}|e^{-t\hat{L}}(\chi_{U_{k}(B(x,r_{2}))}f_{x})(y)|^{2}d\mu(y)\right)^{\frac{1}{2}}$
$\displaystyle\leq
C\frac{\mu(U_{k}(B(x,r_{2})))^{1/2}}{\mu(B(x,t^{1/m}))}\exp\left(-c\left(\frac{2^{k}r_{2}}{t^{1/m}}\right)^{\frac{m}{m-1}}\right)\left(\int_{U_{k}(B(x,r_{2}))}|f(x,w)|^{2}d\mu(w)\right)^{\frac{1}{2}}$
$\displaystyle\leq
C\frac{1}{\mu(B(x,t^{1/m}))^{1/2}}\bigg{(}1+\frac{2^{k}r_{2}}{t^{1/m}}\bigg{)}^{\frac{n}{2}}\exp\left(-c\left(\frac{2^{k}r_{2}}{t^{1/m}}\right)^{\frac{m}{m-1}}\right)\left(\int_{U_{k}(B(x,r_{2}))}|f(x,w)|^{2}d\mu(w)\right)^{\frac{1}{2}}$
(4.12) $\displaystyle\leq
C\frac{1}{\mu(B(x,t^{1/m}))^{1/2}}\exp\left(-c\left(\frac{2^{k}r_{2}}{t^{1/m}}\right)^{\frac{m}{m-1}}\right)\left(\int_{U_{k}(B(x,r_{2}))}|f(x,w)|^{2}d\mu(w)\right)^{\frac{1}{2}}.$

By the doubling condition (1.4),

$\displaystyle\frac{1}{\mu(B(x,t^{1/m}))^{1/2}}\leq\frac{C}{\mu(B(x,r_{1}))^{1/2}}\bigg{(}1+\frac{r_{1}}{t^{1/m}}\bigg{)}^{\frac{n}{2}}\leq\frac{C}{\mu(B(x,r_{1}))^{1/2}}\bigg{(}1+\frac{2^{k}r_{2}}{t^{1/m}}\bigg{)}^{\frac{n}{2}}.$

Noting that the extra term
$\big{(}1+\frac{2^{k}r_{2}}{t^{1/m}}\big{)}^{\frac{n}{2}}$ can be absorbed in
the exponential in the right-hand side of (4.1), we obtain (4.1) in this case.

Case 2: if $r_{1}\leq t^{1/m}$, then we note that inequality (4.1) still holds
when $k\geq 2$ with the same proof. Therefore, it remains to provide a proof
for the case $k=1$. To this end, we divide our proof into two subcases.

Subcase 1: if $2r_{2}\leq t^{1/m}$, then we apply the assumption imposed on
$L$ to get that

$\displaystyle\left(\fint_{B(x,r_{1})}|e^{-t\hat{L}}(\chi_{B(x,2r_{2})}f_{x})(y)|^{2}d\mu(y)\right)^{\frac{1}{2}}$
(4.13) $\displaystyle\leq
C\left(\fint_{B(x,r_{1})}\left(\int_{B(x,2r_{2})}\frac{1}{\mu(B(y,t^{1/m}))^{2}}d\mu(w)\right)d\mu(y)\right)^{\frac{1}{2}}\left(\int_{B(x,2r_{2})}|f(x,w)|^{2}d\mu(w)\right)^{\frac{1}{2}}.$

By inequality (4.1) and using the condition $r_{1}\leq t^{1/m}$, we see that

(4.14)
$\displaystyle\frac{1}{\mu(B(y,t^{1/m}))}\leq\frac{C}{\mu(B(x,t^{1/m}))}\bigg{(}1+\frac{d(x,y)}{t^{1/m}}\bigg{)}^{D}\leq\frac{C}{\mu(B(x,t^{1/m}))}\bigg{(}1+\frac{r_{1}}{t^{1/m}}\bigg{)}^{D}\leq\frac{C}{\mu(B(x,t^{1/m}))}.$

Then by Hölder’s inequality, the right-hand side of (4.1) can be estimated as
follows.

$\displaystyle{\rm RHS}\ {\rm of}\ \eqref{rhso22}$ $\displaystyle\leq
C\frac{\mu(B(x,2r_{2}))^{1/2}}{\mu(B(x,t^{1/m}))^{1/2}}\left(\frac{1}{\mu(B(x,t^{1/m}))}\int_{B(x,2r_{2})}|f(x,w)|^{2}d\mu(w)\right)^{\frac{1}{2}}$
$\displaystyle\leq
C\left(\frac{1}{\mu(B(x,t^{1/m}))}\int_{B(x,2r_{2})}|f(x,w)|^{2}d\mu(w)\right)^{\frac{1}{2}}.$

Subcase 2: if $2r_{2}\geq t^{1/m}$, then we choose $k_{0}\in\mathbb{Z}_{+}$
such that

$2^{k_{0}-1}t^{1/m}\leq 2r_{2}<2^{k_{0}}t^{1/m}$

and apply the assumption imposed on $L$ together with inequality (4.14) to get
that

$\displaystyle\left(\fint_{B(x,r_{1})}|e^{-t\hat{L}}(\chi_{B(x,2r_{2})}f_{x})(y)|^{2}d\mu(y)\right)^{\frac{1}{2}}$
$\displaystyle=C\left(\fint_{B(x,r_{1})}\left|\int_{B(x,2r_{2})}p_{t}(y,w)f(x,w)d\mu(w)\right|^{2}d\mu(y)\right)^{\frac{1}{2}}$
$\displaystyle\leq
C\left(\fint_{B(x,r_{1})}\int_{B(x,2r_{2})}|p_{t}(y,w)|^{2}d\mu(w)d\mu(y)\right)^{\frac{1}{2}}\left(\int_{B(x,2r_{2})}|f(x,w)|^{2}d\mu(w)\right)^{\frac{1}{2}}$
$\displaystyle\leq
C\left(\fint_{B(x,r_{1})}\int_{B(x,2r_{2})}\left|\frac{1}{\mu(B(y,t^{1/m}))}\exp\left(-c\left(\frac{d(y,w)}{t^{1/m}}\right)^{\frac{m}{m-1}}\right)\right|^{2}d\mu(w)d\mu(y)\right)^{\frac{1}{2}}\left(\int_{B(x,2r_{2})}|f(x,w)|^{2}d\mu(w)\right)^{\frac{1}{2}}$
$\displaystyle\leq
C\left(\sum_{j=1}^{k_{0}}\fint_{B(x,r_{1})}\int_{U_{j}(B(x,t^{1/m}))\cap
B(x,2r_{2})}\left|\frac{E_{j}(-c)}{\mu(B(x,t^{1/m}))}\right|^{2}d\mu(w)d\mu(y)\right)^{\frac{1}{2}}\left(\int_{B(x,2r_{2})}|f(x,w)|^{2}d\mu(w)\right)^{\frac{1}{2}}$
$\displaystyle\leq
C\left(\sum_{j=1}^{k_{0}}E_{j}(-c)\frac{\mu(B(x,2^{j}t^{1/m}))^{1/2}}{\mu(B(x,t^{1/m}))^{1/2}}\right)\left(\frac{1}{\mu(B(x,t^{1/m}))}\int_{B(x,2r_{2})}|f(x,w)|^{2}d\mu(w)\right)^{\frac{1}{2}}$
$\displaystyle\leq
C\left(\frac{1}{\mu(B(x,t^{1/m}))}\int_{B(x,2r_{2})}|f(x,w)|^{2}d\mu(w)\right)^{\frac{1}{2}}.$

Combining the estimates in two cases together, we deduce inequality (4.1).

Finally, we turn to the topological conditions (Ti) and (Tii). Note that in
the proof of Theorem 1.1, the assumption (Ti) is only used to ensure the
validity of triangle inequality: for any $F\in\mathcal{S}(\mathbb{R})$ and
$s>0$,

$\displaystyle\left\|\fint_{B(x,r)}|F(\hat{L})f_{x}(y)|^{2}d\mu(y)\right\|_{\mathcal{N}}^{\frac{1}{2}}\leq\sum_{k\geq
1}\left\|\fint_{B(x,r)}|F(\hat{L})(\chi_{U_{k}(B(x,s))}f_{x})(y)|^{2}d\mu(y)\right\|_{\mathcal{N}}^{\frac{1}{2}},$

while the assumption (Tii) is only used to ensure the validity of (2.5), (2.5)
and (3.3). However, these inequalities can be verified easily by using the
tensor separation nature of $\mathcal{N}$, identifying $\mathcal{M}$ as a von
Neumann subalgebra of $B(L_{2}(\mathcal{M}))$ and then applying Lemma 4.2.

This ends the proof of Theorem 4.3. ∎

######  Remark 4.4.

Together with the available $L_{p}$-interpolation theorem at hand, Theorem 4.3
can be regarded as an extension of two previous results. Firstly, note that
the positivity assumption on the heat kernel is only used to ensure that
$BMO_{\mathcal{S}}$ and $BMO_{\mathcal{Q}}$ are (semi-)normed spaces and
$\mathcal{Q}_{\mathcal{N}}$ is a P-metric in $\mathcal{N}$. However, in the
case of $\mathcal{M}=\mathbb{C}$, these facts hold trivially without this
assumption, so Theorem 4.3 goes back to the classical result due to [12],
which showed the sharp endpoint $L_{p}$ boundedness of Schrödinger groups
under (${\rm GE}_{m}$). Secondly, in the case of $X=\mathbb{R}^{n}$ and
$L=-\Delta$, the above theorem goes back to the $n$-dimensional analogous of
[63, Theorem 6.4].

######  Remark 4.5.

Inspired by the classical result in [12], one may expect that the regularity
assumption can be removed from the second statement of Theorem 4.3. To remove
this, it requires an interpolation theorem for the operator-valued BMO space
associated with non-negative self-adjoint operator satisfying (${\rm
GE}_{m}$), which will be addressed in the forthcoming paper [31]. At this
stage we just point out that there are still a large number of concrete
examples fulfilling all the assumptions imposed on Theorem 4.3. This can be
seen easily from the fact that on commutative or semicommutative von Neumann
algebras, a Markov semigroup is regular if it is diffusion (see [57, Remark
1.4]). In particular, Theorem 4.3 holds for the following generators $L$:

(1) Neumann-Laplacian operator $\Delta_{N}$ on $\mathbb{R}^{n}$ (see for
example [21]);

(2) Neumann-Laplacian operator $\Delta_{N_{\pm}}$ on $\mathbb{R}_{\pm}^{n}$
(see for example [21]);

(3) Dunkl-Laplacian operator $\Delta_{{\rm Dunkl}}$ on $\mathbb{R}^{N}$
equipped with Dunkl measure $d\omega$ (see for example [29]);

(4) Laplace-Beltrami operator $\Delta_{g}$ on $n$-dimensional complete
Riemannian manifold $(M,g)$ with non-negative Ricci curvature (see for example
[19, Section 5]).

###  4.2. Quantum Euclidean space

In this subsection we apply our algebraic approach to study Schrödinger groups
in quantum Euclidean space.

At the beginning of this subsection, we recall some concepts and definitions
in quantum Euclidean space. There are several alternative definitions of
quantum Euclidean space (see [35, 38, 57, 62]). We adopt the one from [38,
Section 1] (see also [57, Section 4.3]). To begin with, let $\Theta$ be an
anti-symmetric $\mathbb{R}$-valued $n\times n$ matrix with $n\geq 1$.
$A_{\Theta}$ is defined as the universal $C^{*}$-algebra generated by a family
$\\{u_{j}(s)\\}_{j=1}^{n}$ of strongly continuous one-parameter unitary groups
in $s\in\mathbb{R}$ which satisfies the following $\Theta$-commutation
relations

$u_{j}(s)u_{k}(t)=e^{2\pi i\Theta_{jk}st}u_{k}(t)u_{j}(s).$

In the case of $\Theta=0$, by Stone’s theorem, one can take
$u_{j}(s)=\exp(2\pi is\langle e_{j},\cdot\rangle)$ and $A_{\Theta}$ be the
space consisting of all bounded continuous functions on $\mathbb{R}^{n}$. In
general, for any $\xi=(\xi_{1},\cdots,\xi_{n})\in\mathbb{R}^{n}$, we define
the unitaries $\lambda_{\Theta}(\xi)=u_{1}(\xi_{1})u_{2}(\xi_{2})\cdots
u_{n}(\xi_{n})$. Next, $E_{\Theta}$ is defined as the closure in $A_{\Theta}$
of $\lambda_{\Theta}(L_{1}(\mathbb{R}^{n}))$ with

$f=\int_{\mathbb{R}^{n}}\hat{f}_{\Theta}(\xi)\lambda_{\Theta}(\xi)d\xi.$

Note that if $\Theta=0$, then $E_{\Theta}=\mathcal{C}_{0}(\mathbb{R}^{n})$.
Define

$\displaystyle\tau_{\Theta}(f)=\tau_{\Theta}\left(\int_{\mathbb{R}^{n}}\hat{f}_{\Theta}(\xi)\lambda_{\Theta}(\xi)d\xi\right)=\hat{f}_{\Theta}(0),$

where $\hat{f}_{\Theta}$ is an integrable and smooth function mapping
$\mathbb{R}^{n}$ to $\mathbb{C}$. $\tau_{\Theta}$ extends to a normal
semifinite faithful trace on $E_{\Theta}$. Let $\mathcal{R}_{\Theta}$ be the
von Neumann algebra generated by $E_{\Theta}$ (that is, the double commutant
of $E_{\Theta}$) in the GNS representation of $\tau_{\Theta}$.
$\mathcal{R}_{\Theta}$ is called a quantum Euclidean space associated with
$\Theta$. Note that if $\Theta=0$, then
$\mathcal{R}_{\Theta}=L_{\infty}(\mathbb{R}^{n})$. Consider a normal injective
$*$-homomorphism

$\sigma_{\Theta}:\mathcal{R}_{\Theta}\ni\lambda_{\Theta}(\xi)\mapsto\exp_{\xi}\otimes\lambda_{\Theta}(\xi)\in
L_{\infty}(\mathbb{R}^{n})\bar{\otimes}\mathcal{R}_{\Theta},$

where $\exp_{\xi}$ stands for the Fourier character $\exp(2\pi
i\langle\xi,\cdot\rangle)$. Now we set
$\mathcal{S}_{\Theta}=\big{\\{}f\in\mathcal{R}_{\Theta}:\check{f}_{\Theta}\in\mathcal{S}(\mathbb{R}^{n})\big{\\}}$
be the quantum Schwartz class, which is a weak-$*$ dense subalgebra of
$\mathcal{R}_{\Theta}$ and dense in $L_{p}(\mathcal{M})$ for all $p>0$ ([38,
Proposition 1.13]).

Our first goal is to modify the argument in [57] to construct a natural
P-metric for quantum Euclidean spaces. To begin with, we recall the heat
semigroup $\\{H_{t}\\}_{t\geq 0}=\\{e^{t\Delta}\\}_{t\geq 0}$ on
$\mathbb{R}^{n}$ acting on $f:\mathbb{R}^{n}\rightarrow\mathbb{C}$, which can
be expressed as follows

(4.15) $\displaystyle
H_{t}f(x)=\int_{\mathbb{R}^{n}}\hat{f}(\xi)e^{-t|\xi|^{2}}\exp_{\xi}(x)d\xi.$

This induces a regular semigroup $\mathcal{S}=\\{S_{\Theta,t}\\}_{t\geq 0}$ on
$\mathcal{R}_{\Theta}$ determined by

(4.16) $\displaystyle\sigma_{\Theta}\circ S_{\Theta,t}=(H_{t}\otimes
id_{\mathcal{R}_{\Theta}})\circ\sigma_{\Theta}.$

$S_{\Theta,t}$ gives a Markov semigroup on $\mathcal{R}_{\Theta}$ which
formally acts as

$\displaystyle
S_{\Theta,t}(f)=\int_{\mathbb{R}^{n}}\hat{f}_{\Theta}(\xi)e^{-t|\xi|^{2}}\lambda_{\Theta}(\xi)d\xi.$

Recall that $S_{\Theta,t}$ admits an infinitesimal negative generator

$\Delta_{\Theta}f=\lim_{t\rightarrow
0}\frac{S_{\Theta,t}(f)-f}{t}=-4\pi^{2}\int_{\mathbb{R}^{n}}\hat{f}_{\Theta}(\xi)|\xi|^{2}\lambda_{\Theta}(\xi)d\xi.$

Alternatively, one may define $\Delta_{\Theta}$ via second-order partial
$\Theta$-derivatives. To illustrate this, for $1\leq j\leq n$, define the
$j$-th first-order partial $\Theta$-derivative $\partial_{\Theta,j}$ as the
linear extension of the map

$\partial_{\Theta,j}(\lambda_{\Theta}(\xi))=2\pi
i\xi_{j}\lambda_{\Theta}(\xi)$

over the quantum Schwartz class $\mathcal{S}_{\Theta}$. Then it is easy to see
that $\Delta_{\Theta}=\sum_{j=1}^{n}\partial_{\Theta,j}^{2}$.

Recall from Section 4.1 that the semicommutative extension $H_{t}\otimes
id_{\mathcal{R}_{\Theta}}$ satisfies the following semigroup majorization

(4.17) $\displaystyle(H_{t}\otimes
id_{\mathcal{R}_{\Theta}})(|\xi|^{2})\leq\sum_{j\geq
0}2^{jn}\exp(-c2^{2j})(R_{j,\sqrt{t}}\otimes
id_{\mathcal{R}_{\Theta}})(|\xi|^{2}),\ {\rm for}\ {\rm any}\ \xi\in
L_{\infty}(\mathbb{R}^{n})\bar{\otimes}\mathcal{R}_{\Theta},$

and the metric integrability condition as well as the average domination
condition. Then one can produce a P-metric on $\mathcal{R}_{\Theta}$. To this
end, let $B(r):=B(0,r)$ and consider the projections
$q_{r}=\chi_{B(r)}\otimes{\bf 1}_{\mathcal{R}_{\Theta}}$. Define the
completely positive unital map

$\displaystyle
R_{\Theta,j,r}(f)=\frac{1}{|B(2^{j}r)|}\int_{B(2^{j}r)}\sigma_{\Theta}(f)(x)dx=\frac{1}{|B(2^{j}r)|}\int_{\mathbb{R}^{n}}\hat{\chi}_{B(2^{j}r)}(\xi)\hat{f}_{\Theta}(\xi)\lambda_{\Theta}(\xi)d\xi.$

Then $R_{\Theta,r}:=R_{\Theta,0,r}$ is of the form (2.5) with $\rho_{1}={\bf
1}\otimes\cdot$, $\rho_{2}=\sigma_{\Theta}$, $E_{\rho}=\int\otimes
id_{\mathcal{R}_{\Theta}}$ and with $q_{r}$ being the projection
$\chi_{B(r)}\otimes{\bf 1}_{\mathcal{R}_{\Theta}}$. One could easily check
that

(4.18) $\displaystyle\sigma_{\Theta}\circ R_{\Theta,j,r}=(R_{j,r}\otimes
id_{\mathcal{R}_{\Theta}})\circ\sigma_{\Theta}$

and $R_{\Theta,j,r}f=f$ for any $f$ in the null space of
$BMO_{\mathcal{S}}^{c}(\mathcal{R}_{\Theta})$-seminorm. Note that by
substituting $\xi$ with $\sigma_{\Theta}(f)$ in (4.17) for some
$f\in\mathcal{R}_{\Theta}$ and then using the intertwining identities (4.16)
and (4.18), we conclude that the semigroup majorization

(4.19) $\displaystyle S_{\Theta,t}(|\xi|^{2})\leq C\sum_{j\geq
0}2^{jn}\exp(-c2^{2j})R_{\Theta,j,\sqrt{t}}(|\xi|^{2})$

holds for all $\xi\in\mathcal{R}_{\Theta}$.

Now we claim that for any $x_{0}\in\mathbb{R}^{n}$ and $r>0$, one has

(4.20)
$\displaystyle\left\|\fint_{B(x_{0},r)}|\sigma_{\Theta}(f)(x)|^{2}dx\right\|_{\mathcal{R}_{\Theta}}\leq\left\|\fint_{B(0,r)}|\sigma_{\Theta}(f)(x)|^{2}dx\right\|_{\mathcal{R}_{\Theta}}.$

Indeed, since $\sigma_{\Theta}$ is an injective $*$-homomorphism:
$\mathcal{R}_{\Theta}\rightarrow