# Best constants in the vector-valued Littlewood-Paley-Stein theory

Guixiang Hong  Institute for Advanced Study in Mathematics, Harbin Institute
of Technology, Harbin 150001, China<EMAIL_ADDRESS>,  Zhendong Xu
Laboratoire de Mathématiques, Université de Bourgogne Franche-Comté, 25030
Besançon Cedex, France<EMAIL_ADDRESS>and  Hao Zhang  Department
of Mathematics, University of Illinois Urbana-Champaign, USA
<EMAIL_ADDRESS>
###### Abstract.

Let $L$ be a sectorial operator of type $\alpha$ ($0\leq\alpha<\pi/2$) on
$L^{2}({}^{d})$ with the kernels of $\\{e^{-tL}\\}_{t>0}$ satisfying certain
size and regularity conditions. Define

$S_{q,L}(f)(x)=\left(\int_{0}^{\infty}\int_{|y-x|<t}\left\|tL{e^{-tL}}(f)(y)\right\|_{X}^{q}\,\frac{{\rm
d}y{\rm d}t}{t^{d+1}}\right)^{\frac{1}{q}},$
$G_{q,{L}}(f)=\left(\int_{0}^{\infty}\left\|t{L}{e^{-t{L}}}(f)(y)\right\|_{X}^{q}\,\frac{{\rm
d}t}{t}\right)^{\frac{1}{q}}.$

We show that for $\underline{\mathrm{any}}$ Banach space $X$, $1\leq p<\infty$
and $1<q<\infty$ and $f\in C_{c}(\mathbb{R}^{d})\otimes X$, there hold

$\displaystyle
p^{-\frac{1}{q}}\|S_{q,{\sqrt{\Delta}}}(f)\|_{p}\lesssim_{d,\gamma,\beta}\left\|S_{q,L}(f)\right\|_{p}\lesssim_{d,\gamma,\beta}p^{\frac{1}{q}}\|S_{q,{\sqrt{\Delta}}}(f)\|_{p},$
$\displaystyle
p^{-\frac{1}{q}}\|S_{q,L}(f)\|_{p}\lesssim_{d,\gamma,\beta}\|G_{q,L}(f)\|_{p}\lesssim_{d,\gamma,\beta}p^{\frac{1}{q}}\|S_{q,L}(f)\|_{p},$

where $\Delta$ is the standard Laplacian; moreover all the orders appeared
above are optimal as $p\rightarrow 1$. This, combined with the existing
results in [29, 33], allows us to resolve partially Problem 1.8, Problem A.1
and Conjecture A.4 regarding the optimal Lusin type constant and the
characterization of martingale type in a recent remarkable work due to Xu [48]
.

Several difficulties originate from the arbitrariness of $X$, which excludes
the use of vector-valued Calderón-Zygmund theory. To surmount the obstacles,
we introduce the novel vector-valued Hardy and BMO spaces associated with
sectorial operators; in addition to Mei’s duality techniques and Wilson’s
intrinsic square functions developed in this setting, the key new input is the
vector-valued tent space theory and its unexpected amalgamation with these
‘old’ techniques.

2020 Mathematics Subject Classification: Primary: 42B35, 47D03. Secondary:
46F10, 46E30

Key words: Littlewood-Paley-Stein inequalities; Hardy and BMO spaces; vector-
valued tent spaces; holomorphic functional calculus; martingale type and
cotype; intrinsic square function

##  1\. Introduction

Motivated by Banach space geometry [34, 35] and Stein’s semigroup theory [37],
the investigation of the vector-valued Littlewood-Paley-Stein theory has
started with Xu’s Poisson semigroup on the unit circle [44], and then was
continued in [29, 45, 46] for symmetric Markovian semigroups. Afterwards,
Betancor et al developed this theory in some special cases which are not
Markovian (cf. [5, 6, 7, 3]), such as Schrödinger, Hermite, Laguerre
semigroups etc., see also [1, 2, 4, 24, 33, 39, 22] for related results. In a
recent remarkable paper [48], Xu investigated for the first time the vector-
valued Littlewood-Paley-Stein inequalities for semigroups of regular
contractions $\\{e^{-tL}\\}_{t>0}$ on $L^{p}(\Omega)$ for a fixed
$1<p<\infty$. That is, for a Banach space $X$ of martingale cotype $q$ ($2\leq
q<\infty$), he showed the Lusin cotype of $X$ relative to
$\\{e^{-t\sqrt{L}}\\}_{t>0}$, in other words, there exists a constant $C>0$
such that

(1.1) $\displaystyle\|G_{q,\sqrt{L}}(f)\|_{p}\leq
C\|f\|_{L^{p}(X)},\quad\forall f\in L^{p}(\Omega)\otimes X,$

where

$G_{q,\sqrt{L}}(f)(x)=\left(\int_{0}^{\infty}\left\|t\sqrt{L}{e^{-t\sqrt{L}}}(f)(x)\right\|_{X}^{q}\,\frac{{\rm
d}t}{t}\right)^{\frac{1}{q}}.$

More importantly, by deeply exploring holomorphic functional calculus,
Fendler’s dilation, Calderón-Zygmund theory and Wilson’s intrinsic square
functions, he was able to obtain the sharp bounds depending on the martingale
cotype constant, and the latter in turn enables him to resolve an open problem
posed by Naor and Young [32]. More precisely, let
$\mathsf{L}^{\sqrt{L}}_{c,q,p}(X)$ be the least constant $C$ in (1.1)—the
Lusin cotype constant of $X$, and $\mathsf{M}_{c,q}(X)$ the martingale cotype
$q$ constant of $X$, he obtained

(1.2)
$\displaystyle\mathsf{L}^{\sqrt{L}}_{c,q,p}(X)\lesssim\max\left\\{p^{\frac{1}{q}},p^{\prime}\right\\}\mathsf{M}_{c,q}(X)$

with the order $\max\left\\{p^{\frac{1}{q}},p^{\prime}\right\\}$ being sharp.
We refer the reader to Section 6 for the definition of $\mathsf{M}_{c,q}(X)$
and the martingale type constant $\mathsf{M}_{t,q}(X)$.

By duality, the converse inequality of (1.1) also holds under the condition
that $X$ is of martingale type $q$ ($1<q\leq 2$)

$\|f-\mathbf{F}(f)\|_{L^{p}(X)}\leq C\|G_{q,\sqrt{L}}(f)\|_{p},\quad\forall
f\in L^{p}(\Omega)\otimes X,$

where $\mathbf{F}$ is the obvious vector-valued extension of the projection
from $L^{p}(\Omega)$ onto the fixed point space of $\\{e^{-tL}\\}_{t>0}$, and
the resulting type bounds satisfy

(1.3)
$\displaystyle\mathsf{L}^{\sqrt{L}}_{t,q,p}(X)\lesssim\max\left\\{p,p^{\prime\frac{1}{q^{\prime}}}\right\\}\mathsf{M}_{t,q}(X).$

Nevertheless the order $\max\left\\{p,p^{\prime\frac{1}{q^{\prime}}}\right\\}$
is now very likely to be suboptimal suggested by the special case
$L=\Delta$—the Laplacian on $\mathbb{R}^{d}$, $q=2$ and $X=\mathbb{C}$, where

(1.4) $\displaystyle\mathsf{M}_{t,2}(\mathbb{C})=1,\ \mathrm{and}\
\sqrt{p}\lesssim\mathsf{L}^{\sqrt{\Delta}}_{t,2,p}(\mathbb{C})\lesssim p,$

see for instance [47, Theorem 1]. The sharpness of (1.4) when $p\rightarrow 1$
is essentially equivalent to the fact that $L^{1}(\mathbb{R}^{d})$-norm of the
classical $g$-function controls that of the Lusin square function, which
dominates in turn $L^{1}(\mathbb{R}^{d})$-norm of the function itself; this
involves the deep theory of Hardy/BMO spaces. Other than this special case,
the problem of determining the optimal order of
$\mathsf{L}^{\sqrt{L}}_{t,q,p}(X)$ when $p\rightarrow 1$ in (1.3) has been
left open widely even in the case $L=\Delta$, see e.g. Remark 1.3, Problem 1.8
and Problem 8.4 in the aforementioned paper [48]. For the other endpoint-side,
the optimal order of $\mathsf{L}^{\sqrt{L}}_{t,q,p}(X)$ as
$p\rightarrow\infty$ has been determined in [49] for all symmetric Markovian
semigroups. However it seems much harder to consider the corresponding problem
for a fixed semigroup, and actually the special case
$\mathsf{L}^{\sqrt{\Delta}}_{t,2,p}(\mathbb{C})$ remains open (cf. [47,
Problem 5]).

In the present paper, we will determine the optimal order of
$\mathsf{L}^{{L}}_{t,q,p}(X)$ as $p\rightarrow 1$ in (1.3) for a large class
of approximation identities $\\{e^{-tL}\\}_{t>0}$ on $\mathbb{R}^{d}$, and
thus answer the questions mentioned in [48, Remark 1.3 and Problem 1.8].
Moreover, our result will assert that the Lusin type of $X$ relative to this
class of approximation identities implies the martingale type of $X$, and thus
partially resolves [48, Problem A.1 and Conjecture A.4].

Let $L$ be a sectorial operator of type $\alpha$ ($0\leq\alpha<\pi/2$) on
$L^{2}({}^{d})$, and thus it generates a holomorphic semigroup $e^{-zL}$ with
$0\leq|{\rm Arg}(z)|<\pi/2-\alpha$. Partially inspired by [16, Section 6.2.2],
the kind of approximation identity $\\{e^{-tL}\\}_{t>0}$ that we will be
interested in in the present paper is assumed to have kernel $K(t,x,y)$
satisfying the following three assumptions: there exist positive constants
$0<\beta,\gamma\leq 1$ and $c$ such that for any $t>0$, $x,y,h\in{}^{d}$,

(1.5) $|K(t,x,y)|\leq\frac{ct^{\beta}}{(t+|x-y|)^{d+\beta}},$ (1.6)
$|K(t,x+h,y)-K(t,x,y)|+|K(t,x,y+h)-K(t,x,y)|\leq\frac{c|h|^{\gamma}t^{\beta}}{(t+|x-y|)^{d+\beta+\gamma}}$

whenever $2|h|\leq t+|x-y|$, and

(1.7) $\int_{{}^{d}}K(t,x,y)\,{\rm d}x=\int_{{}^{d}}K(t,x,y)\,{\rm d}y=1.$

One may find these concepts in Section 2. Then it is well-known (see e.g.
[48]) that the semigroup $\\{e^{-tL}\\}_{t>0}$ extends to $L^{p}({}^{d};X)$
($1\leq p\leq\infty$), where $L^{p}({}^{d};X)$ is the space of all strongly
measurable functions $f:{}^{d}\to X$ such that $\|f(x)\|_{X}\in
L^{p}({}^{d})$. The resulting semigroup is still denoted by
$\\{e^{-tL}\\}_{t>0}$ without confusion.

Let $1<q<\infty$, the $q$-variant of Lusin area integral associated with $L$
is defined as follows: for $f\in C_{c}(\mathbb{R}^{d})\otimes X$,

$\displaystyle
S_{q,L}(f)(x)=\left(\int_{0}^{\infty}\int_{|y-x|<t}\left\|tL{e^{-tL}}(f)(y)\right\|_{X}^{q}\,\frac{{\rm
d}y{\rm d}t}{t^{d+1}}\right)^{\frac{1}{q}}.$

Our main result reads as below.

######  Theorem 1.1.

Let $L$ be a sectorial operator of type $\alpha$ ($0\leq\alpha<\pi/2$) on
$L^{2}({}^{d})$ satisfying (1.5), (1.6) and (1.7). Let $1\leq p<\infty$ and
$1<q<\infty$. For ${\underline{any}}$ Banach space $X$ and $f\in
C_{c}(\mathbb{R}^{d})\otimes X$, there hold

(1.8) $\displaystyle
p^{-\frac{1}{q}}\|S_{q,{\sqrt{\Delta}}}(f)\|_{p}\lesssim_{\gamma,\beta}\left\|S_{q,L}(f)\right\|_{p}\lesssim_{\gamma,\beta}p^{\frac{1}{q}}\|S_{q,{\sqrt{\Delta}}}(f)\|_{p},$
(1.9) $\displaystyle
p^{-\frac{1}{q}}\|S_{q,L}(f)\|_{p}\lesssim_{\gamma,\beta}\|G_{q,L}(f)\|_{p}\lesssim_{\gamma,\beta}p^{\frac{1}{q}}\|S_{q,L}(f)\|_{p}.$

Moreover, the orders in both (1.8) and (1.9) are optimal as $p\rightarrow 1$.

When $X=\mathbb{C}$ and $q=2$, the equivalence (1.8) in the case $1<p<\infty$
without explicit orders follows from the classical Littlewood-Paley theory
which in turn relies on Calderón-Zygmund theory; while the case $p=1$ is
deduced from the holomorphic functional calculus, Calderón-Zygmund theory and
the theory of Hardy and BMO spaces associated with differential operators (cf.
[16, Theorem 6.10]). Our estimate (1.8) for any Banach space $X$, any $1\leq
p<\infty$ and any $1<q<\infty$ goes much beyond this and its proof provides a
new approach to the mentioned scalar case with optimal orders as $p\rightarrow
1$. Indeed, the arbitrariness of $X$ presents a surprise and usually one
expects certain property of Banach space geometry to be imposed on the square
function inequalities. For the technical side, the arbitrariness of $X$
prevents us from the use of (vector-valued) Calderón-Zygmund theory. Instead,
we will make use of vector-valued Wilson’s intrinsic square functions as a
media to relate $\Delta$ and $L$, and then exploit the vector-valued tent
space theory such as interpolation, duality as well as atomic decomposition.
Even though both of these two tools have been developed or applied in the
literature, they need to be taken care of in the present setting. For
instance, because our $L$’s are usually not translation invariant or of
scaling structure, we have to introduce Wilson’s intrinsic square functions
via nice functions of two variables satisfying (4.1), (4.2) and (4.3); to
avoid the use of Calderón-Zygmund theory to deal with Wilson’s intrinsic
square functions (cf. [41, 47]), we prove the boundedness of a linear operator
$\mathcal{K}$ on vector-valued tent spaces (see Lemma 3.5); last but not the
least, since our interested $X$ is arbitrary, one cannot establish the basic
theory of vector-valued tent space using Calderón-Zygmund theory as in [26,
27, 28, 23], and we shall adapt the classical arguments (cf. [11]), see
Section 3 for details.

After all the preparing work, the equivalence (1.8) will be an immediate
consequence of Theorem 4.1, where we collect all the intermediate estimates
involving vector-valued Wilson’s square functions.

Regarding another equivalence (1.9), in the special situation $X=\mathbb{C}$
and $q=2$ and $L=\sqrt{\Delta}$, the equivalence for $1<p<\infty$ without
optimal orders comes from the classical Littlewood-Paley theory while the case
$p=1$ constitutes one essential part of the famous real variable theory on
Hardy spaces (cf. [17, 18, 19] ); in particular the upper estimate of (1.9)
follows from harmonicity of Poisson integrals or Calderón-Zygmund theory.
Again, the arbitrariness of $X$ excludes the use of vector-valued Calderón-
Zygmund theory and there is an obvious lack of harmonicity related to general
$L$. To surmount these difficulties, in addition to the application of Theorem
4.1—Wilson’s intrinsic square functions, we will fully develop the duality
theory between vector-valued Hardy and BMO type spaces in Section 5; the
latter is inspired by Mei’s duality arguments [31] (see also [43, 47]). In
turn, part of the theory of vector-valued Hardy and BMO spaces will be deduced
from vector-valued tent spaces, and the projection $\pi_{L}$ (see Lemma 3.6)
will play a key role in passing from the results about tent spaces to those on
Hardy/BMO spaces.

Together with the related results in [29, 33] where the authors showed the
Lusin type $q$ of a Banach space $X$ relative to
$\\{e^{-t\sqrt{\Delta}}\\}_{t>0}$ is equivalent to the martingale type $q$ of
$X$ (see Section 6), our vector-valued tent space theory and Theorem 1.1 imply
the following result, resolving partially [48, Problem 1.8, Problem A.1 and
Conjecture A.4] (see Remark 6.4).

######  Theorem 1.2.

Let $L$ be a sectorial operator of type $\alpha$ ($0\leq\alpha<\pi/2$) on
$L^{2}({}^{d})$ satisfying (1.5), (1.6) and (1.7). Let $1<q\leq 2$. The
followings are equivalent

  1. (i)

$X$ is of martingale type $q$;

  2. (ii)

$X$ is of Lusin type $q$ relative to $\left\\{e^{-tL}\right\\}_{t>0}$.
Moreover, we have the following estimate for the corresponding Lusin type
constant,

$\mathsf{L}^{L}_{t,q,p}(X)\lesssim_{\gamma,\beta}p\mathsf{M}_{t,p}(X),\quad
1<p<\infty.$

Combining the main result in [33], a much stronger result than Theorem 1.2
involving the case $p=1,\infty$ will be presented in Corollary 6.3.

The paper is organized essentially as described above with a rigorous
introduction of vector-valued tent space, Hardy spaces and BMO spaces in the
next section.

Notation: In the following context, $X$ will be an arbitrary fixed Banach
space without further elaboration. $X^{*}$ denotes the dual Banach space of
$X$. Additionally, the positive real interval $\mathbb{R}_{+}=(0,\infty)$ is
equipped with the measure ${\rm d}t/t$ without providing additional
explanations.

We will use the following convention: $A\lesssim B$ (resp.
$A\lesssim_{\alpha}B$) means that $A\leq CB$ (resp. $A\leq C_{\alpha}B$) for
some absolute positive constant C (resp. a positive constant $C_{\alpha}$
depending only on a parameter $\alpha$). $A\approx B$ or $A\approx_{\alpha}B$
means that these inequalities as well as their inverses hold. We also denote
by $\|\cdot\|_{p}$ the norm $\|\cdot\|_{L^{p}({}^{d})}$ and by
$\|\cdot\|_{L^{p}(X)}$ the norm $\|\cdot\|_{L^{p}({}^{d};X)}$ $(1\leq
p\leq\infty)$.

##  2\. Preliminaries

###  2.1. Functional calculus

We start with a brief introduction of some preliminaries around the
holomorphic functional calculus (cf. [30]). Let $0\leq\alpha<\pi$. Define the
closed sector in the complex plane ${\mathbb{C}}$ as

$S_{\alpha}=\left\\{z\in{\mathbb{C}}:|\arg z|\leq\alpha\right\\},$

and $S_{\alpha}^{0}$ is denoted as the interior of $S_{\alpha}$. Let
$\gamma>\alpha$ and denote by $H(S_{\gamma}^{0})$ the space of all holomorphic
functions on $S_{\gamma}^{0}$. Define

$H_{\infty}(S_{\gamma}^{0})=\left\\{b\in
H(S_{\gamma}^{0}):\|b\|_{\infty}<\infty\right\\},$

where $\|b\|_{\infty}=\sup\left\\{|b(z)|:z\in S_{\gamma}^{0}\right\\}$ and

$\Psi(S_{\gamma}^{0})=\left\\{\psi\in H(S_{\gamma}^{0}):\exists\,s>0\text{
s.t. }|\psi(z)|\leq c|z|^{s}(1+|z|^{2s})^{-1}\right\\}.$

A densely defined closed operator $L$ acting on a Banach space $Y$ is called a
_sectorial operator of type $\alpha$_ if for each $\gamma>\alpha$,
$\sigma(L)\subset S_{\gamma}$ and

$\sup\left\\{\|z(z{\rm Id}-L)^{-1}\|_{B(Y)}:z\notin
S_{\gamma}\right\\}<\infty,$

where $\|\cdot\|_{B(Y)}$ denotes the operator norm and ${\rm Id}$ the identity
operator.

Assume that $L$ is a sectorial operator of type $\alpha$. Let
$0\leq\alpha<\theta<\gamma<\pi$ and $\Gamma$ be the boundary of $S_{\theta}$
oriented in the positive sense. For $\psi\in\Psi(S_{\gamma}^{0})$, we define
the operator $\psi(L)$ as

$\psi(L)=\frac{1}{2\pi i}\int_{\Gamma}\psi(z)(z{\rm Id}-L)^{-1}\,{\rm d}z.$

By Cauchy’s theorem, this integral converges absolutely in $B(Y)$ and it is
clear that the definition is independent of the choice of $\theta$. For every
$t>0$, denote by $\psi_{t}(z)=\psi(tz)$ for $z\in S_{\gamma}^{0}$, we have
$\psi_{t}\in\Psi(S_{\gamma}^{0})$. Set

$h(z)=\int_{0}^{\infty}\psi(tz)\,\frac{{\rm d}t}{t},\quad z\in
S_{\gamma}^{0}.$

One gets that $h$ is a constant on $S_{\gamma}^{0}$, hence by the convergence
lemma (cf. [13, Lemma 2.1]),

$h(L)x=\int_{0}^{\infty}\psi(tL)x\,\frac{{\rm d}t}{t}=cx,\quad
x\in{\mathscr{D}}(L)\cap{\rm im}(L).$

By applying a limiting argument, the above identity extends to $\overline{{\rm
im}(L)}$. In particular, take $\psi(z)=z^{2}e^{-2z}$, then

(2.1) $\int_{0}^{\infty}-tLe^{-tL}(-tLe^{-tL})x\,\frac{{\rm
d}t}{t}=\frac{1}{4}x,\quad x\in\overline{{\rm im}(L)},$

which will be useful later. We refer the reader to [21] for more information
on functional calculus.

###  2.2. Main assumptions

Throughout the paper, we assume $L$ is a sectorial operator of type $\alpha$
($0\leq\alpha<\pi/2$) on $L^{2}({}^{d})$ such that the kernels
$\\{K(t,x,y)\\}_{t>0}$ of $\\{e^{-tL}\\}_{t>0}$ satisfy assumptions (1.5),
(1.6) and (1.7) with $\beta>0,0<\gamma\leq 1$. It is well-known that such an
$L$ generates a holomorphic semigroup $e^{-zL}$ with $0\leq|{\rm
Arg}(z)|<\pi/2-\alpha$ (cf. [21, Chapter 3, 3.2]). Let $\\{k(t,x,y)\\}_{t>0}$
be the kernels of $\\{-tLe^{-tL}\\}_{t>0}$ and it is easy to see

$k(t,x,y)=t\partial_{t}K(t,x,y).$

The following lemma is justified in [16, Lemma 6.9]

######  Lemma 2.1.

Let $L$ be an operator satisfying (1.5) and (1.6) with $\beta>0$,
$0<\gamma\leq 1$. Then

(i) there exist constants $0<\beta_{1}<\beta$, $0<\gamma_{1}<\gamma$ and $c>0$
such that

(2.2) $|k(t,x,y)|\leq\frac{ct^{\beta_{1}}}{(t+|x-y|)^{d+\beta_{1}}},$

and

$|k(t,x+h,y)-k(t,x,y)|+|k(t,x,y+h)-k(t,x,y)|\leq\frac{c|h|^{\gamma_{1}}t^{\beta_{1}}}{(t+|x-y|)^{d+\beta_{1}+\gamma_{1}}}$

whenever $2|h|\leq t+|x-y|$;

(ii) for $\alpha<\theta<\pi/2$, there exist positive constants
$0<\beta_{2}<\beta$, $0<\gamma_{2}<\gamma$ and $c>0$ such that for any $|\arg
z|<\pi/2-\theta$,

(2.3) $|K(z,x,y)|\leq\frac{c|z|^{\beta_{2}}}{(|z|+|x-y|)^{d+\beta_{2}}}$

and

$|K(z,x+h,y)-K(z,x,y)|+|K(z,x,y+h)-K(z,x,y)|\leq\frac{c|h|^{\gamma_{2}}|z|^{\beta_{2}}}{(|z|+|x-y|)^{d+\beta_{2}+\gamma_{2}}}$

whenever $2|h|\leq|z|+|x-y|$.

######  Remark 2.2.

By [12, Lemma 2.5], the estimate (2.3) implies that for all
$k\in{\mathbb{N}}$, $t>0$ and almost everywhere $x,y\in{}^{d}$,

(2.4)
$\left|t^{k}\partial^{k}_{t}K(t,x,y)\right|\leq\frac{ct^{\beta_{2}}}{(t+|x-y|)^{d+\beta_{2}}}.$

Convention. To simplify notation, we will write below $\gamma$, $\beta$
instead of $\gamma_{1},\ \beta_{1}$ and $\gamma_{2},\ \beta_{2}$ appearing in
Lemma 2.1, and it should not cause any confusion.

One can verify that $\left\\{e^{-tL}\right\\}_{t>0}$ is a family of regular
operators on $L^{p}({}^{d})$ for $1\leq p\leq\infty$. Then it is well-known
(see e.g. [48]) that the semigroup $\\{e^{-tL}\\}_{t>0}$ extends to
$L^{p}({}^{d};X)$ ($1\leq p\leq\infty$), which is the space of all strongly
measurable functions $f:{}^{d}\to X$ such that $\|f(x)\|_{X}\in
L^{p}({}^{d})$. The resulting semigroup is still denoted by
$\\{e^{-tL}\\}_{t>0}$ without causing confusion. To well define the vector-
valued BMO type spaces, we need more notations. For $\varepsilon>0$, define

${\mathcal{N}}_{\varepsilon}=\left\\{f\in L^{1}_{{\rm
loc}}({}^{d};X):\exists\,c>0\text{ such that
}\int_{{}^{d}}\frac{\|f(x)\|_{X}}{(1+|x|)^{d+\varepsilon}}\,{\rm d}x\leq
c\right\\},$

equipped with norm defined as the infimum of all the possible constant $c$.
Then ${\mathcal{N}}_{\varepsilon}$ is a Banach space (cf. [16]). For a given
generator $L$, let $\Theta(L)=\sup\left\\{\beta_{2}>0:\eqref{pt1}\text{
holds}\right\\}$. Then we define

${\mathcal{N}}=\begin{cases}{\mathcal{N}}_{\Theta(L)},&\ \mathrm{if}\
\Theta(L)<\infty;\\\
\bigcup_{0<\varepsilon<\infty}{\mathcal{N}}_{\varepsilon},&\ \mathrm{if}\
\Theta(L)=\infty.\end{cases}$

It is clear that $L^{p}({}^{d};X)\subset{\mathcal{N}}$ for all $1\leq
p\leq\infty$. Moreover, By the definition of ${\mathcal{N}}$ and Remark 2.2,
we know that the operators $e^{-tL}$ and $tLe^{-tL}$ are well-defined on
${\mathcal{N}}$.

Denote by $\mathbf{F}_{L}$ the fixed point space of $\\{e^{-tL}\\}_{t>0}$ on
${\mathcal{N}}$, namely

$\mathbf{F}_{L}=\left\\{f\in{\mathcal{N}}:e^{-tL}(f)=f,\
\forall\,t>0\right\\}.$

It is well-known that $\mathbf{F}_{L}$ coincides with the null space of
$\\{tLe^{-tL}\\}_{t>0}$, and the resulting quotient space is defined as
${\mathcal{N}}_{L}:={\mathcal{N}}/\mathbf{F}_{L}$. For $1\leq p<\infty$, the
fixed point subspace of $L^{p}({}^{d};X)$ is $\mathbf{F}_{L}\cap
L^{p}({}^{d};X)=\left\\{0\right\\}$ (see [16, Theorem 6.10]); in other words,
the projection from $L^{p}({}^{d};X)$ to the fixed point subspace for all
$1\leq p<\infty$ is $0$. See e.g. [29, 48] for more information on this
projection.

######  Remark 2.3.

Let $L^{*}$ be the adjoint operator of $L$. Then $L^{*}$ is also a sectorial
operator with the kernels of $\\{e^{-tL^{*}}\\}_{t>0}$ satisfying (1.5), (1.6)
and (1.7) (cf. [16, Theorem 6.10]).

###  2.3. Vector-valued tent, Hardy and BMO spaces

In this subsection, we introduce several spaces including vector-valued tent
spaces, vector-valued Hardy and BMO spaces associated with a generator $L$.

####  2.3.1. Vector-valued Tent spaces

We first introduce vector-valued tent spaces. We denote by ${}^{d+1}_{+}$ the
usual upper half-space in d+1 i.e. ${}^{d}\times(0,\infty)$. Let
$\Gamma(x)=\\{(y,t)\in{}^{d+1}_{+}:|y-x|<t\\}$ denote the standard cone with
vertex at $x$. For any closed subset $F\subset{}^{d}$, define
${\mathcal{R}}(F)=\bigcup_{x\in F}\Gamma(x)$. If $O\subset{}^{d}$ is an open
subset, then the tent over $O$, denoted by $\widehat{O}$, is given as
$\widehat{O}=\left({\mathcal{R}}(O^{C})\right)^{C}$.

For any strongly measurable function $f:{}^{d+1}_{+}\to X$, we define two
operators as follows:

${\mathcal{A}}_{q}(f)(x)=\left(\int_{\Gamma(x)}\|f(y,t)\|_{X}^{q}\frac{{\rm
d}y{\rm
d}t}{t^{d+1}}\right)^{\frac{1}{q}},\quad\mathcal{C}_{q}(f)(x)=\sup_{x\in
B}\left(\frac{1}{|B|}\int_{\widehat{B}}\|f(y,t)\|_{X}^{q}\,\frac{{\rm d}y{\rm
d}t}{t}\right)^{\frac{1}{q}},$

where the supremum runs over all balls $B$ in d.

######  Definition 2.4.

Let $1\leq p<\infty$ and $1<q<\infty$. The vector-valued tent space
$T_{q}^{p}({}^{d+1}_{+};X)$ is defined as the subspace consisting of all
strongly measurable functions $f:{}^{d+1}_{+}\to X$ such that

$\|f\|_{T_{q}^{p}(X)}:=\|{\mathcal{A}}_{q}(f)\|_{p}<\infty,$

and $T_{q}^{\infty}({}^{d+1}_{+};X)$ is defined as the subspace of all
strongly measurable functions $g:{}^{d+1}_{+}\to X$ such that

$\|g\|_{T_{q}^{\infty}(X)}:=\|\mathcal{C}_{q}(g)\|_{\infty}<\infty.$

Let $C_{c}({}^{d+1}_{+})\otimes X$ be the space of finite linear combinations
of elements from $C_{c}({}^{d+1}_{+})$ and $X$. The following density follows
from the standard arguments (see e.g. [23]), and we omit the details.

######  Lemma 2.5.

Let $X$ be a Banach space and $1<q<\infty$. Then $C_{c}({}^{d+1}_{+})\otimes
X$ is norm dense in $T^{p}_{q}({}^{d+1}_{+};X)$ for $1\leq p<\infty$, and
weak-$\ast$ dense in
$\left(T^{1}_{q^{\prime}}({}^{d+1}_{+};X^{*})\right)^{*}$.

####  2.3.2. Vector-valued Hardy spaces

Given a function $f\in{\mathcal{N}}_{L}$, the $q$-variant of Lusin area
integral function of $f$ associated with $L$ is defined by

$S_{q,L}(f)(x)=\left(\int_{\Gamma(x)}\|tLe^{-tL}(f)(y)\|_{X}^{q}\,\frac{{\rm
d}y{\rm d}t}{t^{d+1}}\right)^{\frac{1}{q}};$

and the $q$-variant of Littlewood-Paley $g$-function is defined by

$G_{q,L}(f)(x)=\left(\int_{0}^{\infty}\|tLe^{-tL}(f)(x)\|_{X}^{q}\,\frac{{\rm
d}t}{t}\right)^{\frac{1}{q}}.$

######  Definition 2.6.

Let $1\leq p<\infty$ and $1<q<\infty$. We define the vector-valued Hardy space
$H_{q,L}^{p}({}^{d};X)$ associated with $L$ as

$H_{q,L}^{p}({}^{d};X)=\left\\{f\in{\mathcal{N}}_{L}:S_{q,L}(f)\in
L^{p}({}^{d})\right\\},$

equipped with the norm

$\|f\|_{H^{p}_{q,L}(X)}=\|S_{q,L}(f)\|_{p}.$

It is easy to check that $H^{p}_{q,L}({}^{d};X)$ is a Banach space from the
definition of ${\mathcal{N}}_{L}$. The space $H^{p}_{q,L}({}^{d};X)$ has deep
connection with the vector-valued tent space, namely, a strongly measurable
function $f\in{\mathcal{N}}_{L}$ belongs to $H^{p}_{q,L}({}^{d};X)$ if and
only if $\mathcal{Q}(f)\in T^{p}_{q}({}^{d+1}_{+};X)$ where
${\mathcal{Q}}(f)(x,t)=-tLe^{-tL}(f)(x)$. Moreover,

$\|f\|_{H^{p}_{q,L}(X)}=\|{\mathcal{Q}}(f)\|_{T^{p}_{q}(X)}.$

####  2.3.3. Vector-valued BMO spaces

######  Definition 2.7.

Let $1\leq p\leq\infty$ and $1<q<\infty$. We define the vector-valued BMO
space $BMO_{q,L}^{p}({}^{d};X)$ associated with $L$ as

$BMO_{q,L}^{p}({}^{d};X)=\left\\{f\in{\mathcal{N}}_{L}:\|\mathcal{C}_{q}({\mathcal{Q}}(f))\|_{p}<\infty\right\\}$

equipped with the norm

$\|f\|_{BMO_{q,L}^{p}(X)}=\|\mathcal{C}_{q}({\mathcal{Q}}(f))\|_{p}.$

In particular, for $p=\infty$, we denote it by $BMO_{q,L}({}^{d};X)$ for
short.

It is easy to verify that $BMO^{p}_{q,L}({}^{d};X)$ equipped the the norm
$\|\cdot\|_{BMO^{p}_{q,L}(X)}$ is a Banach space from the definition of
${\mathcal{N}}_{L}$.

The vector-valued Hardy and BMO spaces enjoy the similar relationship as the
scalar-valued ones (see e.g. [11]). We collect them below with a brief
explanation.

######  Lemma 2.8.

Let $X$ be any fixed Banach space and $1<q<\infty$. One has for $f\in
C_{c}({}^{d+1}_{+})\otimes X$,

(2.5)
$\|\mathcal{C}_{q}(f)\|_{p}\lesssim\left(\frac{p}{p-q}\right)^{\frac{1}{q}}\|{\mathcal{A}}_{q}(f)\|_{p},\quad
q<p\leq\infty,$

and

(2.6) $\|{\mathcal{A}}_{q}(f)\|_{p}\lesssim
q^{\frac{p}{q}}\|\mathcal{C}_{q}(f)\|_{p},\quad 1\leq p<\infty.$

Therefore, we have for $1\leq p\leq q$,

$BMO_{q,L}^{p}({}^{d};X)\subset H^{p}_{q,L}({}^{d};X)$

and for $q<p<\infty$,

$H^{p}_{q,L}({}^{d};X)=BMO_{q,L}^{p}({}^{d};X)$

with equivalent norms.

###### Proof.

Given an $X$-valued function $f$ defined on ${}^{d+1}_{+}$, we consider the
scalar-valued function $\widetilde{f}(x,t)=\|f(x,t)\|_{X}$. Then one may apply
(2.5) and (2.6) in the case $X=\mathbb{C}$ for $\widetilde{f}$ (see e.g.[11,
Theorem 3]) to obtain (2.5) and (2.6) for general $X$. Thus by using the
operator $\mathcal{Q}$ and the density in Lemma 2.5, for any $f\in
BMO_{q,L}^{p}({}^{d};X)$ ($1\leq p\leq q$), we get

$\|f\|_{H^{p}_{q,L}(X)}=\|{\mathcal{A}}_{q}({\mathcal{Q}}(f))\|_{p}\lesssim
q^{\frac{p}{q}}\|\mathcal{C}_{q}({\mathcal{Q}}(f))\|_{p}=\|f\|_{BMO_{q,L}^{p}(X)},$

and the same argument works for $q<p<\infty$. ∎

######  Remark 2.9.

In particular, $BMO_{q,L}({}^{d};X)$ is closely related to the Carleson
measure. Recall that a scalar-valued measure $\mu$ defined on ${}^{d+1}_{+}$
is a Carleson measure if there exists a constant $c$ such that for all balls
$B$ in d,

$|\mu(\widehat{B})|\leq c|B|,$

where $\widehat{B}$ is the tent over $B$. The norm is defined as

$\|\mu\|_{c}=\sup_{B}|B|^{-1}|\mu(\widehat{B})|,$

where the supremum runs over all the balls in d.

For a vector-valued function $f\in{\mathcal{N}}_{L}$, we define a measure
$\mu_{q,f}$ as

$\mu_{q,f}(x,t)=\frac{\|{\mathcal{Q}}(f)(x,t)\|_{X}^{q}{\rm d}x{\rm d}t}{t}.$

Then $f$ belongs to $BMO_{q,L}({}^{d};X)$ if and only if $\mu_{q,f}$ is a
Carleson measure, and moreover

$\|f\|_{BMO_{q,L}(X)}=\|\mu_{q,f}\|^{\frac{1}{q}}_{c}.$

##  3\. Theory of vector-valued tent spaces and two key linear operators

In this section, we will first present the basic theory of vector-valued tent
spaces such as atomic decomposition, interpolation and duality, and then
introduce two important linear operators $\mathcal{K}$ and $\pi_{L}$ which
enable us to exploit the basic theory of tent spaces to investigate in later
sections vector-valued Wilson’s square functions and Theorem 1.1.

Note that if the underlying Banach space $X$ has some geometric property such
as UMD, then the vector-valued tent space theory have been established in the
literature [26, 27, 28]. In the present paper, we observe that the theory of
vector-valued tent space holds for any Banach space; and this is quite
essential for the applications in the present paper.

###  3.1. Basic theory of vector-valued tent spaces

We begin this subsection by presenting the atomic decomposition of tent space
in the context of vector-valued context. It has been established in [27,
Theorem 4.5], for the completeness of this article, we will attach the proof.
Recall that a strongly measurable function $a:{}^{d+1}_{+}\to X$ is called an
$(X,q)$-atom if

  1. (1)

$\text{supp}\,a\subset\widehat{B}$ where $B$ is a ball in d;

  2. (2)

$\left(\int_{{}^{d+1}_{+}}\|a(x,t)\|_{X}^{q}\,\frac{{\rm d}x{\rm
d}t}{t}\right)^{\frac{1}{q}}\leq|B|^{\frac{1}{q}-1}.$

######  Lemma 3.1.

Let $X$ be any fixed Banach space and $1<q<\infty$. For each $f\in
T_{q}^{1}({}^{d+1}_{+};X)$, there exists a sequence of complex numbers
$\\{\lambda_{k}\\}_{k\geq 1}$ and $(X,q)$-atoms $a_{k}$ such that

$f=\sum_{k\geq 1}\lambda_{k}a_{k},\quad\|f\|_{T_{q}^{1}(X)}\approx\sum_{k\geq
1}|\lambda_{k}|.$

###### Proof.

Let $a$ be an $(X,q)$-atom and $\text{supp}\,a\subset\widehat{B}$ where
$B=B(c_{B},r_{B})$ with center $c_{B}$ and radius $r_{B}$. If
$\Gamma(x)\cap\widehat{B}\neq\emptyset$, there exists
$(y,t)\in\Gamma(x)\cap\widehat{B}$. Then we have
$|x-c_{B}|\leq|x-y|+|y-c_{B}|<t+r_{B}<2r_{B}$. By Hölder’s inequality and
Fubini’s theorem,

$\|a\|_{T^{1}_{q}(X)}=\int_{2B}\left(\int_{\Gamma(x)}\|a(y,t)\|_{X}^{q}\,\frac{{\rm
d}y{\rm d}t}{t^{d+1}}\right)^{\frac{1}{q}}{\rm
d}x\lesssim|2B|^{1-\frac{1}{q}}\left(\int_{{}^{d+1}_{+}}\|a(y,t)\|_{X}^{q}\,\frac{{\rm
d}y{\rm d}t}{t}\right)^{\frac{1}{q}}\lesssim 1.$

Therefore any $(X,q)$-atom belongs to $T^{1}_{q}({}^{d+1}_{+};X)$.

Let $0<\lambda<1/2$. We define two sequences of open sets
$\left\\{O_{k}\right\\}_{k\in{\mathbb{Z}}}$ and
$\left\\{O_{k}^{*}\right\\}_{k\in{\mathbb{Z}}}$ as

$O_{k}=\left\\{x\in{}^{d}:{\mathcal{A}}_{q}(f)(x)>2^{k}\right\\},\quad
O_{k}^{*}=\left\\{x\in{}^{d}:M(\mathbbm{1}_{O_{k}})(x)>1-\lambda\right\\},$

where $M(\mathbbm{1}_{O_{k}})$ is the centered Hardy-Littlewood maximal
function. It is clear that both $O_{k}$ and $O^{*}_{k}$ have finite measure.
Additionally, the following properties hold: $O_{k+1}\subset O_{k}$,
$O^{*}_{k+1}\subset O^{*}_{k}$ and $|O^{*}_{k}|\leq C_{\lambda}|O_{k}|$ (see
e.g. [11]).

We follow a similar construction as in [27]. The Vitali covering lemma and
[28, Lemma 4.2] assert that for each $O_{k}^{*}$, there exist disjoint balls
$B_{k}^{j}\subset O_{k}^{*}$ ($j\geq 1$) such that

$\widehat{O}^{*}_{k}\subset\bigcup_{j\geq
1}\widehat{5B}_{k}^{j},\quad\sum_{j\geq 1}|B_{k}^{j}|\leq|O_{k}^{*}|.$

With this setup, we proceed to define a family of functions $\chi_{k}^{j}$ by
the partition of unity:

$0\leq\chi_{k}^{j}\leq 1,\ \sum_{j\geq 1}\chi_{k}^{j}=1\text{ on
}\widehat{O}_{k}^{*}\ \text{ and }\
\text{supp}\,\chi_{k}^{j}\subset\widehat{5B}_{k}^{j}.$

Therefore

$f=\sum_{k\in{\mathbb{Z}}}f_{k}=\sum_{k\in{\mathbb{Z}}}\sum_{j\geq
1}\chi_{k}^{j}f_{k}=\sum_{k\in{\mathbb{Z}}}\sum_{j\geq
1}\lambda_{k}^{j}a_{k}^{j},$

where

(3.1)
$f_{k}=f\mathbbm{1}_{\widehat{O}^{*}_{k}\setminus\widehat{O}^{*}_{k+1}},\
\lambda_{k}^{j}=|5B_{k}^{j}|^{\frac{1}{q^{\prime}}}\left(\int_{5B_{k}^{j}}{\mathcal{A}}_{q}(f_{k})^{q}(x)\,{\rm
d}x\right)^{\frac{1}{q}},\
a_{k}^{j}=\frac{\chi_{k}^{j}f_{k}}{\lambda_{k}^{j}}.$

Now we only need to show that each $a_{k}^{j}$ is an $(X,q)$-atom and

$\sum_{k\in{\mathbb{Z}}}\sum_{j\geq
1}|\lambda_{k}^{j}|\lesssim\|f\|_{T^{1}_{q}(X)}.$

It is clear that $\text{supp}\,a_{k}^{j}\subset\widehat{5B}_{k}^{j}$.
Furthermore,

$\displaystyle\|a\|_{L^{q}({}^{d+1}_{+};X)}^{q}$
$\displaystyle\leq|5B_{k}^{j}|^{1-q}\|{\mathcal{A}}_{q}(f_{k})\mathbbm{1}_{5B_{k}^{j}}\|_{q}^{-q}\left(\int_{\widehat{5B}_{k}^{j}}\|f_{k}(y,t)\|_{X}^{q}\,\frac{{\rm
d}y{\rm d}t}{t}\right)$
$\displaystyle\leq|5B_{k}^{j}|^{1-q}\|{\mathcal{A}}_{q}(f_{k})\mathbbm{1}_{5B_{k}^{j}}\|_{q}^{-q}\left(\int_{5B_{k}^{j}}\left({\mathcal{A}}_{q}(f_{k})(x)\right)^{q}\,{\rm
d}x\right)$ $\displaystyle=|5B_{k}^{j}|^{1-q}.$

Hence each $a_{k}^{j}$ is an $(X,q)$-atom.

According to [11, Lemma 5], it is known that ${\mathcal{A}}_{q}(f_{k})$ is
supported in $O^{*}_{k}\setminus O_{k+1}$, then we deduce that
${\mathcal{A}}_{q}(f_{k})(x)\leq 2^{k+1}$ by definition. Thus

$\sum_{k\in{\mathbb{Z}}}\sum_{j\geq
1}|\lambda_{k}^{j}|\leq\sum_{k\in{\mathbb{Z}}}\sum_{j\geq
1}|5B_{k}^{j}|^{\frac{1}{q^{\prime}}}2^{k+1}|5B_{k}^{j}|^{\frac{1}{q}}\leq\sum_{k\in{\mathbb{Z}}}2^{k+1}|O_{k}^{*}|\leq\sum_{k\in{\mathbb{Z}}}2^{k+1}C_{\lambda}|O_{k}|.$

However, ${\mathcal{A}}_{q}(f)(x)>2^{(k+m)}$ on $O_{k+m}$, then

$2^{k}|O_{k}|=\int_{O_{k}}2^{k}\,{\rm
d}x=\sum_{m=0}^{\infty}\int_{O_{k+m}\setminus O_{k+m+1}}2^{k}\,{\rm
d}x\leq\sum_{m=0}^{\infty}2^{-m}\int_{O_{k+m}\setminus
O_{k+m+1}}{\mathcal{A}}_{q}(f)(x)\,{\rm d}x.$

Hence

$\sum_{k\in{\mathbb{Z}}}2^{k+1}C_{\lambda}|O_{k}|\leq\sum_{m=0}^{\infty}\sum_{k\in{\mathbb{Z}}}2^{-m+1}C_{\lambda}\int_{O_{k+m}\setminus
O_{k+m+1}}{\mathcal{A}}_{q}(f)(x)\,{\rm d}x\lesssim\|f\|_{T^{1}_{q}(X)}.$

We complete the proof. ∎

######  Remark 3.2.

From the atomic decomposition of $T^{1}_{q}(\mathbb{R}^{d+1}_{1};X)$—Lemma
3.1, one may conclude a molecule decomposition of the corresponding Hardy
space. This might have further applications, and we include it in the
Appendix.

The following lemma is the complex interpolation theory of vector-valued tent
spaces.

######  Lemma 3.3.

Let $X$ be any fixed Banach space, $1<q<\infty$ and $1\leq
p_{1}<p<p_{2}<\infty$ such that $1/p=(1-\theta)/p_{1}+\theta/p_{2}$ with
$0\leq\theta\leq 1$. Then

$[T^{p_{1}}_{q}({}^{d+1}_{+};X),T^{p_{2}}_{q}({}^{d+1}_{+};X)]_{\theta}=T^{p}_{q}({}^{d+1}_{+};X),$

with equivalent norms, where $[\cdot,\cdot]_{\theta}$ is the complex
interpolation space. More precisely, for $f\in C_{c}({}^{d+1}_{+})\otimes X$,
one has

$\|f\|_{T^{p}_{q}(X)}\lesssim\|f\|_{[T^{p_{1}}_{q}({}^{d+1}_{+};X),T^{p_{2}}_{q}({}^{d+1}_{+};X)]_{\theta}}\lesssim
p^{\frac{1}{q}}\|f\|_{T^{p}_{q}(X)}.$

###### Proof.

For the interpolation theory, we introduce two important operators, which
allow us to relate $T^{p}_{q}({}^{d+1}_{+};X)$ with $L^{p}({}^{d};E)$ with $E$
being the Banach space $L^{q}({}^{d+1}_{+};X)$ equipped with the measure ${\rm
d}x{\rm d}t/t^{d+1}$. The first operator is defined as

$i(f)(x,y,t)=\mathbbm{1}_{\Gamma(x)}(y,t)f(y,t),$

for $f\in T^{p}_{q}({}^{d+1}_{+};X)$. Then it is clear that
$\|i(f)\|_{L^{p}(E)}=\|f\|_{T^{p}_{q}(X)}$. Denote by $\widetilde{T}_{q}^{p}$
the range of the operator $i$. Now we introduce another operator $N$ given by

$N(f)(x,y,t)=\mathbbm{1}_{\Gamma(x)}(y,t)\frac{1}{w_{d}t^{d}}\int_{|z-y|<t}f(z,y,t)\,{\rm
d}z,$

where $w_{d}$ is the volume of the $d$-dimensional unit ball. It is known that
$N$ is a continuous projection from $L^{p}({}^{d};E)$ onto itself with range
$\widetilde{T}_{q}^{p}$ for $1<p<\infty$ (cf. [23]). Consider the maximal
operator

$M_{1}(f)(x,y,t)=\sup_{x\in B}\frac{1}{|B|}\int_{B}\|f(z,y,t)\|_{X}\,{\rm
d}z,$

where the supremum is taken over all balls $B$ in d. It is known from the
maximal inequalities (see e.g. [38, Chapter II]) that $M_{1}$ is bounded on
$L^{p}\left({}^{d};L^{q}({}^{d+1};{\rm d}y{\rm d}t/t^{d+1})\right)$ for
$1<p<\infty$; in particular, we view $\|f\|_{X}$ as a scalar-valued function
in $L^{p}\left({}^{d};L^{q}({}^{d+1};{\rm d}y{\rm d}t/t^{d+1})\right)$, then

$\|M_{1}(f)\|_{L^{p}\left({}^{d};L^{q}\left({}^{d+1}_{+},\frac{{\rm d}y{\rm
d}t}{t^{d+1}}\right)\right)}=\|M_{1}(\|f\|_{X})\|_{L^{p}\left({}^{d};L^{q}\left({}^{d+1}_{+},\frac{{\rm
d}y{\rm d}t}{t^{d+1}}\right)\right)}\lesssim
p^{\frac{1}{q}}\|f\|_{L^{p}(E)},\quad q\leq p<\infty.$

Then we deduce from the definition of $N$ that

$\|N(f)(x,y,t)\|_{X}\leq\mathbbm{1}_{\Gamma(x)}(y,t)\frac{1}{|B(y,t)|}\int_{B(y,t)}\|f(z,y,t)\|_{X}\,{\rm
d}z\leq M_{1}(f)(x,y,t).$

Therefore

$\|N(f)\|_{L^{p}(E)}\leq\|M_{1}(f)\|_{L^{p}\left({}^{d};L^{q}\left({}^{d+1}_{+};\frac{{\rm
d}y{\rm d}t}{t^{d+1}}\right)\right)}\lesssim
p^{\frac{1}{q}}\|f\|_{L^{p}(E)},\quad q\leq p<\infty.$

We denote by $F$ the Banach space $L^{q^{\prime}}({}^{d+1}_{+};X^{*})$
equipped with the measure ${\rm d}x{\rm d}t/t^{d+1}$. Then it is clear that
$F\subset E^{*}$ and $F$ is norming for $E$. For $1<p<q$, we have

$\displaystyle\|N(f)\|_{L^{p}(E)}$
$\displaystyle=\sup_{g}\left|\int_{{}^{d}}\int_{{}^{d+1}_{+}}\left\langle
N(f)(x,y,t),g(x,y,t)\right\rangle_{X\times X^{*}}\,\frac{{\rm d}y{\rm
d}t}{t^{d+1}}{\rm d}x\right|$
$\displaystyle=\sup_{g}\left|\int_{{}^{d}}\int_{{}^{d+1}_{+}}\left\langle
f(x,y,t),N(g)(x,y,t)\right\rangle_{X\times X^{*}}\,\frac{{\rm d}y{\rm
d}t}{t^{d+1}}{\rm d}x\right|$
$\displaystyle\leq\|f\|_{L^{p}(E)}\|N(g)\|_{L^{p^{\prime}}(F)}\lesssim
p^{\prime\frac{1}{q^{\prime}}}\|f\|_{L^{p}(E)}\|g\|_{L^{p^{\prime}}(F)},$

where the supremum is taken over all $g$ in the unit ball of
$L^{p^{\prime}}({}^{d};F)$. We conclude

(3.2)
$\|N(f)\|_{L^{p}(E)}\lesssim\max\left\\{p^{\frac{1}{q}},p^{\prime\frac{1}{q^{\prime}}}\right\\}\|f\|_{L^{p}(E)},\quad
1<p<\infty.$

Now we turn to the interpolation theory. The proof of the case
$1<p_{1}<p_{2}<\infty$ follows from [23] by virtue of the immersion $i$ and
the projection $N$.

For the case $p_{1}=1$, we adapt the classical argument as in [11, Lemma 4,
Lemma 5]. Since the immersion $i$ is an isometry, the exactness of the
exponent $\theta$ of complex interpolation functor reads that

$\displaystyle\|i(f)\|_{[L^{1}({}^{d};E),L^{p_{2}}({}^{d};E)]_{\theta}}$
$\displaystyle\leq$ $\displaystyle\|i\|_{T^{1}_{q}({}^{d+1}_{+};X)\to
L^{1}({}^{d};E)}^{1-\theta}\|i\|_{T^{p_{2}}_{q}({}^{d+1}_{+};X)\to
L^{p_{2}}({}^{d};E)}^{\theta}\|f\|_{[T^{1}_{q}({}^{d+1}_{+};X),T^{p_{2}}_{q}({}^{d+1}_{+};X)]_{\theta}}$
$\displaystyle\leq$
$\displaystyle\|f\|_{[T^{1}_{q}({}^{d+1}_{+};X),T^{p_{2}}_{q}({}^{d+1}_{+};X)]_{\theta}}.$

By the interpolation theory of vector-valued $L^{p}$ spaces, (see e.g. [9]),
we have

$\|i(f)\|_{[L^{1}({}^{d};E),L^{p_{2}}({}^{d};E)]_{\theta}}=\|i(f)\|_{L^{p}(E)}=\|f\|_{T^{p}_{q}(X)}.$

Thus

$\|f\|_{T^{p}_{q}(X)}\leq\|f\|_{[T^{1}_{q}({}^{d+1}_{+};X),T^{p_{2}}_{q}({}^{d+1}_{+};X)]_{\theta}}.$

For the reverse direction, let $f\in T^{p}_{q}({}^{d+1}_{+};X)$ and
$\|f\|_{T^{p}_{q}(X)}=1$. By taking into account the atomic decomposition of
$T^{1}_{q}({}^{d+1}_{+};X)$—Lemma 3.1, we define the interpolation functor $F$
as

$F(z)=\sum_{k\in{\mathbb{Z}}}2^{k(\alpha(z)p-1)}f_{k},$

where $\alpha(z)=1-z+z/p_{2}$ and $f_{k}$ is defined in (3.1). We have
$F(\theta)=f$. Then the proof can be then conducted in the same way as in [11,
Lemma 5], we omit the details. ∎

We now provide a characterization of $T^{p}_{q}({}^{d+1}_{+};X)$-norm. It
belongs to the norming subspace theory of vector-valued $L^{p}$-spaces, see
e.g. [15, Chapter II, Section 4]. The proof is in spirit the same as the
scalar-valued case (cf. [23, Theorem 2.4] and [11, Theorem 1], but we include
a proof here to provide explicit orders for later applications.

Let $(\Omega,{\mathcal{F}},\mu)$ be a measure space. Recall that a Banach
space $X$ has the _Radon-Nikodým property_ with respect to
$(\Omega,{\mathcal{F}},\mu)$ if for each $\mu$-continuous vector-valued
measure $\nu:{\mathcal{F}}\to X$ of bounded variation, there exists $g\in
L^{1}(\Omega;X)$ with respect to the measure $\mu$ such that

$\nu(E)=\int_{E}g\,{\rm d}\mu,\quad\forall\,E\in{\mathcal{F}}.$

In the following context, we call a Banach space has the Radon-Nikodým
property for short when there is no ambiguity. We refer readers to [14,
Chapter III] for more details.

######  Lemma 3.4.

Let $X$ be any fixed Banach space and $1<q<\infty$. The space
$T^{p^{\prime}}_{q^{\prime}}({}^{d+1}_{+};X^{*})$ is isomorphically identified
as a subspace of the dual space of $T^{p}_{q}({}^{d+1}_{+};X)$. Moreover, it
is norming for $T^{p}_{q}({}^{d+1}_{+};X)$ in the following sense,

(3.3)
$\displaystyle\|f\|_{T^{p}_{q}(X)}\lesssim\max\left\\{p^{\frac{1}{q}},p^{\prime\frac{1}{q^{\prime}}}\right\\}\sup_{g}\left|\int_{{}^{d+1}_{+}}\left\langle
f(x,t),g(x,t)\right\rangle_{X\times X^{*}}\,\frac{{\rm d}x{\rm
d}t}{t}\right|,\quad 1<p<\infty,$

where the supremum is taken over all $g\in C_{c}({}^{d+1}_{+})\otimes X^{*}$
such that $\|\mathcal{A}_{q^{\prime}}(g)\|_{p^{\prime}}\leq 1$; and similarly,

(3.4)
$\displaystyle\|f\|_{T^{p}_{q}(X)}\lesssim\left(\frac{p(q-1)}{q-p}\right)^{\frac{1}{q^{\prime}}}\sup_{g}\left|\int_{{}^{d+1}_{+}}\left\langle
f(x,t),g(x,t)\right\rangle_{X\times X^{*}}\,\frac{{\rm d}x{\rm
d}t}{t}\right|,\quad 1\leq p<q,$

where the supremum is taken over all $g\in C_{c}({}^{d+1}_{+})\otimes X^{*}$
such that $\|\mathcal{C}_{q^{\prime}}(g)\|_{p^{\prime}}\leq 1$. Furthermore,
if $X^{*}$ has the Radon-Nikodým property, then

$T^{p^{\prime}}_{q^{\prime}}({}^{d+1}_{+};X^{*})=\left(T^{p}_{q}({}^{d+1}_{+};X)\right)^{*},\quad
1\leq p<\infty.$

###### Proof.

We adopt the maps $i$ and $N$ used in the proof of Lemma 3.3. We first prove
the estimate (3.3).

For any $g\in C_{c}({}^{d+1}_{+})\otimes X^{*}$ and $f\in
T^{p}_{q}({}^{d+1}_{+};X)$, we denote by

$g(f)=\int_{{}^{d+1}_{+}}\left\langle f(x,t),g(x,t)\right\rangle_{X\times
X^{*}}\,\frac{{\rm d}x{\rm d}t}{t}.$

Thus we have

(3.5) $\begin{split}|g(f)|&=\left|\int_{{}^{d+1}_{+}}\left\langle
f(y,t),g(y,t)\left(w_{d}^{-1}\int_{|x-y|<t}1\,{\rm
d}x\right)\right\rangle_{X\times X^{*}}\,\frac{{\rm d}y{\rm
d}t}{t^{d+1}}\right|\\\
&=w_{d}^{-1}\left|\int_{{}^{d}}\int_{{}^{d+1}_{+}}\left\langle
i(f)(x,y,t),i(g)(x,y,t)\right\rangle_{X\times X^{*}}\,\frac{{\rm d}y{\rm
d}t}{t^{d+1}}{\rm d}x\right|.\end{split}$

Since $i(f)\in\widetilde{T}^{p}_{q}$, we have $N(i(f))=i(f)$. Then we deduce
that

(3.6)
$\begin{split}\|f\|_{T^{p}_{q}(X)}&=\|i(f)\|_{L^{p}(E)}=\sup_{g}\left|\int_{{}^{d}}\int_{{}^{d+1}_{+}}\left\langle
i(f)(x,y,t),g(x,y,t)\right\rangle_{X\times X^{*}}\,\frac{{\rm d}y{\rm
d}t}{t^{d+1}}{\rm d}x\right|\\\
&=\sup_{g}\left|\int_{{}^{d}}\int_{{}^{d+1}_{+}}\left\langle
N(i(f))(x,y,t),g(x,y,t)\right\rangle_{X\times X^{*}}\,\frac{{\rm d}y{\rm
d}t}{t^{d+1}}{\rm d}x\right|\\\
&=\sup_{g}\left|\int_{{}^{d}}\int_{{}^{d+1}_{+}}\left\langle
i(f)(x,y,t),N(g)(x,y,t)\right\rangle_{X\times X^{*}}\,\frac{{\rm d}y{\rm
d}t}{t^{d+1}}{\rm d}x\right|\end{split}$

where the supremum is taken over all $g$ in the unit ball of
$L^{p^{\prime}}({}^{d};F)$. Notice that $N(g)=i\left(i^{-1}(N(g))\right)$ and
by (3.2)

$\|i^{-1}(N(g))\|_{T^{p^{\prime}}_{q^{\prime}}(X^{*})}=\|N(g)\|_{L^{p^{\prime}}(F)}\lesssim\max\left\\{p^{\frac{1}{q}},{p^{\prime}}^{\frac{1}{q^{\prime}}}\right\\}\|g\|_{L^{p^{\prime}}(F)}.$

Consequently, combining (3.5) and (3.6), we obtain

$\|f\|_{T^{p}_{q}(X)}\lesssim\max\left\\{p^{\frac{1}{q}},{p^{\prime}}^{\frac{1}{q^{\prime}}}\right\\}\sup_{g}|g(f)|,\quad
1<p<\infty,$

where the supremum is taken over all $g\in C_{c}({}^{d+1}_{+})\otimes X^{*}$
such that $\|\mathcal{A}_{q^{\prime}}(g)\|_{p^{\prime}}\leq 1$, and we
actually exploit a limiting argument: since not only the subset of
$C_{c}({}^{d+1}_{+})\otimes X^{*}$ with norm
$\|\mathcal{A}_{q^{\prime}}(g)\|_{p^{\prime}}\leq 1$ is contained in the unit
ball of $T^{p^{\prime}}_{q^{\prime}}({}^{d+1}_{+};X^{*})$, but also its
closure contains the unit sphere, and thus one concludes that this subset is
still norming for $T^{p}_{q}({}^{d+1}_{+};X)$.

Now we deal with the estimate (3.4) in the case $1<p<q$. Let $g\in
L^{p^{\prime}}({}^{d};F)$. By definition we have

$\displaystyle\|i^{-1}(N(g))(y,t)\|_{X^{*}}^{q^{\prime}}$
$\displaystyle\leq\left(\frac{1}{|B(y,t)|}\int_{B(y,t)}\|g(z,y,t)\|_{X^{*}}\,{\rm
d}z\right)^{q^{\prime}}$
$\displaystyle\leq\frac{1}{|B(y,t)|}\int_{|z-y|<t}\|g(z,y,t)\|_{X^{*}}^{q^{\prime}}\,{\rm
d}z.$

For a ball $B$ in d, we observe

$\displaystyle\int_{\widehat{B}}\|i^{-1}(N(g))(y,t)\|_{X^{*}}^{q^{\prime}}\,\frac{{\rm
d}y{\rm d}t}{t}$
$\displaystyle\lesssim\int_{\widehat{B}}\int_{|z-y|<t}\|g(z,y,t)\|_{X^{*}}^{q^{\prime}}\,{\rm
d}z\frac{{\rm d}y{\rm d}t}{t^{d+1}}$
$\displaystyle\leq\int_{2B}\int_{{}^{d+1}_{+}}\|\mathbbm{1}_{\widehat{B}}(y,t)g(z,y,t)\|_{X^{*}}^{q^{\prime}}\,\frac{{\rm
d}y{\rm d}t}{t^{d+1}}{\rm d}z$ $\displaystyle=\int_{2B}H^{q^{\prime}}(z)\,{\rm
d}z,$

where

$H(z)=\left(\int_{{}^{d+1}_{+}}\|\mathbbm{1}_{\widehat{B}}(y,t)g(z,y,t)\|_{X^{*}}^{q^{\prime}}\,\frac{{\rm
d}y{\rm d}t}{t^{d+1}}\right)^{\frac{1}{q^{\prime}}}.$

Then we have

$\mathcal{C}_{q^{\prime}}\left[i^{-1}(N(g))\right](x)\lesssim\left({\mathcal{M}}(H^{q^{\prime}})(x)\right)^{\frac{1}{q^{\prime}}},$

where ${\mathcal{M}}$ is the Hardy-Littlewood maximal operator. Therefore when
$q^{\prime}<p^{\prime}\leq\infty$, we obtain

$\|\mathcal{C}_{q^{\prime}}\left[i^{-1}(N(g))\right]\|_{p^{\prime}}\lesssim\|{\mathcal{M}}(H^{q^{\prime}})^{\frac{1}{q^{\prime}}}\|_{p^{\prime}}\lesssim\left(\frac{p(q-1)}{q-p}\right)^{\frac{1}{q^{\prime}}}\|H\|_{p^{\prime}}\leq\left(\frac{p(q-1)}{q-p}\right)^{\frac{1}{q^{\prime}}}\|g\|_{L^{p^{\prime}}(F)}.$

Thus we observe

$\|f\|_{T^{p}_{q}(X)}\lesssim\left(\frac{p(q-1)}{q-p}\right)^{\frac{1}{q^{\prime}}}\sup_{g}|g(f)|,\quad
1<p<q,$

with the supremum being taken over all $g\in C_{c}({}^{d+1}_{+})\otimes X^{*}$
such that $\|\mathcal{C}_{q^{\prime}}(g)\|_{p^{\prime}}\leq 1$.

For the endpoint case $p=1$ of (3.4), because of the failure of vector-valued
Calderón-Zygmund theory, the above arguments adapted from [23, Theorem 2.4] do
not work any more. Instead, by using the atomic decomposition of
$T^{1}_{q}({}^{d+1}_{+};X)$—Lemma 3.1, one may carry out the classical
arguments as in [11, Theorem 1] in the present vector-valued setting, and we
leave the details to the interested reader.

When the Banach space $X^{*}$ has the Radon-Nikodým property, one gets
$F=E^{*}$ (cf. [25, Theorem 1.3.10]). Then the duality follows from then an
analogous argument in [23] for $1<p<\infty$. Again, the duality in the case
$p=1$ can be deduced as in the scalar-valued case [11, Theorem 1], and we
leave the details to the interested reader. ∎

###  3.2. The two linear operators $\mathcal{K}$ and $\pi_{L}$

Let ${\mathcal{K}}:{}^{d+1}_{+}\times{}^{d+1}_{+}\to\real$ be a reasonable
real-valued function such that for $f\in C_{c}({}^{d+1}_{+})\otimes X$, the
linear operator ${\mathcal{K}}$ is well defined as below,

${\mathcal{K}}(f)(x,t):=\int_{{}^{d+1}_{+}}{\mathcal{K}}_{t,s}(x,y)f(y,s)\,\frac{{\rm
d}y{\rm d}s}{s}.$

######  Lemma 3.5.

Let $X$ be any fixed Banach space and $1<q<\infty$. Assume that the kernel
${\mathcal{K}}_{t,s}(x,y)$ satisfies the following estimation: there exist
positive constants $\kappa$, $\varepsilon$, $C$ such that

(3.7)
$|{\mathcal{K}}_{t,s}(x,y)|\leq\frac{C\min\left\\{\frac{s}{t},\frac{t}{s}\right\\}^{\varepsilon}\min\left\\{\frac{1}{t},\frac{1}{s}\right\\}^{d}}{\left(1+\min\left\\{\frac{1}{t},\frac{1}{s}\right\\}|x-y|\right)^{d+\kappa}}.$

Then the linear operator ${\mathcal{K}}$ initially defined on
$C_{c}({}^{d+1}_{+})\otimes X$ extends to a bounded linear operator on
$T^{p}_{q}({}^{d+1}_{+};X)$ for $1\leq p<\infty$. More precisely,

$\|{\mathcal{K}}(f)\|_{T^{p}_{q}(X)}\lesssim_{\varepsilon,\kappa}p^{\frac{1}{q}}\|f\|_{T^{p}_{q}(X)},\quad\forall\,f\in
T^{p}_{q}({}^{d+1}_{+};X),\ 1\leq p<\infty.$

Furthermore, for any $f\in C_{c}({}^{d+1}_{+})\otimes X$, we have

$\|\mathcal{C}_{q}\left({\mathcal{K}}(f)\right)\|_{p}\lesssim_{\varepsilon,\kappa}\|\mathcal{C}_{q}(f)\|_{p},\quad
1\leq p\leq\infty.$

###### Proof.

Fix $f\in C_{c}({}^{d+1}_{+})\otimes X$. Without loss of generality, we can
assume $\kappa<\varepsilon$ from (3.7). We first deal with the case $p=q$. By
Hölder’s inequality, we have

$\displaystyle\int_{{}^{d+1}_{+}}\|{\mathcal{K}}(f)(x,t)\|_{X}^{q}\,\frac{{\rm
d}y{\rm d}t}{t}$
$\displaystyle=\int_{{}^{d+1}_{+}}\left\|\int_{{}^{d+1}_{+}}{\mathcal{K}}_{t,s}(y,w)f(w,s)\,\frac{{\rm
d}w{\rm d}s}{s}\right\|_{X}^{q}\,\frac{{\rm d}y{\rm d}t}{t}$
$\displaystyle\leq\int_{{}^{d+1}_{+}}\left(\int_{{}^{d+1}_{+}}|{\mathcal{K}}_{t,s}(y,w)|\,\frac{{\rm
d}w{\rm d}s}{s}\right)^{\frac{q}{q^{\prime}}}$
$\displaystyle\qquad\cdot\left(\int_{{}^{d+1}_{+}}|{\mathcal{K}}_{t,s}(y,w)|\|f(w,s)\|_{X}^{q}\,\frac{{\rm
d}w{\rm d}s}{s}\right)\,\frac{{\rm d}y{\rm d}t}{t}.$

We obtain that

$\displaystyle\int_{0}^{\infty}\int_{{}^{d}}|{\mathcal{K}}_{t,s}(y,w)|\,\frac{{\rm
d}w{\rm d}s}{s}$
$\displaystyle\leq\int_{0}^{t}\int_{{}^{d}}\frac{Cs^{\varepsilon}t^{-\varepsilon}t^{-d}}{\left(1+t^{-1}|y-w|\right)^{d+\kappa}}\,\frac{{\rm
d}w{\rm d}s}{s}$
$\displaystyle\qquad+\int_{t}^{\infty}\int_{{}^{d}}\frac{Ct^{\varepsilon}s^{-\varepsilon}s^{-d}}{\left(1+s^{-1}|y-w|\right)^{d+\kappa}}\,\frac{{\rm
d}w{\rm d}s}{s}$
$\displaystyle\lesssim_{\varepsilon,\kappa}\int_{{}^{d}}\frac{Ct^{-d}}{\left(1+t^{-1}|y-w|\right)^{d+\kappa}}\,{\rm
d}w+\int_{t}^{\infty}t^{\varepsilon}s^{-\varepsilon-1}\,{\rm d}s$
$\displaystyle\lesssim_{\varepsilon,\kappa}1.$

It is clear that in the assumption of ${\mathcal{K}}_{t,s}(y,w)$, $(w,s)$
plays the same role as $(y,t)$. Thus

(3.8)
$\|{\mathcal{K}}(f)\|_{L^{q}({}^{d+1}_{+};X)}^{q}\lesssim_{\varepsilon,\kappa}\int_{{}^{d+1}_{+}}\|f(w,s)\|_{X}^{q}\,\frac{{\rm
d}w{\rm d}s}{s}=\|f\|_{L^{q}({}^{d+1}_{+};X)}^{q}.$

Then the case $p=q$ is done since
$\|f\|_{T^{q}_{q}(X)}\approx{\|f\|_{L^{q}({}^{d+1}_{+};X)}}$. Moreover, from
the proof we observe that ${\mathcal{K}}$ is always bounded on
$L^{p}({}^{d+1}_{+};X)$ for $1\leq p\leq\infty$.

For $1\leq p<q$, by the interpolation—Lemma 3.3, it suffices to show the case
$p=1$. By the atomic decomposition—Lemma 3.1, It suffices to show that

(3.9) $\|{\mathcal{K}}(a)\|_{T^{1}_{q}(X)}\lesssim_{\varepsilon,\kappa}1,$

where $a$ is an $(X,q)$-atom with $\text{supp}\,a\subset\widehat{B}$ and
$B=B(c_{B},r_{B})$. One can write

$\displaystyle\|{\mathcal{A}}_{q}[{\mathcal{K}}(a)]\|_{1}$
$\displaystyle=\int_{4B}{\mathcal{A}}_{q}[{\mathcal{K}}(a)](x)\,{\rm
d}x+\int_{(4B)^{C}}{\mathcal{A}}_{q}[{\mathcal{K}}(a)](x)\,{\rm d}x$
$\displaystyle=I+II.$

From (3.8) we obtain

(3.10)
$\|{\mathcal{A}}_{q}[{\mathcal{K}}(a)]\|_{q}^{q}\lesssim_{\varepsilon,\kappa}\int_{{}^{d+1}_{+}}\|a(w,s)\|_{X}^{q}\,\frac{{\rm
d}w{\rm d}s}{s}\leq|B|^{1-q}.$

Then we can estimate the term $I$:

(3.11)
$I\leq|4B|^{\frac{1}{q^{\prime}}}\|{\mathcal{A}}_{q}[{\mathcal{K}}(a)]\|_{q}\lesssim_{\varepsilon,\kappa}1.$

Now we handle the second term $II$. By Hölder’s inequality, we observe

$\displaystyle\left({\mathcal{A}}_{q}[{\mathcal{K}}(a)](x)\right)^{q}$
$\displaystyle\leq\int_{0}^{\infty}\int_{|y-x|<t}\left(\int_{\widehat{B}}|{\mathcal{K}}_{t,s}(y,w)|^{q^{\prime}}\,\frac{{\rm
d}w{\rm
d}s}{s}\right)^{\frac{q}{q^{\prime}}}\cdot\left(\int_{\widehat{B}}\|a(w,s)\|_{X}^{q}\,\frac{{\rm
d}w{\rm d}s}{s}\right)\,\frac{{\rm d}y{\rm d}t}{t^{d+1}}$
$\displaystyle\lesssim|B|^{1-q}\int_{0}^{\infty}\int_{|y-x|<t}\left(\int_{\widehat{B}}|{\mathcal{K}}_{t,s}(y,w)|^{q^{\prime}}\,\frac{{\rm
d}w{\rm d}s}{s}\right)^{\frac{q}{q^{\prime}}}\,\frac{{\rm d}y{\rm
d}t}{t^{d+1}}$
$\displaystyle=|B|^{1-q}\int_{0}^{r_{B}}\int_{|y-x|<t}\left(\int_{\widehat{B}}|{\mathcal{K}}_{t,s}(y,w)|^{q^{\prime}}\,\frac{{\rm
d}w{\rm d}s}{s}\right)^{\frac{q}{q^{\prime}}}\,\frac{{\rm d}y{\rm
d}t}{t^{d+1}}$
$\displaystyle\qquad+|B|^{1-q}\int_{r_{B}}^{\infty}\int_{|y-x|<t}\left(\int_{\widehat{B}}|{\mathcal{K}}_{t,s}(y,w)|^{q^{\prime}}\,\frac{{\rm
d}w{\rm d}s}{s}\right)^{\frac{q}{q^{\prime}}}\,\frac{{\rm d}y{\rm
d}t}{t^{d+1}}$ $\displaystyle=:J_{1}+J_{2}.$

When $x\in(4B)^{C},\ w\in B$, we have

$r_{B}<|x-w|\leq|x-y|+|y-w|<t+|y-w|,$

hence

$|x-c_{B}|\leq|x-w|+|w-c_{B}|<2(t+|y-w|)\leq
2(\max\left\\{t,s\right\\}+|y-w|).$

Therefore we observe from (3.7) that

$\displaystyle|{\mathcal{K}}_{t,s}(y,w)|$
$\displaystyle\lesssim_{\varepsilon,\kappa}\frac{\min\left\\{\frac{s}{t},\frac{t}{s}\right\\}^{\varepsilon}\min\left\\{\frac{1}{t},\frac{1}{s}\right\\}^{d}}{\left(\max\left\\{t,s\right\\}+|y-w|\right)^{d+\kappa}\min\left\\{\frac{1}{t},\frac{1}{s}\right\\}^{d+\kappa}}$
$\displaystyle\lesssim_{\varepsilon,\kappa}\frac{\min\left\\{\frac{s}{t},\frac{t}{s}\right\\}^{\varepsilon}\min\left\\{\frac{1}{t},\frac{1}{s}\right\\}^{-\kappa}}{|x-c_{B}|^{d+\kappa}}=\frac{\min\left\\{s^{\varepsilon}t^{\kappa-\varepsilon},t^{\varepsilon}s^{\kappa-\varepsilon}\right\\}}{|x-c_{B}|^{d+\kappa}}.$

Then

$\displaystyle J_{1}$
$\displaystyle\lesssim_{\varepsilon,\kappa}\frac{|B|^{1-q}|B|^{\frac{q}{q^{\prime}}}}{|x-c_{B}|^{q(d+\kappa)}}\int_{0}^{r_{B}}\left(\int_{0}^{r_{B}}\min\left\\{s^{q^{\prime}\varepsilon}t^{q^{\prime}(\kappa-\varepsilon)},t^{q^{\prime}\varepsilon}s^{q^{\prime}(\kappa-\varepsilon)}\right\\}\,\frac{{\rm
d}s}{s}\right)^{\frac{q}{q^{\prime}}}\,\frac{{\rm d}t}{t}$
$\displaystyle=\frac{1}{|x-c_{B}|^{q(d+\kappa)}}\int_{0}^{r_{B}}\left(\int_{0}^{t}t^{q^{\prime}(\kappa-\varepsilon)}s^{q^{\prime}\varepsilon}\,\frac{{\rm
d}s}{s}+\int_{t}^{r_{B}}s^{q^{\prime}(\kappa-\varepsilon)}t^{q^{\prime}\varepsilon}\,\frac{{\rm
d}s}{s}\right)^{\frac{q}{q^{\prime}}}\frac{{\rm d}t}{t}$
$\displaystyle\lesssim_{\varepsilon,\kappa}\frac{r_{B}^{q\kappa}}{|x-c_{B}|^{q(d+\kappa)}}.$

For $J_{2}$, since $t\geq r_{B}\geq s$, and $|x-c_{B}|<2(t+|y-w|)$,

$\displaystyle J_{2}$
$\displaystyle\lesssim_{\varepsilon,\kappa}\frac{|B|^{1-q}|B|^{\frac{q}{q^{\prime}}}}{|x-c_{B}|^{q(d+\kappa)}}\int_{r_{B}}^{\infty}\left(\int_{0}^{r_{B}}s^{q^{\prime}\varepsilon}t^{q^{\prime}(\kappa-\varepsilon)}\,\frac{{\rm
d}s}{s}\right)^{\frac{q}{q^{\prime}}}\,\frac{{\rm d}t}{t}$
$\displaystyle\lesssim_{\varepsilon,\kappa}\frac{r_{B}^{q\kappa}}{|x-c_{B}|^{q(d+\kappa)}}.$

Thus

${\mathcal{A}}_{q}[{\mathcal{K}}(a)](x)\lesssim_{\varepsilon,\kappa}\frac{r_{B}^{\kappa}}{|x-c_{B}|^{d+\kappa}},\quad
x\in(4B)^{C}.$

Since

$\displaystyle\int_{(4B)^{C}}\frac{r_{B}^{\kappa}}{|x-c_{B}|^{d+\kappa}}\,{\rm
d}x$ $\displaystyle=\sum_{m=2}^{\infty}\int_{2^{m+1}B\setminus
2^{m}B}\frac{r_{B}^{\kappa}}{|x-c_{B}|^{d+\kappa}}\,{\rm
d}x\leq\sum_{m=2}^{\infty}\int_{2^{m+1}B}\frac{r_{B}^{\kappa}}{2^{m(d+\kappa)}r_{B}^{d+\kappa}}\,{\rm
d}x$
$\displaystyle\lesssim\sum_{m=2}^{\infty}\frac{(2^{m+1}r_{B})^{d}}{2^{m(d+\kappa)}r_{B}^{d}}\lesssim\sum_{m=2}^{\infty}2^{-m\kappa}\lesssim_{\kappa}1,$

we obtain

(3.12) $II\lesssim_{\varepsilon,\kappa}1.$

For the case $q<p<+\infty$, we denote by ${\mathcal{K}}^{*}$ the adjoint
operator. It is clear that the kernel of ${\mathcal{K}}^{*}$ has the same
estimation as that of ${\mathcal{K}}$. For $f\in C_{c}({}^{d+1}_{+})\otimes
X$, we obtain from Lemma 3.4 that

$\displaystyle\|{\mathcal{K}}(f)\|_{T^{p}_{q}(X)}$ $\displaystyle\lesssim
p^{\frac{1}{q}}\sup_{g}\left|\int_{{}^{d+1}_{+}}\left\langle{\mathcal{K}}(f)(x,t),g(x,t)\right\rangle_{X\times
X^{*}}\,\frac{{\rm d}x{\rm d}t}{t}\right|$
$\displaystyle=p^{\frac{1}{q}}\sup_{g}\left|\int_{{}^{d+1}_{+}}\left\langle
f(x,t),{\mathcal{K}}^{*}(g)(x,t)\right\rangle_{X\times X^{*}}\,\frac{{\rm
d}x{\rm d}t}{t}\right|$ $\displaystyle\leq
p^{\frac{1}{q}}\|f\|_{T^{p}_{q}(X)}\|{\mathcal{K}}^{*}(g)\|_{T^{p^{\prime}}_{q^{\prime}}(X^{*})}\lesssim
p^{\frac{1}{q}}\|f\|_{T^{p}_{q}(X)},$

where the supremum is taken over all $g\in C_{c}({}^{d+1}_{+})\otimes X$ such
that $\|{\mathcal{A}}_{q^{\prime}}(g)\|_{p^{\prime}}\leq 1$. Consequently, we
observe that ${\mathcal{K}}$ extends to a bounded linear operator on
$T^{p}_{q}({}^{d+1}_{+};X)$ for $1\leq p<\infty$. More precisely,

$\|{\mathcal{K}}(f)\|_{T^{p}_{q}(X)}\lesssim_{\varepsilon,\kappa}p^{\frac{1}{q}}\|f\|_{T^{p}_{q}(X)},\quad\forall\,f\in
T^{p}_{q}({}^{d+1}_{+};X).$

Now we prove the second assertion of this lemma. Fix $f\in
C_{c}({}^{d+1}_{+})\otimes X$, take a ball $B$ in d, we can write

$\displaystyle\left(\int_{\widehat{B}}\|{\mathcal{K}}(f)(x,t)\|_{X}^{q}\,\frac{{\rm
d}x{\rm d}t}{t}\right)^{\frac{1}{q}}$
$\displaystyle=\sup_{g}\left|\int_{\widehat{B}}\left\langle{\mathcal{K}}(f)(x,t),g(x,t)\right\rangle_{X\times
X^{*}}\,\frac{{\rm d}x{\rm d}t}{t}\right|$
$\displaystyle=\sup_{g}\left|\int_{\widehat{B}}\left\langle
f(x,t),{\mathcal{K}}^{*}(g)(x,t)\right\rangle_{X\times X^{*}}\,\frac{{\rm
d}x{\rm d}t}{t}\right|$
$\displaystyle\leq\sup_{g}\|{\mathcal{K}}^{*}(g)\|_{L^{q^{\prime}}(\widehat{B};X^{*})}\left(\int_{\widehat{B}}\|f(x,t)\|_{X}^{q}\,\frac{{\rm
d}x{\rm d}t}{t}\right)^{\frac{1}{q}},$

where the supremum is taken over all $g$ in the unit ball of
$L^{q^{\prime}}(\widehat{B};X^{*})$. From (3.8) we know that

$\|{\mathcal{K}}^{*}(g)\|_{L^{q^{\prime}}({}^{d+1}_{+};X^{*})}\lesssim_{\varepsilon,\kappa}\|g\|_{L^{q^{\prime}}(\widehat{B};X^{*})}.$

Thus for any $x\in{}^{d}$,

$\displaystyle\left(\mathcal{C}_{q}\left[{\mathcal{K}}(f)\right](x)\right)^{q}$
$\displaystyle=\sup_{x\in
B}\frac{1}{|B|}\int_{\widehat{B}}\|{\mathcal{K}}(f)(x,t)\|_{X}^{q}\,\frac{{\rm
d}x{\rm d}t}{t}$ $\displaystyle\lesssim_{\varepsilon,\kappa}\sup_{x\in
B}\frac{1}{|B|}\int_{\widehat{B}}\|f(x,t)\|_{X}^{q}\,\frac{{\rm d}x{\rm
d}t}{t}=\left(\mathcal{C}_{q}(f)(x)\right)^{q}.$

Therefore we obtain

$\|\mathcal{C}_{q}\left[{\mathcal{K}}(f)\right]\|_{p}\lesssim_{\varepsilon,\kappa}\|\mathcal{C}_{q}\left(f\right)\|_{p},\quad
1\leq p\leq\infty.$

Moreover, from (2.5) we also observe

$\|\mathcal{C}_{q}\left[{\mathcal{K}}(f)\right]\|_{p}\lesssim_{\varepsilon,\kappa}\left(\frac{p}{p-q}\right)^{\frac{1}{q}}\|{\mathcal{A}}_{q}(f)\|_{p},\quad
q<p\leq\infty.$

The proof is completed. ∎

Now we come to the second important linear operator, which will relate the
tent space $T^{p}_{q}({}^{d+1}_{+};X)$ to the Hardy space
$H^{p}_{q,L}({}^{d};X)$.

Recall the operator ${\mathcal{Q}}(f)(x,t)=-tLe^{-tL}(f)(x)$. Define the
operator $\pi_{L}$ acting on $C_{c}({}^{d+1}_{+})\otimes X$ as

$\pi_{L}(f)(x)=\int_{0}^{\infty}{\mathcal{Q}}(f(\cdot,t))(x,t)\,\frac{{\rm
d}t}{t},\quad\forall\,x\in{}^{d}.$

It is easy to verify that $\pi_{L}$ is well-defined. The following lemma
asserts that $\pi_{L}$ extends to a bounded linear operator from
$T^{p}_{q}({}^{d+1}_{+};X)$ to $H^{p}_{q,L}({}^{d};X)$. We will denote it by
$\pi_{L}$ as well.

######  Lemma 3.6.

Let $X$ be any fixed Banach space and $1<q<\infty$. The operator $\pi_{L}$
initially defined on $C_{c}({}^{d+1}_{+})\otimes X$ extends to a bounded
linear operator from $T^{p}_{q}({}^{d+1}_{+};X)$ to $H^{p}_{q,L}({}^{d};X)$
for $1\leq p<\infty$. More precisely,

$\|\pi_{L}(f)\|_{H^{p}_{q,L}(X)}\lesssim_{\beta}p^{\frac{1}{q}}\|f\|_{T^{p}_{q}(X)},\quad\forall\,f\in
T^{p}_{q}({}^{d+1}_{+};X),\ 1\leq p<\infty.$

Furthermore, for any $f\in C_{c}({}^{d+1}_{+})\otimes X$, we have

$\|\pi_{L}(f)\|_{BMO^{p}_{q,L}(X)}\lesssim_{\beta}\|\mathcal{C}_{q}(f)\|_{p},\quad
1\leq p\leq\infty.$

###### Proof.

Let $f\in C_{c}({}^{d+1}_{+})\otimes X$. Recall that $k(t,x,y)$ is the kernel
of the operator ${\mathcal{Q}}$, then

(3.13)
$\begin{split}{\mathcal{Q}}[\pi_{L}(f)](x,t)&=\int_{{}^{d}}k(t,y,z)\pi_{L}(f)(z)\,{\rm
d}z\\\
&=\int_{{}^{d}}k(t,y,z)\left(\int_{{}^{d+1}_{+}}k(s,z,w)f(w,s)\,\frac{{\rm
d}w{\rm d}s}{s}\right)\,{\rm d}z\\\
&=\int_{{}^{d+1}_{+}}\left(\int_{{}^{d}}k(t,y,z)k(s,z,w)\,{\rm
d}z\right)f(w,s)\,\frac{{\rm d}w{\rm d}s}{s}.\end{split}$

We denote by

$\Phi_{t,s}(y,w)=\int_{{}^{d}}k(t,y,z)k(s,z,w)\,{\rm d}z.$

Note that $k(t,\cdot,\cdot)$ is the kernel of the operator
${\mathcal{Q}}=-te^{-tL}$, thus $\Phi_{t,s}(\cdot,\cdot)$ is the kernel of
$-tLe^{-tL}\circ(-sLe^{-sL})=tsL^{2}e^{-(t+s)L}$. On the other hand,
$\partial^{2}_{r}(e^{-rL})|_{r=t+s}=L^{2}e^{-(t+s)L}$ which has the kernel
$\partial^{2}_{r}K(r,\cdot,\cdot)|_{r=t+s}$. Then by (2.4), we obtain

$\displaystyle|\Phi_{t,s}(y,w)|\lesssim_{d,\beta}\frac{ts}{(t+s)^{2-\beta}(t+s+|y-w|)^{d+\beta}}\lesssim_{\beta}\frac{\min\left\\{\frac{s}{t},\frac{t}{s}\right\\}\min\left\\{\frac{1}{t},\frac{1}{s}\right\\}^{d}}{\left(1+\min\left\\{\frac{1}{t},\frac{1}{s}\right\\}|x-y|\right)^{d+\beta}}.$

Denote by

(3.14) $\displaystyle\mathcal{P}=4\mathcal{Q}\circ\pi_{L}.$

From Lemma 3.5, we conclude that $\mathcal{P}$ initially defined on
$C_{c}({}^{d+1}_{+})\otimes X$ extends to a bounded linear operator on
$T^{p}_{q}({}^{d+1}_{+};X)$. Moreover,

$\left\|\mathcal{P}(f)\right\|_{T^{p}_{q}(X)}\lesssim_{\beta}p^{\frac{1}{q}}\left\|f\right\|_{T^{p}_{q}(X)}.$

Therefore

$\|\pi_{L}(f)\|_{H^{p}_{q,L}}=4^{-1}\|\mathcal{P}(f)\|_{T^{p}_{q}(X)}\lesssim_{\beta}p^{\frac{1}{q}}\|f\|_{T^{p}_{q}(X)},$

which is the desired assertion.

For the second part, we obtain the desired assertion from Lemma 3.5
immediately. ∎

######  Remark 3.7.

One can verify that $\mathcal{P}\circ\mathcal{P}=\mathcal{P}$, thus
$\mathcal{P}$ serves as a continuous projection from
$T^{p}_{q}\left({}^{d+1}_{+};X\right)$ onto itself. Indeed, we can also obtain
this lemma under the assumption that $L$ is a sectotrial operator satisfying
only (2.3).

##  4\. vector-valued intrinsic square functions

In this section, we begin with the introduction of vector-valued intrinsic
square functions, originally presented by Wilson in [42] in the case of
convolution operators. We then proceed to compare them with the $q$-variant of
Lusin area integral associated with a generator $L$.

Recall that $L$ is assumed to be a sectorial operator of type $\alpha$
($0\leq\alpha<\pi/2$) satisfying assumptions (1.5), (1.6) and (1.7) with
$\beta>0,0<\gamma\leq 1$. Define $\mathcal{H}_{\gamma,\beta}$ as the family of
functions $\varphi:{}^{d}\times{}^{d}\to\real$ such that

(4.1) $|\varphi(x,y)|\leq\frac{1}{(1+|x-y|)^{d+\beta}},$ (4.2)
$|\varphi(x+h,y)-\varphi(x,y)|+|\varphi(x,y+h)-\varphi(x,y)|\leq\frac{|h|^{\gamma}}{(1+|x-y|)^{d+\beta+\gamma}}$

whenever $2|h|\leq 1+|x-y|$ and

(4.3) $\int_{{}^{d}}\varphi(x,y)\,{\rm d}x=\int_{{}^{d}}\varphi(x,y)\,{\rm
d}y=0.$

For $\varphi\in\mathcal{H}_{\gamma,\beta}$, define
$\varphi_{t}(x,y)=t^{-d}\varphi(t^{-1}x,t^{-1}y)$.

Let $f\in C_{c}({}^{d})\otimes X$. We define

$A_{\gamma,\beta}(f)(x,t)=\sup_{\varphi\in\mathcal{H}_{\gamma,\beta}}\left\|\int_{{}^{d}}\varphi_{t}(x,y)f(y)\,{\rm
d}y\right\|_{X},\quad\forall\,(x,t)\in{}^{d+1}_{+}.$

Then the intrinsic square functions of $f$ are defined as

$S_{q,\gamma,\beta}(f)(x)=\left(\int_{\Gamma(x)}\left(A_{\gamma,\beta}(f)(y,t)\right)^{q}\,\frac{{\rm
d}y{\rm d}t}{t^{d+1}}\right)^{\frac{1}{q}},$

and

$G_{q,\gamma,\beta}(f)(x)=\left(\int_{0}^{\infty}\left(A_{\gamma,\beta}(f)(x,t)\right)^{q}\,\frac{{\rm
d}t}{t}\right)^{\frac{1}{q}}.$

######  Theorem 4.1.

Let $X$ be any fixed Banach space, $1<q<\infty$ and $1\leq p<\infty$. Let $L$
be any fixed sectorial operator $L$ satisfying (1.5), (1.6) and (1.7). For any
$f\in C_{c}(\mathbb{R}^{d})\otimes X$, we have

(4.4) $\displaystyle
S_{q,\gamma,\beta}(f)(x)\approx_{\gamma,\beta}G_{q,\gamma,\beta}(f)(x),$

(4.5) $\displaystyle S_{q,L}(f)(x)\lesssim S_{q,\gamma,\beta}(f)(x),\quad
G_{q,L}(f)(x)\lesssim G_{q,\gamma,\beta}(f)(x),$

and

(4.6)
$\displaystyle\|S_{q,\gamma,\beta}(f)\|_{p}\lesssim_{\gamma,\beta}p^{\frac{1}{q}}\|S_{q,L}(f)\|_{p}.$

######  Remark 4.2.

The following $g$-function version of (4.6) holds also

(4.7)
$\displaystyle\|G_{q,\gamma,\beta}(f)\|_{p}\lesssim_{\gamma,\beta}{p^{\frac{2}{q}}}\|G_{q,L}(f)\|_{p}.$

But its proof is much more involved and depends in turn on Theorem 1.1 that
will be concluded in the next section.

As in the classical case [41], the assertions (4.4) and (4.5) can be deduced
easily from the following facts on $\mathcal{H}_{\gamma,\beta}$.

######  Lemma 4.3.

Let $\varphi\in\mathcal{H}_{\gamma,\beta}$. The following properties hold:

(i) if $t\geq 1$, then
$t^{-d-\gamma}\varphi_{t}\in\mathcal{H}_{\gamma,\beta}$;

(ii) if $|z|\leq 1$, $t\geq 1$, then
$(2t)^{-d-\gamma-\beta}\left(\varphi^{(z)}\right)_{t}\in\mathcal{H}_{\gamma,\beta}$,
where $\varphi^{(z)}(x,y)=\varphi(x-z,y)$.

###### Proof.

The proof is similar to the case of Wilson [41], while the present setting is
non-convolutive, let us give the sketch. The claim (i) is easy by definition.
For the claim (ii), notice that

$2^{-1}(1+|x-y|)\leq 1+|(x-z)-y|\leq 2(1+|x-y|).$

By definition, we have

$|\varphi^{(z)}(x,y)|=|\varphi(x-z,y)|\leq\frac{1}{(1+|(x-z)-y|)^{d+\beta}}\leq\frac{2^{d+\beta}}{(1+|x-y|)^{d+\beta}}.$

and

$\displaystyle|\varphi^{(z)}(x+h,y)-\varphi^{(z)}(x,y)|$
$\displaystyle=|\varphi(x-z+h,y)-\varphi(x-z,y)|$
$\displaystyle\leq\frac{|h|^{\gamma}}{(1+|(x-z)-y|)^{d+\beta+\gamma}}$
$\displaystyle\leq\frac{2^{d+\beta+\gamma}|h|^{\gamma}}{(1+|x-y|)^{d+\beta+\gamma}}.$

The same Hölder continuity estimation holds for the variable $y$. Thus we
obtain $2^{-d-\beta-\gamma}\varphi^{(z)}\in\mathcal{H}_{\gamma,\beta}$. Then
the claim (ii) follows from the claim (i). ∎

With Lemma 4.3, the assertions (4.4) and (4.5) will follow easily. The most
challenging part of Theorem 4.4 lies in (4.6). In addition to the
interpolation and duality theory on the (vector-valued) tent space that have
been built in Section 3, the following pointwise estimate is another technical
part in the proof of estimate (4.6).

Recall that $k(t,x,y)$ is the kernel of the operator ${\mathcal{Q}}$. Let
${\theta\in\mathcal{H}_{\gamma,\beta}}$, define

${\mathcal{L}}^{\theta}_{t,s}(y,w)=\int_{{}^{d}}\theta_{t}(y,z)k(s,z,w)\,{\rm
d}z.$

######  Lemma 4.4.

Let $\nu=2^{-1}\min\left\\{\gamma,\beta\right\\}$ and
$\zeta=(d+2^{-1}\beta)(d+\beta)^{-1}$, then

$\sup_{\theta\in\mathcal{H}_{\gamma,\beta}}|{\mathcal{L}}^{\theta}_{t,s}(y,w)|\lesssim_{\gamma,\beta}\frac{\min\left\\{\frac{s}{t},\frac{t}{s}\right\\}^{(1-\zeta)\nu}\min\left\\{\frac{1}{t},\frac{1}{s}\right\\}^{d}}{\left(1+\min\left\\{\frac{1}{t},\frac{1}{s}\right\\}|y-w|\right)^{d+\frac{1}{2}\beta}}.$

###### Proof.

To estimate the kernel ${\mathcal{L}}^{\theta}_{t,s}(y,w)$, we follow a
similar argument presented in [20, Chapter 8, 8.6.3].

Let $\theta\in\mathcal{H}_{\gamma,\beta}$, we have

(4.8)
$|\theta_{t}(y,z)|\leq\frac{t^{-d}}{(1+t^{-1}|y-z|)^{d+\beta}},\quad\forall\,y,z\in{}^{d},\
t>0.$

For $2|z-z^{\prime}|<t+|y-z|$, we have $t^{-1}|z-z^{\prime}|<1+t^{-1}|y-z|$,
then

$\displaystyle|\theta_{t}(y,z)-\theta_{t}(y,z^{\prime})|$
$\displaystyle\leq\frac{t^{-d-\gamma}|z-z^{\prime}|^{\gamma}}{(1+t^{-1}|y-z|)^{d+\beta+\gamma}}\leq\frac{\min\left\\{(t^{-1}|z-z^{\prime}|)^{\gamma},\left(1+t^{-1}|y-z|\right)^{\gamma}\right\\}}{t^{d}(1+t^{-1}|y-z|)^{d+\beta+\gamma}}$
$\displaystyle\lesssim\frac{\min\left\\{1,(t^{-1}|z-z^{\prime}|)^{\gamma}\right\\}}{t^{d}}.$

For $2|z-z^{\prime}|\geq t+|y-z|$, we have $t^{-1}|z-z^{\prime}|\geq 1/2$,
then

$|\theta_{t}(y,z)-\theta_{t}(y,z^{\prime})|\leq|\theta_{t}(y,z)|+|\theta_{t}(y,z^{\prime})|\leq
2t^{-d}\lesssim\frac{\min\left\\{1,(t^{-1}|z-z^{\prime}|)^{\gamma}\right\\}}{t^{d}}.$

Hence

$|\theta_{t}(y,z)-\theta_{t}(y,z^{\prime})|\lesssim\frac{\min\left\\{1,(t^{-1}|z-z^{\prime}|)^{\gamma}\right\\}}{t^{d}},\quad\forall\,y,z,z^{\prime}\in{}^{d},\
t>0.$

On the other hand, Lemma 2.1 asserts that there exists a positive constant
$C_{k}$ such that
$C_{k}^{-1}(k(s,\cdot,\cdot))_{s^{-1}}\in\mathcal{H}_{\gamma,\beta}$ (see also
the Convention afterwards). Thus, one gets for all $w,z,z^{\prime}\in{}^{d},\
s>0$,

$|k(s,z,w)|\lesssim\frac{C_{k}s^{-d}}{(1+s^{-1}|z-w|)^{d+\beta}},\quad|k(s,z,w)-k(s,z^{\prime},w)|\lesssim\frac{C_{k}\min\left\\{1,(s^{-1}|z-z^{\prime}|)^{\gamma}\right\\}}{s^{d}}.$

Now we start to deal with the kernel ${\mathcal{L}}^{\theta}_{t,s}(y,w)$. By
symmetry, it suffices to handle the case $s\leq t$. First we observe the
following estimate,

$\displaystyle\int_{{}^{d}}\frac{s^{-d}\min\left\\{1,(t^{-1}|u|)^{\gamma}\right\\}}{(1+s^{-1}|u|)^{d+\beta}}\,{\rm
d}u$
$\displaystyle=\int_{|u|<t}\frac{s^{-d}(t^{-1}|u|)^{\gamma}}{(1+s^{-1}|u|)^{d+\beta}}\,{\rm
d}u+\int_{|u|>t}\frac{s^{-d}}{(1+s^{-1}|u|)^{d+\beta}}\,{\rm d}u$
$\displaystyle\leq\int_{|v|<t/s}\left(\frac{s}{t}\right)^{\gamma}\frac{|v|^{\gamma}}{(1+|v|)^{d+\beta}}\,{\rm
d}v+\int_{|u|>t}s^{\beta}|u|^{-d-\beta}\,{\rm d}u$
$\displaystyle=:J_{1}+J_{2}$

Taking $\nu=2^{-1}\min\left\\{\gamma,\beta\right\\}$, and we have
$|v|^{\gamma}<(t/s)^{\gamma-\nu}|v|^{\nu}$. Then we obtain

$J_{1}\leq\left(\frac{s}{t}\right)^{\nu}\int_{{}^{d}}\frac{|v|^{\nu}}{(1+|v|)^{d+\beta}}\,{\rm
d}v\lesssim_{\gamma,\beta}\left(\frac{s}{t}\right)^{\nu}.$

For $J_{2}$, we have

$J_{2}\lesssim\int_{t}^{\infty}s^{\beta}r^{-\beta-1}\,{\rm
d}r\lesssim_{\beta}\left(\frac{s}{t}\right)^{\beta}\leq\left(\frac{s}{t}\right)^{\nu}.$

Thus for $s\leq t$, by the vanishing property (4.3) of $k(s,\cdot,w)$, one
gets

$\displaystyle|{\mathcal{L}}^{\theta}_{t,s}(y,w)|$
$\displaystyle\leq\left|\int_{{}^{d}}\left[\theta_{t}(y,z)-\theta_{t}(y,w)\right]k(s,z,w)\,{\rm
d}z\right|$ $\displaystyle\leq
C_{k}\int_{{}^{d}}\frac{\min\left\\{1,(t^{-1}|z-w|)^{\gamma}\right\\}}{t^{d}}\frac{s^{-d}}{(1+s^{-1}|z-w|)}\,{\rm
d}z$
$\displaystyle\lesssim_{\gamma,\beta}t^{-d}\left(\frac{s}{t}\right)^{\nu}\leq\min\left\\{\frac{1}{t},\frac{1}{s}\right\\}^{d}\min\left\\{\frac{s}{t},\frac{t}{s}\right\\}^{\nu}.$

On the other hand,

$|{\mathcal{L}}^{\theta}_{t,s}(y,w)|\leq\int_{{}^{d}}|\theta_{t}(y,z)||k(s,z,w)|\,{\rm
d}z\lesssim_{\beta}\frac{\min\left\\{\frac{1}{t},\frac{1}{s}\right\\}^{d}}{\left(1+\min\left\\{\frac{1}{t},\frac{1}{s}\right\\}|y-w|\right)^{d+\beta}}.$

Let $\zeta=(d+2^{-1}\beta)(d+\beta)^{-1}$, we then get

$|{\mathcal{L}}^{\theta}_{t,s}(y,w)|=|{\mathcal{L}}^{\theta}_{t,s}(y,w)|^{1-\zeta}|{\mathcal{L}}^{\theta}_{t,s}(y,w)|^{\zeta}\lesssim_{\gamma,\beta}\frac{\min\left\\{\frac{s}{t},\frac{t}{s}\right\\}^{(1-\zeta)\nu}\min\left\\{\frac{1}{t},\frac{1}{s}\right\\}^{d}}{\left(1+\min\left\\{\frac{1}{t},\frac{1}{s}\right\\}|y-w|\right)^{d+\frac{1}{2}\beta}}.$

It is clear that the estimation of ${\mathcal{L}}^{\theta}_{t,s}(y,w)$ is
independent of the choice of $\theta$, and thus the desired estimate is
obtained. ∎

Now let us prove Theorem 4.1.

###### Proof.

The pointwise estimate (4.4) follows from Lemma 4.3 (ii). Indeed, for
$|x-y|<t$, let $w=(x-y)/t$; then for any
$\varphi\in\mathcal{H}_{\gamma,\beta}$, we have
$2^{-d-\beta-\gamma}\varphi^{(w)}\in\mathcal{H}_{\gamma,\beta}$. Hence

$\displaystyle A_{\gamma,\beta}(f)(x,t)$
$\displaystyle=\sup_{\varphi\in\mathcal{H}_{\gamma,\beta}}\left\|\int_{{}^{d}}\varphi_{t}(x,z)f(z)\,{\rm
d}z\right\|_{X}$ $\displaystyle\leq
2^{d+\beta+\gamma}\sup_{\varphi^{(w)}\in\mathcal{H}_{\gamma,\beta}}\left\|\int_{{}^{d}}\left(\varphi^{(w)}\right)_{t}(x,z)f(z)\,{\rm
d}z\right\|_{X}$ $\displaystyle=2^{d+\beta+\gamma}A_{\gamma,\beta}(f)(y,t).$

Exchanging $x$ and $y$ and taking $-w$ in place of $w$, the reverse inequality
is also true. Then (4.4) follows immediately.

Now we turn to the pointwise estimates (4.5). Lemma 2.1 asserts that there
exists a positive constant $C_{k}$ such that
$C_{k}^{-1}(k(t,\cdot,\cdot))_{t^{-1}}\in\mathcal{H}_{\gamma,\beta}$ (see also
the Convention afterwards). Consequently, for all $x\in{}^{d},\ t>0$, we have

(4.9) $\displaystyle\|{\mathcal{Q}}(f)(x,t)\|_{X}$
$\displaystyle=\left\|\int_{{}^{d}}k(t,x,y)f(y)\,{\rm
d}y\right\|_{X}=C_{k}\left\|\int_{{}^{d}}\left(C_{k}^{-1}(k(t,x,y))_{t^{-1}}\right)_{t}f(y)\,{\rm
d}y\right\|_{X}$ $\displaystyle\leq
C_{k}\sup_{\varphi\in\mathcal{H}_{\gamma,\beta}}\left\|\int_{{}^{d}}\varphi_{t}(x,y)f(y)\,{\rm
d}y\right\|_{X}=C_{k}A_{\gamma,\beta}(f)(x,t).$

Then the estimates (4.5) follows trivially.

Below we explain the proof of (4.6). Let $h\in C_{c}({}^{d+1}_{+})\otimes X$,
we have

$\displaystyle A_{\gamma,\beta}[\pi_{L}(h)](y,t)$
$\displaystyle=\sup_{\theta\in\mathcal{H}_{\gamma,\beta}}\left\|\int_{{}^{d}}\theta_{t}(y,z)\left(\int_{{}^{d+1}_{+}}k(s,z,w)h(w,s)\,\frac{{\rm
d}w{\rm d}s}{s}\right)\,{\rm d}z\right\|_{X}$
$\displaystyle=\sup_{\theta\in\mathcal{H}_{\gamma,\beta}}\left\|\int_{{}^{d+1}_{+}}\left(\int_{{}^{d}}\theta_{t}(y,z)k(s,z,w)\,{\rm
d}z\right)h(w,s)\,\frac{{\rm d}w{\rm d}s}{s}\right\|_{X}$
$\displaystyle=\sup_{\theta\in\mathcal{H}_{\gamma,\beta}}\left\|\int_{{}^{d+1}_{+}}{\mathcal{L}}^{\theta}_{t,s}(y,w)h(w,s)\,\frac{{\rm
d}w{\rm d}s}{s}\right\|_{X}$
$\displaystyle\leq\int_{{}^{d+1}_{+}}\left(\sup_{\theta\in\mathcal{H}_{\gamma,\beta}}|{\mathcal{L}}^{\theta}_{t,s}(y,w)|\right)\|h(w,s)\|_{X}\,\frac{{\rm
d}w{\rm d}s}{s}$ $\displaystyle=:{\mathcal{L}}(\|h\|_{X})(y,t),$

where the linear operator ${\mathcal{L}}$ has the kernel

${\mathcal{L}}_{t,s}(y,w)=\sup_{\theta\in\mathcal{H}_{\gamma,\beta}}|{\mathcal{L}}^{\theta}_{t,s}(y,w)|.$

Then by Lemma 4.4 and Lemma 3.5 in the case $X={\mathbb{C}}$, one obtaines

$\|{\mathcal{L}}(\|h\|_{X})\|_{T^{p}_{q}({\mathbb{C}})}\lesssim_{\gamma,\beta}p^{\frac{1}{q}}\|\|h\|_{X}\|_{T^{p}_{q}({\mathbb{C}})}=p^{\frac{1}{q}}\|h\|_{T^{p}_{q}(X)},\quad
1\leq p<\infty.$

Therefore

$\|A_{\gamma,\beta}[\pi_{L}(h)]\|_{T^{p}_{q}({\mathbb{C}})}\lesssim_{\gamma,\beta}p^{\frac{1}{q}}\|h\|_{T^{p}_{q}(X)},\quad
1\leq p<\infty.$

Let $f\in C_{c}(\mathbb{R}^{d})\otimes X$, then we have ${\mathcal{Q}}(f)\in
T^{p}_{q}({}^{d+1}_{+};X)$; moreover from the formula (2.1) and the fact that
the fixed point subspace of $L^{p}(\mathbb{R}^{d};X)$ is $0$ (see the
statement before Remark 2.3), the following Calderón identity holds

(4.10)
$f=4\int_{0}^{\infty}{\mathcal{Q}}\left[{\mathcal{Q}}(f)(\cdot,t)\right](\cdot,t)\,\frac{{\rm
d}t}{t}.$

Therefore, one has that for $1\leq p<\infty$,

$\displaystyle\|S_{q,\gamma,\beta}(f)\|_{p}$
$\displaystyle=\|A_{\gamma,\beta}(f)\|_{T^{p}_{q}({\mathbb{C}})}=4\|A_{\gamma,\beta}\left[\pi_{L}({\mathcal{Q}}(f))\right]\|_{T^{p}_{q}({\mathbb{C}})}$
$\displaystyle\lesssim_{\gamma,\beta}p^{\frac{1}{q}}\|{\mathcal{Q}}(f)\|_{T^{p}_{q}(X)}=p^{\frac{1}{q}}\|S_{q,L}(f)\|_{p},$

which is the desired inequality. ∎

######  Remark 4.5.

For any $f\in C_{c}({}^{d})\otimes X$, by Lemma 3.5, we also obtain

$\|A_{\gamma,\beta}(f)\|_{T^{\infty}_{q}({\mathbb{C}})}=4\|A_{\gamma,\beta}\left[\pi_{L}({\mathcal{Q}}(f))\right]\|_{T^{\infty}_{q}({\mathbb{C}})}\lesssim_{\gamma,\beta}\|{\mathcal{Q}}(f)\|_{T^{\infty}_{q}(X)}=\|f\|_{BMO_{q,L}(X)}.$

Together with the pointwise estimate (4.9), one gets the BMO–version of
Theorem 1.1: Let $L$ be a generator as in Theorem 1.1, then

(4.11)
$\|f\|_{BMO_{q,L}(X)}\approx_{\gamma,\beta}\|f\|_{BMO_{q,\sqrt{\Delta}}(X)}.$

##  5\. Proof of the main Theorem

As pointed out in the introduction, the equivalence (1.8) in Theorem 1.1 is an
easy consequence of Theorem 4.1; but for another equivalence (1.9), we need to
develop fully Mei’s duality arguments between vector-valued Hardy and BMO type
spaces [31]. This will be accomplished in the present section by combining the
theory of vector-valued tent spaces and vector-valued Wilson’s square
functions—Theorem 4.1.

First of all, based on the duality between tent spaces—Lemma 3.4, the
boundedness of the projection $\pi_{L}$—Lemma 3.6—yields the following vector-
valued Fefferman-Stein duality theorem.

######  Theorem 5.1.

Let $X$ be any fixed Banach space and $1<q<\infty$. Let $L$ be as in Theorem
1.1. Both the spaces $BMO_{q^{\prime},L^{*}}^{p^{\prime}}({}^{d};X^{*})$ and
$H^{p^{\prime}}_{q^{\prime},L^{*}}({}^{d};X^{*})$ are isomorphically
identified as subspaces of the dual space of $H^{p}_{q,L}({}^{d};X)$.
Moreover, they are norming for $H^{p}_{q,L}({}^{d};X)$ in the following sense,

$\|f\|_{H^{p}_{q,L}}\lesssim_{\beta}\max\left\\{p^{\frac{1}{q}}p^{\prime\frac{1}{q^{\prime}}},p^{\prime\frac{2}{q^{\prime}}}\right\\}\sup_{g}\left|\int_{{}^{d}}\left\langle
f(x),g(x)\right\rangle_{X\times X^{*}}\,{\rm d}x\right|,\quad 1<p<\infty,$

where the supremum is taken over all $g\in C_{c}({}^{d})\otimes X^{*}$ such
that $\|g\|_{H^{p^{\prime}}_{q^{\prime},L^{*}}(X^{*})}\leq 1$, and similarly,

$\|f\|_{H^{p}_{q,L}}\lesssim_{\beta}\left(\frac{p(q-1)}{q-p}\right)^{\frac{1}{q^{\prime}}}\sup_{g}\left|\int_{{}^{d}}\left\langle
f(x),g(x)\right\rangle_{X\times X^{*}}\,{\rm d}x\right|,\quad 1\leq p<q,$

where the supremum is taken over all $g\in C_{c}({}^{d})\otimes X^{*}$ such
that $\|g\|_{BMO^{p^{\prime}}_{q^{\prime},L^{*}}(X^{*})}\leq 1$. Furthermore,
if $X^{*}$ has the Radon-Nikodým property. Then

$\displaystyle
BMO^{p^{\prime}}_{q^{\prime},L^{*}}(X^{*})=\left(H^{p}_{q,L}({}^{d};X)\right)^{*},\quad
1\leq p<q;$ $\displaystyle
H^{p^{\prime}}_{q^{\prime},L^{*}}(X^{*})=\left(H^{p}_{q,L}({}^{d};X)\right)^{*},\quad
1<p<\infty.$

######  Remark 5.2.

Indeed, we can also obtain this duality theorem under the assumption that $L$
be a sectotrial operator satisfying only (2.3), see Remark 3.7.

The more essential auxiliary result is the following duality property, which
is inspired by [31, Theorem 2.4] (see also [43, 47]).

######  Proposition 5.3.

Let $X$ be any fixed Banach space and $1\leq p<q$. Let $L$ be any fixed
sectorial operator satisfying (1.5), (1.6) and (1.7). Then for any $f\in
C_{c}({}^{d})\otimes X$ and $g\in C_{c}({}^{d})\otimes X^{*}$, one has

(5.1) $\left|\int_{{}^{d}}\left\langle f(x),g(x)\right\rangle_{X\times
X^{*}}dx\right|\lesssim_{\gamma,\beta}\|G_{q,L}(f)\|_{p}^{\frac{p}{q}}\|S_{q,L}(f)\|_{p}^{1-\frac{p}{q}}\|g\|_{BMO^{p^{\prime}}_{q^{\prime},L^{*}}(X^{*})}.$

###### Proof.

Fixing $f\in C_{c}({}^{d})\otimes X$ and $g\in C_{c}({}^{d})\otimes X^{*}$, we
consider truncated versions of $G_{q,L}(f)(x)$ as follows:

$G(x,t):=\left(\int_{t}^{\infty}\|{\mathcal{Q}}(f)(x,s)\|_{X}^{q}\frac{dxds}{s}\right)^{\frac{1}{q}},\quad
x\in{}^{d},t>0.$

By approximation, we can assume that $G(x,t)$ is strictly positive. The
operator $-tL^{*}e^{-tL^{*}}$ is denoted by ${\mathcal{Q}}^{*}$. By the
Calderón identity—(4.10), we have

$\displaystyle\left|\int_{{}^{d}}\left\langle f(x),g(x)\right\rangle_{X\times
X^{*}}\,{\rm d}x\right|$
$\displaystyle=4\left|\int_{{}^{d+1}_{+}}\left\langle{\mathcal{Q}}(f)(x,t),{\mathcal{Q}}^{*}(g)(x,t)\right\rangle_{X\times
X^{*}}\,\frac{{\rm d}x{\rm d}t}{t}\right|$
$\displaystyle=4\left|\int_{{}^{d+1}_{+}}\left\langle
G^{\frac{p-q}{q}}(x,t){\mathcal{Q}}(f)(x,t),G^{\frac{q-p}{q}}(x,t){\mathcal{Q}}^{*}(g)(x,t)\right\rangle_{X\times
X^{*}}\,\frac{{\rm d}x{\rm d}t}{t}\right|$
$\displaystyle\lesssim\left(\int_{{}^{d+1}_{+}}G^{p-q}(x,t)\|{\mathcal{Q}}(f)(x,t)\|_{X}^{q}\,\frac{{\rm
d}x{\rm d}t}{t}\right)^{\frac{1}{q}}$
$\displaystyle\qquad\cdot\left(\int_{{}^{d+1}_{+}}G^{\frac{q-p}{q-1}}(x,t)\|{\mathcal{Q}}^{*}(g)(x,t)\|_{X^{*}}^{q^{\prime}}\,\frac{{\rm
d}x{\rm d}t}{t}\right)^{\frac{1}{q^{\prime}}}$ $\displaystyle=I\cdot II.$

The term $I$ is estimated as below,

$\displaystyle I^{q}$
$\displaystyle=-\int_{{}^{d}}\int_{0}^{\infty}G^{p-q}(x,t)\partial_{t}\left(G^{q}(x,t)\right)\,{\rm
d}t{\rm d}x$
$\displaystyle=-q\int_{{}^{d}}\int_{0}^{\infty}G^{p-1}(x,t)\partial_{t}G(x,t)\,{\rm
d}t{\rm d}x$
$\displaystyle\leq-q\int_{{}^{d}}\int_{0}^{\infty}G^{p-1}(x,0)\partial_{t}G(x,t)\,{\rm
d}t{\rm d}x$ $\displaystyle=q\int_{{}^{d}}G^{p}(x,0)\,{\rm d}t{\rm
d}x=q\|G_{q,L}(f)\|_{p}^{p},$

since $G(x,t)$ is decreasing in $t$, and $G(x,0)=G_{q,L}(f)(x)$.

For the term $II$, we introduce two more variants of $S_{q,\gamma,\beta}(f)$
(cf. [47]). The first is defined similarly to $G(\cdot,t)$:

$S(x,t)=\left(\int_{t}^{\infty}\int_{|y-x|<s-\frac{t}{2}}\left(A_{\gamma,\beta}(f)(y,s)\right)^{q}\,\frac{{\rm
d}y{\rm d}s}{s^{d+1}}\right)^{\frac{1}{q}},\quad x\in{}^{d},\ t>0.$

To introduce the second one, let ${\mathscr{D}}_{k}$ be the family of dyadic
cubes in d of side length $2^{-k}$, that is,

${\mathscr{D}}_{k}=\left\\{2^{-k}\prod_{j=1}^{d}[m_{j},m_{j}+1):m_{j}\in{\mathbb{Z}},k\in{\mathbb{Z}}\right\\}.$

Denote $c_{Q}$ as the center of a cube $Q$. Then, we define

$\mathcal{S}(x,k)=\left(\int_{\sqrt{d}2^{-k}}^{\infty}\int_{|y-c_{Q}|<s}\left(A_{\gamma,\beta}(f)(y,s)\right)^{q}\,\frac{{\rm
d}y{\rm d}s}{s^{d+1}}\right)^{\frac{1}{q}},\quad\text{if }x\in
Q\in{\mathscr{D}}_{k},\ k\in{\mathbb{Z}}.$

By definition, we have the following properties,

  * (i)

$\mathcal{S}(\cdot,k)$ is increasing in $k$,

  * (ii)

$\mathcal{S}(\cdot,k)$ is constant on every cube $Q\in{\mathscr{D}}_{k}$,

  * (iii)

$\mathcal{S}(x,-\infty)=0$ and
$\mathcal{S}(x,\infty)=S(x,0)=S_{q,\gamma,\beta}(f)(x)$.

If $s\geq t\geq\sqrt{d}2^{-k}$ and $x\in Q\in{\mathscr{D}}_{k}$, then
$B(x,s-\frac{t}{2})\subset B(c_{Q},s)$, where $B(x,t)$ denotes the ball with
center $x$ and radius $t$. This implies

$S(x,t)\leq\mathcal{S}(x,k),\quad x\in Q\in{\mathscr{D}}_{k}\text{ whenever
}t\geq\sqrt{d}2^{-k}.$

Using (4.4) and (4.5) we have

$G_{q,L}(f)(x)\lesssim
G_{q,\gamma,\beta}(f)(x)\approx_{\gamma,\beta}S_{q,\gamma,\beta}(f)(x),$

and similarly,

(5.2) $G(x,t)\lesssim_{\gamma,\beta}S(x,t).$

Now we proceed to estimate the term $B$ based on these observations. Applying
(5.2) to $II$, we have

$\displaystyle II^{q^{\prime}}$
$\displaystyle\lesssim_{\gamma,\beta}\int_{{}^{d+1}_{+}}S^{\frac{q-p}{q-1}}(x,t)\|{\mathcal{Q}}^{*}(g)(x,t)\|_{X^{*}}^{q^{\prime}}\,\frac{{\rm
d}x{\rm d}t}{t}$
$\displaystyle=\sum_{k=-\infty}^{\infty}\sum_{Q\in{\mathscr{D}}_{k}}\int_{Q}\int_{\sqrt{d}2^{-k}}^{\sqrt{d}2^{-k+1}}S^{\frac{q-p}{q-1}}(x,t)\|{\mathcal{Q}}^{*}(g)(x,t)\|_{X^{*}}^{q^{\prime}}\,\frac{{\rm
d}t}{t}{\rm d}x$
$\displaystyle\leq\sum_{k=-\infty}^{\infty}\sum_{Q\in{\mathscr{D}}_{k}}\int_{Q}\int_{\sqrt{d}2^{-k}}^{\sqrt{d}2^{-k+1}}\mathcal{S}^{\frac{q-p}{q-1}}(x,k)\|{\mathcal{Q}}^{*}(g)(x,t)\|_{X^{*}}^{q^{\prime}}\,\frac{{\rm
d}t}{t}{\rm d}x$
$\displaystyle=\int_{{}^{d}}\sum_{k=-\infty}^{\infty}\sum_{j=-\infty}^{k}D(x,j)\int_{\sqrt{d}2^{-k}}^{\sqrt{d}2^{-k+1}}\|{\mathcal{Q}}^{*}(g)(x,t)\|_{X^{*}}^{q^{\prime}}\,\frac{{\rm
d}t}{t}{\rm d}x$

where
$D(x,j)=\mathcal{S}^{\frac{q-p}{q-1}}(x,j)-\mathcal{S}^{\frac{q-p}{q-1}}(x,j-1)$.
Then $D(x,j)$ is constant on every cube $Q\in{\mathscr{D}}_{j}$. Thus

$\displaystyle II^{q^{\prime}}$
$\displaystyle\lesssim_{\gamma,\beta}\int_{{}^{d}}\sum_{j=-\infty}^{\infty}D(x,j)\left(\sum_{k=j}^{\infty}\int_{\sqrt{d}2^{-k}}^{\sqrt{d}2^{-k+1}}\|{\mathcal{Q}}^{*}(g)(x,t)\|_{X^{*}}^{q^{\prime}}\,\frac{{\rm
d}t}{t}\right){\rm d}x$
$\displaystyle=\sum_{j=-\infty}^{\infty}\sum_{Q\in{\mathscr{D}}_{j}}\int_{Q}D(x,j)\int_{0}^{\sqrt{d}2^{-j+1}}\|{\mathcal{Q}}^{*}(g)(x,t)\|_{X^{*}}^{q^{\prime}}\,\frac{{\rm
d}t}{t}{\rm d}x$
$\displaystyle=\sum_{j=-\infty}^{\infty}\sum_{Q\in{\mathscr{D}}_{j}}D(x,j)\mathbbm{1}_{Q}(x)\int_{Q}\int_{0}^{2\sqrt{d}\ell(Q)}\|{\mathcal{Q}}^{*}(g)(x,t)\|_{X^{*}}^{q^{\prime}}\,\frac{{\rm
d}t}{t}{\rm d}x,$

where $\ell(Q)$ denotes the length of $Q$. There exists a ball $B$ such that
$Q\subset B$, $Q\times(0,2\sqrt{d}\ell(Q)]\subset\widehat{B}$ and
$|B|\lesssim|Q|$. Then we deduce that

$\int_{Q}\int_{0}^{2\sqrt{d}\ell(Q)}\|{\mathcal{Q}}^{*}(g)(x,t)\|_{X^{*}}^{q^{\prime}}\,\frac{{\rm
d}t}{t}{\rm d}x\leq\inf_{y\in
B}\left\\{\mathcal{C}_{q^{\prime}}\left[{\mathcal{Q}}^{*}(g)\right](y)\right\\}^{q^{\prime}}|B|\lesssim\inf_{y\in
Q}\left\\{\mathcal{C}_{q^{\prime}}\left[{\mathcal{Q}}^{*}(g)\right](y)\right\\}^{q^{\prime}}|Q|.$

Therefore

$\displaystyle II^{q^{\prime}}$
$\displaystyle\lesssim_{\gamma,\beta}\sum_{j=-\infty}^{\infty}\sum_{Q\in{\mathscr{D}}_{j}}D(x,j)\mathbbm{1}_{Q}(x)\inf_{y\in
Q}\left\\{\mathcal{C}_{q^{\prime}}\left[{\mathcal{Q}}^{*}(g)\right](y)\right\\}^{q^{\prime}}|Q|$
$\displaystyle\leq\sum_{j=-\infty}^{\infty}\sum_{Q\in{\mathscr{D}}_{j}}\int_{Q}D(x,j)\left(\mathcal{C}_{q^{\prime}}\left[{\mathcal{Q}}^{*}(g)\right](x)\right)^{q^{\prime}}\,{\rm
d}x\leq\int_{{}^{d}}\sum_{j=-\infty}^{\infty}D(x,j)\left(\mathcal{C}_{q^{\prime}}\left[{\mathcal{Q}}^{*}(g)\right](x)\right)^{q^{\prime}}\,{\rm
d}x$
$\displaystyle=\int_{{}^{d}}\mathcal{S}^{\frac{q-p}{q-1}}(x,\infty)\left(\mathcal{C}_{q^{\prime}}\left[{\mathcal{Q}}^{*}(g)\right](x)\right)^{q^{\prime}}\,{\rm
d}x=\|S^{\frac{q-p}{q-1}}_{q,\gamma,\beta}(f)\|_{r}\left\|\left(\mathcal{C}_{q^{\prime}}\left[{\mathcal{Q}}^{*}(g)\right]\right)^{q^{\prime}}\right\|_{r^{\prime}}$
$\displaystyle=\|S_{q,\gamma,\beta}(f)\|_{p}^{\frac{q-p}{q-1}}\|\mathcal{C}_{q^{\prime}}\left[{\mathcal{Q}}^{*}(g)\right]\|_{p^{\prime}}^{q^{\prime}},$

where $1/r=1-q^{\prime}/p^{\prime}=(q-p)/(qp-p)$.

Combining the estimates of $I$ and $II$ with Theorem 4.1, we get the desired
assertion. ∎

Finally, we arrive at the proof of our main theorem.

###### Proof of Theorm 1.1.

The first part (1.8) of Theorem 1.1 is a consequence of Theorem 4.1. Indeed,
suppose $L$ be a generator such that the kernels of the generating semigroup
satisfy (1.5) , (1.6) and (1.7) with $0<\beta,\gamma\leq 1$, then the
classical Poisson semigroup generated by $\sqrt{\Delta}$ satisfy obviously the
same assumptions. Then

$\|S_{q,L}(f)\|_{p}\lesssim\|S_{q,\gamma,\beta}(f)\|_{p}\lesssim_{\gamma,\beta}p^{\frac{1}{q}}\|S_{q,\sqrt{\Delta}}(f)\|_{p},\quad
1\leq p<\infty.$

Similarly we obtain

$\|S_{q,\sqrt{\Delta}}(f)\|_{p}\lesssim_{\gamma,\beta}p^{\frac{1}{q}}\|S_{q,L}(f)\|_{p},\quad
1\leq p<\infty.$

As for another part (1.9), one side is easy by Theorem 4.1,

$\|G_{q,L}(f)\|_{p}\lesssim\|G_{q,\gamma,\beta}(f)\|_{p}\approx_{\gamma,\beta}\|S_{q,\gamma,\beta}(f)\|_{p}\lesssim_{\gamma,\beta}p^{\frac{1}{q}}\|S_{q,L}(f)\|_{p},\quad
1\leq p<\infty.$

For the reverse direction, by Theorem 5.1 and Proposition 5.3, we have for
$1\leq p<(1+q)/2$,

$\displaystyle\|f\|_{H^{p}_{q,L}(X)}$
$\displaystyle\lesssim\left(\frac{p(q-1)}{q-p}\right)^{\frac{1}{q^{\prime}}}\sup_{g}\left|\int_{{}^{d}}\left\langle
f(x),g(x)\right\rangle_{X\times X^{*}}\,{\rm d}x\right|$
$\displaystyle\lesssim_{\gamma,\beta}\sup_{g}\|G_{q,L}(f)\|_{p}^{\frac{p}{q}}\|S_{q,L}(f)\|_{p}^{1-\frac{p}{q}}\|g\|_{BMO^{p^{\prime}}_{q^{\prime},L^{*}}(X^{*})}$
$\displaystyle\lesssim_{\gamma,\beta}\|G_{q,L}(f)\|_{p}^{\frac{p}{q}}\|S_{q,L}(f)\|_{p}^{1-\frac{p}{q}},$

where the supremum is taken over all $g\in C_{c}(\mathbb{R}^{d})\otimes X^{*}$
such that its $BMO_{q^{\prime},L^{*}}^{p^{\prime}}(X^{*})$-norm is not more
than $1$. Hence

$\|S_{q,L}(f)\|_{p}\lesssim_{\gamma,\beta}\|G_{q,L}(f)\|_{p},\quad 1\leq
p<\frac{1+q}{2}.$

Now we deal with the case $(1+q)/2\leq p<\infty$. Let $f\in
C_{c}({}^{d})\otimes X$, we deduce from Theorem 5.1 that

$\displaystyle\|f\|_{H^{p}_{q,L}(X)}$
$\displaystyle\lesssim\max\left\\{p^{\prime\frac{1}{q^{\prime}}}p^{\frac{1}{q}},p^{\prime\frac{2}{q^{\prime}}}\right\\}\sup_{h}\left|\int_{{}^{d}}\left\langle
f(x),h(x)\right\rangle_{X\times X^{*}}\,{\rm d}x\right|$
$\displaystyle\lesssim
p^{\frac{1}{q}}\sup_{h}\left|\int_{{}^{d+1}_{+}}\left\langle{\mathcal{Q}}(f)(x,t),{\mathcal{Q}}^{*}(h)(x,t)\right\rangle_{X\times
X^{*}}\,\frac{{\rm d}x{\rm d}t}{t}\right|$ $\displaystyle\lesssim
p^{\frac{1}{q}}\sup_{h}\|G_{q,L}(f)\|_{p}\|G_{q^{\prime},L^{*}}(h)\|_{p^{\prime}}$
$\displaystyle\lesssim_{\gamma,\beta}p^{\frac{1}{q}}p^{\prime\frac{1}{q^{\prime}}}\sup_{h}\|G_{q,L}(f)\|_{p}\|S_{q^{\prime},L^{*}}(h)\|_{p^{\prime}}$
$\displaystyle\lesssim p^{\frac{1}{q}}\|G_{q,L}(f)\|_{p},$

where the supremum is taken over all $h\in C_{c}({}^{d})\otimes X^{*}$ such
that its $H^{p^{\prime}}_{q^{\prime},L^{*}}(X^{*})$-norm is not more than $1$.

Combining the estimations above we conclude that

$p^{-\frac{1}{q}}\|S_{q,L}(f)\|_{p}\lesssim_{\gamma,\beta}\|G_{q,L}(f)\|_{p}\lesssim_{\gamma,\beta}p^{\frac{1}{q}}\|S_{q,L}(f)\|_{p},\quad
1\leq p<\infty.$

We complete the proof. ∎

##  6\. Applications

In this section, we first recall the previous related results in [33, 29].
These, together with the tent space theory and Theorem 1.1, will enable us to
obtain the optimal Lusin type constants and the characterization of martingale
type. In particular, this resolves partially Problem 1.8, Problem A.1 and
Conjecture A.4 in the recent paper of Xu [48].

Some notions and notations need to be presented. We first introduce the
vector-valued atomic Hardy space $H^{1}_{{\rm at}}({}^{d};X)$. A measurable
function $a\in L^{\infty}({}^{d};X)$ is called an $X$-valued atom if

$\text{supp}\,a\subset B,\ \int_{{}^{d}}a(x)\,{\rm d}x=0,\
\|a\|_{L^{\infty}(X)}\leq|B|^{-1},$

where $B$ is a ball in d. The atomic Hardy space $H^{1}_{{\rm at}}({}^{d};X)$
is defined as the function space consisting of all functions $f$ which admits
an expression of the form

$f=\sum_{j=1}^{\infty}\lambda_{j}a_{j},\quad\sum_{j=1}^{\infty}|\lambda_{j}|<\infty,$

where $a_{j}$ is an $X$-valued atom. The norm of $H^{1}_{{\rm at}}({}^{d};X)$
is defined as

$\|f\|_{H^{1}_{{\rm
at}}(X)}=\inf\left\\{\sum_{j=1}^{\infty}|\lambda_{j}|:f(x)=\sum_{j=1}^{\infty}\lambda_{j}a_{j}(x)\right\\}.$

This is a Banach space.

The BMO space $BMO({}^{d};X)$ is defined as the space of all $f\in L^{1}_{{\rm
loc}}({}^{d};X)$ equipped with the semi-norm

$\|f\|_{BMO(X)}=\sup_{B}\frac{1}{|B|}\int_{B}\|f-f_{B}\|_{X}\,{\rm
d}x<\infty,$

where the supremum runs over all the balls in d and $f_{B}$ represents the
average of $f$ over $B$. $BMO({}^{d};X)$ is a Banach space modulo constants.

It is well-known that $BMO({}^{d};X^{*})$ is isomorphically identified as a
subspace of the dual space of $H^{1}_{{\rm at}}({}^{d};X)$ (cf. [10]) and it
is norming in the following sense

$\|f\|_{H^{1}_{{\rm at}}(X)}\approx\sup\left\\{|\left\langle
f,g\right\rangle|:g\in BMO({}^{d};X^{*}),\ \|g\|_{BMO(X)}\leq 1\right\\}$

with universal constants. Furthermore, if the Banach space $X^{*}$ has the
Radon-Nikodým property, then (cf. [8])

(6.1) $\displaystyle(H^{1}_{{\rm at}}({}^{d};X))^{*}=BMO({}^{d};X^{*}),$

with equivalent norms.

We recall the following definitions on the geometric properties for Banach
spaces. A Banach space $X$ is said to be of _martingale type_ $q$ (with
$1<q\leq 2$) if there exists a positive constant $c$ such that every finite
$X$-valued $L^{q}$-martingale $(f_{n})_{n\geq 0}$, the following inequality
holds

$\sup_{n\geq 0}{\mathbb{E}}\|f_{n}\|_{X}^{q}\leq c^{q}\sum_{n\geq
1}{\mathbb{E}}\|f_{n}-f_{n-1}\|_{X}^{q},$

where ${\mathbb{E}}$ denotes the underlying expectation; and the least
constant $c$ is called the martingale type constant, denoted as
$\mathsf{M}_{t,q}(X)$. While $X$ is said to be of _martingale cotype_ $q$
(with $2\leq q<\infty$) if the reverse inequalities holds with $c^{-1}$ in
place of $c$ and the corresponding martingale cotype constant is denoted by
$\mathsf{M}_{c,q}(X)$. Pisier’s famous renorming theorem shows that $X$ is of
martingale cotype (respectively, type) $q$ if and only if $X$ admits an
equivalent $q$-uniform convex (respectively, smooth) norm. We refer the reader
to [34, 35, 36] for more details.

Let $X$ be a Banach space and $1<q\leq 2$. The authors in [29] showed that the
assertion that $X$ is of martingale type $q$ is equivalent to the one that for
any $1<p<\infty$, there exists a constant $c_{p}$ such that for any $f\in
C_{c}({}^{d})\otimes X$,

(6.2) $\displaystyle\|f\|_{L^{p}(X)}\leq c_{p}\|S_{q,\sqrt{\Delta}}(f)\|_{p}.$

Later on, in [33] the authors investigated the relationships between
$H^{1}_{{\rm at}}({}^{d};X)$ and $H^{1}_{q,\sqrt{\Delta}}({}^{d};X)$ as well
as the ones between $BMO({}^{d};X)$ and $BMO_{q,\sqrt{\Delta}}({}^{d};X)$, and
provided insights into the geometric properties of the underlying Banach space
$X$.

######  Theorem 6.1.

Let $X$ be a Banach space and $1<q\leq 2$. The followings are equivalent

  1. (i)

$X$ is of martingale type $q$;

  2. (ii)

there exists a positive constant c such that for any $f\in
C_{c}({}^{d})\otimes X$,

$\|f\|_{H^{1}_{\rm at}(X)}\leq c\|S_{q,\sqrt{\Delta}}(f)\|_{1};$

  3. (iii)

there exists a positive constant c such that for any $f\in
C_{c}({}^{d})\otimes X$,

$\|f\|_{BMO(X)}\leq c\|f\|_{BMO_{q,\sqrt{\Delta}}(X)}.$

Moreover, the constants in (ii) and (iii) are majored by
$\mathsf{M}_{t,q}(X)$.

The following theorem follows from the interpolation theory between vector-
valued tent spaces—Lemma 3.3—and the boundedness of the projection
$\pi_{L}$—Lemma 3.6. See for instance the general interpolation theory of
complemented subspaces (cf. [40, Section 1.17]), and we omit the details.

######  Theorem 6.2.

Let $X$ be any fixed Banach space, $1<q<\infty$ and $1\leq
p_{1}<p<p_{2}<\infty$ such that $1/p=(1-\theta)/p_{1}+\theta/p_{2}$ with
$0\leq\theta\leq 1$. Let $L$ be as in Theorem 1.1. Then

$[H^{p_{1}}_{q,L}({}^{d};X),H^{p_{2}}_{q,L}({}^{d};X)]_{\theta}=H^{p}_{q,L}({}^{d};X),$

with equivalent norms, where $[\cdot,\cdot]_{\theta}$ is the complex
interpolation space. More precisely, for $f\in C_{c}({}^{d})\otimes X$, one
has

$\|f\|_{H^{p}_{q,L}(X)}\lesssim\|f\|_{[H^{p_{1}}_{q,L}({}^{d};X),H^{p_{2}}_{q,L}({}^{d};X)]_{\theta}}\lesssim
p^{\frac{2}{q}}\|f\|_{H^{p}_{q,L}(X)}.$

Now we are at the position to give the applications.

######  Corollary 6.3.

Let $X$ be a Banach space and $1<q\leq 2$. Let $L$ be as in Theorem 1.1. The
followings are equivalent

  1. (i)

$X$ is of martingale type $q$;

  2. (ii)

for any $f\in C_{c}({}^{d})\otimes X$,

$\|f\|_{H^{1}_{{\rm
at}}(X)}\lesssim_{\gamma,\beta}\mathsf{M}_{t,q}(X)\|G_{q,L}(f)\|_{1};$

  3. (iii)

for any $1<p<\infty$ and $f\in C_{c}({}^{d})\otimes X$,

$\|f\|_{L^{p}(X)}\lesssim_{\gamma,\beta}p\mathsf{M}_{t,q}(X)\|G_{q,L}(f)\|_{p};$

  4. (iv)

for any $f\in C_{c}({}^{d})\otimes X$,

$\|f\|_{BMO(X)}\lesssim_{\gamma,\beta}\mathsf{M}_{t,q}(X)\|f\|_{BMO_{q,L}(X)}.$

###### Proof.

(i)$\Leftrightarrow$(ii). This follows immediately from Theorem 1.1 and
Theorem 6.1.

(iii)$\Rightarrow$(i). This is deduced from Theorem 1.1 and (6.2).

(i)$\Rightarrow$(iii). In the case $1<p<q$, by Theorem 1.1, it suffices to
show that

(6.3)
$\displaystyle\|f\|_{L^{p}(X)}\lesssim\mathsf{M}_{t,q}(X)\|S_{q,\sqrt{\Delta}}\|_{p}.$

Keeping in mind (6.2) and Theorem 6.1 (ii), we consider

$[H^{1}_{q,\sqrt{\Delta}}({}^{d};X),H^{q}_{q,\sqrt{\Delta}}({}^{d};X)]_{\theta}\subset[H^{1}_{{\rm
at}}({}^{d};X),L^{q}({}^{d};X)]_{\theta};$

then combining Theorem 6.2 with the interpolation between $H^{1}_{{\rm
at}}({}^{d};X)$ and $L^{q}({}^{d};X)$ (cf. [9, Theorem A]), one gets for any
$f\in C_{c}({}^{d})\otimes X$, $1/p=1-\theta+\theta/q$,

$\|f\|_{L^{p}(X)}\lesssim\|f\|_{[H^{1}_{{\rm
at}}({}^{d};X),L^{q}({}^{d};X)]_{\theta}}\lesssim\mathsf{M}_{t,q}(X)\|f\|_{[H^{1}_{q,\sqrt{\Delta}}({}^{d};X),H^{q}_{q,\sqrt{\Delta}}({}^{d};X)]_{\theta}}\lesssim\mathsf{M}_{t,q}(X)\|f\|_{H^{p}_{q,\sqrt{\Delta}}(X)}.$

This is the desired (6.3). Combining it with the related result for $q\leq
p<\infty$ in [48], we conclude

$\|f\|_{L^{p}(X)}\lesssim_{\gamma,\beta}p\mathsf{M}_{t,q}(X)\|G_{q,L}(f)\|_{p},\quad
1<p<\infty.$

(i)$\Leftrightarrow$(iv). This follows from Remark 4.5 and Theorem 6.1 (iii).
∎

######  Remark 6.4.

(1). Taking $L=\sqrt{\Delta}$ in the assertion (iii), we get

$\mathsf{L}_{t,q,p}^{\sqrt{\Delta}}(X)\lesssim p\mathsf{M}_{t,q}(X),\quad
1<p<\infty,$

where the order is optimal as $p$ tends to $1$. This solves partially [48,
Problem 1.8].

(2). The implication (iii)$\rightarrow$(i) says that a Banach space $X$ which
is Lusin type $q$ relative to $\\{e^{-tL}\\}_{t>0}$ implies the martingale
type $q$ for a large class of generators $L$. This answers partially [48,
Problem A.1 and Conjecture A.4].

##  7\. Appendix

From the atomic decomposition of $T^{1}_{q}({}^{d+1}_{+};X)$, we derive the
following molecular decomposition for $H^{1}_{q,L}({}^{d};X)$ for any Banach
space $X$, which might have further applications.

######  Theorem 7.1.

Let $X$ be any fixed Banach space and $1<q<\infty$. For any $f\in
H^{1}_{q,L}({}^{d};X)$, there exist a sequence of complex numbers
$\left\\{\lambda_{j}\right\\}_{j\geq 1}$ and corresponding molecules
$\alpha_{j}=\pi_{L}(a_{j})$ with $a_{j}(x,t)$ being an $(X,q)$-atom such that

$f=\sum_{j\geq
1}\lambda_{j}\alpha_{j},\quad\|f\|_{H^{1}_{q,L}(X)}\approx\sum_{j\geq
1}|\lambda_{j}|.$

###### Proof.

Let $f\in H^{1}_{q,L}({}^{d};X)$. It follows that ${\mathcal{Q}}(f)\in
T^{1}_{q}({}^{d+1}_{+};X)$. Hence ${\mathcal{Q}}(f)$ admits an atomic
decomposition by Lemma 3.1. More precisely, there exist a sequence of complex
numbers $\\{c_{j}\\}_{j\geq 1}$ and $(X,q)$-atoms $a_{j}$ such that

${\mathcal{Q}}(f)=\sum_{j=1}^{\infty}c_{j}a_{j},\quad\|f\|_{H^{1}_{q,L}(X)}=\|{\mathcal{Q}}(f)\|_{T^{1}_{q}(X)}\approx\sum_{j=1}^{\infty}|c_{j}|.$

Then by Lemma 3.6, it follows that $\pi_{L}(a_{j})=\alpha_{j}\in
H^{1}_{q,L}({}^{d};X)$ for all $j\geq 1$. Recall below the Calderón
identity—(4.10),

$f(x)=4\int_{0}^{\infty}{\mathcal{Q}}[{\mathcal{Q}}(f)(\cdot,t)](x,t)\,\frac{{\rm
d}t}{t}.$

This further deduce that

$f(x)=4\sum_{j=1}^{\infty}c_{j}\int_{0}^{\infty}{\mathcal{Q}}[a_{j}(\cdot,t)](x,t))\,\frac{{\rm
d}t}{t}=4\sum_{j=1}^{\infty}c_{j}\alpha_{j}(x),$

and thus we obtain the desired molecular decomposition. ∎

Acknowledgments.The subject of this article was proposed by Professor Quanhua
Xu when he was completing his paper [48]. The authors are very grateful to his
suggestions, encouragement, support as well as his careful reading over the
whole paper. We also would like to thank Professor Éric Ricard for his
comments and suggestions. G. Hong was supported by National Natural Science
Foundation of China (No. 12071355, No. 12325105, No. 12031004) and Fundamental
Research Funds for the Central Universities (No. 2042022kf1185).

## References

  * [1] I. Abu-Falahah, P. Stinga, and J. Torrea.  Square functions associated to Schrödinger operators.  Stud. Math., 2(203):171–194, 2011.
  * [2] J. J. Betancor, A. J. Castro, J. C. Fariña, and L. Rodríguez-Mesa.  UMD Banach spaces and square functions associated with heat semigroups for Schrödinger, Hermite and Laguerre operators.  Math. Nachr., 289(4):410–435, 2016.
  * [3] J. J. Betancor, A. J. Castro, and L. Rodríguez-Mesa.  Characterization of uniformly convex and smooth Banach spaces by using Carleson measures in Bessel settings.  J. Convex Anal., 20:763–811, 2013.
  * [4] J. J. Betancor, J. C. Fariña, V. Galli, and S. M. Molina.  Uniformly convex and smooth Banach spaces and $l_{p}$-boundedness properties of Littlewood-Paley and area functions associated with semigroups.  J. Math. Anal. Appl., 482(1):123534, 2020.
  * [5] J. J. Betancor, J. C. Fariña, T. Martínez, and J. L. Torrea.  Riesz transform and $g$-function associated with Bessel operators and their appropriate Banach spaces.  Isr. J. Math., 157(1):259–282, 2007.
  * [6] J. J. Betancor, J. C. Fariña, L. Rodríguez-Mesa, A. Sanabria, and J. L. Torrea.  Lusin type and cotype for Laguerre $g$-functions.  Isr. J. Math., 182:1–30, 2011.
  * [7] J. J. Betancor, S. M. Molina, and L. Rodríguez-Mesa.  Area littlewood–paley functions associated with hermite and laguerre operators.  Potential Anal., 4(34):345–369, 2011.
  * [8] O. Blasco.  Hardy spaces of vector-valued functions: duality.  Trans. Am. Math. Soc., 308(2):495–507, 1988.
  * [9] O. Blasco.  Interpolation between ${H}^{1}_{B_{0}}$ and ${L}^{p}_{B_{1}}$.  Stud. Math., 92(3):295–210, 1989.
  * [10] J. Bourgain.  Vector-valued singular integrals and the ${H}^{1}$-${BMO}$ duality.  Probability theory and harmonic analysis, pages 1–19, 1986.
  * [11] R. R. Coifman, Y. Meyer, and E. M. Stein.  Some new function spaces and their applications to harmonic analysis.  J. Funct. Anal., 62(2):304–335, 1985.
  * [12] T. Coulhon and X. T. Duong.  Maximal regularity and kernel bounds: Observations on a theorem by Hieber and Prüss.  Adv. Differ. Equ., 5:343–368, 2000.
  * [13] M. Cowling, I. Doust, A. Micintosh, and A. Yagi.  Banach space operators with a bounded ${H}_{\infty}$ functional calculus.  J. Aust. Math. Soc., 60(1):51–89, 1996.
  * [14] J. Diestel and J. J. Uhl.  Vector Measures.  Mathematical surveys and monographs. American Mathematical Society, 1977\.
  * [15] N. Dinculeanu.  Vector measures.  Elsevier, 2014.
  * [16] X. T. Duong and L. Yan.  Duality of Hardy and ${BMO}$ spaces associated with operators with heat kernel bounds.  J. Am. Math. Soc., 18(4):943–973, 2005.
  * [17] C. Fefferman.  Characterizations of bounded mean oscillation.  Bull. Am. Math. Soc., 77(4):587–588, 1971.
  * [18] C. Fefferman and E. M. Stein.  Some maximal inequalities.  Am. J. Math., 93(1):107–115, 1971.
  * [19] C. Fefferman and E. M. Stein.  ${H}_{p}$ spaces of several variables.  Acta Math., 129(1):137–195, 1972.
  * [20] L. Grafakos.  Modern fourier analysis, volume 250.  Springer, New York, 2009.
  * [21] M. Haase.  The Functional Calculus of Sectorial Operators.  Birkhäuser Basel, 1st edition, 2005.
  * [22] E. Harboure, J. L. Torrea, and B. Viviani.  Vector-valued extensions of operators related to the Ornstein-Uhlenbeck semigroup.  J. Anal. Math., 91(1):1–29, 2003.
  * [23] E. Harboure, J. L. Torrea, and B. E. Viviani.  A vector-valued approach to tent spaces.  J. Anal. Math., 56:125–140, 1991.
  * [24] T. Hytönen.  Littlewood-Paley-Stein theory for semigroups in UMD spaces.  Rev. Mat. Iberoam., 23(3):973–1009, 2007.
  * [25] T. Hytönen, J. Van Neerven, M. Veraar, and L. Weis.  Analysis in Banach spaces, volume 12.  Springer, Berlin, 2016.
  * [26] T. Hytönen, J. M. A. M. van Neerven, and P. Portal.  Conical square function estimates in UMD banach spaces and applications to ${H}_{\infty}$-functional calculi.  J. Anal. Math., 106:317–351, 2008.
  * [27] M. Kemppainen.  The vector-valued tent spaces $T_{1}$ and $T_{\infty}$.  J. Aust. Math. Soc., 97(1):107–126, 2014.
  * [28] M. Kemppainen.  On vector-valued tent spaces and Hardy spaces associated with non-negative self-adjoint operators.  Glasg. Math. J., 58(3):689–716, 2016.
  * [29] T. Martínez, J. Torrea, and Q. Xu.  Vector-valued Littlewood-Paley-Stein theory for semigroups.  Adv. Math., 203(2):430–475, 2006.
  * [30] A. McIntosh.  Operators which have an ${H}_{\infty}$ functional calculus.  In Miniconference on operator theory and partial differential equations, volume 14, pages 210–232. Australian National University, Mathematical Sciences Institute, 1986.
  * [31] T. Mei.  Operator valued Hardy spaces.  American Mathematical Soc., 2007.
  * [32] A. Naor and R. Young.  Foliated corona decompositions.  Acta Math., 229(1):55–200, 2022.
  * [33] C. Ouyang and Q. Xu.  BMO functions and Carleson measures with values in uniformly convex spaces.  Can. J. Math., 62(4):827–844, 2010.
  * [34] G. Pisier.  Martingales with values in uniformly convex spaces.  Isr. J. Math., 20(3):326–350, 1975.
  * [35] G. Pisier.  Probabilistic methods in the geometry of Banach spaces.  In Probability and analysis, pages 167–241. Springer, 1986.
  * [36] G. Pisier.  Martingales in Banach spaces, volume 155.  Cambridge University Press, 2016.
  * [37] E. M Stein.  Topics in Harmonic Analysis, Related to the Littlewood-Paley Theory.  Princeton University Press, 1970.
  * [38] E. M. Stein and T. S. Murphy.  Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, volume 3.  Princeton University Press, 1993.
  * [39] J. L. Torrea and C. Zhang.  Fractional vector-valued Littlewood-Paley-Stein theory for semigroups.  Proc. R. Soc. Edinb. A Math., 144(3):637–667, 2014.
  * [40] H. Triebel.  Interpolation Theory, Function Spaces and Differential Operators.  North-Holland, Amsterdam, 1978.
  * [41] J. M. Wilson.  The intrinsic square function.  Rev. Mat. Iberoam., 23(3):771–791, 2007.
  * [42] J. M. Wilson.  Weighted Littlewood-Paley theory and exponential-square integrability.  Number 1924. Springer Science & Business Media, 2008.
  * [43] R. Xia, X. Xiong, and Q. Xu.  Characterizations of operator-valued Hardy spaces and applications to harmonic analysis on quantum tori.  Adv. Math., 291:183–227, 2016.
  * [44] Q. Xu.  Littlewood-Paley theory for functions with values in uniformly convex spaces.  J. Reine Angew. Math., 1998:195–226, 1998.
  * [45] Q. Xu.  ${H}_{\infty}$ functional calculus and maximal inequalities for vector-valued semigroups of contractions on ${L}_{p}$-spaces.  Int. Math. Res. Not., 14:5715-5732, 2015.
  * [46] Q. Xu.  Vector-valued Littlewood-Paley-Stein theory for semigroups II.  Int. Math. Res. Not., 21:7769–7791, 2020.
  * [47] Q. Xu.  Optimal orders of the best constants in the Littlewood-Paley inequalities.  J. Funct. Anal., page 109570, 2022.
  * [48] Q. Xu.  Holomorphic functional calculus and vector-valued Littlewood-Paley-Stein theory for semigroups.  arXiv:2105.12175. To appear in J. Euro. Math. Soc., 2024.
  * [49] Z. Xu, and H. Zhang.  From the Littlewood-Paley-Stein inequality to the Burkholder-Gundy inequality.  Trans. Am. Math. Soc., 376(01):371–389, 2023.