# Holomorphic functional calculus and vector-valued Littlewood-Paley-Stein
theory for semigroups

Quanhua Xu  Laboratoire de Mathématiques, Université de Bourgogne Franche-
Comté, 25030 Besançon Cedex, France<EMAIL_ADDRESS>
###### Abstract.

We study vector-valued Littlewood-Paley-Stein theory for semigroups of regular
contractions $\\{T_{t}\\}_{t>0}$ on $L_{p}(\Omega)$ for a fixed $1<p<\infty$.
We prove that if a Banach space $X$ is of martingale cotype $q$, then there is
a constant $C$ such that

$\left\|\left(\int_{0}^{\infty}\big{\|}t\frac{\partial}{\partial
t}P_{t}(f)\big{\|}_{X}^{q}\,\frac{dt}{t}\right)^{\frac{1}{q}}\right\|_{L_{p}(\Omega)}\leq
C\,\big{\|}f\big{\|}_{L_{p}(\Omega;X)}\,,\quad\forall\,f\in L_{p}(\Omega;X),$

where $\\{P_{t}\\}_{t>0}$ is the Poisson semigroup subordinated to
$\\{T_{t}\\}_{t>0}$. Let $\mathsf{L}^{P}_{c,q,p}(X)$ be the least constant
$C$, and let $\mathsf{M}_{c,q}(X)$ be the martingale cotype $q$ constant of
$X$. We show

$\mathsf{L}^{P}_{c,q,p}(X)\lesssim\max\big{(}p^{\frac{1}{q}},\,p^{\prime}\big{)}\mathsf{M}_{c,q}(X).$

Moreover, the order $\max\big{(}p^{\frac{1}{q}},\,p^{\prime}\big{)}$ is
optimal as $p\to 1$ and $p\to\infty$. If $X$ is of martingale type $q$, the
reverse inequality holds. If additionally $\\{T_{t}\\}_{t>0}$ is analytic on
$L_{p}(\Omega;X)$, the semigroup $\\{P_{t}\\}_{t>0}$ in these results can be
replaced by $\\{T_{t}\\}_{t>0}$ itself.

Our new approach is built on holomorphic functional calculus. Compared with
all the previous, this new one is more powerful in several aspects: a) it
permits us to go much further beyond the setting of symmetric submarkovian
semigroups; b) it yields the optimal orders of growth on $p$ for most of the
relevant constants; c) it gives new insights into the scalar case for which
our orders of the best constants in the classical Littlewood-Paley-Stein
inequalities for symmetric submarkovian semigroups are better than the
previous by Stein.

In particular, we resolve a problem of Naor and Young on the optimal order of
the best constant in the above inequality when $X$ is of martingale cotype $q$
and $\\{P_{t}\\}_{t>0}$ is the classical Poisson and heat semigroups on
$\mathbb{R}^{d}$.

2000 Mathematics Subject Classification: Primary: 47D05, 46B20. Secondary:
42B25, 47D06.

Key words: Littlewood-Paley-Stein inequalities, analytic semigroups of regular
contractions, martingale type and cotype, Lusin type and cotype, holomorphic
functional calculus, $\ell_{q}$-boundedness

Dedicated to the memory of Elias M. Stein

##  1\. Introduction

This article pursues our investigation on the vector-valued Littlewood-Paley-
Stein theory that was initiated in [57] and further carried out in [34, 58,
59]. My personal research in this domain has been profoundly influenced by
Stein’s monograph [48] and developed according to two parallel directions: on
the one hand, for the Banach space valued case, and on the other hand, for the
noncommutative extension as in [27, 32] for maximal function inequalities and
in [26] for square function inequalities.

Note that Betancor and coauthors studied this theory for some special
semigroups (cf. [6, 7, 8, 9]); see also [1, 4, 5, 20, 23, 42, 49] for related
results. Recently, the theory has found applications to Lipschitz embedding of
metric spaces into Banach spaces, and to approximation of Lipschitz maps by
linear maps, see, for instance, the papers by Hytönen and Naor [24], Lafforgue
and Naor [30], Naor and Young [39].

First, we recall the famous Littlewood-Paley-Stein inequality that is the
starting point of all our research in the domain. Let
$(\Omega,{\mathcal{A}},\mu)$ be a $\sigma$-finite measure space and
$\\{T_{t}\\}_{t>0}$ a symmetric diffusion semigroup on
$(\Omega,{\mathcal{A}},\mu)$ in Stein’s sense [48, section III.1]. Namely,
$\\{T_{t}\\}_{t>0}$ satisfies the following conditions

  * $\bullet$

$T_{t}$ is a contraction on $L_{p}(\Omega)$ for every $1\leq p\leq\infty$,

  * $\bullet$

$T_{t}T_{s}=T_{t+s}$,

  * $\bullet$

$\lim_{t\to 0}T_{t}(f)=f$ in $L_{2}(\Omega)$ for every $f\in L_{2}(\Omega)$,

  * $\bullet$

$T_{t}$ is positive (i.e. positivity preserving),

  * $\bullet$

$T_{t}$ is selfadjoint on $L_{2}(\Omega)$,

  * $\bullet$

$T_{t}(1)=1$.

The last condition is the markovianity; the next to the last is the symmetry.
Thus such a semigroup is also called a symmetric markovian semigroup. A
semigroup satisfying all the above conditions except markovianity is usually
called a symmetric submarkovian semigroup (the submarkovianity means
$T_{t}(1)\leq 1$).

It is a classical fact that the orthogonal projection $\mathsf{F}$ from
$L_{2}(\Omega)$ onto the fixed point subspace of $\\{T_{t}\\}_{t>0}$ extends
to a contractive projection on $L_{p}(\Omega)$ for every $1\leq p\leq\infty$.
Then $\mathsf{F}$ is also positive and $\mathsf{F}\big{(}L_{p}(\Omega)\big{)}$
is the fixed point subspace of $\\{T_{t}\\}_{t>0}$ on $L_{p}(\Omega)$.

Stein’s celebrated extension of the classical Littlewood-Paley inequality
asserts that for every symmetric diffusion semigroup $\\{T_{t}\\}_{t>0}$ and
every $1<p<\infty$

(1.1)
$\|f-\mathsf{F}(f)\|_{L_{p}(\Omega)}\approx_{p}\left\|\left(\int_{0}^{\infty}\Big{|}t\frac{\partial}{\partial
t}T_{t}(f)\Big{|}^{2}\,\frac{dt}{t}\right)^{\frac{1}{2}}\right\|_{L_{p}(\Omega)}\,,\quad
f\in L_{p}(\Omega).$

The classical inequality corresponds to the case where $\\{T_{t}\\}_{t>0}$ is
the Poisson semigroup on the torus ${\mathbb{T}}$ or the Euclidean space
${\mathbb{R}}^{d}$. Stein’s inequality above is the core of [48] in which
Stein developed a beautiful general theory. Later, Cowling [11] presented an
elegant alternative approach to Stein’s theory for symmetric submarkovian
semigroups; Cowling’s goal is to show that the negative generator of
$\\{T_{t}\\}_{t>0}$ has a bounded holomorphic functional calculus, then to
deduce the maximal inequality on $\\{T_{t}\\}_{t>0}$ which is another
fundamental result of Stein.

In the present article we are concerned with the vector-valued case. Given a
Banach space $X$ let $L_{p}(\Omega;X)$ denote the $L_{p}$-space of strongly
measurable functions from $\Omega$ to $X$. It is a well known elementary fact
that if $T$ is a positive bounded operator on $L_{p}(\Omega)$ with $1\leq
p\leq\infty$, then $T\otimes{\rm Id}_{X}$ is bounded on $L_{p}(\Omega;X)$ with
the same norm. For notational convenience, throughout this article, we will
denote $T\otimes{\rm Id}_{X}$ by $T$ too. Thus $\\{T_{t}\\}_{t>0}$ is also a
semigroup of contractions on $L_{p}(\Omega;X)$ for any Banach space $X$ with
$\mathsf{F}\big{(}L_{p}(\Omega;X)\big{)}$ as its fixed point subspace.

The vector-valued Littlewood-Paley-Stein theory consists in investigating
(1.1) for $f\in L_{p}(\Omega;X)$ (with the absolute value on the right hand
side replaced by the norm of $X$). It is not hard to show that the equivalence
(1.1) continues to hold in the $X$-valued setting for the Poisson semigroup on
${\mathbb{T}}$ iff $X$ is isomorphic to a Hilbert space (cf. [19, 57]).
However, if one requires only the validity of one of the two one-sided
inequalities, the corresponding family of Banach spaces is much larger: the
upper estimate corresponds to 2-uniformly smooth spaces while the lower to
2-uniformly convex spaces (up to a renorming).

These geometrical properties of Banach spaces can be characterized by
martingale inequalities. Recall that a Banach space $X$ is of _martingale
cotype_ $q$ (with $2\leq q<\infty$) if there exists a positive constant $c$
such that every finite $X$-valued $L_{q}$-martingale $(f_{n})$ satisfies the
following inequality

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

where $\mathbb{E}$ denotes the underlying expectation. $X$ is of _martingale
type_ $q$ (with $1<q\leq 2$) if the reverse inequality holds (with $c^{-1}$ in
place of $c$). The corresponding best constant will be denoted by
$\mathsf{M}_{c,q}(X)$ for the martingale cotype $q$ and by
$\mathsf{M}_{t,q}(X)$ for the martingale type $q$. Pisier’s famous renorming
theorem asserts that $X$ is of martingale type (resp. cotype) $q$ iff $X$
admits an equivalent norm that is $q$-uniformly smooth (resp. convex). We
refer the reader to [44, 46, 47] for more information.

Note that in the study of one-sided inequalities in the vector-valued case,
the index $2$ on the right hand side of (1.1) plays no special role and can be
replaced by $1<q<\infty$, $q\leq 2$ for the upper estimate and $q\geq 2$ for
the lower. Now we can summarize the main results of [57, 34, 59] as follows:

Theorem A. _Let $X$ be a Banach space and $1<q<\infty$._

  1. (i)

$X$_is of martingale cotype_ $q$ iff for every symmetric diffusion semigroup
$\\{T_{t}\\}_{t>0}$ and for every $1<p<\infty$ $($equivalently, for some
$1<p<\infty)$ there exists a constant $c$ such that

$\left\|\left(\int_{0}^{\infty}\big{\|}t\frac{\partial}{\partial
t}T_{t}(f)\big{\|}_{X}^{q}\,\frac{dt}{t}\right)^{\frac{1}{q}}\right\|_{L_{p}(\Omega)}\leq
c\,\big{\|}f\big{\|}_{L_{p}(\Omega;X)}\,,\quad f\in L_{p}(\Omega;X).$

  2. (ii)

$X$_is of martingale type_ $q$ iff for every symmetric diffusion semigroup
$\\{T_{t}\\}_{t>0}$ and for every $1<p<\infty$ $($equivalently, for some
$1<p<\infty)$ there exists a constant $c$ such that

$\big{\|}f-\mathsf{F}(f)\big{\|}_{L_{p}(\Omega;X)}\leq
c\left\|\left(\int_{0}^{\infty}\big{\|}t\frac{\partial}{\partial
t}T_{t}(f)\big{\|}_{X}^{q}\,\frac{dt}{t}\right)^{\frac{1}{q}}\right\|_{L_{p}(\Omega)}\,,\quad
f\in L_{p}(\Omega;X).$

Both “if” parts are proved in [57]; for that purpose we need only the case
where $\\{T_{t}\\}_{t>0}$ is the usual Poisson semigroup on ${\mathbb{T}}$ (or
${\mathbb{R}}^{d}$ as in [34]). This is the easy direction thanks to the
classical link between Poisson integral and Brownian motion. The other
direction is harder. It is first proved in [57] for the Poisson semigroup on
the unit circle, then in [34] for the Poisson semigroup subordinated to any
symmetric diffusion semigroup $\\{T_{t}\\}_{t>0}$. Left as an open problem in
[34], the statement for $\\{T_{t}\\}_{t>0}$ itself as above was finally
settled in [59]. Note that like in [48], the key tool in [34, 59] is Rota’s
martingale dilation of a symmetric diffusion semigroup that allows us to adapt
the scalar Littlewood-Paley-Stein theory developed in [48].

The use of Rota’s dilation prevented us from weakening the assumption on a
symmetric diffusion semigroup. Cowling’s approach in [11] does not use Rota’s
dilation but it requires the semigroup in consideration to be symmetric and
submarkovian. It has been an open problem of establishing the results of [34]
or [59] in Cowling’s setting. In fact, since a long time it has been a desire
to extend all previous results to more general semigroups. This was done in
some special cases (Hermite, Laguerre and Bessel semigroups) by Betancor and
coauthors (cf. [6, 7, 8, 9]).

The objective of the present article is to resolve the above problems. We will
develop a vector-valued Littlewood-Paley-Stein theory for semigroups of
regular operators on $L_{p}(\Omega)$ for a single $1<p<\infty$, thereby going
considerably beyond Stein-Cowling’s setting.

Recall that an operator $T$ on $L_{p}(\Omega)$ ($1\leq p\leq\infty$) is
regular (more precisely, contractively regular) if

$\big{\|}\sup_{k}|T(f_{k})|\big{\|}_{p}\leq\big{\|}\sup_{k}|f_{k}|\big{\|}_{p}$

for all finite sequences $\\{f_{k}\\}_{k\geq 1}$ in $L_{p}(\Omega)$.
Obviously, any positive contraction is regular. It is well known that,
conversely, if $T$ is regular, then there exists a positive contraction $S$ on
$L_{p}(\Omega)$ such that $|T(f)|\leq S(|f|)$ for any $f\in L_{p}(\Omega)$;
$S$ is called the absolute value of $T$ and denoted by $|T|$ (see [36, Chapter
1]).

It is well known (and easy to check) that if $T$ is a contraction on
$L_{p}(\Omega)$ for every $1\leq p\leq\infty$, then $T$ is regular on
$L_{p}(\Omega)$. Like positive operators, a regular operator $T$ extends to a
contraction on $L_{p}(\Omega;X)$ for any Banach space $X$. This extension will
be denoted by $T$ too.

Now let $\\{T_{t}\\}_{t>0}$ be a strongly continuous semigroup of regular
operators on $L_{p}(\Omega)$ with $1<p<\infty$. Extended to $L_{p}(\Omega;X)$,
$\\{T_{t}\\}_{t>0}$ remains to be a strongly continuous semigroup of
contractions on $L_{p}(\Omega;X)$. Let again $\mathsf{F}$ be the projection
from $L_{p}(\Omega)$ onto the fixed point subspace of $\\{T_{t}\\}_{t>0}$.
Then $\mathsf{F}$ is also regular, so extends to a contractive projection on
$L_{p}(\Omega;X)$. Note that $\mathsf{F}(L_{p}(\Omega;X))$ coincides with the
fixed point subspace of $\\{T_{t}\\}_{t>0}$ on $L_{p}(\Omega;X)$.

Let $\\{P_{t}\\}_{t>0}$ be the Poisson semigroup subordinated to
$\\{T_{t}\\}_{t>0}$:

(1.2)
$P_{t}(f)=\frac{1}{\sqrt{\pi}}\,\int_{0}^{\infty}\frac{e^{-s}}{\sqrt{s}}\,T_{\frac{t^{2}}{4s}}(f)ds.$

Recall that if $A$ denotes the negative infinitesimal generator of
$\\{T_{t}\\}_{t>0}$ (i.e., $T_{t}=e^{-tA}$), then $P_{t}=e^{-t\sqrt{A}}$.
Instead of the square root, one can, of course, consider other subordinated
semigroups $e^{-tA^{\alpha}}$ for $0<\alpha<1$; but we will not deal with the
latter.

To proceed further, we need to introduce some notions. Define

$\mathcal{G}^{T}_{q}(f)=\Big{(}\int_{0}^{\infty}\big{\|}t\frac{\partial}{\partial
t}T_{t}(f)\big{\|}^{q}_{X}\frac{dt}{t}\Big{)}^{\frac{1}{q}}$

for $f$ in the definition domain of $A$ in $L_{p}(\Omega;X)$. $X$ is said to
be of Lusin cotype $q$ relative to $\\{T_{t}\\}_{t>0}$ if there exists a
constant $c$ such that

$\big{\|}\mathcal{G}^{T}_{q}(f)\big{\|}_{L_{p}(\Omega)}\leq
c\big{\|}f-\mathsf{F}(f)\big{\|}_{L_{p}(\Omega;X)}$

for all $f$ as above. The smallest $c$ is denoted by
$\mathsf{L}^{T}_{c,q,p}(X)$. Similarly, we define the Lusin type $q$ of $X$ by
reversing the above inequality and changing $c$ to $c^{-1}$, the corresponding
type $q$ constant is denoted by $\mathsf{L}^{T}_{t,q,p}(X)$. See section 4
below for more information.

######  Remark 1.1.

The subordination formula (1.2) immediately implies the pointwise inequality
$\mathcal{G}^{P}_{q}(f)\lesssim\mathcal{G}^{T}_{q}(f)$ for any $f$. It then
follows that

$\mathsf{L}^{T}_{c,q,p}(X)\gtrsim\mathsf{L}^{P}_{c,q,p}(X)\;\text{ and
}\;\mathsf{L}^{T}_{t,q,p}(X)\lesssim\mathsf{L}^{P}_{t,q,p}(X).$

In [57], the Lusin type and cotype relative to the Poisson semigroup on the
unit circle are shown to be equivalent to the martingale type and cotype,
respectively. Theorem A above extends this to symmetric diffusion semigroups.

In the sequel, we will use the following convention: $A\lesssim B$ (resp.
$A\lesssim_{\varepsilon}B$) means that $A\leq CB$ (resp. $A\leq
C_{\varepsilon}B$) for some absolute positive constant $C$ (resp. a positive
constant $C_{\varepsilon}$ depending only on $\varepsilon$). $A\approx B$ or
$A\approx_{\varepsilon}B$ means that these inequalities as well as their
inverses hold. The index $p$ will be assumed $1<p<\infty$ and $p^{\prime}$
will denote its conjugate index.

Below is our first principal result.

######  Theorem 1.2.

Let $X$ be a Banach space and $1<p,q<\infty$. Let $\\{T_{t}\\}_{t>0}$ be a
strongly continuous semigroup of regular operators on $L_{p}(\Omega)$ and
$\\{P_{t}\\}_{t>0}$ its subordinated Poisson semigroup.

  1. (i)

If $X$ is of martingale cotype $q$, then $X$ is of Lusin cotype $q$ relative
to $\\{P_{t}\\}_{t>0}$ and

$\mathsf{L}^{P}_{c,q,p}(X)\lesssim\max\big{(}p^{\frac{1}{q}},\,p^{\prime}\big{)}\mathsf{M}_{c,q}(X).$

  2. (ii)

If $X$ is of martingale type $q$, then $X$ is of Lusin type $q$ relative to
$\\{P_{t}\\}_{t>0}$ and

$\mathsf{L}^{P}_{t,q,p}(X)\lesssim\max\big{(}p,\,p^{\prime\frac{1}{q^{\prime}}}\big{)}\mathsf{M}_{t,q}(X).$

The two above inequalities can be reformulated in another (clearer) way, for
instance, the first one reads as

$\displaystyle\mathsf{L}^{P}_{c,q,p}(X)\lesssim\left\\{\begin{array}[]{lcl}\displaystyle
p^{\frac{1}{q}}\,\mathsf{M}_{c,q}(X)&\textrm{ if}&p\geq q,\\\ \displaystyle
p^{\prime}\,\mathsf{M}_{c,q}(X)&\textrm{ if}&p<q.\end{array}\right.$

######  Remark 1.3.

All the growth orders, except the one on $\mathsf{L}^{P}_{t,q,p}(X)$ as $p\to
1$, are optimal since they are already so in the scalar case $X={\mathbb{C}}$.
More precisely,

  1. (i)

$\mathsf{L}^{P}_{c,q,p}({\mathbb{C}})\gtrsim\max\big{(}p^{\frac{1}{q}},\,p^{\prime}\big{)}$
for all $1<p<\infty$ when $\\{P_{t}\\}_{t>0}$ is the classical Poisson
semigroup on ${\mathbb{R}}$ (see Proposition 8.5 below);

  2. (ii)

$\mathsf{L}^{P}_{t,q,p}({\mathbb{C}})\gtrsim p$ as $p\to\infty$ when
$\\{P_{t}\\}_{t>0}$ is the Poisson semigroup subordinated to a symmetric
diffusion semigroup $\\{T_{t}\\}_{t>0}$, as shown by Zhendong Xu and Hao Zhang
[61]; in fact, they proved the stronger inequality
$\mathsf{L}^{T}_{t,q,p}({\mathbb{C}})\gtrsim p$ as $p\to\infty$ for a
symmetric diffusion semigroup $\\{T_{t}\\}_{t>0}$.

Part (i) of the above theorem cannot hold for the semigroup
$\\{T_{t}\\}_{t>0}$ itself without additional assumption (see Remark 4.4
below). It turns out that the missing condition is the analyticity of
$\\{T_{t}\\}_{t>0}$ on $L_{p}(\Omega;X)$. Recall that $\\{T_{t}\\}_{t>0}$ is
analytic on $L_{p}(\Omega;X)$ if $\\{T_{t}\\}_{t>0}$ extends to a bounded
analytic function from an open sector
$\Sigma_{\beta_{0}}=\big{\\{}z\in{\mathbb{C}}:|{\rm
arg}(z)|<\beta_{0}\big{\\}}$ to $B\big{(}L_{p}(\Omega;X)\big{)}$ for some
$0<\beta_{0}\leq\frac{\pi}{2}$, where $B(Y)$ denotes the space of bounded
linear operators on a Banach space $Y$. In this case,

(1.4)
$\mathsf{T}_{\beta_{0}}=\sup\big{\\{}\big{\|}T_{z}\big{\|}_{B(L_{p}(\Omega;X))}:z\in\Sigma_{\beta_{0}}\big{\\}}<\infty.$

######  Theorem 1.4.

Let $X$ and $p,q$ be as above.

  1. (i)

If $X$ is of martingale type $q$, then $X$ is of Lusin type $q$ relative to
$\\{T_{t}\\}_{t>0}$ and

$\mathsf{L}^{T}_{t,q,p}(X)\lesssim\max\big{(}p,\,p^{\prime\frac{1}{q^{\prime}}}\big{)}\mathsf{M}_{t,q}(X).$

  2. (ii)

Assume additionally that $\\{T_{t}\\}_{t>0}$ satisfies (1.4). Let
$\beta_{q}=\beta_{0}\min(\frac{p}{q}\,,\frac{p^{\prime}}{q^{\prime}})$. If $X$
is of martingale cotype $q$, then $X$ is of Lusin cotype $q$ relative to
$\\{T_{t}\\}_{t>0}$ and

$\mathsf{L}^{T}_{c,q,p}(X)\lesssim\beta_{q}^{-3}\,\mathsf{T}_{\beta_{0}}^{\min(\frac{p}{q},\,\frac{p^{\prime}}{q^{\prime}})}\,\max\big{(}p^{\frac{2}{q}},\,p^{\prime
1+\frac{1}{q^{\prime}}}\big{)}\mathsf{M}_{c,q}(X).$

The two previous theorems considerably improve Theorem A. Firstly, the
semigroup $\\{T_{t}\\}_{t>0}$ now acts on $L_{p}(\Omega)$ for a single $p$.
Secondly, the markovianity or submarkovianity is not assumed (in fact,
$T_{t}(1)$ is even not defined if the measure on $\Omega$ is infinite).
Thirdly, the symmetry is neither needed since the semigroup does not act on
$L_{2}(\Omega)$ if $p\neq 2$.

A more improvement concerns the precise estimates on the best constants, the
present estimates are much better than all previously known ones, even in the
scalar case (see section 8 below for historical comments). Moreover, except
one case, they give the optimal orders of growth as $p\to 1$ and $p\to\infty$,
as already pointed out in Remark 1.3. This is perhaps a major novelty of our
method.

The aforementioned optimality allows us to answer a question raised Naor and
Young about the optimal orders of $\mathsf{L}^{P}_{c,q,p}(X)$ and
$\mathsf{L}^{T}_{c,q,p}(X)$ when $\\{T_{t}\\}_{t>0}$ is the heat semigroup on
${\mathbb{R}}^{d}$ (see the appendix of [40]). In fact, we will show a much
stronger result. Let $\varphi:{\mathbb{R}}^{d}\to{\mathbb{C}}$ be a function
satisfying

(1.5)
$\left\\{\begin{array}[]{lcl}\displaystyle|\varphi(x)|\leq\frac{1}{(1+|x|)^{d+\varepsilon}},&x\in{\mathbb{R}}^{d}\\\
\displaystyle|\varphi(x)-\varphi(y)|\leq\frac{|x-y|^{\delta}}{(1+|x|)^{d+\varepsilon+\delta}}+\frac{|x-y|^{\delta}}{(1+|y|)^{d+\varepsilon+\delta}},&x,y\in{\mathbb{R}}^{d}\\\
\displaystyle\int_{{\mathbb{R}}^{d}}\varphi(x)dx=0\end{array}\right.$

for some positive constants $\varepsilon$ and $\delta$.

We will also need $\varphi$ to be nondegenerate in the sense that there exists
another function $\psi$ satisfying (1.5) such that

(1.6)
$\int_{0}^{\infty}\widehat{\varphi}(t\xi)\,\widehat{\psi}(t\xi)\,\frac{dt}{t}=1,\quad\forall\xi\in{\mathbb{R}}^{d}\setminus\\{0\\}.$

This nondegeneracy allows us to use the Calderón reproducing formula. There
exist plenty of functions satisfying these conditions, for instance, the heat
and Poisson kernels as well as any Schwartz function $\varphi$ verifying that
for any $\xi\in{\mathbb{R}}^{d}\setminus\\{0\\}$ there is $t>0$ such that
$\widehat{\varphi}(t\xi)\neq 0$

Let $\varphi_{t}(x)=\frac{1}{t^{d}}\varphi(\frac{x}{t})$ for
$x\in{\mathbb{R}}^{d}$ and $t>0$. We define

(1.7)
$G_{q,\varphi}(f)(x)=\Big{(}\int_{0}^{\infty}\|\varphi_{t}*f(x)\|_{X}^{q}\,\frac{dt}{t}\Big{)}^{\frac{1}{q}},\quad
x\in{\mathbb{R}}^{d}$

for any (reasonable) function $f:{\mathbb{R}}^{d}\to X$. Let
$\mathsf{L}^{\varphi}_{c,q,p}(X)$ be the best constant $c$ such that

$\big{\|}G_{q,\varphi}(f)\big{\|}_{L_{p}({\mathbb{R}}^{d})}\leq
c\,\big{\|}f\big{\|}_{L_{p}({\mathbb{R}}^{d};X)},\quad f\in
L_{p}({\mathbb{R}}^{d};X).$

Similarly, we introduce $\mathsf{L}^{\varphi}_{t,q,p}(X)$ for the reverse
inequality (with $c^{-1}$ instead of $c$).

######  Theorem 1.5.

Let $X$ be a Banach space and $1<p,q<\infty$. Assume that $\varphi$ satisfies
(1.5).

  1. (i)

If $X$ is of martingale cotype $q$, then

$\mathsf{L}^{\varphi}_{c,q,p}(X)\lesssim_{d,\varepsilon,\delta}\max\big{(}p^{\frac{1}{q}},\,p^{\prime}\big{)}\mathsf{M}_{c,q}(X).$

  2. (ii)

Assume additionally that $\varphi$ is nondegenerate. If $X$ is of martingale
type $q$, then

$\mathsf{L}^{\varphi}_{t,q,p}(X)\lesssim_{d,\varepsilon,\delta}\max\big{(}p,\,p^{\prime\frac{1}{q^{\prime}}}\big{)}\mathsf{M}_{t,q}(X).$

Let $\\{\mathbb{H}_{t}\\}_{t>0}$ be the classical heat semigroup on
${\mathbb{R}}^{d}$ whose convolution kernel is given by

$\mathbb{H}_{t}(x)=(4\pi t)^{-\frac{d}{2}}e^{-\frac{|x|^{2}}{4t}}\,.$

Its subordinated Poisson semigroup is the usual Poisson semigroup
$\\{\mathbb{P}_{t}\\}_{t>0}$ with the convolution kernel

$\mathbb{P}_{t}(x)=\frac{c_{d}\,t}{(|x|^{2}+t^{2})^{(d+1)/2}}\,.$

The above theorem immediately implies the following corollary that resolves
Naor and Young’s problem.

######  Corollary 1.6.

We have

$\displaystyle\mathsf{L}^{\mathbb{P}}_{c,q,p}(X)\lesssim\max\big{(}p^{\frac{1}{q}},\,p^{\prime}\big{)}\mathsf{M}_{c,q}(X)\;\text{
and
}\;\mathsf{L}^{\mathbb{H}}_{c,q,p}(X)\lesssim_{d}\max\big{(}p^{\frac{1}{q}},\,p^{\prime}\big{)}\mathsf{M}_{c,q}(X).$

Moreover,

$\mathsf{L}^{\mathbb{H}}_{c,q,p}(X)\gtrsim\mathsf{L}^{\mathbb{P}}_{c,q,p}({\mathbb{C}})\gtrsim\max\big{(}p^{\frac{1}{q}},\,p^{\prime}\big{)}\;\text{
and
}\;\mathsf{L}^{\mathbb{H}}_{c,q,q}(X)\gtrsim\mathsf{L}^{\mathbb{P}}_{c,q,q}(X)\gtrsim\mathsf{M}_{c,q}(X).$

######  Remark 1.7.

It is worth to point out that the estimate on
$\mathsf{L}^{\mathbb{P}}_{c,q,p}(X)$ is independent of $d$ thanks to Theorem
1.2 (i). It would be interesting to have a dimension free estimate for the
heat semigroup too. This is related to another problem whether the analyticity
constant of $\\{\mathbb{H}_{t}\\}_{t>0}$ on $L_{p}({\mathbb{R}}^{d};X)$ can be
controlled by a dimension free constant (see Example A.2 below).

Apart from the inequality
$\big{\|}{\mathcal{G}}^{P}_{q}(f)\big{\|}_{L_{p}(\Omega)}\leq\mathsf{L}^{P}_{c,q,p}(X)\big{\|}f\big{\|}_{L_{p}(\Omega;X)}$,
the following variant is also useful:

$\Big{(}\int_{0}^{\infty}\big{\|}t\frac{\partial}{\partial
t}\,P_{t}(f)\big{\|}^{r}_{L_{p}(\Omega;X)}\,\frac{dt}{t}\Big{)}^{\frac{1}{r}}\leq
c\big{\|}f\big{\|}_{L_{p}(\Omega;X)}$

when $X$ is of martingale cotype $q$ (see, for instance, [30]). Inequalities
of this type are less delicate than the previous. It is well known that if $X$
is of martingale cotype $q$, then $L_{p}(\Omega;X)$ is of martingale cotype
$\max(p,q)$, so the above inequality can hold only for $r=\max(p,q)$. We can,
of course, consider similar variants in the situation of Theorem 1.4 and
Theorem 1.5 as well as their reverse inequalities when $X$ is of martingale
type $q$; but we will concentrate our attention only on the above inequality
and on the case $1<p\leq q$ for illustration.

Theorem 1.2 (i) easily implies the following

######  Corollary 1.8.

Assume that $X$ is of martingale cotype $q$ and $1<p\leq q$. Then

$\Big{(}\int_{0}^{\infty}\big{\|}t\frac{\partial}{\partial
t}\,P_{t}(f)\big{\|}^{q}_{L_{p}(\Omega;X)}\,\frac{dt}{t}\Big{)}^{\frac{1}{q}}\lesssim\max\big{(}p^{\prime\frac{1}{q}},\,\mathsf{M}_{c,q}(X)\big{)}\big{\|}f\big{\|}_{L_{p}(\Omega;X)}$

for all $f\in L_{p}(\Omega;X)$. Moreover, the constant on the right hand side
is optimal as $p\to 1$.

Like the martingale type and cotype, the Lusin type and cotype behave well
with duality as shown by Theorem 4.5 below. This duality theorem allows us to
treat the type case from the cotype one. In contrast with the martingale case,
the proof of Theorem 4.5 is much harder and depends on a bounded projection on
a certain vector-valued radial tent space. Let ${\mathbb{R}}_{+}$ be equipped
with the measure $\frac{dt}{t}$. The radial space is
$L_{p}(\Omega;L_{q}({\mathbb{R}}_{+};X))$ whose elements $h$ are functions of
two variables $\omega\in\Omega$ and $t\in{\mathbb{R}}_{+}$, i.e.,
$h:(\omega,t)\mapsto h_{t}(\omega)$. The desired projection maps
$L_{p}(\Omega;L_{q}({\mathbb{R}}_{+};X))$ onto the subspace of all $h$ of the
form $h_{t}=t\frac{\partial}{\partial t}T_{t}(f)$ for some $f\in
L_{p}(\Omega;X)$; formally, it is given by

(1.8) $\mathcal{T}(h)_{s}=4\int_{0}^{\infty}st\,\frac{\partial}{\partial
s}T_{s}\frac{\partial}{\partial t}T_{t}(h_{t})\,\frac{dt}{t}\,,\quad s>0.$

Here the expression $\frac{\partial}{\partial t}T_{t}(h_{t})$ is interpreted
as $\frac{\partial}{\partial t}T_{t}(f)$ with $f=h_{t}$. Note that
$\mathcal{T}(h)$ is well-defined for nice functions $h\in
L_{p}(\Omega;L_{q}({\mathbb{R}}_{+};X))$, for instance, for all compactly
supported continuous functions from ${\mathbb{R}}_{+}$ to the definition
domain of the generator of $\\{T_{t}\\}_{t>0}$ in $L_{p}(\Omega;X)$.
Similarly, we define $\mathcal{P}$ associated to the subordinated Poisson
semigroup $\\{P_{t}\\}_{t>0}$.

The following is the key result for the duality argument.

######  Theorem 1.9.

Let $X$ be a Banach space and $1<p<\infty$.

  1. (i)

The map $\mathcal{P}$ extends to a bounded projection on
$L_{p}(\Omega;L_{q}({\mathbb{R}}_{+};X))$ with norm majorized by
$C\max(p^{\prime 1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}})$ for
any $1\leq q\leq\infty$.

  2. (ii)

Assume additionally that $1<q<\infty$ and $\\{T_{t}\\}_{t>0}$ satisfies (1.4)
for some $0<\beta_{0}\leq\frac{\pi}{2}$. Then $\mathcal{T}$ extends to a
bounded projection on $L_{p}(\Omega;L_{q}({\mathbb{R}}_{+};X))$ with norm
majorized by

$C\beta_{q}^{-4}\,\mathsf{T}_{\beta_{0}}^{\min(\frac{p}{q},\,\frac{p^{\prime}}{q^{\prime}})}\,\max(p^{\prime
1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}})\;\text{ with
}\;\beta_{q}=\beta_{0}\min(\frac{p}{q}\,,\frac{p^{\prime}}{q^{\prime}}).$

It is remarkable that the first part of the theorem above holds for any Banach
space $X$ and any subordinated Poisson semigroup. Under the stronger
assumption that $\\{T_{t}\\}_{t>0}$ be a symmetric diffusion semigroup,
assertion (i) above is [34, Theorem 3.2]. However, the proof in [34] contains
a gap which lies in the reduction of Theorem 3.2 to Lemma 3.3 in [34] via
Rota’s dilation theorem. Recall that if $\\{P_{t}\\}_{t>0}$ is the Poisson
semigroup on the torus, assertion (i) above was proved in [57] by using
Calderón-Zygmund singular integral theory.

######  Remark 1.10.

The analyticity assumption in Theorem 1.4 and Theorem 1.9 (ii) is unremovable.
Recall that $\\{T_{t}\\}_{t>0}$ is analytic on $L_{p}(\Omega;X)$ with
$1<p<\infty$ in one of the following cases:

  * $\bullet$

$\\{T_{t}\\}_{t>0}$ is a symmetric diffusion semigroup and $X$ is
superreflexive ([45]);

  * $\bullet$

$\\{T_{t}\\}_{t>0}$ is a convolution semigroup induced by symmetric
probability measures on a locally compact abelian group and $X$ is K-convex
([45]);

  * $\bullet$

$\\{T_{t}\\}_{t>0}$ is an analytic semigroup of regular operators on
$L_{p}(\Omega)$ and $X$ is $\theta$-Hilbertian, i.e., a complex interpolation
space of a Hilbert space and another Banach space ([58]).

Many classical semigroups are analytic on $L_{p}(\Omega;X)$ for any $X$ (see
Appendix A below). On the other hand, $\\{T_{t}\\}_{t>0}$ is analytic on
$L_{p}(\Omega;X)$ iff its adjoint semigroup $\\{T^{*}_{t}\\}_{t>0}$ is
analytic on $L_{p^{\prime}}(\Omega;X^{*})$. Thus the class of Banach spaces
$X$ such that $\\{T_{t}\\}_{t>0}$ is analytic on $L_{p}(\Omega;X)$ is stable
under the passage to duals, subspaces and quotient spaces.

On the other hand, it is well known that the subordinated Poisson semigroup
$\\{P_{t}\\}_{t>0}$ is always analytic on $L_{p}(\Omega;X)$ for any Banach
space $X$ since its negative generator is a sectorial operator of type
$\frac{\pi}{4}$ (see section 2 for more information).

A summary of the main techniques and the contents seems to be in order. Our
approach is different from all the previous ones. It is based on holomorphic
functional calculus, which constitutes perhaps one of the major ideas of this
article. In this regard, it shares some common points with Cowling’s approach
that deals with the bounded $H^{\infty}$ functional calculus of the generator
of a symmetric submarkovian semigroup and the related maximal inequality. We
need, however, to adapt McIntosh’s $H^{\infty}$ functional calculus for our
purpose. This is done in the preparatory section 2 in which we introduce a key
notion of the article: the $\ell_{q}$-boundedness of a family of operators on
$L_{p}(\Omega;X)$; it gives rise to the definitions of $\ell_{q}$-sectorial
operators and $\ell_{q}$-analytic semigroups. We transfer to this setting some
well known results about sectorial operators and analytic semigroups. After
this preparation, we prove Theorem 1.9 in section 3. This projection theorem
is a crucial ingredient for the duality studied in section 4. Theorem 4.5
establishes our duality result between the Lusin cotype of $X$ and the Lusin
type of the dual space $X^{*}$; this result is as nice as the corresponding
one in the martingale case, except the links between the involved constants
(compare the constants in Theorem 4.5 and those in (6.2) below). This section
also contains some general properties of the Lusin type and cotype, in
particular, a characterization by lacunary discrete differences (Theorem 4.7).

As Rota’s dilation is no longer available in the present situation, we use
instead Fendler’s dilation for semigroups of regular operators. Fendler’s
theorem transfers Theorem 1.2 (i) to the special case where
$\\{T_{t}\\}_{t>0}$ is the translation group of ${\mathbb{R}}$. This allows us
to exploit techniques from harmonic analysis. Our strategy is built, in a
crucial way, on Calderón-Zygmund singular integral theory and modern real-
variable Littlewood-Paley theory. We present all this in the preparatory
section 5 that will be needed for the proofs of Theorem 1.2 and Theorem 1.5.
These proofs constitute the most heavy and technical part of the article. The
proof of Theorem 1.5 is done in section 6. We then use transference to show
Theorem 1.2 in section 7. To that end, we first need to represent the
$g$-function associated to the Poisson semigroup subordinated to the
translation group of ${\mathbb{R}}$ as a singular integral operator. Theorem
1.4 will follow from Theorem 1.2 by functional calculus; Corollary 1.8 is an
easy consequence of Theorem 1.2 (i).

An additional major significant aspect of the new approach is the fact that it
improves the growth orders on $p$ of the relevant best constants even in the
scalar Littlewood-Paley-Stein inequalities, see section 8; moreover, except
the case of the Lusin type constant as $p\to 1$, it yields the optimal orders,
which is not the case by the previous methods of Stein and Cowling (see the
historical comments in section 8). This shows, to a certain extent, that our
method is optimal. Section 8 also contains the optimality of the best
constants in Corollary 1.6. We end the article by an appendix that gives some
examples of semigroups.

The techniques developed this article allow one to simplify and extend many
recent results in the scalar Littlewood-Paley-Stein theory, in particular,
those on positive operators on $L_{2}$ with kernels satisfying Gaussian upper
estimates. On the other hand, they are also applicable to the noncommutative
setting. We will carry out all this elsewhere.

Throughout the article, $X$ will be a Banach space, $1<p<\infty$ and $1\leq
q\leq\infty$ (but $1<q<\infty$ in most of time). Unless explicitly stated
otherwise, $\\{T_{t}\\}_{t>0}$ will be a strongly continuous semigroup of
regular operators on $L_{p}(\Omega)$ and $\\{P_{t}\\}_{t>0}$ its subordinated
Poisson semigroup. These semigroups are extended to $L_{p}(\Omega;X)$. $A$
will denote the negative generator of $\\{T_{t}\\}_{t>0}$, so $T_{t}=e^{-tA}$
and $P_{t}=e^{-t\sqrt{A}}$.

##  2\. The $\ell_{q}$-boundedness

This section is the preparatory part of the article. We will introduce the
notion of $\ell_{q}$-boundedness that is the direct extension to the vector-
valued setting of the $R_{q}$-boundedness introduced by Weis [52]. In fact,
though not explicitly stated, this notion appeared before in harmonic analysis
with regard to vector-valued inequalities of classical operators. Most results
below are the $\ell_{q}$-boundedness analogues of well known results or those
of Kunstmann and Ullmann [29] in the scalar case. I learnt the existence of
[52, 29] after having posted the first version of this article in arXiv, and I
thank Emiel Lorist for pointing out these references to me.

We start with a brief introduction to holomorphic functional calculus in order
to fix notation (see [12] for more information). Recall that a densely defined
closed operator $B$ on a Banach space $Y$ is called a _sectorial operator of
type_ $\alpha$ with $0\leq\alpha<\pi$ if
${\mathbb{C}}\setminus\overline{\Sigma_{\gamma}}$ is contained in the
resolvent set of $B$ for any $\gamma>\alpha$ and

$\sup\big{\\{}\big{\|}z(z-B)^{-1}\big{\|}_{B(Y)}\,:z\notin\overline{\Sigma_{\gamma}}\big{\\}}<\infty,$

where $\Sigma_{\gamma}$ is the open sector $\big{\\{}z\in{\mathbb{C}}:|{\rm
arg}(z)|<\gamma\big{\\}}$ in the complex plane. Let $\beta>\gamma$ and
$H^{\infty}(\Sigma_{\beta})$ denote the space of bounded analytic functions in
$\Sigma_{\beta}$ and $H^{\infty}_{0}(\Sigma_{\beta})$ its subspace consisting
of all $\varphi$ satisfying

$|\varphi(z)|\leq\frac{c|z|^{\delta}}{1+|z|^{2\delta}}\;\text{ for some
}\;c>0\text{ and }\delta>0.$

Let $\Gamma$ be the boundary of $\Sigma_{\beta}$, oriented in the positive
sense. Then for any $\varphi\in H^{\infty}_{0}(\Sigma_{\beta})$ the integral

$\varphi(B)=\frac{1}{2\pi{\rm i}}\int_{\Gamma}\varphi(z)(z-B)^{-1}dz$

defines a bounded operator on $Y$, where the integral absolutely converges in
$B(Y)$.

The following resolution of the identity will be useful later. Let $\psi\in
H^{\infty}_{0}(\Sigma_{\beta})$ such that

$\int_{0}^{\infty}\psi(t)\frac{dt}{t}=1.$

Then

(2.1) $y=\int_{0}^{\infty}\psi(tB)(y)\frac{dt}{t}\,,\quad y\in\overline{{\rm
im}B}.$

Its verification is easy. Indeed, let $\varphi\in
H^{\infty}_{0}(\Sigma_{\beta})$. Then

$\varphi(z)=\int_{0}^{\infty}\psi(tz)\varphi(z)\frac{dt}{t}\,,\quad
z\in\Sigma_{\beta}.$

Thus for any $y\in Y$

$\varphi(B)(y)=\int_{0}^{\infty}\psi(tB)\varphi(B)(y)\frac{dt}{t}\,.$

Choose

$\varphi(z)=\frac{n^{2}z}{(n+z)(1+nz)}\,.$

Then $\varphi(B)(y)\to y$ in $Y$ as $n\to\infty$ for any $y\in\overline{{\rm
im}B}$ (see [12, Theorem 3.8]), whence (2.1) by virtue of the convergence
lemma ([12, Lemma 2.1]).

######  Definition 2.1.

A family ${\mathcal{F}}\subset B(L_{p}(\Omega;X))$ is said to be
$\ell_{q}$-bounded if there exists a constant $c$ such that

$\Big{\|}\Big{(}\sum_{k}\|A_{k}(f_{k})\|^{q}\Big{)}^{\frac{1}{q}}\Big{\|}_{L_{p}(\Omega)}\leq
c\Big{\|}\Big{(}\sum_{k}\|f_{k}\|^{q}\Big{)}^{\frac{1}{q}}\Big{\|}_{L_{p}(\Omega)}$

for all finite sequences $\\{A_{k}\\}\subset{\mathcal{F}}$ and
$\\{f_{k}\\}\subset L_{p}(\Omega;X)$.

######  Remark 2.2.

It is easy to show that the absolutely convex hull of an $\ell_{q}$-bounded
family is again $\ell_{q}$-bounded with the same constant.

Accordingly, we introduce the $\ell_{q}$-boundedness versions of sectoriality
of operators and analyticity of semigroups. Recall that a semigroup
$\\{S_{t}\\}_{t>0}$ on a Banach space $Y$ is said to be analytic if it extends
to an analytic function from $\Sigma_{\beta}$ to $B(Y)$ for some
$0<\beta\leq\frac{\pi}{2}$ and bounded in any smaller sector. In this case, we
call $\\{S_{t}\\}_{t>0}$ an _analytic semigroup of type_ $\beta$.

######  Definition 2.3.

  1. (i)

A densely defined closed operator $B$ on $L_{p}(\Omega;X)$ is called an
$\ell_{q}$-sectorial operator of type $\alpha$ with $0\leq\alpha<\pi$ if
${\mathbb{C}}\setminus\overline{\Sigma_{\gamma}}$ is contained in the
resolvent set of $B$ for any $\gamma>\alpha$ and the family
$\big{\\{}z(z-B)^{-1}:z\notin\overline{\Sigma_{\gamma}}\big{\\}}$ is
$\ell_{q}$-bounded on $L_{p}(\Omega;X)$.

  2. (ii)

A semigroup $\\{S_{t}\\}_{t>0}$ on $L_{p}(\Omega;X)$ is called an
$\ell_{q}$-analytic semigroup of type $\beta$ with $0<\beta\leq\frac{\pi}{2}$
if $\\{S_{t}\\}_{t>0}$ extends to an analytic function from $\Sigma_{\beta}$
to $B(L_{p}(\Omega;X))$ and for any $\nu<\beta$ the family
$\big{\\{}S_{z}:z\in\Sigma_{\nu}\big{\\}}$ is $\ell_{q}$-bounded on
$L_{p}(\Omega;X)$.

The following is the $\ell_{q}$-boundedness analogue of a classical
characterization of analytic semigroups.

######  Proposition 2.4.

Let $\\{S_{t}\\}_{t>0}$ be a strongly continuous bounded semigroup on
$L_{p}(\Omega;X)$ and $B$ its negative generator. Then the following
statements are equivalent:

  1. (i)

$\\{S_{t}\\}_{t>0}$ is $\ell_{q}$-analytic of type $\beta$ for some
$0<\beta\leq\frac{\pi}{2}$;

  2. (ii)

$B$ is $\ell_{q}$-sectorial of type $\alpha$ for some $\alpha<\frac{\pi}{2}$;

  3. (iii)

$\\{S_{t},tBS_{t}\\}_{t>0}$ is $\ell_{q}$-bounded on $L_{p}(\Omega;X)$.

###### Proof.

The proof is a straightforward adaptation of the classical argument (cf. e.g.
the proof of [43, Theorem 5.2]). As we want to track the links between the
different constants involved, we give below an outline.

(i) $\Rightarrow$ (ii). Let $\alpha=\frac{\pi}{2}-\beta$. For
$\alpha<\gamma<\frac{\pi}{2}$ choose $0<\nu<\beta$ such that
$\gamma+\nu>\frac{\pi}{2}$, for instance, we can take
$\nu=\beta-\frac{\gamma-\alpha}{2}$ so that
$\gamma+\nu=\frac{\pi}{2}+\frac{\gamma-\alpha}{2}$. Then for any $z=re^{{\rm
i}\theta}\notin\Sigma_{\gamma}$ and $z\neq 0$, we have

$z(z-B)^{-1}=z\int_{0}^{\infty}e^{tz}S_{t}dt=ze^{{\rm i}{\rm
sgn}(\theta)\nu}\int_{0}^{\infty}e^{tre^{{\rm i}(\theta+{\rm
sgn}(\theta)\nu)}}S_{te^{{\rm i}{\rm sgn}(\theta)\nu}}dt.$

If $C_{\nu}$ denotes the $\ell_{q}$-boundedness constant of the family
$\big{\\{}S_{\zeta}:\zeta\in\overline{\Sigma_{\nu}}\big{\\}}$, then by Remark
2.2 we deduce that $\big{\\{}z(z-B)^{-1}:z\notin\Sigma_{\gamma},\,z\neq
0\big{\\}}$ is $\ell_{q}$-bounded with constant $C_{\gamma}$ given by

$C_{\gamma}\leq C_{\nu}|z|\int_{0}^{\infty}\big{|}e^{tre^{{\rm i}(\theta+{\rm
sgn}(\theta)\nu)}}\big{|}dt=\frac{C_{\nu}}{|\cos(\theta+{\rm
sgn}(\theta)\nu)|}\leq\frac{C_{\nu}}{|\cos(\gamma+\nu)|}\,.$

(ii) $\Rightarrow$ (iii). Let $\alpha<\gamma<\frac{\pi}{2}$ and $\Gamma$ be
the boundary of $\Sigma_{\gamma}$ oriented in the positive sense. Then

$tBS_{t}=\frac{1}{2\pi{\rm i}}\int_{\Gamma}t\lambda
e^{-t\lambda}(\lambda-B)^{-1}d\lambda,\quad t>0.$

Thus $\\{tBS_{t}\\}_{t>0}$ is $\ell_{q}$-bounded with constant

$C^{\prime}_{d}\leq\frac{C_{\gamma}}{\pi}\int_{0}^{\infty}te^{-tr\cos\gamma}dr=\frac{C_{\gamma}}{\pi\cos\gamma}.$

To show the $\ell_{q}$-boundedness of $\\{S_{t}\\}_{t>0}$ we need to slightly
modify the contour $\Gamma$. Let $\Gamma^{\prime}$ be the union of the part of
$\Gamma$ with $|\lambda|\geq 1$ and the arc in
${\mathbb{C}}\setminus\Sigma_{\gamma}$ of the circle with the origin as center
and radius $1$. Then we have

$S_{t}=\frac{1}{2\pi{\rm
i}}\int_{\Gamma^{\prime}}e^{-t\lambda}(\lambda-B)^{-1}d\lambda,\quad t>0.$

The change of variables $\zeta=t\lambda$ yields

$S_{t}=\frac{1}{2\pi{\rm
i}}\int_{t\Gamma^{\prime}}e^{-\zeta}\,\frac{1}{t}\,(\frac{\zeta}{t}-B)^{-1}d\zeta,$

where $t\Gamma^{\prime}$ is the union of the part of $\Gamma$ with
$|\lambda|\geq t$ and the arc in ${\mathbb{C}}\setminus\Sigma_{\gamma}$ of the
circle with the origin as center and radius $t$. However, the Cauchy formula
insures that we can go back to $\Gamma^{\prime}$:

$S_{t}=\frac{1}{2\pi{\rm
i}}\int_{\Gamma^{\prime}}e^{-\zeta}\,\frac{1}{t}\,(\frac{\zeta}{t}-B)^{-1}d\zeta.$

This implies that $\\{S_{t}\\}_{t>0}$ is $\ell_{q}$-bounded with constant

$C^{\prime\prime}_{d}\leq\frac{C_{\gamma}}{\pi}\int_{1}^{\infty}e^{-r\cos\gamma}\frac{dr}{r}+\frac{C_{\gamma}}{2\pi}\int_{\gamma}^{2\pi-\gamma}e^{-\cos\theta}d\theta\leq\frac{CC_{\gamma}}{\cos\gamma}\,.$

(iii) $\Rightarrow$ (i). Let $C_{d}$ denote the $\ell_{q}$-boundedness
constant of $\\{S_{t},tBS_{t}\\}_{t>0}$. The function $t\mapsto S_{t}$ is
infinitely derivable in ${\mathbb{R}}_{+}$ and for any positive integer $n$

$(S_{t})^{(n)}=\big{(}S^{\prime}_{\frac{t}{n}}\big{)}^{n}\,.$

This shows that $\\{t^{n}(S_{t})^{(n)}\\}_{t>0}$ is $\ell_{q}$-bounded with
constant $(nC_{d})^{n}$. Let $\beta=\arctan\frac{1}{eC_{d}}$. Then
$\\{S_{t}\\}_{t>0}$ becomes an $\ell_{q}$-analytic semigroup of type $\beta$
thanks to the following formula

$S_{z}=\sum_{n=0}^{\infty}\frac{(S_{t})^{(n)}}{n!}\,(z-t)^{n}\,,\quad
z\in\Sigma_{\beta}$

and for any $\nu<\beta$ the family
$\big{\\{}S_{z}:z\in\overline{\Sigma_{\nu}}\big{\\}}$ is $\ell_{q}$-bounded
with constant

$C_{\nu}\leq\frac{1}{1-eC_{d}\tan\nu}\,.$

The proof is thus complete. ∎

The following is again the $\ell_{q}$-boundedness version of an elementary
result on sectorial operators. The case used later concerns only $\sqrt{B}$.

######  Proposition 2.5.

Let $B$ be an $\ell_{q}$-sectorial operator of type $\alpha$ on
$L_{p}(\Omega;X)$ with $\alpha<\pi$. Let $\theta>0$ such that
$\theta\alpha<\pi$. Then $B^{\theta}$ is an $\ell_{q}$-sectorial operator of
type $\theta\alpha$ on $L_{p}(\Omega;X)$.

###### Proof.

Let $\gamma>\gamma^{\prime}>\theta\alpha$. Given
$z\notin\overline{\Sigma_{\gamma}}$, we write

$z(z-B^{\theta}\,)^{-1}=z^{\theta^{-1}}(z^{\theta^{-1}}-B)^{-1}+\varphi(B),$

where
$\varphi(\lambda)=(z^{\theta^{-1}}\lambda^{\theta}-z\lambda)(z-\lambda^{\theta})^{-1}(z^{\theta^{-1}}-\lambda)^{-1}$.
Note that $\varphi$ is analytic in $\Sigma_{\gamma\theta^{-1}}$. Let $\Gamma$
be the boundary of $\Sigma_{\gamma^{\prime}\theta^{-1}}$. Then

$\varphi(B)=\frac{1}{2\pi{\rm
i}}\int_{\Gamma}\varphi(\lambda)(\lambda-B)^{-1}d\lambda.$

The change of variables $\zeta=z^{-\theta^{-1}}\lambda$ yields

$\int_{\Gamma}|\varphi(\lambda)|\,\frac{|d\lambda|}{|\lambda|}\lesssim\theta^{-1}+(\gamma-\gamma^{\prime})^{-2}\,.$

Hence by Remark 2.2, we deduce the assertion. ∎

Now we return back to our distinguished semigroup $\\{T_{t}\\}_{t>0}$ of
regular operators on $L_{p}(\Omega)$. We have extended $\\{T_{t}\\}_{t>0}$ to
$L_{p}(\Omega;X)$. Recall the fact that $T$ is regular on $L_{p}(\Omega)$ iff

$\big{\|}\sum_{k}|T(f_{k})|\big{\|}_{p}\leq\big{\|}\sum_{k}|f_{k}|\big{\|}_{p}$

for all finite sequences $\\{f_{k}\\}$ in $L_{p}(\Omega)$ (see [33]).
Consequently, $T$ is regular on $L_{p}(\Omega)$ iff its adjoint $T^{*}$ is
regular on $L_{p^{\prime}}(\Omega)$. In particular, $\\{T^{*}_{t}\\}_{t>0}$ is
a strongly continuous semigroup of regular operators on
$L_{p^{\prime}}(\Omega)$.

######  Lemma 2.6.

Let

$M_{t}=\frac{1}{t}\int_{0}^{t}T_{s}ds,\quad t>0.$

The family $\\{M_{t}\\}_{t>0}$ is $\ell_{q}$-bounded on $L_{p}(\Omega;X)$ with
constant $\max(p^{\prime
1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}})$.

###### Proof.

The celebrated theorem of Akcoglu [2] asserts that $\\{M_{t}\\}_{t>0}$
satisfies the following maximal ergodic inequality

$\big{\|}\sup_{t>0}|M_{t}(f)|\,\big{\|}_{L_{p}(\Omega)}\leq
p^{\prime}\big{\|}f\big{\|}_{L_{p}(\Omega)}\,,\quad f\in L_{p}(\Omega);$

see also [28, Theorem 5.2.5]. The regularity of $\\{M_{t}\\}_{t>0}$ insures
that this inequality remains valid for any $f\in L_{p}(\Omega;X)$. Thus for
any finite sequences $\\{t_{k}\\}\subset{\mathbb{R}}_{+}$ and
$\\{f_{k}\\}\subset L_{p}(\Omega;X)$, we have

$\displaystyle\big{\|}\sup_{k}\|M_{t_{k}}(f_{k})\|_{X}\big{\|}_{L_{p}(\Omega)}$
$\displaystyle\leq\big{\|}\sup_{k}|M_{t_{k}}|(\|f_{k}\|_{X})\big{\|}_{L_{p}(\Omega)}$
$\displaystyle\leq\big{\|}\sup_{k}|M_{t_{k}}|(\sup_{j}\|f_{j}\|_{X})\big{\|}_{L_{p}(\Omega)}$
$\displaystyle\leq
p^{\prime}\big{\|}\sup_{j}\|f_{j}\|_{X}\,\big{\|}_{L_{p}(\Omega)}\,.$

This means that $\\{M_{t}\\}_{t>0}$ is $\ell_{\infty}$-bounded on
$L_{p}(\Omega;X)$ with constant $p^{\prime}$. On the other hand,
$\\{M_{t}\\}_{t>0}$ is bounded, so $\ell_{p}$-bounded on $L_{p}(\Omega;X)$
with constant $1$. Thus by complex interpolation, $\\{M_{t}\\}_{t>0}$ is
$\ell_{q}$-bounded on $L_{p}(\Omega;X)$ with constant $p^{\prime
1-\frac{p}{q}}$ for $q>p$.

The case of $q<p$ is treated by duality. Applying the previous discussion to
the adjoint semigroup $\\{T^{*}_{t}\\}_{t>0}$, we deduce that
$\\{M^{*}_{t}\\}_{t>0}$ is $\ell_{\infty}$-bounded on
$L_{p^{\prime}}(\Omega;X^{*})$ with constant $p$, so $\\{M_{t}\\}_{t>0}$ is
$\ell_{1}$-bounded on $L_{p}(\Omega;X)$ with constant $p$. The assertion for
$q<p$ then follows by complex interpolation once more. ∎

######  Remark 2.7.

The above lemma and the subordination formula (1.2) imply that the Poisson
subordinated semigroup $\\{P_{t}\\}_{t>0}$ is $\ell_{q}$-bounded on
$L_{p}(\Omega;X)$ with constant $C\max(p^{\prime
1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}})$.

It is a classical result that the negative generator $A$ of
$\\{T_{t}\\}_{t>0}$ on $L_{p}(\Omega;X)$ is a sectorial operator of type
$\frac{\pi}{2}$. The following shows that it is moreover $\ell_{q}$-sectorial.

######  Proposition 2.8.

The negative generator $A$ of $\\{T_{t}\\}_{t>0}$ is an $\ell_{q}$-sectorial
operator of type $\frac{\pi}{2}$ on $L_{p}(\Omega;X)$. More precisely, the
family

$\big{\\{}z(z-A)^{-1}:z\notin\overline{\Sigma_{\alpha}}\,\big{\\}}$

is $\ell_{q}$-bounded on $L_{p}(\Omega;X)$ with constant
$C_{\alpha}\max(p^{\prime
1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}})$ for any
$\frac{\pi}{2}<\alpha<\pi$.

Consequently, $\sqrt{A}$ is an $\ell_{q}$-sectorial operator of type
$\frac{\pi}{4}$ on $L_{p}(\Omega;X)$. Moreover, the $\ell_{q}$-boundedness
constant of $\\{z(z-\sqrt{A}\,)^{-1}:z\notin\overline{\Sigma_{\alpha}}\\}$ is
majorized by $C_{\alpha}\max(p^{\prime
1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}})$ for any
$\frac{\pi}{4}<\alpha<\pi$.

###### Proof.

Let $z\in{\mathbb{C}}$ with ${\rm Re}z<0$. Then

$(z-A)^{-1}=\int_{0}^{\infty}e^{tz}T_{t}dt=-z\int_{0}^{\infty}te^{tz}M_{t}dt.$

Thus by Lemma 2.6 and Remark 2.2, we deduce that

$\Big{\\{}\frac{({\rm Re}z)^{2}}{|z|}(z-A)^{-1}:{\rm Re}z<0\Big{\\}}$

is $\ell_{q}$-bounded on $L_{p}(\Omega;X)$ with constant
$C_{\alpha}\max(p^{\prime
1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}})$. This implies the
assertion on $A$. The second part on $\sqrt{A}$ then follows from Proposition
2.5. ∎

######  Proposition 2.9.

Assume that $\\{T_{t}\\}_{t>0}$ satisfies (1.4) for some
$0<\beta_{0}\leq\frac{\pi}{2}$. Let $1<q<\infty$ and
$\beta_{q}=\beta_{0}\min(\frac{p}{q}\,,\frac{p^{\prime}}{q^{\prime}})$. Then
$\\{T_{t}\\}_{t>0}$ is an $\ell_{q}$-analytic semigroup of type $\beta_{q}$ on
$L_{p}(\Omega;X)$. More precisely, for any $0<\beta<\beta_{q}$ the family
$\big{\\{}T_{z}:z\in\Sigma_{\beta}\big{\\}}$ is $\ell_{q}$-bounded on
$L_{p}(\Omega;X)$ with constant majorized by

$C(\beta_{q}-\beta)^{-1}\,\mathsf{T}_{\beta_{0}}^{\min(\frac{p}{q},\,\frac{p^{\prime}}{q^{\prime}})}\max(p^{\prime
1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}}).$

Consequently, $A$ is $\ell_{q}$-sectorial of type
$\alpha_{q}=\frac{\pi}{2}-\beta_{q}$ on $L_{p}(\Omega;X)$. More precisely, for
any $\alpha_{q}<\alpha<\frac{\pi}{2}$ the family
$\big{\\{}z(z-B)^{-1}:z\notin\overline{\Sigma_{\alpha}}\big{\\}}$ is
$\ell_{q}$-bounded on $L_{p}(\Omega;X)$ with the relevant constant majorized
by

$C(\beta_{q}-\beta)^{-2}\,\mathsf{T}_{\beta_{0}}^{\min(\frac{p}{q},\,\frac{p^{\prime}}{q^{\prime}})}\max(p^{\prime
1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}})\;\text{with
}\;\beta=\frac{\pi}{2}-\alpha.$

###### Proof.

Define

$M_{z}=\frac{1}{z}\int_{0}^{z}T_{\lambda}d\lambda,\quad
z\in\Sigma_{\beta_{0}},$

where the integral is taken along the segment $[0,\,z]$. Clearly, $M$ is
analytic in $\Sigma_{\beta_{0}}$. By Lemma 2.6, $\\{M_{t}:t>0\\}$ is
$\ell_{\infty}$-bounded (resp. $\ell_{1}$-bounded) on $L_{p}(\Omega;X)$ with
constant $p^{\prime}$ (resp. $p$). On the other hand, (1.4) means that
$\\{M_{z}:z\in\Sigma_{\beta_{0}}\\}$ is $\ell_{p}$-bounded on
$L_{p}(\Omega;X)$ with constant $\mathsf{T}_{\beta_{0}}$. Then by complex
interpolation, we see that $\\{M_{z}:z\in\Sigma_{\beta_{q}}\\}$ is
$\ell_{q}$-bounded on $L_{p}(\Omega;X)$ with constant
$\mathsf{T}_{\beta_{0}}^{\frac{p}{q}}\,p^{\prime 1-\frac{p}{q}}$ for $p<q$ and
$\mathsf{T}_{\beta_{0}}^{\frac{p^{\prime}}{q^{\prime}}}\,p^{1-\frac{p^{\prime}}{q^{\prime}}}$
for $p>q$.

We use the identity $T_{z}=M_{z}+zM_{z}^{\prime}$ to pass from $M_{z}$ to
$T_{z}$, so it remains to show that $\\{zM_{z}^{\prime}:z\in\Sigma_{\beta}\\}$
is $\ell_{q}$-bounded. To this end, let $\delta=\frac{1}{2}(\beta+\beta_{q})$.
For any $z=re^{{\rm i}\theta}\in\Sigma_{\beta}$ let ${\mathcal{C}}$ be the
circle with center $z$ and radius $r\sin(\delta-|\theta|)$. Note that one of
the two rays limiting $\Sigma_{\delta}$ is a tangent of ${\mathcal{C}}$. By
the Cauchy integral formula, we have

$zM_{z}^{\prime}=\frac{z}{2\pi{\rm
i}}\int_{\mathcal{C}}\frac{M_{\lambda}\,d\lambda}{(\lambda-z)^{2}}\,.$

Since

$\frac{|z|}{2\pi}\int_{\mathcal{C}}\frac{|d\lambda|}{|\lambda-z|^{2}}=\frac{1}{\sin(\delta-|\theta|)}\leq\frac{1}{\sin\frac{1}{2}(\beta_{q}-\beta)}\,,$

The $\ell_{q}$-boundedness of $\\{M_{z}:z\in\Sigma_{\delta}\\}$ and Remark 2.2
imply that $\\{zM_{z}^{\prime}:z\in\Sigma_{\beta}\\}$ is $\ell_{q}$-bounded on
$L_{p}(\Omega;X)$ with constant majorized by

$C(\beta_{q}-\beta)^{-1}\,\mathsf{T}_{\beta_{0}}^{\min(\frac{p}{q},\,\frac{p^{\prime}}{q^{\prime}})}\max(p^{\prime
1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}}).$

The last part on the $\ell_{q}$-sectoriality of $A$ follows from the proof of
the implication (i)$\,\Rightarrow\,$(ii) of Proposition 2.4, $\beta$ and $\nu$
there being respectively $\beta_{q}$ and $\beta$ now. ∎

##  3\. Proof of Theorem 1.9

Armed with the preparation in section 2, we will follow the proof of [26,
Theorem 4.14]. In the sequel, we will use the abbreviation that
$\partial=\frac{\partial}{\partial t}$. Recall that ${\mathbb{R}}_{+}$ is
equipped with the measure $\frac{dt}{t}$.

We first show part (i) concerning the subordinated Poisson semigroup
$\\{P_{t}\\}_{t>0}$. Fix $\frac{\pi}{4}<\alpha<\beta<\frac{\pi}{2}$. Let
$\Gamma$ be the boundary of $\Sigma_{\alpha}$. Define $F(z)=-ze^{-z}$. Then
$F\in H^{\infty}_{0}(\Sigma_{\beta})$. For any $t>0$ we have

(3.1) $t\partial
P_{t}=F(t\sqrt{A}\,)=\frac{1}{2\pi\textrm{i}}\int_{\Gamma}F(tz)R(z)dz,$

where $R(z)=(z-\sqrt{A}\,)^{-1}$. Recall that the map $\mathcal{P}$ is defined
by (1.8) (with $\\{P_{t}\\}_{t>0}$ instead of $\\{T_{t}\\}_{t>0}$ there). It
can be rewritten as

(3.2)
$\mathcal{P}(h)_{s}=\frac{2}{\pi\textrm{i}}\int_{\Gamma}\int_{0}^{\infty}F(sz)F(tz)zR(z)(h_{t})\,\frac{dt}{t}\,\frac{dz}{z}\,,\quad
s>0.$

Let $\Gamma$ be equipped with the measure $\frac{|dz|}{|z|}$. We define three
maps as follows:

  * $\bullet$

$\Phi_{1}:L_{p}(\Omega;L_{q}({\mathbb{R}}_{+};X))\to
L_{p}(\Omega;L_{q}(\Gamma;X))$ by

$\Phi_{1}(h)_{z}=\int_{0}^{\infty}F(tz)h_{t}\,\frac{dt}{t}\,,\quad
z\in\Gamma,\;h\in L_{p}(\Omega;L_{q}({\mathbb{R}}_{+};X)),$

  * $\bullet$

$\Phi_{2}:L_{p}(\Omega;L_{q}(\Gamma;X))\to
L_{p}(\Omega;L_{q}({\mathbb{R}}_{+};X))$ by

$\Phi_{2}(g)_{s}=\int_{\Gamma}F(sz)g_{z}\,\frac{dz}{z}\,,\quad s>0,\;g\in
L_{p}(\Omega;L_{q}(\Gamma;X))$

  * $\bullet$

$\Phi:L_{p}(\Omega;L_{q}(\Gamma;X))\to L_{p}(\Omega;L_{q}(\Gamma;X))$ by

$\Phi(g)_{z}=\frac{2}{\pi\textrm{i}}\,zR(z)(g_{z}),\quad z\in\Gamma,\;g\in
L_{p}(\Omega;L_{q}(\Gamma;X)).$

Then $\mathcal{P}=\Phi_{2}\Phi\Phi_{1}$. Thus it remains to show that the
three newly defined maps are all bounded. Consider first the case $q<\infty$.
By the Hölder inequality, we have

$\big{\|}\Phi_{1}(h)_{z}\big{\|}_{X}^{q}\leq\Big{(}\int_{0}^{\infty}|F(tz)|\,\frac{dt}{t}\Big{)}^{q-1}\int_{0}^{\infty}|F(tz)|\,\big{\|}h_{t}\big{\|}^{q}_{X}\,\frac{dt}{t}\,.$

Note that for $z=re^{\pm{\rm i}\alpha}\in\Gamma$

$\int_{0}^{\infty}|F(tz)|\,\frac{dt}{t}=\int_{0}^{\infty}|F(te^{\pm{\rm
i}\alpha})|\,\frac{dt}{t}=\frac{1}{\cos\alpha}\,.$

On the other hand, for any $t>0$

$\int_{\Gamma}|F(tz)|\frac{|dz|}{|z|}=\frac{2}{\cos\alpha}\,.$

We then deduce that

$\big{\|}\Phi_{1}(h)\big{\|}_{L_{p}(\Omega;L_{q}(\Gamma;X))}\leq\frac{2^{\frac{1}{q}}}{\cos\alpha}\,\big{\|}h\big{\|}_{L_{p}(\Omega;L_{q}({\mathbb{R}}_{+};X))}\,.$

Thus

$\big{\|}\Phi_{1}\big{\|}\leq\frac{2^{\frac{1}{q}}}{\cos\alpha}\,.$

The same upper estimate holds for $\big{\|}\Phi_{2}\big{\|}$ too. Finally, the
boundedness of $\Phi$ is just a reformulation of the $\ell_{q}$-boundedness of
$\big{\\{}\frac{2}{\pi\textrm{i}}\,zR(z):z\in\Gamma\setminus\\{0\\}\big{\\}}$.
Thus by Proposition 2.8,

$\big{\|}\Phi\big{\|}\lesssim\frac{1}{\cos^{2}\alpha}\max(p^{\prime
1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}}).$

This finishes the proof of the first assertion for $q<\infty$ (choosing
$\alpha$ close to $\frac{\pi}{4}$). The boundedness of $\mathcal{P}$ for
$q=\infty$ is obtained from that for $q=1$ by duality.

Let us show that $\mathcal{P}$ is a projection. Let $h\in
L_{p}(\Omega;L_{q}({\mathbb{R}}_{+};X))$ be given by $h_{t}=t\partial
P_{t}(f)$ for some $f\in L_{p}(\Omega;X)$. Then by (3.2)

$\displaystyle\mathcal{P}(h)_{s}$
$\displaystyle=\frac{2}{\pi\textrm{i}}\int_{\Gamma}\int_{0}^{\infty}F(sz)F(tz)F(tz)R(z)(f)\,\frac{dt}{t}\,dz$
$\displaystyle=\frac{2}{\pi\textrm{i}}\int_{\Gamma}F(sz)R(z)(f)dz\int_{0}^{\infty}(F(t))^{2}\frac{dt}{t}$
$\displaystyle=\frac{1}{2\pi\textrm{i}}\int_{\Gamma}F(sz)R(z)(f)dz$
$\displaystyle=F(s\sqrt{A}\,)(f)=s\partial P_{s}(f).$

Thus $\mathcal{P}(h)=h$, so $\mathcal{P}$ is a projection. This shows the
first assertion (i).

Assertion (ii) on the semigroup $\\{T_{t}\\}_{t>0}$ itself is proved exactly
in the same way. Indeed, letting $\alpha_{q}=\frac{\pi}{2}-\beta_{q}$, by
Proposition 2.9, $A$ is $\ell_{q}$-sectorial of type $\alpha_{q}$. Let
$\beta=\frac{\beta_{q}}{2}$ and $\alpha=\frac{\pi}{2}-\beta$. Then
$\alpha_{q}<\alpha<\frac{\pi}{2}$ and

$\frac{1}{\cos^{2}\alpha}\approx\frac{1}{\beta_{q}^{2}},\quad\frac{1}{(\beta_{q}-\beta)^{2}}\approx\frac{1}{\beta_{q}^{2}}.$

Thus using the estimate on the $\ell_{q}$-sectoriality constant of $A$ and
repeating the above argument, we show that $\mathcal{T}$ is bounded with the
announced norm estimate.

##  4\. Lusin type and cotype

In this section we study Banach spaces that are of Lusin coype or type. Before
proceeding we pause for a brief discussion about the projection $\mathsf{F}$
onto the fixed point subspace of $\\{T_{t}\\}_{t>0}$ (equivalently, of
$\\{P_{t}\\}_{t>0}$). By the mean ergodic theorem, $\mathsf{F}$ is given by

$\mathsf{F}(f)=\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}T_{s}(f)ds,\quad f\in
L_{p}(\Omega).$

Thus $\mathsf{F}$ is also regular, so extends to a contractive projection on
$L_{p}(\Omega;X)$. Then the above formula remains valid for $f\in
L_{p}(\Omega;X)$ and $\mathsf{F}(L_{p}(\Omega;X))$ coincides with the fixed
point subspace of $\\{T_{t}\\}_{t>0}$ on $L_{p}(\Omega;X)$. It follows that
$L_{p}(\Omega;X)$ admits the following direct sum decomposition:

(4.1) $L_{p}(\Omega;X)=\mathsf{F}(L_{p}(\Omega;X))\oplus\ker\mathsf{F}.$

On the other hand, $\ker\mathsf{F}$ is the closure of $\big{\\{}({\rm
Id}-T_{t})(L_{p}(\Omega;X)):t>0\big{\\}}$. Moreover,

(4.2) $\mathsf{F}(L_{p}(\Omega;X))=\ker A=\ker\sqrt{A}\;\text{ and
}\;\ker\mathsf{F}=\overline{{\rm im}A}=\overline{{\rm im}\sqrt{A}}.$

By the paragraph before Lemma 2.6, we know that the adjoint semigroup
$\\{T^{*}_{t}\\}_{t>0}$ is regular on $L_{p^{\prime}}(\Omega)$ too. Thus the
above discussion also applies to the semigroup $\\{T^{*}_{t}\\}_{t>0}$ that is
extended to $L_{p^{\prime}}(\Omega;X^{*})$ again. Consequently, (4.1) and
(4.2) transfer to this dual setting. We should draw the reader’s attention to
the fact that $L_{p^{\prime}}(\Omega;X^{*})$ is in general not the dual of
$L_{p}(\Omega;X)$ but an isometric subspace. With this in mind, we have

${\rm Id}_{X^{*}}\otimes T_{t}^{*}=\big{(}{\rm Id}_{X}\otimes
T_{t})^{*}\Big{|}_{L_{p^{\prime}}(\Omega;X^{*})}\,.$

A similar formula holds for the negative generator $A$ of $\\{T_{t}\\}_{t>0}$
on $L_{p}(\Omega;X)$ and the negative generator $A^{*}$ of
$\\{T^{*}_{t}\\}_{t>0}$ on $L_{p^{\prime}}(\Omega;X^{*})$, that is, the
restriction to $L_{p^{\prime}}(\Omega;X^{*})$ of the adjoint of the former
coincides with the latter. Moreover,
$\mathsf{F}^{*}\big{|}_{L_{p^{\prime}}(\Omega;X^{*})}$ is the fixed point
projection associated to $\\{T^{*}_{t}\\}_{t>0}$ on
$L_{p^{\prime}}(\Omega;X^{*})$. All this allows us to use duality arguments
without any problem as if $L_{p}(\Omega;X)^{*}=L_{p^{\prime}}(\Omega;X^{*})$
(which is the case for reflexive $X$).

According to [57], we introduce the following definition already mentioned
before Theorem 1.2.

######  Definition 4.1.

Let $1\leq q\leq\infty$.

  1. (i)

Define

(4.3) $\mathcal{G}^{T}_{q}(f)=\Big{(}\int_{0}^{\infty}\|t\partial
T_{t}(f)\|^{q}_{X}\frac{dt}{t}\Big{)}^{\frac{1}{q}}\,,\quad f\in
L_{p}(\Omega;X).$

  2. (ii)

$X$ is said to be of Lusin cotype $q$ relative to $\\{T_{t}\\}_{t>0}$ if there
exists a constant $c$ such that

(4.4) $\big{\|}\mathcal{G}^{T}_{q}(f)\big{\|}_{L_{p}(\Omega)}\leq
c\big{\|}f\big{\|}_{L_{p}(\Omega;X)}\,,\quad f\in L_{p}(\Omega;X).$

The smallest $c$ is denoted by $\mathsf{L}^{T}_{c,q,p}(X)$.

  3. (iii)

$X$ is said to be of Lusin type $q$ relative to $\\{T_{t}\\}_{t>0}$ if there
exists a constant $c$ such that

(4.5) $\big{\|}f-\mathsf{F}(f)\big{\|}_{L_{p}(\Omega;X)}\leq
c\big{\|}\mathcal{G}^{T}_{q}(f)\big{\|}_{L_{p}(\Omega)}\,,\quad f\in
L_{p}(\Omega;X).$

The smallest $c$ is denoted by $\mathsf{L}^{T}_{t,q,p}(X)$.

In (4.3), $f$ is implicitly assumed to belong to the definition domain of $A$
in order to guarantee the derivability of $T_{t}(f)$ in $t$. Note that if
$\\{T_{t}\\}_{t>0}$ is analytic on $L_{p}(\Omega;X)$, $\mathcal{G}^{T}_{q}(f)$
is defined for any $f\in L_{p}(\Omega;X)$. When it is defined,
$\mathcal{G}^{T}_{q}(f)$ is a positive measurable function on $\Omega$ but may
not belong to $L_{p}(\Omega)$ in which case
$\|\mathcal{G}^{T}_{q}(f)\|_{L_{p}(\Omega)}$ is interpreted as $\infty$ (then
(4.5) is trivially verified for such $f$). On the other hand, the definition
above implicitly depends on $p$, this dependence is reflected in the constants
$\mathsf{L}^{T}_{c,q,p}(X)$ and $\mathsf{L}^{T}_{t,q,p}(X)$.

######  Remark 4.2.

Without additional assumption on $\\{T_{t}\\}_{t>0}$, the definition may be
insignificant. For instance, if $\\{T_{t}\\}_{t>0}$ is the translation group
of ${\mathbb{R}}$, it is easy to check that

$\Big{\|}\Big{(}\int_{0}^{\infty}|t\partial
T_{t}(f)|^{q}\frac{dt}{t}\Big{)}^{\frac{1}{q}}\Big{\|}^{p}_{L_{p}({\mathbb{R}})}=\int_{{\mathbb{R}}}\Big{(}\int_{0}^{\infty}|tf^{\prime}(s+t)|^{q}\frac{dt}{t}\Big{)}^{\frac{p}{q}}ds=\infty$

for any $1\leq q\leq\infty$ and for any $f\in L_{p}({\mathbb{R}})$ with
$f^{\prime}$ not identically zero. Thus ${\mathbb{C}}$ is not of Lusin cotype
$q$ for any $q$ relative to the translation group of ${\mathbb{R}}$.

This remark shows that to have a meaningful theory of Lusin type and cotype
some minimal condition should be imposed to $\\{T_{t}\\}_{t>0}$. It turns out
that this minimal condition is the analyticity of $\\{T_{t}\\}_{t>0}$ on
$L_{p}(\Omega;X)$. As shown in section 2, the subordinated Poisson semigroup
$\\{P_{t}\\}_{t>0}$ always satisfies this condition.

It is sometimes convenient to have a discrete version of
$\mathcal{G}^{T}_{q}(f)$. Recall that if $\\{T_{t}\\}_{t>0}$ is analytic on
$L_{p}(\Omega)$, we have the following maximal inequality

(4.6)
$\big{\|}\sup_{t>0}|T_{t}(f)|\big{\|}_{L_{p}(\Omega)}\leq\mathsf{T}_{\max}\|f\|_{L_{p}(\Omega)}\,,\quad
f\in L_{p}(\Omega)$

for some constant $\mathsf{T}_{\max}$ (see [32]). A similar inequality holds
for the adjoint semigroup $\\{T^{*}_{t}\\}_{t>0}$, the relevant constant being
denoted by $\mathsf{T}^{*}_{\max}$.

######  Proposition 4.3.

Assume that $\\{T_{t}\\}_{t>0}$ is analytic on $L_{p}(\Omega)$. Let $1\leq
q\leq\infty$ and $a>1$. Then for any $f\in L_{p}(\Omega;X)$

$c_{T,q,a}^{-1}\big{\|}\mathcal{G}^{T}_{q}(f)\big{\|}_{L_{p}(\Omega)}\leq\Big{\|}\Big{(}\sum_{k\in{\mathbb{Z}}}\big{\|}a^{k}\partial
T_{a^{k}}(f)\big{\|}_{X}^{q}\Big{)}^{\frac{1}{q}}\Big{\|}_{L_{p}(\Omega)}\leq
C_{T,q,a}\big{\|}\mathcal{G}^{T}_{q}(f)\big{\|}_{L_{p}(\Omega)}\,,$

where

$\displaystyle
c_{T,q,a}=q^{-\frac{1}{q}}(a^{q}-1)^{\frac{1}{q}}\max(\mathsf{T}_{\max}^{1-\frac{p}{q}},\,{\mathsf{T}^{*}}_{\max}^{1-\frac{p^{\prime}}{q^{\prime}}}),$
$\displaystyle
C_{T,q,a}=q^{\frac{1}{q}}(1-a^{-q})^{-\frac{1}{q}}\max(\mathsf{T}_{\max}^{1-\frac{p}{q}},\,{\mathsf{T}^{*}}_{\max}^{1-\frac{p^{\prime}}{q^{\prime}}})\,.$

Similar inequalities hold for $\\{P_{t}\\}_{t>0}$ in place of
$\\{T_{t}\\}_{t>0}$ without any additional assumption on $\\{T_{t}\\}_{t>0}$,
the corresponding constants being given by

$\displaystyle
c_{P,q,a}=C^{-1}q^{-\frac{1}{q}}(a^{q}-1)^{\frac{1}{q}}\max(p^{\prime
1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}}),$ $\displaystyle
C_{P,q,a}=Cq^{\frac{1}{q}}(1-a^{-q})^{-\frac{1}{q}}\max(p^{\prime
1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}})\,.$

###### Proof.

Using (4.6), its adjoint version and repeating the proof of Lemma 2.6, we show
that $\\{T_{t}\\}_{t>0}$ is $\ell_{q}$-bounded on $L_{p}(\Omega;X)$ with
constant
$\max(\mathsf{T}_{\max}^{1-\frac{p}{q}},\,{\mathsf{T}^{*}}_{\max}^{1-\frac{p^{\prime}}{q^{\prime}}})$.
Write

$\displaystyle\mathcal{G}^{T}_{q}(f)^{q}=\sum_{k\in{\mathbb{Z}}}\int_{a^{k}}^{a^{k+1}}\big{\|}t\partial
T_{t}(f)\big{\|}_{X}^{q}\frac{dt}{t}=\sum_{k\in{\mathbb{Z}}}\int_{1}^{a}\big{\|}a^{k}t\partial
T_{a^{k}t}(f)\big{\|}_{X}^{q}\frac{dt}{t}\,.$

Using $\partial T_{t+s}=T_{s}\partial T_{t}$, we have $\partial
T_{a^{k}t}(f)=T_{a^{k}(t-1)}\partial T_{a^{k}}(f)$. Then the
$\ell_{q}$-boundedness of $\\{T_{t}\\}_{t>0}$ on $L_{p}(\Omega;X)$ yields

$\displaystyle\big{\|}\mathcal{G}^{T}_{q}(f)\big{\|}_{L_{p}(\Omega)}$
$\displaystyle\leq\max(\mathsf{T}_{\max}^{1-\frac{p}{q}},\,{\mathsf{T}^{*}}_{\max}^{1-\frac{p^{\prime}}{q^{\prime}}})\Big{\|}\Big{(}\sum_{k\in{\mathbb{Z}}}\int_{1}^{a}\big{\|}a^{k}t\partial
T_{a^{k}}(f)\big{\|}_{X}^{q}\frac{dt}{t}\Big{)}^{\frac{1}{q}}\Big{\|}_{L_{p}(\Omega)}$
$\displaystyle=q^{-\frac{1}{q}}(a^{q}-1)^{\frac{1}{q}}\max(\mathsf{T}_{\max}^{1-\frac{p}{q}},\,{\mathsf{T}^{*}}_{\max}^{1-\frac{p^{\prime}}{q^{\prime}}})\Big{\|}\Big{(}\sum_{k\in{\mathbb{Z}}}\big{\|}a^{k}\partial
T_{a^{k}}(f)\big{\|}_{X}^{q}\Big{)}^{\frac{1}{q}}\Big{\|}_{L_{p}(\Omega)}\,.$

For the converse inequality, we write

$\big{\|}a^{k}\partial
T_{a^{k}}(f)\big{\|}_{X}^{q}=q(1-a^{-q})^{-1}\int_{a^{-1}}^{1}\big{\|}a^{k}tT_{a^{k}(1-t)}\partial
T_{a^{k}t}(f)\big{\|}_{X}^{q}\frac{dt}{t}\,.$

As above, we then deduce

$\Big{\|}\Big{(}\sum_{k\in{\mathbb{Z}}}\big{\|}a^{k}\partial
T_{a^{k}}(f)\big{\|}_{X}^{q}\Big{)}^{\frac{1}{q}}\Big{\|}_{L_{p}(\Omega)}\leq
q^{\frac{1}{q}}(1-a^{-q})^{-\frac{1}{q}}\max(\mathsf{T}_{\max}^{1-\frac{p}{q}},\,{\mathsf{T}^{*}}_{\max}^{1-\frac{p^{\prime}}{q^{\prime}}})\big{\|}\mathcal{G}^{T}_{q}(f)\big{\|}_{L_{p}(\Omega)}\,.$

The second part on $\\{P_{t}\\}_{t>0}$ is just a particular case with
$\mathsf{P}_{\max}=Cp^{\prime}$ and $\mathsf{P}^{*}_{\max}=Cp$ by virtue of
Remark 2.7. ∎

The following remark immediately follows from the above result; it shows in
particular that the analyticity of $\\{T_{t}\\}_{t>0}$ on $L_{p}(\Omega;X)$ is
necessary for $X$ to be of Lusin cotype $q$ relative to $\\{T_{t}\\}_{t>0}$
for some $q$.

######  Remark 4.4.

Assume that $\\{T_{t}\\}_{t>0}$ is analytic on $L_{p}(\Omega)$. If $X$ is of
Lusin cotype $($resp. type$)$ $q$ relative to $\\{T_{t}\\}_{t>0}$, then $X$ is
of Lusin cotype $($resp. type$)$ $r$ relative to $\\{T_{t}\\}_{t>0}$ for any
$r>q$ ($($resp. $r<q)$. Moreover, if $X$ is of Lusin cotype $\infty$ relative
to $\\{T_{t}\\}_{t>0}$, then $\\{T_{t}\\}_{t>0}$ must be analytic on
$L_{p}(\Omega;X)$.

The following is a main result of this section.

######  Theorem 4.5.

Let $X$ be a Banach space and $1\leq q\leq\infty$.

  1. (i)

$X$ is of Lusin cotype $q$ relative to $\\{P_{t}\\}_{t>0}$ iff $X^{*}$ is of
Lusin type $q^{\prime}$ relative to $\\{P^{*}_{t}\\}_{t>0}$.
Moreover, the relevant constants satisfy

$\mathsf{L}^{P^{*}}_{t,q^{\prime},p^{\prime}}(X^{*})\lesssim\mathsf{L}^{P}_{c,q,p}(X)\lesssim\max(p^{\prime
1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}})\mathsf{L}^{P^{*}}_{t,q^{\prime},p^{\prime}}(X^{*})\,.$

  2. (ii)

Assume additionally that $1<q<\infty$ and $\\{T_{t}\\}_{t>0}$ satisfies (1.4)
for some $0<\beta_{0}\leq\frac{\pi}{2}$. Then $X$ is of Lusin cotype $q$
relative to $\\{T_{t}\\}_{t>0}$ iff $X^{*}$ is of Lusin type $q^{\prime}$
relative to $\\{T^{*}_{t}\\}_{t>0}$. Moreover, the relevant constants satisfy

$\mathsf{L}^{T^{*}}_{t,q^{\prime},p^{\prime}}(X^{*})\lesssim\mathsf{L}^{T}_{c,q,p}(X)\lesssim\beta_{q}^{-4}\,\mathsf{T}_{\nu^{\prime}}^{\min(\frac{p}{q},\,\frac{p^{\prime}}{q^{\prime}})}\,\max(p^{\prime
1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}})\,\mathsf{L}^{T^{*}}_{t,q^{\prime},p^{\prime}}(X^{*})$

with $\beta_{q}=\beta_{0}\min(\frac{p}{q},\,\frac{p^{\prime}}{q^{\prime}})$.

###### Proof.

(i) Assume that $X$ is of Lusin cotype $q$. Let $g\in
L_{p^{\prime}}(\Omega;X^{*})$ with $\mathsf{F^{*}}(g)=0$. Let $f\in
L_{p}(\Omega;X)$. We want to estimate $\langle f,\,g\rangle$, where the
duality bracket is that between $L_{p}(\Omega;X)$ and
$L_{p^{\prime}}(\Omega;X^{*})$. By (4.1) and its dual version, we can assume
$\mathsf{F}(f)=0$, which, together with (4.2), implies that
$f\in\ker\mathsf{F}=\overline{{\rm im}\sqrt{A}}$. With $F(z)=-ze^{-z}$ and by
(2.1) we have

$f=4\int_{0}^{\infty}F(t\sqrt{A}\,)F(t\sqrt{A}\,)(f)\,\frac{dt}{t}\,.$

Thus by the Hölder inequality and the Lusin cotype $q$ of $X$

$\displaystyle\big{|}\langle f,\,g\rangle\big{|}$
$\displaystyle=4\left|\int_{0}^{\infty}\langle
F(t\sqrt{A}\,)(f),\,F(t\sqrt{A^{*}})(g)\rangle\,\frac{dt}{t}\right|$
$\displaystyle\leq
4\big{\|}\mathcal{G}^{P}_{q}(f)\big{\|}_{L_{p}(\Omega)}\,\big{\|}\mathcal{G}^{P^{*}}_{q^{\prime}}(g)\big{\|}_{L_{p^{\prime}}(\Omega)}$
$\displaystyle\leq
4\,\mathsf{L}^{P}_{c,q,p}(X)\big{\|}f\big{\|}_{L_{p}(\Omega;X)}\,\big{\|}\mathcal{G}^{P^{*}}_{q^{\prime}}(g)\big{\|}_{L_{p^{\prime}}(\Omega)}\,.$

Taking the supremum over $f$ with norm 1, we show that $X^{*}$ is of Lusin
type $q^{\prime}$ with

$\mathsf{L}^{P^{*}}_{t,q^{\prime},p^{\prime}}(X^{*})\leq
4\,\mathsf{L}^{P}_{c,q,p}(X).$

To show the converse implication, let $f\in L_{p}(\Omega;X)$ and $h\in
L_{p^{\prime}}(\Omega;L_{q^{\prime}}({\mathbb{R}}_{+};X^{*}))$ (recalling that
${\mathbb{R}}_{+}$ is equipped with $\frac{dt}{t}$). We have

$\int_{0}^{\infty}\langle t\partial
P_{t}(f),\,h_{t}\rangle\,\frac{dt}{t}=\int_{0}^{\infty}\langle f,\,t\partial
P^{*}_{t}(h_{t})\rangle\,\frac{dt}{t}=\langle f,\,g\rangle,$

where

$g=\int_{0}^{\infty}t\partial P^{*}_{t}(h_{t})\,\frac{dt}{t}.$

Applying Theorem 1.9 (i) to $\\{P^{*}_{t}\\}_{t>0}$ on
$L_{p^{\prime}}(\Omega;X^{*})$, we have

$\big{\|}\mathcal{G}^{P^{*}}_{q^{\prime}}(g)\big{\|}_{L_{p^{\prime}}(\Omega)}\lesssim\max(p^{\prime
1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}})\big{\|}h\big{\|}_{L_{p^{\prime}}(\Omega;L_{q^{\prime}}({\mathbb{R}}_{+};X^{*}))}\,.$

Combining the above inequalities, we get

$\left|\int_{0}^{\infty}\langle t\partial
P_{t}(f),\,h_{t}\rangle\,\frac{dt}{t}\right|\lesssim\max(p^{\prime
1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}})\mathsf{L}^{P^{*}}_{t,q^{\prime}}(X^{*})\big{\|}f\big{\|}_{L_{p}(\Omega;X)}\,\big{\|}h\big{\|}_{L_{p^{\prime}}(\Omega;L_{q^{\prime}}({\mathbb{R}}_{+};X^{*}))}\,,$

which implies the Lusin cotype $q$ of $X$ with

$\mathsf{L}^{P}_{c,q,p}(X)\lesssim\max(p^{\prime
1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}})\mathsf{L}^{P^{*}}_{t,q^{\prime},p^{\prime}}(X^{*})\,.$

(ii) The proof of this part is done in a similar way by using Theorem 1.9
(ii). ∎

######  Corollary 4.6.

Any Banach space $X$ is of Lusin type $1$ relative to $\\{P_{t}\\}_{t>0}$, so
relative to $\\{T_{t}\\}_{t>0}$ too. If $\\{T_{t}\\}_{t>0}$ is analytic on
$L_{p}(\Omega;X)$, then $X$ is of Lusin cotype $\infty$ relative to
$\\{T_{t}\\}_{t>0}$, so $X$ is always of Lusin cotype $\infty$ relative to the
subordinated Poisson semigroup $\\{P_{t}\\}_{t>0}$.

###### Proof.

Indeed, let $f\in L_{p}(\Omega;X)$ such that $\mathsf{F}(f)=0$. Then by (4.2)
and (2.1) we have

$f=-\int_{0}^{\infty}t\partial P_{t}(f)\,\frac{dt}{t}\,,$

whence

$\big{\|}f\big{\|}_{L_{p}(\Omega;X)}\leq\big{\|}\mathcal{G}^{P}_{1}(f)\big{\|}_{L_{p}(\Omega)}\,.$

Thus $X$ is of Lusin type $1$ relative to $\\{P_{t}\\}_{t>0}$, hence also
relative to $\\{T_{t}\\}_{t>0}$ by virtue of Remark 1.1. Passing to duality by
means of Theorem 4.5, we see that $X$ is of Lusin cotype $\infty$ relative to
$\\{T_{t}\\}_{t>0}$ under the analyticity assumption of $\\{T_{t}\\}_{t>0}$ on
$L_{p}(\Omega;X)$. ∎

The following formulation of the Littlewood-Paley function
$\mathcal{G}^{P}_{q}(f)$ by discrete lacunary differences is of interest for
its own right.

######  Theorem 4.7.

Let $X$ be a Banach space, $1\leq q\leq\infty$ and $a>1$.

  1. (i)

$X$ is of Lusin cotype $q$ relative to $\\{P_{t}\\}_{t>0}$ iff there exists a
constant $c$ such that

(4.7)
$\Big{\|}\Big{(}\sum_{k\in{\mathbb{Z}}}\big{\|}(P_{a^{k}t}-P_{a^{k+1}t})(f)\big{\|}_{X}^{q}\Big{)}^{\frac{1}{q}}\Big{\|}_{L_{p}(\Omega)}\leq
c\big{\|}f\big{\|}_{L_{p}(\Omega;X)}$

for all $1\leq t\leq a$ and $f\in L_{p}(\Omega;X)$. Moreover, the best $c$ and
$\mathsf{L}^{P}_{c,q,p}(X)$ are linked by

$(\log a)^{-\frac{1}{q^{\prime}}}c\leq\mathsf{L}^{P}_{c,q,p}(X)\lesssim(\log
a)^{\frac{1}{q}}\max(p^{\prime
1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}})c.$

  2. (ii)

$X$ is of Lusin type $q$ relative to $\\{P_{t}\\}_{t>0}$ iff there exists a
constant $c$ such that

(4.8) $\big{\|}f-\mathsf{F}(f)\big{\|}_{L_{p}(\Omega;X)}\leq
c\Big{\|}\Big{(}\int_{1}^{a}\sum_{k\in{\mathbb{Z}}}\big{\|}(P_{a^{k}t}-P_{a^{k+1}t})(f)\big{\|}_{X}^{q}dt\Big{)}^{\frac{1}{q}}\Big{\|}_{L_{p}(\Omega)}$

for all $f\in L_{p}(\Omega;X)$. Moreover, the best $c$ and
$\mathsf{L}^{P}_{t,q,p}(X)$ are linked by

$a^{-\frac{1}{q}}(\log
a)^{-\frac{1}{q^{\prime}}}c\leq\mathsf{L}^{P}_{t,q,p}(X)\lesssim\frac{a+1}{a-1}\max(p^{\prime
1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}})c.$

  3. (iii)

Similar statements hold for $\\{T_{t}\\}_{t>0}$ under the additional
assumption that $\\{T_{t}\\}_{t>0}$ be analytic on $L_{p}(\Omega;X)$ and
$1<q<\infty$.

###### Proof.

(i) We have

$\big{\|}(P_{a^{k}t}-P_{a^{k+1}t})(f)\big{\|}_{X}^{q}=\Big{\|}\int_{a^{k}t}^{a^{k+1}t}\partial
P_{s}(f)ds\Big{\|}_{X}^{q}\leq(\log
a)^{\frac{q}{q^{\prime}}}\int_{a^{k}t}^{a^{k+1}t}\big{\|}s\partial
P_{s}(f)\big{\|}_{X}^{q}\frac{ds}{s}\,,$

which implies

$\sum_{k\in{\mathbb{Z}}}\big{\|}(P_{a^{k}t}-P_{a^{k+1}t})(f)\big{\|}_{X}^{q}\leq(\log
a)^{\frac{q}{q^{\prime}}}\mathcal{G}^{P}_{q}(f)^{q}\,.$

Thus if $X$ is of Lusin cotype $q$ relative to $\\{P_{t}\\}_{t>0}$, then (4.7)
holds with $c\leq(\log a)^{\frac{1}{q^{\prime}}}\mathsf{L}^{P}_{c,q,p}(X)\,.$

To show the converse implication, let $b=\frac{1}{2}(1+a)$. We use an idea
from [24] (see also [59]) to write

$\partial P_{t}=\sum_{k=0}^{\infty}\big{(}\partial P_{b^{k}t}-\partial
P_{b^{k+1}t}\big{)}=\sum_{k=0}^{\infty}\partial
P_{b^{k}2^{-1}t}\big{(}P_{b^{k}2^{-1}t}-P_{ab^{k}2^{-1}t}\big{)}.$

Hence

$\displaystyle\big{\|}\mathcal{G}^{P}_{q}(f)\big{\|}_{L_{p}(\Omega)}$
$\displaystyle\leq\sum_{k=0}^{\infty}\Big{\|}\big{\\{}t\partial
P_{b^{k}2^{-1}t}\big{(}P_{b^{k}2^{-1}t}-P_{ab^{k}2^{-1}t}\big{)}(f)\big{\\}}_{t>0}\Big{\|}_{L_{p}(\Omega;L_{q}({\mathbb{R}}_{+};X))}$
$\displaystyle=2\sum_{k=0}^{\infty}b^{-k}\Big{\|}\big{\\{}t\partial
P_{t}\big{(}P_{t}-P_{at}\big{)}(f)\big{\\}}_{t>0}\Big{\|}_{L_{p}(\Omega;L_{q}({\mathbb{R}}_{+};X))}$
$\displaystyle=\frac{2(a+1)}{a-1}\Big{\|}\big{\\{}t\partial
P_{t}\big{(}P_{t}-P_{at}\big{)}(f)\big{\\}}_{t>0}\Big{\|}_{L_{p}(\Omega;L_{q}({\mathbb{R}}_{+};X))}\,.$

By Propositions 2.4 and 2.8, the family $\\{t\partial P_{t}\\}_{t>0}$ is
$\ell_{q}$-bounded with constant $C\max(p^{\prime
1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}})$. Therefore,

$\Big{\|}\big{\\{}t\partial
P_{t}\big{(}P_{t}-P_{at}\big{)}(f)\big{\\}}_{t>0}\Big{\|}_{L_{p}(\Omega;L_{q}({\mathbb{R}}_{+};X))}\lesssim\max(p^{\prime
1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}})\Big{\|}\big{\\{}\big{(}P_{t}-P_{at}\big{)}(f)\big{\\}}_{t>0}\Big{\|}_{L_{p}(\Omega;L_{q}({\mathbb{R}}_{+};X))}\,.$

To estimate the norm on the right hand side, we write

$\displaystyle\int_{0}^{\infty}\big{\|}\big{(}P_{t}-P_{at}\big{)}(f)\big{\|}^{q}_{X}\,\frac{dt}{t}$
$\displaystyle=\sum_{k\in{\mathbb{Z}}}\int_{a^{k}}^{a^{k+1}}\big{\|}\big{(}P_{t}-P_{at}\big{)}(f)\big{\|}^{q}_{X}\,\frac{dt}{t}$
$\displaystyle=\int_{1}^{a}\sum_{k\in{\mathbb{Z}}}\big{\|}\big{(}P_{a^{k}t}-P_{a^{k+1}t}\big{)}(f)\big{\|}^{q}_{X}\,\frac{dt}{t}\,.$

Note that the function

$t\mapsto\Big{(}\sum_{k\in{\mathbb{Z}}}\big{\|}\big{(}P_{a^{k}t}-P_{a^{k+1}t}\big{)}(f)\big{\|}^{q}_{X}\Big{)}^{\frac{1}{q}}$

is continuous from ${\mathbb{R}}_{+}$ to $L_{p}(\Omega)$, so there exists
$t_{0}\in[1,\,a]$ such that

$\int_{1}^{a}\sum_{k\in{\mathbb{Z}}}\big{\|}\big{(}P_{a^{k}t}-P_{a^{k+1}t}\big{)}(f)\big{\|}^{q}_{X}\,\frac{dt}{t}\leq\log
a\sum_{k\in{\mathbb{Z}}}\big{\|}\big{(}P_{a^{k}t_{0}}-P_{a^{k+1}t_{0}}\big{)}(f)\big{\|}^{q}_{X}\,.$

We then deduce the Lusin cotype $q$ of $X$ from (4.7) with

$\mathsf{L}^{P}_{c,q,p}(X)\lesssim(\log a)^{\frac{1}{q}}\max(p^{\prime
1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}})c.$

(ii) The above argument yields the following discretization of
$\mathcal{G}^{P}_{q}(f)$:

$\displaystyle\frac{a-1}{a+1}$ $\displaystyle\big{(}\max(p^{\prime
1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}})\big{)}^{-1}\big{\|}\mathcal{G}^{P}_{q}(f)\big{\|}_{L_{p}(\Omega)}$
$\displaystyle\;\;\lesssim\Big{\|}$
$\displaystyle\Big{(}\int_{1}^{a}\sum_{k\in{\mathbb{Z}}}\big{\|}\big{(}P_{a^{k}t}-P_{a^{k+1}t}\big{)}(f)\big{\|}^{q}_{X}\,dt\Big{)}^{\frac{1}{q}}\Big{\|}_{L_{p}(\Omega)}$
$\displaystyle\leq a^{\frac{1}{q}}(\log
a)^{\frac{1}{q^{\prime}}}\big{\|}\mathcal{G}^{P}_{q}(f)\big{\|}_{L_{p}(\Omega)}\,.$

This immediately implies assertion (ii).

(iii) This part is proved similarly by virtue of Propositions 2.4 and 2.9. ∎

We have seen that the proofs of Theorem 4.5 and Theorem 1.9 are based on
functional calculus for the special function $F(z)=-ze^{-z}$. It is known that
functional calculus allows us to use more general functions.

######  Definition 4.8.

Let $B$ be an $\ell_{q}$-sectorial operator of type $\alpha$ on
$L_{p}(\Omega;X)$ with $\alpha<\pi$. Let $\beta>\alpha$ and $\varphi\in
H^{\infty}_{0}(\Sigma_{\beta})$ be a nonzero function. Define

$\mathcal{G}^{B}_{q,\varphi}(f)=\left(\int_{0}^{\infty}\big{\|}\varphi(tB)(f)\big{\|}_{X}^{q}\,\frac{dt}{t}\right)^{\frac{1}{q}}\,,\quad
f\in L_{p}(\Omega;X).$

The following result is a variant of [35, Theorem 5] (see also [59, Lemma
20]).

######  Proposition 4.9.

Let $\varphi$ and $\psi$ be two nonzero functions in
$H^{\infty}_{0}(\Sigma_{\beta})$. Then

$\big{\|}\mathcal{G}^{B}_{q,\varphi}(f)\big{\|}_{L_{p}(\Omega)}\leq
C_{B,q,\varphi,\psi}\big{\|}\mathcal{G}^{B}_{q,\psi}(f)\big{\|}_{L_{p}(\Omega)}\,,\quad
f\in L_{p}(\Omega;X).$

###### Proof.

Let

$a=\int_{0}^{\infty}\psi(t)^{2}\,\frac{dt}{t}\,.$

Then

$1=\frac{1}{a}\int_{0}^{\infty}\psi(tz)^{2}\,\frac{dt}{t}\,,\quad
z\in\Sigma_{\beta}.$

Combined with (2.1), this implies

$f=\frac{1}{a}\int_{0}^{\infty}\psi(tB)^{2}(f)\,\frac{dt}{t}\,,\quad
f\in\overline{{\rm im}B},$

whence

$\varphi(sB)(f)=\frac{1}{a}\int_{0}^{\infty}\varphi(sB)\psi(tB)\big{(}\psi(tB)(f)\big{)}\,\frac{dt}{t}\,,\quad
s>0.$

Let $\alpha<\gamma<\beta$ and $\Gamma$ be the boundary of $\Sigma_{\gamma}$.
We then deduce

$\varphi(sB)(f)=\frac{1}{2a\pi{\rm
i}}\int_{\Gamma}\int_{0}^{\infty}\varphi(sz)\psi(tz)z(z-B)^{-1}\big{(}\psi(tB)(f)\big{)}\,\frac{dt}{t}\,\frac{dz}{z}\,.$

These equalities are the analogues of (3.1) and (3.2) with
$h_{t}=\psi(tB)(f)$. It remains to repeat the proof of Theorem 1.9 to conclude

$\big{\|}\mathcal{G}^{B}_{q,\varphi}(f)\big{\|}_{L_{p}(\Omega)}\leq\frac{1}{2|a|\pi}\,C_{B,\gamma}C_{\varphi,\gamma}C_{\psi,\gamma}\big{\|}\mathcal{G}^{B}_{q,\psi}(f)\big{\|}_{L_{p}(\Omega)}\,,$

where $C_{B,\gamma}$ is the $\ell_{q}$-boundedness constant of
$\big{\\{}z(z-B)^{-1}:z\in\Gamma\setminus\\{0\\}\big{\\}}$,

$C_{\varphi,\gamma}=\max_{\varepsilon=\pm 1}\int_{0}^{\infty}|\varphi(te^{{\rm
i}\varepsilon\gamma})|\,\frac{dt}{t}$

and $C_{\psi,\gamma}$ is similarly defined. ∎

In particular, combining the previous proposition with the results in section
2, we obtain the following

######  Corollary 4.10.

Let $X$ be a Banach space and $1\leq q\leq\infty$.

  1. (i)

$X$ is of Lusin cotype $q$ relative to $\\{P_{t}\\}_{t>0}$ iff for every
nonzero $\varphi\in H_{0}^{\infty}(\Sigma_{\beta})$ with $\beta>\pi/4$
$($equivalently, for some nonzero $\varphi\in H_{0}^{\infty}(\Sigma_{\beta}))$
there exists a constant $c$ such that

$\big{\|}\mathcal{G}^{\sqrt{A}}_{q,\varphi}(f)\big{\|}_{L_{p}(\Omega)}\leq
c\,\big{\|}f\big{\|}_{L_{p}(\Omega;X)}\,,\quad f\in L_{p}(\Omega;X).$

  2. (ii)

$X$ is of Lusin type $q$ relative to $\\{P_{t}\\}_{t>0}$ iff for every nonzero
$\varphi\in H_{0}^{\infty}(\Sigma_{\beta})$ with $\beta>\pi/4$
$($equivalently, for some nonzero $\varphi\in H_{0}^{\infty}(\Sigma_{\beta}))$
there exists a constant $c$ such that

$\big{\|}f-\mathsf{F}(f)\big{\|}_{L_{p}(\Omega;X)}\leq
c\,\big{\|}\mathcal{G}^{\sqrt{A}}_{q,\varphi}(f)\big{\|}_{L_{p}(\Omega)}\,,\quad
f\in L_{p}(\Omega;X).$

  3. (iii)

Similar statements hold for $\\{T_{t}\\}_{t>0}$ when $\\{T_{t}\\}_{t>0}$ is
analytic on $L_{p}(\Omega;X)$ and $1<q<\infty$.

We conclude this section by some remarks on general $\ell_{q}$-sectorial
operators on $L_{p}(\Omega;X)$ for which we have defined the $g$-function in
Definition 4.8. In fact, what we have done so far for semigroups can be
developed for these operators too.

######  Definition 4.11.

Let $B$ be an $\ell_{q}$-sectorial operator of type $\alpha$ on
$L_{p}(\Omega;X)$ with $\alpha<\pi$.

  1. (i)

$X$ is said to be of Lusin cotype $q$ relative to $B$ if there exists a
constant $c$ such that

$\big{\|}\mathcal{G}^{B}_{q,\varphi}(f)\big{\|}_{L_{p}(\Omega)}\leq
c\big{\|}f\big{\|}_{L_{p}(\Omega;X)}\,,\quad f\in\overline{{\rm im}B}.$

  2. (ii)

$X$ is said to be of Lusin type $q$ relative to $B$ if there exists a constant
$c$ such that

$\big{\|}f\big{\|}_{L_{p}(\Omega;X)}\leq
c\big{\|}\mathcal{G}^{B}_{q,\varphi}(f)\big{\|}_{L_{p}(\Omega)}\,,\quad
f\in\overline{{\rm im}B}.$

Proposition 4.9 shows that the above definition is independent of the choice
of $\varphi$. Assume additionally that $B$ admits a dual operator $B^{\prime}$
on $L_{p^{\prime}}(\Omega;X^{*})$ in the sense of [12], namely,

$\langle B(f),\,g\rangle=\langle f,\,B^{\prime}(g)\rangle,\quad f\in{\rm
Dom}B,\;g\in{\rm Dom}B^{\prime}.$

Assume further that $B^{\prime}$ is also $\ell_{q}$-sectorial of type $\alpha$
on $L_{p^{\prime}}(\Omega;X^{*})$.

######  Proposition 4.12.

Under the above assumption, $X$ is of Lusin cotype $q$ relative to $B$ iff
$X^{*}$ is of Lusin type $q^{\prime}$ relative to $B^{\prime}$.

###### Proof.

Noting that Theorem 1.9 transfers to the present setting, we can repeat the
proof of Theorem 4.5, and so omit the details. ∎

It would be interesting to investigate the Lusin type and cotype relative to
$B$ as above. Guided by Theorems 1.2 and 1.4, one would like to know those
operators $B$ such that the Lusin type or cotype relative to $B$ is implied by
the martingale type or cotype. It seems that more structure should be imposed
to $B$ in order to get significant results. We have seen that this is indeed
the case if $B$ is the negative generator of $\\{P_{t}\\}_{t>0}$ or
$\\{T_{t}\\}_{t>0}$ on $L_{p}(\Omega;X)$. On the other hand, we have the
following proposition that is contained (essentially) in [51]. Note that [51]
can be viewed as the particular case of our discussion where $\Omega$ is a
singleton (the $\ell_{q}$-boundedness then simply becomes the usual
boundedness).

The notion of type and cotype referred in the next proposition is the usual
Rademacher type and cotype.

######  Proposition 4.13.

Assume that $B$ has a bounded $H^{\infty}$ functional calculus. If $X$ is of
cotype $($resp. type$)$ $q$, then $X$ is of Lusin cotype $($resp. Lusin type
$)$ $q$ relative to $B$.

###### Proof.

As in [51], this is a simple consequence of Kalton-Weis’ theorem on the
unconditionality of bounded $H^{\infty}$ functional calculus (see Theorem
10.4.6 or its discrete version, Theorem 10.4.4. in [25]). ∎

######  Remark 4.14.

The assumption on the $H^{\infty}$ functional calculus of $B$ seems too strong
since it implies that $X$ is a UMD space in many cases, for instance, if $B$
is the negative generator of $\\{T_{t}\\}_{t>0}$ on $L_{p}({\mathbb{R}};X)$
when $\\{T_{t}\\}_{t>0}$ is the heat or Poisson semigroup on ${\mathbb{R}}$.

##  5\. Dyadic martingales and singular integrals

The proofs of Theorem 1.2 (i) and Theorem 1.5 heavily rely on tools from
harmonic analysis, notably from modern real-variable Littlewood-Paley theory.
This section is a preparation for these tools. We will mainly follow Wilson’s
beautiful treatment in [55] (see also [53, 54]) for the part on Littlewood-
Paley theory.

###  5.1. Dyadic martingales

All cubes in ${\mathbb{R}}^{d}$ considered in the sequel are bounded and with
sides parallel to the axes. $|Q|$ and $\ell(Q)$ denote respectively the volume
and side length of a cube $Q$; $tQ$ stands for the cube with the same center
as $Q$ and $\ell(tQ)=t\ell(Q)$ for $t>0$.

Let ${\mathcal{D}}$ be the family of all dyadic cubes of ${\mathbb{R}}^{d}$,
${\mathcal{D}}_{k}\subset{\mathcal{D}}$ the subfamily of all cubes with side
length $2^{-k}$ for $k\in{\mathbb{Z}}$. Let ${\mathcal{A}}_{k}$ be the
$\sigma$-algebra generated by ${\mathcal{D}}_{k}$ and ${\mathbb{E}}_{k}$ the
associated conditional expectation. For $f\in L_{p}({\mathbb{R}}^{d};X)$,

${\mathbb{E}}_{k}(f)=\sum_{Q\in{\mathcal{D}}_{k}}\Big{(}\frac{1}{|Q|}\,\int_{Q}f\Big{)}{\mathds{1}}_{Q}.$

Let $d_{k}(f)={\mathbb{E}}_{k}(f)-{\mathbb{E}}_{k-1}(f)$ and

$S_{q}(f)=\Big{(}\sum_{k\in{\mathbb{Z}}}\big{\|}d_{k}(f)\big{\|}_{X}^{q}\Big{)}^{\frac{1}{q}}.$

$S_{q}(f)$ is the $q$-variant of the usual martingale square function of $f$.
It is useful to note that $d_{k}(f)$ is of vanishing mean on every
$Q\in{\mathcal{D}}_{k-1}$ and constant on every $R\in{\mathcal{D}}_{k}$.

Thus if $X$ is of martingale type $q$, then

$\big{\|}f\big{\|}_{L_{q}({\mathbb{R}}^{d};X)}\leq\mathsf{M}_{t,q}(X)\big{\|}S_{q}(f)\big{\|}_{L_{q}({\mathbb{R}}^{d})}\,,\quad
f\in L_{q}({\mathbb{R}}^{d};X).$

We will need dyadic-like families of cubes that Wilson calls good families.
${\mathcal{F}}$ is such a family if

  * a)

for $Q\in{\mathcal{F}}$, all its $2^{d}$ immediate dyadic subcubes belong to
${\mathcal{F}}$ too;

  * b)

every $Q\in{\mathcal{F}}$ is one of the $2^{d}$ immediate dyadic subcubes of
another one in ${\mathcal{F}}$;

  * c)

for all $Q,R\in{\mathcal{F}}$, $Q\subset R$, $R\subset Q$, or $Q\cap
R=\emptyset$.

For ${\mathcal{F}}$ a dyadic-like family of cubes, we define the associated
$S_{q,{\mathcal{F}}}$:

$S_{q,{\mathcal{F}}}(f)=\Big{(}\sum_{Q\in{\mathcal{F}}}\,\sum_{R\in{\mathcal{F}},\,R\subset
Q,\,\ell(R)=\frac{\ell(Q)}{2}}\big{\|}\frac{1}{|R|}\,\int_{R}f-\frac{1}{|Q|}\,\int_{Q}f\big{\|}^{q}_{X}\,{\mathds{1}}_{R}\Big{)}^{\frac{1}{q}}.$

It is easy to see that given a finite number of cubes in ${\mathcal{F}}$, we
can bring the subfamily consisting of those cubes in ${\mathcal{F}}$ that are
contained in one of the given cubes to a subfamily of ${\mathcal{D}}$ after
appropriate translation and rescaling. Thus we have the following

######  Lemma 5.1.

Let ${\mathcal{F}}$ be a dyadic-like family of cubes. If $X$ is of martingale
type $q$, then

$\big{\|}f\big{\|}_{L_{q}({\mathbb{R}}^{d};X)}\leq\mathsf{M}_{t,q}(X)\big{\|}S_{q,{\mathcal{F}}}(f)\big{\|}_{L_{q}({\mathbb{R}}^{d})}$

for all $f\in L_{q}({\mathbb{R}}^{d};X)$ supported by cubes from
${\mathcal{F}}$.

An important case needed later concerns the family
$\\{3Q\\}_{Q\in{\mathcal{D}}}$. The following is due to Wilson [53].

######  Lemma 5.2.

The family $\\{3Q\\}_{Q\in{\mathcal{D}}}$ is a disjoint union of $3^{d}$
dyadic-like families.

It suffices to consider the case $d=1$. Then every $3Q$ can be written in the
form $[\frac{3j+s}{2^{k}},\,\frac{3(j+1)+s}{2^{k}})$ with $j,k\in{\mathbb{Z}}$
and $s\in\\{0,1,2\\}$. Let ${\mathcal{F}}^{k}_{s}$ be the collection of all
such intervals for given $k$ and $s$. Then the desired union is
$\cup_{s=0}^{2}{\mathcal{F}}^{k}_{2^{|k|}s\;{\rm mod}\;3}$.

Let ${\mathcal{F}}$ be a dyadic-like family of cubes and $\delta>0$. Consider
a family $\\{a_{Q}\\}_{Q\in{\mathcal{F}}}$ of $X$-valued functions satisfying
the following conditions:

(5.1) ${\rm supp}(a_{Q})\subset Q,\quad\int a_{Q}=0\;\text{ and
}\;\|a_{Q}(x)-a_{Q}(y)\|_{X}\leq|Q|^{-\frac{1}{q}}\,\big{(}\frac{|x-y|}{\ell(Q)}\big{)}^{\delta}.$

These are smooth atoms. Let $\\{\lambda_{Q}\\}_{Q\in{\mathcal{F}}}$ be a
finite family of complex numbers and
$f=\sum_{Q\in{\mathcal{F}}}\lambda_{Q}\,a_{Q}$.

The following is the adaptation of a lemma due to Wilson to the presenting
setting. We include its proof for the convenience of the reader.

######  Lemma 5.3.

Under the above assumption, we have

$S_{q,{\mathcal{F}}}(f)\lesssim_{d,\delta}\Big{(}\sum_{Q\in{\mathcal{F}}}\frac{|\lambda_{Q}|^{q}}{|Q|}\,{\mathds{1}}_{Q}\Big{)}^{\frac{1}{q}}.$

###### Proof.

Without loss of generality, we assume that ${\mathcal{F}}={\mathcal{D}}$, so
$S_{q,{\mathcal{F}}}(f)=S_{q}(f)$. Since $a_{Q}$ is of vanishing mean,
$d_{k}(a_{Q})=0$ whenever $k\leq k(Q)$, where $2^{-k(Q)}=\ell(Q)$ . Let
$R\in{\mathcal{D}}_{k-1}$ with $k>k(Q)$ and $R\subset Q$. Then on $R$,

$\displaystyle d_{k}(a_{Q})$
$\displaystyle=\sum_{I\in{\mathcal{D}}_{k},\,I\subset
R}\big{(}\frac{1}{|I|}\,\int_{I}a_{Q}-\frac{1}{|R|}\,\int_{R}a_{Q}\big{)}{\mathds{1}}_{I}$
$\displaystyle=\sum_{I\in{\mathcal{D}}_{k},\,I\subset
R}\frac{1}{|I|\,|R|}\,\int_{I\times
R}\big{(}a_{Q}(x)-a_{Q}(y)\big{)}dx\,dy\,{\mathds{1}}_{I}.$

Thus by the last condition of (5.1),

$\|d_{k}(a_{Q})\|_{X}\lesssim_{d,\delta}|Q|^{-\frac{1}{q}}\,\big{(}\frac{\ell(R)}{\ell(Q)}\big{)}^{\delta};$

hence on $R$,

$\displaystyle\|d_{k}(f)\|_{X}$
$\displaystyle\lesssim_{d,\delta}\Big{(}\sum_{Q:Q\supset
R}\frac{|\lambda_{Q}|^{q}}{|Q|}\,\big{(}\frac{\ell(R)}{\ell(Q)}\big{)}^{\delta}\Big{)}^{\frac{1}{q}}\Big{(}\sum_{Q:Q\supset
R}\big{(}\frac{\ell(R)}{\ell(Q)}\big{)}^{\delta}\Big{)}^{\frac{1}{q^{\prime}}}$
$\displaystyle\lesssim_{d,\delta}\Big{(}\sum_{Q:Q\supset
R}\frac{|\lambda_{Q}|^{q}}{|Q|}\,\big{(}\frac{\ell(R)}{\ell(Q)}\big{)}^{\delta}\Big{)}^{\frac{1}{q}}.$

It then follows that

$\displaystyle S_{q}(f)^{q}(x)$
$\displaystyle=\sum_{k\in{\mathbb{Z}}}\,\sum_{R\in{\mathcal{D}}_{k-1}}\|d_{k}(f)\|^{q}_{X}\,{\mathds{1}}_{R}(x)$
$\displaystyle\lesssim_{d,\delta}\sum_{R:x\in R}\,\sum_{Q:Q\supset
R}\frac{|\lambda_{Q}|^{q}}{|Q|}\,\big{(}\frac{\ell(R)}{\ell(Q)}\big{)}^{\delta}$
$\displaystyle\lesssim_{d,\delta}\sum_{Q}\frac{|\lambda_{Q}|^{q}}{|Q|}\;\sum_{R:x\in
R\subset Q}\,\big{(}\frac{\ell(R)}{\ell(Q)}\big{)}^{\delta}$
$\displaystyle\lesssim_{d,\delta}\sum_{Q}\frac{|\lambda_{Q}|^{q}}{|Q|}\,{\mathds{1}}_{Q}(x)\,.$

This gives the desired assertion. ∎

###  5.2. Singular integrals

Given $\varepsilon>0$ and $\delta>0$, let ${\mathcal{H}}_{\varepsilon,\delta}$
be the class of all functions $\varphi$ on ${\mathbb{R}}^{d}$ satisfying
(1.5). Let $\varphi\in{\mathcal{H}}_{\varepsilon,\delta}$. We consider the
vector-valued kernel $K$ defined by $K(x)=\\{\varphi_{t}(x)\\}_{t>0}$ for
$x\in{\mathbb{R}}^{d}$, that is, $K$ is a function from ${\mathbb{R}}^{d}$ to
$L_{q}({\mathbb{R}}_{+})$. With a slight abuse of notation, we use $K$ to
denote the associated singular integral too:

$K(f)=\int_{\mathbb{R}}K(x-y)f(y)dy.$

Then

$G_{q,\varphi}(f)(x)=\big{\|}K(f)(x)\big{\|}_{L_{q}({\mathbb{R}}_{+};X)}\,,\quad
x\in{\mathbb{R}}^{d}.$

######  Lemma 5.4.

The kernel $K$ satisfies the regularity properties:

$\big{\|}K(x)\big{\|}_{L_{q}({\mathbb{R}}_{+})}\lesssim_{\varepsilon}\frac{1}{|x|^{d}}\;\text{
and
}\;\big{\|}K(x+y)-K(x)\big{\|}_{L_{q}({\mathbb{R}}_{+})}\lesssim_{\varepsilon}\frac{|y|^{\delta}}{|x|^{d+\delta}},\quad
x,y\in{\mathbb{R}}^{d},\;|x|>2|y|.$

###### Proof.

Let $x\in{\mathbb{R}}\setminus\\{0\\}$. Then by (1.5),

$\displaystyle\big{\|}K(x)\big{\|}_{L_{q}({\mathbb{R}}_{+})}^{q}$
$\displaystyle=\int_{0}^{\infty}|\varphi_{t}(x)|^{q}\,\frac{dt}{t}\leq\int_{0}^{\infty}\big{[}\frac{1}{t^{d}}\,\frac{1}{\big{(}1+\frac{|x|}{t}\big{)}^{d+\varepsilon}}\big{]}^{q}\,\frac{dt}{t}$
$\displaystyle=\frac{1}{|x|^{dq}}\int_{0}^{\infty}\frac{t^{\varepsilon
q}}{(1+t)^{(d+\varepsilon)q}}\,\frac{dt}{t}\lesssim_{\varepsilon}\frac{1}{|x|^{dq}}$

Similarly,

$\displaystyle\big{\|}K(x+y)-K(x)\big{\|}_{L_{q}({\mathbb{R}}_{+})}^{q}$
$\displaystyle\lesssim|y|^{\delta
q}\int_{0}^{\infty}\big{[}\frac{1}{t^{d+\delta}}\,\frac{1}{\big{(}1+\frac{|x|}{t}\big{)}^{d+\varepsilon+\delta}}\big{]}^{q}\,\frac{dt}{t}$
$\displaystyle=\frac{|y|^{\delta
q}}{|x|^{(d+\delta)q}}\int_{0}^{\infty}\frac{t^{\varepsilon
q}}{(1+t)^{(d+\varepsilon+\delta)q}}\,\frac{dt}{t}\lesssim_{\varepsilon}\frac{|y|^{\delta
q}}{|x|^{(d+\delta)q}}$

This lemma is proved. ∎

###  5.3. A quasi-orthogonal decomposition

Beside ${\mathcal{H}}_{\varepsilon,\delta}$ introduced in the previous
subsection, we will need its subclass of functions supported in the unit ball.
More precisely, let ${\mathcal{H}}^{0}_{\delta}$ be the class of functions
$\varphi$ on ${\mathbb{R}}^{d}$ such that

(5.2) ${\rm supp}(\varphi)\subset
B(0,1),\quad|\varphi(x)-\varphi(y)|\leq|x-y|^{\delta},\quad\int_{{\mathbb{R}}^{d}}\varphi(x)dx=0.$

Here $B(x,t)$ denotes the ball of ${\mathbb{R}}^{d}$ with center $x$ and
radius $t$.

Any function in ${\mathcal{H}}_{\varepsilon,\delta}$ can be decomposed into a
series of functions in ${\mathcal{H}}^{0}_{\delta}$ thanks to the following
lemma due to Uchiyama [50] (see also [54]).

######  Lemma 5.5.

Let $\varphi\in{\mathcal{H}}_{\varepsilon,\delta}$. Then there exist a
positive constant $C_{\varepsilon,\delta}$ and a sequence of functions
$\psi^{(k)}\in{\mathcal{H}}^{0}_{\delta}$ such that

$\varphi=C_{\varepsilon,\delta}\sum_{k=0}^{\infty}2^{-\varepsilon
k}(\psi^{(k)})_{2^{k}}\,.$

The proof is elementary. Let $\eta$ be a smooth function supported in
$\\{x\in{\mathbb{R}}^{d}:\frac{1}{2}<|x|<2\\}$ such that

$\sum_{k\in{\mathbb{Z}}}\eta(2^{-k}x)=1,\quad\forall\,x\in{\mathbb{R}}^{d}\setminus\\{0\\}.$

Define

$\rho_{0}(x)=\sum_{j\leq-1}\eta(2^{-j}|x|),\quad\rho_{k}(x)=\eta(2^{-k+1}|x|)\;\text{
for}\;k\geq 1$

and

$\zeta_{k}=\frac{\sum_{0\leq j\leq
k}\int_{{\mathbb{R}}}\rho_{j}\varphi}{\int_{{\mathbb{R}}}\rho_{k}}\,\rho_{k}\,.$

Then the desired decomposition is given by

$\varphi=(\varphi\rho_{0}-\zeta_{0})+\sum_{k=1}^{\infty}(\varphi\rho_{k}-\zeta_{k}+\zeta_{k-1})=C_{\varepsilon,\delta}\sum_{k=0}^{\infty}2^{-\varepsilon
k}(\psi^{(k)})_{2^{k}}.$

By our convention that ${\mathbb{R}}_{+}$ is equipped with the measure
$\frac{dt}{t}$, the upper half space ${\mathbb{R}}^{d+1}_{+}$ is equipped with
the product measure $\frac{dx\,dt}{t}$. Consistent with our convention before,
we write a function $h:{\mathbb{R}}^{d+1}_{+}\to X$ as $h(x,t)=h_{t}(x)$ for
$x\in{\mathbb{R}}^{d}$ and $t\in{\mathbb{R}}_{+}$. Let
$\varphi\in{\mathcal{H}}_{\varepsilon,\delta}$ and $h\in
L_{q}({\mathbb{R}}^{d+1}_{+};X)$ with compact support. Consider the following
function

$g(x)=\int_{{\mathbb{R}}^{d+1}_{+}}\varphi_{t}(y-x)h_{t}(y)\,\frac{dy\,dt}{t}.$

We will decompose $g$ into a series of smooth atoms

$g=\sum_{i}\lambda_{i}a_{i},$

where the $a_{i}$’s satisfy (5.1) relative to $\\{3Q\\}_{Q\in{\mathcal{D}}}$
and the $\lambda_{i}$’s are reals such that

$\Big{(}\sum_{i}|\lambda_{i}|^{q}\big{)}^{\frac{1}{q}}\lesssim_{d,\varepsilon,\delta}\big{\|}h\big{\|}_{L_{q}({\mathbb{R}}^{d+1}_{+};X)}.$

First, using Lemma 5.5, we reduce our problem to the case where $\varphi$ is
supported in the unit ball:

$\displaystyle g(x)$
$\displaystyle=C_{\varepsilon,\delta}\sum_{k=0}^{\infty}2^{-\varepsilon
k}\int_{{\mathbb{R}}^{d+1}_{+}}(\psi^{(k)})_{2^{k}t}(y-x)h_{t}(y)\,\frac{dy\,dt}{t}$
$\displaystyle=C_{\varepsilon,\delta}\sum_{k=0}^{\infty}2^{-\varepsilon
k}\int_{{\mathbb{R}}^{d+1}_{+}}(\psi^{(k)})_{t}(y-x)h_{2^{-k}t}(y)\,\frac{dy\,dt}{t}.$

Note that $h_{2^{-k}\,\cdot}$ has the same norm as $h$ in
$L_{q}({\mathbb{R}}^{d+1}_{+};X)$. Thus it suffices to do the decomposition
for each $\psi^{(k)}$ in place of $\varphi$.

In the following, we will assume that $\varphi$ itself belongs to
${\mathcal{H}}^{0}_{\delta}$. The argument below is classical. For
$Q\in{\mathcal{D}}$, let $T_{Q}=\\{(y,t):y\in
Q,\;\frac{\ell(Q)}{2}<t\leq\ell(Q)\\}$. Then $\\{T_{Q}\\}_{Q\in{\mathcal{D}}}$
is a partition of ${\mathbb{R}}^{d+1}_{+}$. So

$g(x)=\sum_{Q\in{\mathcal{D}}}\int_{T_{Q}}\varphi_{t}(y-x)h_{t}(y)\,\frac{dy\,dt}{t}\;{\mathop{=}^{\rm
def}}\,\sum_{Q\in{\mathcal{D}}}\lambda_{Q}a_{Q}(x)$

with

$\lambda_{Q}=\Big{(}\int_{T_{Q}}\|h_{t}(y)\|_{X}^{q}\,\frac{dy\,dt}{t}\Big{)}^{\frac{1}{q}}.$

Clearly,

$\sum_{Q\in{\mathcal{D}}}|\lambda_{Q}|^{q}=\int_{{\mathbb{R}}^{d+1}_{+}}\|h_{t}(y)\|_{X}^{q}\,\frac{dy\,dt}{t}=\big{\|}h\big{\|}^{q}_{L_{q}({\mathbb{R}}^{d+1}_{+};X)}.$

Since $\varphi$ is supported in the unit ball and of vanishing mean, we see
that $a_{Q}$ is supported in $3Q$ and of vanishing mean too. On the other
hand, since $\varphi$ is in the Hölder class ${\mathcal{H}}^{0}_{\delta}$, by
the Hölder inequality,

$\displaystyle\|a_{Q}(x)-a_{Q}(x^{\prime})\|_{X}$
$\displaystyle\leq\Big{(}\int_{T_{Q}}|\varphi_{t}(y-x)-\varphi_{t}(y-x^{\prime})|^{q^{\prime}}\,\frac{dy\,dt}{t}\Big{)}^{\frac{1}{q^{\prime}}}$
$\displaystyle\leq\Big{(}\int_{T_{Q}}\big{[}\frac{1}{t^{d}}\,\big{(}\frac{|x-x^{\prime}|}{t}\big{)}^{\delta}\big{]}^{q^{\prime}}\,\frac{dy\,dt}{t}\Big{)}^{\frac{1}{q^{\prime}}}$
$\displaystyle\lesssim_{d,\delta}|Q|^{-\frac{1}{q}}\,\big{(}\frac{|x-x^{\prime}|}{\ell(Q)}\big{)}^{\delta}.$

Thus $a_{Q}$ is a smooth atom. This yields the desired quasi-orthogonal
decomposition.

Combining the above discussion with Lemmas 5.1 – 5.3, we get the following

######  Lemma 5.6.

Keep the above notation and assume that $X$ is of martingale type $q$. Then

$\big{\|}g\big{\|}_{L_{q}({\mathbb{R}}^{d};X)}\lesssim_{d,\varepsilon,\delta}\mathsf{M}_{t,q}(X)\,\big{\|}h\big{\|}_{L_{q}({\mathbb{R}}^{d+1}_{+};X)}.$

##  6\. Proof of Theorem 1.5

With the preparation in section 5, we are in a position to show Theorem 1.5.
For clarity, we divide the proof into several steps. $X$ will be assumed of
martingale cotype $q$ in the first four steps, and of martingale type $q$ in
the last.

Step 1: A weighted norm inequality. Let $\varphi$ be a function satisfying
(1.5), i.e., $\varphi\in{\mathcal{H}}_{\varepsilon,\delta}$. Beside the
$g$-function defined by (1.7), we will need the Lusin integral function

$S_{q,\varphi}(f)(x)=\Big{(}\int_{|y-x|<t}\big{\|}\varphi_{t}*f(y)\big{\|}_{X}^{q}\,\frac{dydt}{t^{d+1}}\Big{)}^{\frac{1}{q}}$

for nice $f:{\mathbb{R}}^{d}\to X$. The key of this proof is the following
weighted norm inequality:

For any locally integrable nonnegative function $w$ on ${\mathbb{R}}^{d}$ and
any $f\in L_{q}({\mathbb{R}}^{d};X)$

(6.1)
$\Big{(}\int_{{\mathbb{R}}^{d}}\big{(}S_{q,\varphi}(f)(x)\big{)}^{q}w(x)dx\Big{)}^{\frac{1}{q}}\lesssim_{d,\varepsilon,\delta}\mathbf{M}_{c,q}(X)\Big{(}\int_{{\mathbb{R}}^{d}}\big{\|}f(x)\big{\|}_{X}^{q}M(w)(x)dx\Big{)}^{\frac{1}{q}},$

where $M(w)$ denotes the Hardy-Littlewood maximal function of $w$:

$M(w)(x)=\sup_{x\in Q}\frac{1}{|Q|}\,\int_{Q}w.$

First consider the unweighted case, i.e., $w\equiv 1$. By the Fubini theorem,
we have

$\int_{{\mathbb{R}}^{d}}\big{(}S_{q,\varphi}(f)(x)\big{)}^{q}dx=c_{d}\int_{{\mathbb{R}}^{d+1}_{+}}\|\varphi_{t}*f(x)\|^{q}_{X}\,\frac{dx\,dt}{t}\,.$

Let $h:{\mathbb{R}}^{d+1}_{+}\to X^{*}$ be a compactly supported smooth
function such that

$\int_{{\mathbb{R}}^{d+1}_{+}}\|h_{t}(x)\|^{q^{\prime}}_{X^{*}}\,\frac{dx\,dt}{t}\leq
1.$

Then

$\displaystyle\int_{{\mathbb{R}}^{d+1}_{+}}\langle\varphi_{t}*f(y),\,h_{t}(y)\rangle\,\frac{dy\,dt}{t}=\int_{{\mathbb{R}}^{d}}\langle
f(x),\,g(x)\rangle dx\,,$

where

$g(x)=\int_{{\mathbb{R}}^{d+1}_{+}}\varphi_{t}(y-x)h_{t}(y)\,\frac{dy\,dt}{t}.$

Recall that it is well known (and easy to check) that $X$ is of martingale
type $q$ iff $X^{*}$ is of martingale cotype $q^{\prime}$ with the following
relation between the relevant constants:

(6.2) $\mathsf{M}_{t,q}(X)\leq\mathsf{M}_{c,q^{\prime}}(X^{*})\leq
2\mathsf{M}_{t,q}(X).$

Thus applying Lemma 5.6 to $h$ and $g$ with $(X^{*},q^{\prime})$ in place of
$(X,q)$ there, we get

$\displaystyle\|g\|_{L_{q^{\prime}}({\mathbb{R}};X^{*})}\lesssim_{d,\varepsilon,\delta}\mathbf{M}_{t,q^{\prime}}(X^{*})\,\big{\|}h\big{\|}_{L_{q^{\prime}}({\mathbb{R}}^{d+1}_{+};X^{*})}\lesssim_{d,\varepsilon,\delta}\mathbf{M}_{c,q}(X).$

Then taking the supremum over all $h$ in the unit ball of
$L_{q^{\prime}}({\mathbb{R}}^{d+1}_{+};X^{*})$, we deduce

$\big{\|}S_{q,\varphi}(f)\big{\|}_{L_{p}({\mathbb{R}}^{d})}\lesssim_{d,\varepsilon,\delta}\mathbf{M}_{c,q}(X)\big{\|}f\big{\|}_{L_{p}({\mathbb{R}}^{d};X)}.$

Namely, the unweighted version of (6.1) holds.

We will deduce the weighted version by a trick from [10]. By Lemma 5.5, we can
assume that $\varphi$ is supported in the unit ball of ${\mathbb{R}}^{d}$.
Given a weight $w$, we write (recalling that $B(y,t)$ denotes the ball of
center $y$ and radius $t$)

$\int_{{\mathbb{R}}^{d}}\big{(}S_{q,\varphi}(f)(x)\big{)}^{q}w(x)dx=c_{d}\int_{{\mathbb{R}}^{d+1}_{+}}\|\varphi_{t}*f(y)\|_{X}^{q}\,\Big{(}\frac{1}{|B(y,t)|}\int_{B(y,t)}w(x)dx\Big{)}\,\frac{dy\,dt}{t}.$

Let

$F_{k}=\big{\\{}(y,t)\in{\mathbb{R}}^{d+1}_{+}:2^{k}<\frac{1}{|B(y,t)|}\int_{B(y,t)}w(x)dx\leq
2^{k+1}\big{\\}},\quad k\in{\mathbb{Z}}.$

Clearly, $(y,t)\in F_{k}$ implies $B(y,t)\subset E_{k}=\\{x:M(w)(x)>2^{k}\\}$.
Together with the fact that $\varphi_{t}(y-\cdot)$ is supported in $B(y,t)$,
this implies $\varphi_{t}*f(y)=\varphi_{t}*(f{\mathds{1}}_{E_{k}})(y)$
whenever $(y,t)\in F_{k}$. Thus using the unweighted version already proved
(applied to $f{\mathds{1}}_{E_{k}}$), we deduce

$\displaystyle\int_{{\mathbb{R}}^{d}}\big{(}S_{q,\varphi}(f)(x)\big{)}^{q}w(x)dx$
$\displaystyle\lesssim_{d}\sum_{k\in{\mathbb{Z}}}2^{k}\int_{F_{k}}\|\varphi_{t}*f(y)\|_{X}^{q}\,\frac{dy\,dt}{t}$
$\displaystyle\lesssim_{d}\sum_{k\in{\mathbb{Z}}}2^{k}\int_{F_{k}}\|\varphi_{t}*(f{\mathds{1}}_{E_{k}})(y)\|_{X}^{q}\,\frac{dy\,dt}{t}$
$\displaystyle\lesssim_{d,\varepsilon,\delta}\mathbf{M}_{c,q}(X)^{q}\sum_{k\in{\mathbb{Z}}}2^{k}\int_{E_{k}}\|f(x)\|_{X}^{q}\,dx$
$\displaystyle\lesssim_{d,\varepsilon,\delta}\mathbf{M}_{c,q}(X)^{q}\
\int_{{\mathbb{R}}^{d}}\|f(x)\|_{X}^{q}M(w)(x)\,dx.$

Thus (6.1) is proved.

Step 2: A more weighted norm inequality. We need to show that (6.1) remains
valid for $G_{q,\varphi}$ instead of $S_{q,\varphi}$:

(6.3)
$\Big{(}\int_{{\mathbb{R}}^{d}}\big{(}G_{q,\varphi}(f)(x)\big{)}^{q}w(x)dx\Big{)}^{\frac{1}{q}}\lesssim_{d,\varepsilon,\delta}\mathbf{M}_{c,q}(X)\Big{(}\int_{{\mathbb{R}}^{d}}\big{\|}f(x)\big{\|}_{X}^{q}M(w)(x)dx\Big{)}^{\frac{1}{q}},$

To this end, we have to control the $g$-function by the Lusin area function.
If $\varphi$ is the Poisson kernel, this is a classical fact thanks to
harmonicity. In the presenting setting, we need a little bit more efforts.

If additionally all partial derivatives of $\varphi$ with order up to $d$
belong to ${\mathcal{H}}_{\varepsilon,\delta}$, then we can show

$G_{q,\varphi}(f)(x)\lesssim_{d}\sum_{|\alpha|\leq
d}S_{q,D^{\alpha}\varphi}(f)(x),\quad x\in{\mathbb{R}}^{d},$

where $D^{\alpha}=\frac{\partial^{\alpha_{1}}}{\partial
x_{1}^{\alpha_{1}}}\cdots\frac{\partial^{\alpha_{d}}}{\partial
x_{d}^{\alpha_{d}}}$ for $\alpha=(\alpha_{1},\cdots,\alpha_{d})$ and
$|\alpha|=\alpha_{1}+\cdots+\alpha_{d}$. The proof of this inequality is
elementary (see [56, Lemma 4.3]). Thus (6.3) holds for such $\varphi$.

For a general $\varphi$, we need to adapt the arguments of [54] to the present
setting by introducing the vector-valued $q$-variants of Wilson’s intrinsic
square functions:

$\displaystyle S_{q,\varepsilon,\delta}(f)(x)^{q}$
$\displaystyle=\int_{|y-x|<t}\,\sup_{\varphi\in{\mathcal{H}}_{\varepsilon,\delta}}\big{\|}\varphi_{t}*f(y)\big{\|}_{X}^{q}\,\frac{dy\,dt}{t^{d+1}},$
$\displaystyle G_{q,\varepsilon,\delta}(f)(x)^{q}$
$\displaystyle=\int_{0}^{\infty}\,\sup_{\varphi\in{\mathcal{H}}_{\varepsilon,\delta}}\big{\|}\varphi_{t}*f(x)\big{\|}_{X}^{q}\,\frac{dt}{t}.$

One can show, quite easily, that
$S_{q,\varepsilon,\delta}(f)(x)\approx_{d,\varepsilon,\delta}G_{q,\varepsilon,\delta}(f)(x)$
for every $x\in{\mathbb{R}}^{d}$ (see [54] for more details).

On the other hand, for a compactly supported smooth function
$h:{\mathbb{R}}^{d+1}_{+}\to X^{*}$ choose a family
$\\{\varphi^{(y,t)}\\}_{(y,t)\in K}\subset{\mathcal{H}}_{\varepsilon,\delta}$
($K$ being the support of $h$) such that

$\big{\|}\varphi^{(y,t)}_{t}*f(y)\big{\|}_{X}\geq\frac{1}{2}\,\sup_{\varphi\in{\mathcal{H}}_{\varepsilon,\delta}}\big{\|}\varphi_{t}*f(y)\big{\|}_{X}\,,\quad(y,t)\in
K.$

Then as in step 1, one can estimate the integral

$\displaystyle\int_{{\mathbb{R}}^{d+1}_{+}}\langle\varphi^{(y,t)}_{t}*f(y),\,h_{t}(y)\rangle\,\frac{dy\,dt}{t}=\int_{{\mathbb{R}}^{d}}\langle
f(x),\,\int_{{\mathbb{R}}^{d+1}_{+}}\varphi^{(y,t)}_{t}(y-x)h_{t}(y)\,\frac{dy\,dt}{t}\rangle
dx\,$

to conclude that

$\big{\|}S_{q,\varepsilon,\delta}(f)\big{\|}_{L_{q}({\mathbb{R}}^{d})}\lesssim_{d,\varepsilon,\delta}\mathbf{M}_{c,q}(X)\big{\|}f\big{\|}_{L_{q}({\mathbb{R}}^{d};X)}.$

This implies (6.1) with $S_{q,\psi}$ replaced by $S_{q,\varepsilon,\delta}$ by
the passage from the unweighted case to the weighted one. Then the pointwise
equivalence
$S_{q,\varepsilon,\delta}(f)\approx_{d,\varepsilon,\delta}G_{q,\varepsilon,\delta}(f)$
shows that (6.3) holds for $G_{q,\varepsilon,\delta}$ instead of
$G_{q,\varphi}$, whence (6.3) for every
$\varphi\in{\mathcal{H}}_{\varepsilon,\delta}$.

Step 3: Proof of Theorem 1.5 (i) for $p\geq q$. We can now easily prove part
(i) of Theorem 1.5 for $p\geq q$. Indeed, the case $p=q$ is just the
unweighted version of (6.3). For $p>q$, let $w$ be a nonnegative function on
${\mathbb{R}}^{d}$ with $L_{r}$-norm equal to 1, where $r$ is the conjugate
index of $\frac{p}{q}$. Then for $f\in L_{p}({\mathbb{R}};X)$,

$\displaystyle\int_{{\mathbb{R}}^{d}}\big{(}G_{q,\varphi}(f)(x)\big{)}^{q}w(x)dx$
$\displaystyle\lesssim_{d,\varepsilon,\delta}\mathbf{M}_{c,q}(X)^{q}\,\int_{{\mathbb{R}}^{d}}\big{\|}f(x)\big{\|}_{X}^{q}M(w)(x)dx$
$\displaystyle\lesssim_{d,\varepsilon,\delta}\mathbf{M}_{c,q}(X)^{q}\,\big{\|}f\big{\|}_{L_{p}({\mathbb{R}}^{d};X)}^{q}\,\big{\|}M(w)\big{\|}_{L_{r}({\mathbb{R}}^{d})}$
$\displaystyle\lesssim_{d,\varepsilon,\delta}r^{\prime}\,\mathbf{M}_{c,q}(X)^{q}\,\big{\|}f\big{\|}^{q}_{L_{p}({\mathbb{R}}^{d};X)}.$

Taking the supremum over all $w$, we get

$\big{\|}G_{q,\varphi}(f)\big{\|}_{L_{p}({\mathbb{R}}^{d})}\lesssim_{d,\varepsilon,\delta}p^{\frac{1}{q}}\,\mathbf{M}_{c,q}(X)\,\big{\|}f\big{\|}_{L_{p}({\mathbb{R}}^{d};X)},$

whence
$\mathbf{L}^{\varphi}_{c,q,p}(X)\lesssim_{d,\varepsilon,\delta}p^{\frac{1}{q}}\,\mathbf{M}_{c,q}(X)$.

Step 4: Proof of Theorem 1.5 (i) for $p<q$. We deal with the case $p<q$ by
singular integrals. Let $K$ be the singular integral associated to $\varphi$
as in subsection 5.2. We reduce to showing that $K$ is bounded from
$L_{p}({\mathbb{R}}^{d};X)$ to
$L_{p}({\mathbb{R}}^{d};L_{q}({\mathbb{R}}_{+};X))$. The previous step insures
this boundedness for $p=q$. On the other hand, Lemma 5.4 shows that $K$ is a
regular Calderón-Zygmund kernel. Thus $K$ satisfies the assumption of [15,
Theorem V.3.4]. Note that [15, Theorem V.3.4] is formulated for kernels
satisfying the regularities in Lemma 5.4 with $\delta=1$; however, it is well
known that [15, Theorem V.3.4] remains valid for any kernel as in Lemma 5.4
with the same proof. Therefore, $K$ is of weak type $(1,1)$, so by the
Marcinkiewicz interpolation theorem, $K$ is bounded from
$L_{p}({\mathbb{R}}^{d};X)$ to
$L_{p}({\mathbb{R}}^{d};L_{q}({\mathbb{R}}_{+};X))$ with norm controlled by
$C_{d,\varepsilon,\delta}\,p^{\prime}\mathsf{M}_{c,q}(X)$ for $1<p<q$. This
finishes the proof of Theorem 1.5 (i).

Step 5: Proof of Theorem 1.5 (ii). In this last step we show part (ii) by
duality. Let $\psi\in{\mathcal{H}}_{\varepsilon,\delta}$ such that (1.6)
holds. Let $f\in L_{p}({\mathbb{R}}^{d};X)$ and $g\in
L_{p^{\prime}}({\mathbb{R}}^{d};X^{*})$. Then (1.6) implies

$\int_{{\mathbb{R}}^{d}}\langle
f(x),\,g(x)\rangle\,dx=\int_{{\mathbb{R}}^{d+1}_{+}}\langle\varphi_{t}*f(x),\,\psi_{t}*g(x)\rangle\,\frac{dxdt}{t}.$

In the scalar case, this Calderón reproducing formula is proved by taking
Fourier transforms of the both sides. Then by linearity, the formula extends
to the vector-valued case too when both $f$ and $g$ take values in finite
dimensional subspaces, which can be assumed by approximation. Therefore,

$\Big{|}\int_{{\mathbb{R}}^{d}}\langle
f(x),\,g(x)\rangle\,dx\Big{|}\leq\big{\|}G_{q,\varphi}(f)\big{\|}_{L_{p}({\mathbb{R}}^{d})}\,\big{\|}G_{q^{\prime},\psi}(g)\big{\|}_{L_{p^{\prime}}({\mathbb{R}}^{d})}.$

Since we are in part (ii) of Theorem 1.5, $X$ is of martingale type $q$, so
$X^{*}$ is of martingale cotype $q^{\prime}$ and
$\mathsf{M}_{c,q^{\prime}}(X^{*})\leq 2\mathsf{M}_{t,q}(X)$ by (6.2). Thus by
part (i) already proved, we have

$\displaystyle\big{\|}G_{q^{\prime},\psi}(g)\big{\|}_{L_{p^{\prime}}({\mathbb{R}}^{d})}$
$\displaystyle\leq\mathsf{L}^{\psi}_{c,q^{\prime},p^{\prime}}(X^{*})\,\big{\|}g\big{\|}_{L_{p^{\prime}}({\mathbb{R}}^{d};X^{*})}$
$\displaystyle\lesssim_{d,\varepsilon,\delta}\max\big{(}p^{\prime\frac{1}{q^{\prime}}},\,p\big{)}\mathsf{M}_{c,q^{\prime}}(X^{*})\,\big{\|}g\big{\|}_{L_{p^{\prime}}({\mathbb{R}}^{d};X^{*})}.$

Hence

$\Big{|}\int_{{\mathbb{R}}^{d}}\langle
f(x),\,g(x)\rangle\,dx\Big{|}\lesssim_{d,\varepsilon,\delta}\max\big{(}p,\,p^{\prime\frac{1}{q^{\prime}}}\big{)}\mathsf{M}_{t,q}(X)\,\big{\|}G_{q,\varphi}(f)\big{\|}_{L_{p}({\mathbb{R}}^{d})}\,\big{\|}g\big{\|}_{L_{p^{\prime}}({\mathbb{R}}^{d};X^{*})}.$

Taking the supremum over all $g$ with
$\big{\|}g\big{\|}_{L_{p^{\prime}}({\mathbb{R}}^{d};X^{*})}\leq 1$, we deduce

$\mathsf{L}^{\varphi}_{t,q,p}(X)\lesssim_{d,\varepsilon,\delta}\max\big{(}p,\,p^{\prime\frac{1}{q^{\prime}}}\big{)}\mathsf{M}_{t,q}(X)$

as desired. So the proof of Theorem 1.5 is complete.

##  7\. Proofs of Theorem 1.2, Theorem 1.4 and Corollary 1.8

In this section we will first prove Theorem 1.2. Theorem 1.4 and Corollary 1.8
will then follow quite easily. Our strategy for the proof of Theorem 1.2 is to
reduce part (i), via transference, to the special case of the translation
group to which we can apply Theorem 1.5.

###### Proof of Theorem 1.2.

Again, we divide this proof into several steps. $X$ will be assumed to be of
martingale cotype $q$ in the first two steps, and of martingale type $q$ in
the last.

Step 1: The case of the translation group. For any $t\in{\mathbb{R}}$ let
$\tau_{t}$ be the translation by $t$ on $L_{p}({\mathbb{R}})$, i.e.,
$\tau_{t}(f)(s)=f(s+t)$. Then $\\{\tau_{t}\\}_{t\in{\mathbb{R}}}$ is a
strongly continuous group of positive isometries on $L_{p}({\mathbb{R}})$. As
usual, $\\{\tau_{t}\\}_{t\in{\mathbb{R}}}$ extends to a group of isometries on
$L_{p}({\mathbb{R}};X)$ too. Let $\\{P_{t}^{\tau}\\}_{t>0}$ be the associated
Poisson subordinated semigroup. Our aim in this step is to show

(7.1)
$\mathsf{L}^{P^{\tau}}_{c,q,p}(X)\lesssim\max\big{(}p^{\frac{1}{q}},\,p^{\prime}\big{)}\mathsf{M}_{c,q}(X).$

We need to express $t\partial P^{\tau}_{t}$ as a convolution operator:

$\sqrt{t}\,\partial
P^{\tau}_{\sqrt{t}}(f)(x)=\int_{\mathbb{R}}\phi_{t}(x-y)f(y)dy=\phi_{t}*f(x),\quad
x\in{\mathbb{R}},\;t>0.$

Then

(7.2)
$\mathcal{G}^{P^{\tau}}_{q}(f)(x)=2^{-\frac{1}{q}}\Big{(}\int_{0}^{\infty}\|\phi_{t}*f(x)\|^{q}_{X}\frac{dt}{t}\Big{)}^{\frac{1}{q}}=2^{-\frac{1}{q}}G_{q,\phi}(f)(x)\,.$

Elementary computations show that $\phi$ is the function with Fourier
transform

(7.3) $\widehat{\phi}(\xi)=-\sqrt{-2\pi{\rm i}\,\xi}\,e^{-\sqrt{-2\pi{\rm
i}\,\xi}\,}=-\sqrt{2\pi|\xi|}\,e^{-\frac{{\rm i}\,{\rm
sgn}(\xi)\pi}{4}}\exp\big{(}-\sqrt{2\pi|\xi|}\,e^{-\frac{{\rm i}\,{\rm
sgn}(\xi)\pi}{4}}\big{)}\,.$

We are going to show that $\phi$ belongs to the class
${\mathcal{H}}_{\frac{1}{2},1}$ introduced in subsection 5.2. More precisely,
$\phi$ satisfies the following estimates:

(7.4) $|\phi(x)|\lesssim\frac{1}{(1+|x|)^{\frac{3}{2}}}\;\text{ and
}\;|\phi^{\prime}(x)|\lesssim\frac{1}{(1+|x|)^{\frac{5}{2}}},\quad
x\in{\mathbb{R}}.$

Since $\xi^{k}\widehat{\phi}(\xi)$ is integrable on ${\mathbb{R}}$ for any
nonnegative integer $k$, $\phi$ is of class $C^{\infty}$ with bounded
derivatives of any order. Thus it suffices to prove the estimates for $|x|\geq
1$.

Let $\eta$ be a $C^{\infty}$ even function on ${\mathbb{R}}$, supported in
$\\{\xi:\frac{1}{2}<|\xi|<2\\}$, such that

$\sum_{j\in{\mathbb{Z}}}\eta(2^{-j}\xi)=1,\quad\xi\in{\mathbb{R}}\setminus\\{0\\}.$

Let $m_{j}(\xi)=\widehat{\phi}(\xi)\eta(2^{-j}\xi)$ and $\phi^{(j)}$ be
defined by $\widehat{\phi^{(j)}}=m_{j}$. Then

$\phi=\sum_{j\in{\mathbb{Z}}}\phi^{(j)}.$

Using (7.3), one easily shows

$\int_{2^{j-1}\leq|\xi|\leq
2^{j+1}}\Big{|}\frac{d^{k}}{d\xi^{k}}\,\widehat{\phi}(\xi)\Big{|}d\xi\lesssim
2^{j(1-k)}\sum_{\ell=0}^{k}(\sqrt{2^{j}}\,)^{\ell+1}e^{-\sqrt{\pi
2^{j-1}}}\,,\quad 0\leq k\leq 3,$

This implies

(7.5)
$\int_{{\mathbb{R}}}\Big{|}\frac{d^{k}}{d\xi^{k}}\,m_{j}(\xi)\Big{|}d\xi\lesssim
2^{j(1-k)}\sum_{\ell=0}^{k}(\sqrt{2^{j}}\,)^{\ell+1}e^{-\sqrt{\pi
2^{j-1}}}\,,\quad 0\leq k\leq 3.$

Let $x\in{\mathbb{R}}$ with $|x|>1$. We consider $j$ according to two cases,
I: $2^{j}|x|\leq 1$ and II: $2^{j}|x|>1$.

In Case I, we must have $j\leq-1$. Using (7.5) for $k=1$, we then have

$|x\phi^{(j)}(x)|\lesssim\sqrt{2^{j}}\,.$

Thus

$\sum_{j\in{\rm I}}|\phi^{(j)}(x)|\lesssim\frac{1}{|x|^{\frac{3}{2}}}\,.$

On the other hand, if $j\in{\rm II}$, we use (7.5) for $k=3$ to get

$|x^{3}\phi^{(j)}(x)|\lesssim 2^{-\frac{3}{2}j}\,.$

We deduce again

$\sum_{j\in{\rm II}}|\phi^{(j)}(x)|\lesssim\frac{1}{|x|^{\frac{3}{2}}}\,.$

Hence, the first estimate of (7.4) is proved.

The second is shown in a similar way. Indeed, since
$\widehat{\phi^{\prime}}(\xi)=2\pi{\rm i}\,\xi\widehat{\phi}(\xi)$, we have

$\int_{2^{j-1}\leq|\xi|\leq
2^{j+1}}\Big{|}\frac{d^{k}}{d\xi^{k}}\,\widehat{\phi^{\prime}}(\xi)\Big{|}d\xi\lesssim
2^{j(2-k)}\sum_{\ell=0}^{k}(\sqrt{2^{j}}\,)^{\ell+1}e^{-\sqrt{\pi
2^{j-1}}}\,,\quad 0\leq k\leq 3,$

It then remains to repeat the above argument with $\phi$ replaced by
$\phi^{\prime}$.

Thus by (7.2), Theorem 1.5 implies (7.1). Let us note that for the kernel
$\phi$ here, we can avoid Wilson’s intrinsic square functions considered in
step 2 of the proof of Theorem 1.5 since
$\phi^{\prime}\in{\mathcal{H}}_{\frac{1}{2},1}$ too. Indeed, repeating the
proof of (7.4), we show

$|\phi^{\prime\prime}(x)|\lesssim\frac{1}{(1+|x|)^{\frac{7}{2}}}.$

Consequently, as pointed out in step 2 of the proof of Theorem 1.5, we have

$G_{q,\phi}(f)(x)\lesssim S_{q,\phi}(f)(x)+S_{q,\phi^{\prime}}(f)(x),\quad
x\in{\mathbb{R}}.$

The proof of this inequality is very easy. It suffices to consider $x=0$. Let
$y\in{\mathbb{R}}$ such that $|y|<t$. We write

$\|\phi_{t}*f(y)\|^{q}_{X}-\|\phi_{t}*f(0)\|^{q}_{X}=\int_{0}^{1}\frac{d}{ds}\,\|\phi_{t}*f(sy)\|^{q}_{X}ds.$

Then

$\displaystyle\Big{|}\|\phi_{t}*f(y)\|^{q}_{X}-\|\phi_{t}*f(0)\|^{q}_{X}\Big{|}$
$\displaystyle\leq
q\int_{0}^{1}\|\phi_{t}*f(sy)\|^{q-1}_{X}\,\|\phi^{\prime}_{t}*f(sy)\|_{X}\,\frac{|y|}{t}\,ds$
$\displaystyle\leq
q\int_{0}^{|y|}\|\phi_{t}*f(s\frac{y}{|y|})\|^{q-1}_{X}\,\|\phi^{\prime}_{t}*f(s\frac{y}{|y|})\|_{X}\frac{ds}{t}$
$\displaystyle\leq
q\Big{(}\int_{0}^{|y|}\big{(}\|\phi_{t}*f(s\frac{y}{|y|})\|^{q}_{X}+\|\phi^{\prime}_{t}*f(s\frac{y}{|y|})\|^{q}_{X}\big{)}\frac{ds}{t}\Big{)}.$

Thus

$\displaystyle\|\phi_{t}*f(0)\|^{q}_{X}\leq\|\phi_{t}*f(y)\|^{q}_{X}+q\Big{(}\int_{0}^{|y|}\big{(}\|\phi_{t}*f(s\frac{y}{|y|})\|^{q}_{X}+\|\phi^{\prime}_{t}*f(s\frac{y}{|y|})\|^{q}_{X}\big{)}\frac{ds}{t}\Big{)}.$

Integrating both sides against $\frac{dydt}{t^{2}}$ over the cone
$\\{(y,t)\in{\mathbb{R}}^{2}_{+}:|y|<t\\}$, we deduce the desired inequality.

Step 2: Transference. We now use the transference principle to bring the
general case to the special one of the translation group. To that end, we
first need to dilate our semigroup $\\{T_{t}\\}_{t>0}$ to a group of
isometries. Fendler’s dilation theorem is at our disposal for this purpose. It
insures that there exist another larger measure space
$(\widetilde{\Omega},\widetilde{\mu})$, a strongly continuous group
$\\{\widetilde{T}_{t}\\}_{t\in{\mathbb{R}}}$ of regular isometries on
$L_{p}(\widetilde{\Omega})$, a positive isometric embedding $D$ from
$L_{p}(\Omega)$ into $L_{p}(\widetilde{\Omega})$ and a regular projection $P$
from $L_{p}(\widetilde{\Omega})$ onto $L_{p}(\Omega)$ such that

$T_{t}=P\widetilde{T}_{t}D,\quad\forall\;t>0.$

This theorem is proved in [16] for positive $T_{t}$ and then extended to
regular $T_{t}$ in [17].

To prove part (i) of Theorem 1.2, it suffices to show

(7.6) $\mathsf{L}^{P}_{c,q,p}(X)\leq\mathsf{L}^{P^{\tau}}_{c,q,p}(X).$

By the above dilation, we can assume that $\\{T_{t}\\}_{t>0}$ itself is a
group of regular isometries on $L_{p}(\Omega)$. So its extension to
$L_{p}(\Omega;X)$ is a group of isometries too. Recall that
$\\{M_{t}\\}_{t>0}$ denote the ergodic averages of $\\{T_{t}\\}_{t>0}$ in
Lemma 2.6. We use $\\{M^{\tau}_{t}\\}_{t>0}$ to denote the corresponding
averages of the translation group $\\{\tau_{t}\\}_{t>0}$. By (1.2), we have

$P_{t}=\frac{1}{2\sqrt{\pi}}\,\int_{0}^{\infty}\frac{t}{s^{\frac{3}{2}}}\exp\big{(}-\frac{t^{2}}{4s}\big{)}\,T_{s}ds.$

Thus

$\displaystyle t\partial P_{t}$
$\displaystyle=\frac{1}{2\sqrt{\pi}}\,\int_{0}^{\infty}\big{(}\frac{t}{s^{\frac{3}{2}}}-\frac{t^{3}}{2s^{\frac{5}{2}}}\big{)}\exp\big{(}-\frac{t^{2}}{4s}\big{)}\,T_{s}ds$
$\displaystyle=\frac{1}{2\sqrt{\pi}}\,\int_{0}^{\infty}\big{(}\frac{t}{s^{\frac{3}{2}}}-\frac{t^{3}}{s^{\frac{5}{2}}}\big{)}\exp\big{(}-\frac{t^{2}}{4s}\big{)}\,(sM_{s})^{\prime}ds$
$\displaystyle=\int_{0}^{\infty}\varphi(\frac{t}{\sqrt{s}})M_{s}\,\frac{ds}{s}$
$\displaystyle=\int_{0}^{\infty}\varphi(\frac{1}{\sqrt{s}})M_{t^{2}s}\,\frac{ds}{s},$

where

$\varphi(x)=\frac{1}{16\sqrt{\pi}}\,(12x-12x^{3}+x^{5})e^{-\frac{x^{2}}{4}}.$

Let $f\in L_{p}(\Omega;X)$ be an element of norm $1$. Let $a>0$ (large). Then

$\int_{a}^{\infty}|\varphi(\frac{1}{\sqrt{s}})|\,\frac{ds}{s}\leq\frac{C}{\sqrt{a}}\,.$

Thus for any $t>0$

$\displaystyle\Big{\|}\int_{a}^{\infty}\varphi(\frac{1}{\sqrt{s}})M_{t^{2}s}(f)\,\frac{ds}{s}\Big{\|}_{X}\leq\int_{a}^{\infty}|\varphi(\frac{1}{\sqrt{s}})|\big{\|}M_{t^{2}s}(f)\big{\|}_{X}\,\frac{ds}{s}\leq\frac{C}{\sqrt{a}}\,M^{*}(f),$

where $M^{*}(f)=\sup_{v>0}\big{\|}M_{v}(f)\big{\|}_{X}$. By Lemma 2.6,

$\big{\|}M^{*}(f)\big{\|}_{L_{p}(\Omega)}\leq
p^{\prime}\big{\|}f\big{\|}_{L_{p}(\Omega;X)}=p^{\prime}.$

Let $b$ be another large number. Then

$\displaystyle\left\|\Big{(}\int_{b^{-1}}^{b}\big{\|}t\partial
P_{t}(f)\big{\|}_{X}^{q}\,\frac{dt}{t}\Big{)}^{\frac{1}{q}}\right\|_{L_{p}(\Omega)}\leq\left\|\Big{(}\int_{b^{-1}}^{b}\Big{\|}\int_{0}^{a}\varphi(\frac{1}{\sqrt{s}})M_{t^{2}s}(f)\,\frac{ds}{s}\Big{\|}_{X}^{q}\,\frac{dt}{t}\Big{)}^{\frac{1}{q}}\right\|_{L_{p}(\Omega)}+\frac{C_{p,q,b}}{\sqrt{a}},$

where $C_{p,q,b}=Cp^{\prime}q^{-\frac{1}{q}}(2\log b)^{\frac{1}{q}}$. Denote
the first term on the right hand side by I. Using the fact that $\\{T_{t}\\}$
is a group of isometries on $L_{p}(\Omega;X)$, we introduce an additional
variable $u$ in the integrand of I:

${\rm
I}\leq\left\|\Big{(}\int_{b^{-1}}^{b}\Big{\|}\int_{0}^{a}\varphi(\frac{1}{\sqrt{s}})M_{t^{2}s}T_{u}(f)\,\frac{ds}{s}\Big{\|}_{X}^{q}\,\frac{dt}{t}\Big{)}^{\frac{1}{q}}\right\|_{L_{p}(\Omega)}\,,\quad
u>0.$

Let $c>0$. Now define $g:{\mathbb{R}}\to L_{p}(\Omega;X)$ by
$g(u)={\mathds{1}}_{(0,\,ab^{2}+c]}(u)T_{u}(f)$. We easily verify that

$M_{t^{2}s}T_{u}(f)=M^{\tau}_{t^{2}s}(g)(u)\,,\quad 0<s\leq a,\;0<t\leq
b,\;0<u\leq c.$

Hence

$\displaystyle{\rm I}^{p}$
$\displaystyle\leq\frac{1}{c}\int_{0}^{c}\int_{\Omega}\Big{(}\int_{b^{-1}}^{b}\Big{\|}\int_{0}^{a}\varphi(\frac{1}{\sqrt{s}})M^{\tau}_{t^{2}s}(g)(u)\,\frac{ds}{s}\Big{\|}_{X}^{q}\,\frac{dt}{t}\Big{)}^{\frac{p}{q}}d\omega\,du$
$\displaystyle\leq\frac{1}{c}\left\|\Big{(}\int_{b^{-1}}^{b}\Big{\|}\int_{0}^{a}\varphi(\frac{1}{\sqrt{s}})M^{\tau}_{t^{2}s}(g)\,\frac{ds}{s}\Big{\|}_{X}^{q}\,\frac{dt}{t}\Big{)}^{\frac{1}{q}}\right\|^{p}_{L_{p}({\mathbb{R}}\times\Omega)}\,.$

Let $(M^{\tau})^{*}(g)=\sup_{v>0}\big{\|}M^{\tau}_{v}(g)\big{\|}_{X}$, so

$\big{\|}(M^{\tau})^{*}(g)\big{\|}_{L_{p}({\mathbb{R}}\times\Omega)}\leq
p^{\prime}\big{\|}g\big{\|}_{L_{p}({\mathbb{R}}\times\Omega;X)}\leq
p^{\prime}(ab^{2}+c)^{\frac{1}{p}}.$

Reversing the preceding procedure with $\\{P_{t}\\}_{t>0}$ replaced by
$\\{P^{\tau}_{t}\\}_{t>0}$, we have

$\displaystyle{\rm I}$ $\displaystyle\leq
c^{-\frac{1}{p}}\left\|\Big{(}\int_{b^{-1}}^{b}\Big{\|}\int_{0}^{\infty}\varphi(\frac{1}{\sqrt{s}})M^{\tau}_{t^{2}s}(g)\,\frac{ds}{s}\Big{\|}_{X}^{q}\,\frac{dt}{t}\Big{)}^{\frac{1}{q}}\right\|_{L_{p}({\mathbb{R}}\times\Omega)}+\frac{C_{p,q,b}}{\sqrt{a}}\,\Big{(}\frac{ab^{2}+c}{c}\Big{)}^{\frac{1}{p}}$
$\displaystyle\leq
c^{-\frac{1}{p}}\big{\|}\mathcal{G}^{P^{\tau}}_{q}(g)\big{\|}_{L_{p}({\mathbb{R}}\times\Omega;X)}+\frac{C_{p,q,b}}{\sqrt{a}}\,\Big{(}\frac{ab^{2}+c}{c}\Big{)}^{\frac{1}{p}}\,.$

However,

$\big{\|}\mathcal{G}^{P^{\tau}}_{q}(g)\big{\|}_{L_{p}({\mathbb{R}}\times\Omega)}\leq\mathsf{L}^{P^{\tau}}_{c,q,p}(X)\big{\|}g\big{\|}_{L_{p}({\mathbb{R}}\times\Omega;X)}\leq\mathsf{L}^{P^{\tau}}_{c,q,p}(X)(ab^{2}+c)^{\frac{1}{p}}\,.$

Combining all inequalities obtained so far, we finally deduce

$\left\|\Big{(}\int_{b^{-1}}^{b}\big{\|}t\partial
P_{t}(f)\big{\|}_{X}^{q}\,\frac{dt}{t}\Big{)}^{\frac{1}{q}}\right\|_{L_{p}(\Omega)}\leq\mathsf{L}^{P^{\tau}}_{c,q,p}(X)\Big{(}\frac{ab^{2}+c}{c}\Big{)}^{\frac{1}{p}}+\frac{C_{p,q,b}}{\sqrt{a}}\left(1+\Big{(}\frac{ab^{2}+c}{c}\Big{)}^{\frac{1}{p}}\right)\,.$

Letting successively $c\to\infty$, $a\to\infty$ and $b\to\infty$, we get

$\big{\|}\mathcal{G}^{P}_{q}(f)\big{\|}_{L_{p}(\Omega)}\leq\mathsf{L}^{P^{\tau}}_{c,q,p}(X)\,,$

whence (7.6).

Step 3: Duality. Assertion (ii) follows from (i), Theorem 4.5 and (6.2). ∎

######  Remark 7.1.

The step 1 of the previous proof of Theorem 1.2 can be largely shortened for
the case $p\geq q$. This alternate proof does not rely on the heavy
Littlewood-Paley theory. Its key point is to show the boundedness of
$\mathcal{G}^{P}_{q}$ on $L_{q}(\Omega;X)$, i.e., for $p=q$ (see the following
remark). Assuming this boundedness, we can then use the singular integral as
in the proof of the step 4 of Theorem 1.5 to conclude the case $p>q$.
Unfortunately, this proof yields $p$ as the order of
$\mathsf{L}^{P}_{c,q,p}(X)$ instead of the optimal $p^{\frac{1}{q}}$.

######  Remark 7.2.

The boundedness of $\mathcal{G}^{P}_{q}$ on $L_{q}(\Omega;X)$ can be proved by
using Theorem 4.7 and the following inequality from [21]: For a Banach space
$X$ of martingale cotype $q$, we have

$\Big{\|}\Big{(}\sum_{k\in{\mathbb{Z}}}\big{\|}\big{(}M^{\tau}_{2^{k}}-M^{\tau}_{2^{k+1}}\big{)}(f)\big{\|}_{X}^{q}\Big{)}^{\frac{1}{q}}\Big{\|}_{L_{q}({\mathbb{R}})}\lesssim\mathsf{M}_{c,q}(X)\big{\|}f\big{\|}_{L_{q}({\mathbb{R}};X)}\,,\quad
f\in L_{q}({\mathbb{R}};X).$

By the discussion in the previous remark, the validity of the above inequality
characterizes the martingale cotype $q$ of $X$. More generally, let
$1<p<\infty$ and $2\leq q<\infty$. Then $X$ is of martingale cotype $q$ iff
there exists a constant $c$ such that

$\Big{\|}\Big{(}\sum_{k\in{\mathbb{Z}}}\big{\|}\big{(}M^{\tau}_{2^{k}}-M^{\tau}_{2^{k+1}}\big{)}(f)\big{\|}_{X}^{q}\Big{)}^{\frac{1}{q}}\Big{\|}_{L_{p}({\mathbb{R}})}\leq
c\,\big{\|}f\big{\|}_{L_{p}({\mathbb{R}};X)}\,,\quad f\in
L_{p}({\mathbb{R}};X).$

See [22] for related results.

###### Proof of Theorem 1.4.

By Remark 1.1, $\mathsf{L}^{T}_{t,q,p}(X)\lesssim\mathsf{L}^{P}_{t,q,p}(X)$.
Thus assertion (i) follows from Theorem 1.2 (ii).

Assertion (ii) is an easy consequence of Theorem 1.2 and Corollary 4.10. But
we will use Proposition 4.9 in order to explicitly track the relevant
constants. Let $\varphi(z)=-ze^{-z}$ and $\psi(z)=-\sqrt{z}\,e^{-\sqrt{z}}$.
Then

$\mathcal{G}_{q,\varphi}^{A}(f)=\mathcal{G}_{q}^{T}(f)\quad\text{and
}\quad\mathcal{G}_{q,\psi}^{A}(f)=\sqrt{2}\,\mathcal{G}_{q}^{P}(f).$

By Proposition 2.9 and the notation there, $A$ is $\ell_{q}$-sectorial of type
$\alpha_{q}=\frac{\pi}{2}-\beta_{q}$. Let $0<\beta<\beta_{q}$ and
$\alpha=\frac{\pi}{2}-\beta$. Then by Proposition 4.9 and the estimates
obtained in its proof, we get

$\big{\|}\mathcal{G}_{q}^{T}(f)\big{\|}_{L_{p}(\Omega)}\lesssim(\cos\alpha\cos\frac{\alpha}{2})^{-1}(\beta_{q}-\beta)^{-2}\,\mathsf{T}_{\beta_{0}}^{\min(\frac{p}{q},\,\frac{p^{\prime}}{q^{\prime}})}\max(p^{\prime
1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}})\big{\|}\mathcal{G}_{q}^{P}(f)\big{\|}_{L_{p}(\Omega)}\,.$

Choosing $\beta=\frac{\beta_{q}}{2}$ yields

$\big{\|}\mathcal{G}_{q}^{T}(f)\big{\|}_{L_{p}(\Omega)}\lesssim
D\,\big{\|}\mathcal{G}_{q}^{P}(f)\big{\|}_{L_{p}(\Omega)},$

where

$D=\beta_{q}^{-3}\,\mathsf{T}_{\beta_{0}}^{\min(\frac{p}{q}\,,\frac{p^{\prime}}{q^{\prime}})}\,\max\big{(}p^{\prime
1-\frac{p}{q}},\,p^{1-\frac{p^{\prime}}{q^{\prime}}}\big{)}.$

We then deduce Theorem 1.4 (ii) from Theorem 1.2 (i). ∎

###### Proof of Corollary 1.8.

By [37, 38],

$\mathsf{M}_{c,q}(L_{p}(\Omega;X))\lesssim\max\big{(}p^{\prime\frac{1}{q}},\,\mathsf{M}_{c,q}(X)\big{)}.$

Fix $f_{0}\in L_{q}(\Omega)$ with norm 1. Given $f\in L_{p}(\Omega;X)$ let
$\widetilde{f}=f\otimes f_{0}$. We view $\widetilde{f}$ as a function from
$\Omega$ to $L_{p}(\Omega;X)$ by $\omega\mapsto f(\omega)f_{0}$. Applying
Theorem 1.2 (i) to this function with $X$ replaced by $L_{p}(\Omega;X)$ and
$p=q$, we get

$\displaystyle\left\|\Big{(}\int_{0}^{\infty}\big{\|}t\frac{\partial}{\partial
t}P_{t}(f)\otimes
f_{0}\big{\|}^{q}_{L_{p}(\Omega;X)}\frac{dt}{t}\Big{)}^{\frac{1}{q}}\right\|_{L_{q}(\Omega)}$
$\displaystyle\lesssim\max\big{(}p^{\prime\frac{1}{q}},\,\mathsf{M}_{c,q}(X)\big{)}\big{\|}f\otimes
f_{0}\big{\|}_{L_{q}(\Omega;L_{p}(\Omega;X))}$
$\displaystyle=\max\big{(}p^{\prime\frac{1}{q}},\,\mathsf{M}_{c,q}(X)\big{)}\big{\|}f\big{\|}_{L_{p}(\Omega;X)}.$

The left hand side is qual to

$\displaystyle\Big{(}\int_{0}^{\infty}\big{\|}t\frac{\partial}{\partial
t}P_{t}(f)\otimes
f_{0}\big{\|}^{q}_{L_{q}(\Omega;L_{p}(\Omega;X))}\frac{dt}{t}\Big{)}^{\frac{1}{q}}=\Big{(}\int_{0}^{\infty}\big{\|}t\frac{\partial}{\partial
t}P_{t}(f)\big{\|}^{q}_{L_{p}(\Omega;X)}\frac{dt}{t}\Big{)}^{\frac{1}{q}}.$

Combining the above estimates we get the desired inequality of the corollary.

To show the optimality of the constant, we consider the special case where
$X={\mathbb{C}}$ and the classical Poisson semigroup
$\\{\mathbb{P}_{t}\\}_{t>0}$ on ${\mathbb{R}}$. Let $f=\mathbb{P}_{1}$. Then

$\big{\|}f\big{\|}_{L_{p}({\mathbb{R}})}\approx 1\;\text{and
}\;\big{\|}t\frac{\partial}{\partial
t}\mathbb{P}_{t}(f)\big{\|}_{L_{p}({\mathbb{R}})}\approx\frac{t}{(t+1)^{1+\frac{1}{p^{\prime}}}}.$

Thus

$\Big{(}\int_{0}^{\infty}\big{\|}t\frac{\partial}{\partial
t}\mathbb{P}_{t}(f)\big{\|}^{q}_{L_{p}({\mathbb{R}})}\,\frac{dt}{t}\big{)}^{\frac{1}{q}}\approx
p^{\prime\frac{1}{q}}.$

We then deduce the announced optimality. ∎

##  8\. The scalar case revisited and optimality

The approach previously presented gives new insights even in the scalar case
with regard to the involved best constants. Let

$\mathsf{L}^{T}_{c,p}=\mathsf{L}_{c,2,p}({\mathbb{C}})\;\text{ and
}\;\mathsf{L}^{T}_{t,p}=\mathsf{L}_{t,2,p}({\mathbb{C}}).$

Thus $\mathsf{L}^{T}_{c,p}$ and $\mathsf{L}^{T}_{t,p}$ are the best constants
in the following inequalities

$(\mathsf{L}^{T}_{t,p})^{-1}\|f-\mathsf{F}(f)\|_{L_{p}(\Omega)}\leq\big{\|}\mathcal{G}^{T}_{2}(f)\big{\|}_{L_{p}(\Omega)}\leq\mathsf{L}^{T}_{c,p}\|f-\mathsf{F}(f)\|_{L_{p}(\Omega)},\quad
f\in L_{p}(\Omega).$

Let us restate Theorems 1.2 and 1.4 for $X={\mathbb{C}}$ and $q=2$.

######  Theorem 8.1.

Let $\\{T_{t}\\}_{t>0}$ be a semigroup of regular contractions on
$L_{p}(\Omega)$ with $1<p<\infty$ and $\\{P_{t}\\}_{t>0}$ its subordinated
Poisson semigroup.

  1. (i)

We have

$\mathsf{L}^{P}_{c,p}\lesssim\max(\sqrt{p},\,p^{\prime})\quad\text{ and
}\quad\mathsf{L}^{P}_{t,p}\lesssim\max(p,\,\sqrt{p^{\prime}}).$

  2. (ii)

Assume that $\\{T_{t}\\}_{t>0}$ satisfies (1.4) for $X={\mathbb{C}}$. Let
$\beta_{p}=\beta_{0}\min(p\,,p^{\prime})$. Then

$\mathsf{L}^{T}_{c,p}\lesssim\beta_{p}^{-3}\,\mathsf{T}_{\beta_{0}}^{\frac{1}{2}\,\min(p,\,p^{\prime})}\max\big{(}p,\,p^{\prime\frac{3}{2}}\big{)}\quad\text{
and }\quad\mathsf{L}^{T}_{t,p}\lesssim\max(p,\,\sqrt{p^{\prime}}).$

For symmetric diffusion semigroups we have the following more precise orders
than those in part (ii) above.

######  Corollary 8.2.

Let $\\{T_{t}\\}_{t>0}$ be a semigroup of contractions on $L_{p}(\Omega)$ for
every $1\leq p\leq\infty$. Assume that $\\{T_{t}\\}_{t>0}$ is strongly
continuous on $L_{2}(\Omega)$ and each $T_{t}$ is a selfadjoint operator on
$L_{2}(\Omega)$. Then

$\mathsf{L}^{T}_{c,p}\lesssim\max\big{(}p^{4},\,p^{\prime\frac{9}{2}}\big{)}\quad\text{
and }\quad\mathsf{L}^{T}_{t,p}\lesssim\max(p,\,\sqrt{p^{\prime}}).$

###### Proof.

First note that $T_{t}$ is a regular contraction on $L_{p}(\Omega)$ for any
$1\leq p\leq\infty$ and any $t>0$. The selfadjointness of $T_{t}$ implies that
$\\{T_{t}\\}_{t>0}$ is an analytic semigroup of type $\frac{\pi}{2}$ on
$L_{2}(\Omega)$ with constant $1$. Then by Stein’s interpolation (see [48,
section III.2]), $\\{T_{t}\\}_{t>0}$ is analytic of type
$\beta_{p}^{\prime}=\frac{\pi}{2}-\pi|\frac{1}{p}-\frac{1}{2}|$ with constant
$1$ on $L_{p}(\Omega)$ for $1<p<\infty$. It remains to apply Theorem 8.1 (ii)
with $\beta_{0}=\beta_{p}^{\prime}$. The corresponding $\beta_{p}$ is now
equal to $\pi\min(\frac{p^{\prime}}{p},\,\frac{p}{p^{\prime}})$. We then
deduce the desired assertion from Theorem 8.1. ∎

Historical comments. (i) Theorem 8.1 was proved in [32] without any explicit
estimates on the best constants; in fact, their growth obtained there is more
than exponential.

(ii) If $\\{P_{t}\\}_{t>0}$ is the Poisson semigroup on a compact Lie group,
Stein’s proof in [48, section II.3] yields that
$\mathsf{L}^{P}_{c,p}\lesssim\max(p,\,p^{\prime})$ and