# Maximal operator, Littlewood-Paley functions and variation operators
associated with nonsymmetric Ornstein-Uhlenbeck operators.

Víctor Almeida  ,  Jorge J. Betancor  ,  Pablo Quijano  and  Lourdes
Rodríguez-Mesa  Víctor Almeida, Jorge J. Betancor, Lourdes Rodríguez-Mesa
Departamento de Análisis Matemático, Universidad de La Laguna,
Campus de Anchieta, Avda. Astrofísico Sánchez, s/n,
38721 La Laguna (Sta. Cruz de Tenerife), Spain<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Pablo Quijano
Instituto de Matemática Aplicada del Litora, Santa Fe-Argentina
<EMAIL_ADDRESS>
###### Abstract.

In this paper we establish $L^{p}$ boundedness properties for maximal
operators, Littlewood-Paley functions and variation operators involving
Poisson semigroups and resolvent operators associated with nonsymmetric
Ornstein-Uhlenbeck operators. We consider the Ornstein-Uhlenbeck operators
defined by the identity as the covariance matrix and having a drift given by
the matrix $-\lambda(I+R)$, being $\lambda>0$ and $R$ a skew-adjoint matrix.
The semigroup associated with these Ornstein-Uhlenbeck operators are the basic
building blocks of all the normal Ornstein-Uhlenbeck semigroups.

###### Key words and phrases:

Maximal operators, Littlewood-Paley functions,variation operator, nonsymmetric
Ornstein-Uhlenbeck.

###### 2020 Mathematics Subject Classification:

47B90, 42B25, 42B20

The authors are partially supported by grant PID2019-106093GB-I00 from the
Spanish Government

##  1\. Introduction

In this paper we are concerned with maximal operators, Littlewood-Paley
functions and variation operators defined by Poisson semigroups and resolvent
operators for nonsymmetric Ornstein-Uhlenbeck operators.

We denote by $Q$ a real, symmetric and positive definite $d\times d$ matrix
and by $B$ a nonzero real $d\times d$ matrix having eigenvalues with negative
real parts, being $d\in\mathbb{N}$, $d\geq 1$. We now introduce the Ornstein-
Uhlenbeck semigroup defined by $Q$, named the covariance matrix, and $B$
called the drift matrix. For every $t\in(0,\infty]$ we consider the matrix
$Q_{t}$ given by

$Q_{t}=\int_{0}^{t}e^{sB}Qe^{sB^{*}}ds,$

and the Gaussian measure $\gamma_{t}$ on $\mathbb R^{d}$ having mean zero and
covariance matrix $Q_{t}$ defined by

$d\gamma_{t}(x)=(2\pi)^{-\frac{d}{2}}(\det
Q_{t})^{-\frac{1}{2}}e^{-\frac{1}{2}\langle Q^{-1}_{t}x,x\rangle}dx.$

The Ornstein-Uhlenbeck semigroup is $\\{\mathcal{H}^{Q,B}_{t}\\}_{t>0}$,
where, for every $t>0$,

(1.1) $\mathcal{H}^{Q,B}_{t}(f)(x)=\int_{\mathbb
R^{d}}f\left(e^{tB}x-y\right)d\gamma_{t}(y),\;\;\;x\in\mathbb R^{d},$

where $f$ belongs to the space $\mathcal{C}_{b}(\mathbb R^{d})$ of bounded
continuous functions in $\mathbb R^{d}$.

The semigroup $\\{\mathcal{H}^{Q,B}_{t}\\}_{t>0}$ is the transition semigroup
of the Ornstein-Uhlenbeck process on $\mathbb R^{d}$ ([8]). The measure
$\gamma_{\infty}$ is the unique invariant measure for $\mathcal{H}_{t}^{Q,B}$,
$t>0$. Furthermore, the equality (1.1) defines a semigroup of positive
contractions in $L^{p}(\mathbb R^{d},\gamma_{\infty})$, for every $1\leq
p<\infty$.

The Ornstein-Uhlenbeck operator $\mathcal{L}^{Q,B}$ is defined by

$\mathcal{L}^{Q,B}f(x)=\frac{1}{2}\text{tr}(Q\nabla^{2}f)(x)+\langle Bx,\nabla
f(x)\rangle,\;\;\;x\in\mathbb R^{d},$

when $f\in\mathcal{C}^{2}(\mathbb R^{d})$, the set of differentiable functions
with continuity up to order two . Here, $\nabla$ denotes the gradient and
$\nabla^{2}$ represents the Hessian. Thus, $-\mathcal{L}^{Q,B}$ is an elliptic
operator. If $1\leq p<\infty$, by naming $-\mathcal{L}^{Q,B}_{p}$ the
infinitesimal generator of $\\{\mathcal{H}^{Q,B}_{t}\\}_{t>0}$ in
$L^{p}(\mathbb R^{d},\gamma_{\infty})$, we have that
$\mathcal{L}^{Q,B}_{p}f=\mathcal{L}^{Q,B}f$, $f\in S(\mathbb R^{d})$, the set
of Schwartz functions, and $S(\mathbb R^{d})$ is dense in the domain
$\mathcal{D}(\mathcal{L}_{p}^{Q,B})$ of $\mathcal{L}_{p}^{Q,B}$. In [31], it
was proved that $\mathcal{D}(\mathcal{L}_{p}^{Q,B})$ coincides with the
Sobolev space $W^{2,p}(\gamma_{\infty})$.

Harmonic analysis associated with the symmetric Ornstein-Uhlenbeck operator
has been much investigated over the last twenty five years. When $Q=-B=I$,
where $I$ denotes the identity matrix, the operator $\mathcal{L}$ reduces to
the classical Ornstein-Uhlenbeck operator
$\mathcal{L}^{I,-I}=\frac{1}{2}\Delta-x\nabla$ and the Hermite polynomials are
an orthonormal basis in $L^{2}(\mathbb R^{d},\gamma_{\infty})$ of
eigenfunctions of $\mathcal{L}^{I,-I}$. Muckenhoupt ([34]) studied maximal
operator and Riesz transforms in the one dimensional
$\mathcal{L}^{I,-I}$-setting . Sjögren ([40]) extended to higher dimensions
Muckenhoupt’s results about the maximal operator defined by
$\\{\mathcal{H}^{I,-I}_{t}\\}_{t>0}$. Harmonic analysis operators associated
with $\mathcal{L}^{I,-I}$ were studied in [16] and [30] (maximal operators);
in [13], [36] and [38] (Littlewood-Paley functions); in [12], [17], [37] and
[38] (Riesz transforms); in [15] and [18] (spectral multipliers) and in [21]
(variation and oscillation operators). Gutiérrez, Segovia and Torrea ([20])
and Gutiérrez ([19]) studied Riesz transforms defined by the operator
$\mathcal{L}^{I,B}$ when $B$ is symmetric.

Mauceri and Noselli proved $L^{p}$ boundedness properties for maximal
operators ([29]) and Riesz transforms ([28]) when $Q=I$ and $B=-\lambda(I+R)$,
with $\lambda>0$ and $R$ is a skew-adjoint matrix. The semigroups
$\\{\mathcal{H}^{I,-\lambda(I+R)}_{t}\\}_{t>0}$ are the basic building blocks
of normal Ornstein-Uhlenbeck semigroups because, after a change of variables,
any normal Ornstein-Uhlenbeck semigroup can be written as a product of
commuting semigroups of that form.

Recently, Casarino, Ciatti and Sjögren ([6], [7] and [5]) have extended the
results about maximal operators and Riesz transforms due to Mauceri and
Noselli ([28] and [29]).

Our objective in this paper is to establish $L^{p}$ boundedness properties of
some maximal operators, Littlewood-Paley functions and variation operators
involving the Poisson semigroups and the resolvent operators associated with
the nonsymmetric Ornstein-Uhlenbeck operator considered by Mauceri and Noselli
([28] and [29]).

Assume that $Q=I$ and $B=-\lambda(I+R)$ where $\lambda>0$ and $R$ is a skew-
adjoint matrix as in [28] and [29]. After making a change of variables in
(1.1) we get

$\mathcal{H}^{Q,B}_{t}(f)(x)=\int_{\mathbb{R}^{d}}\widetilde{h}^{Q,B}_{t}(x,y)f(y)d\gamma_{\infty,\lambda}(y),\quad
x\in\mathbb{R}^{d}\text{ and }t>0,$

where
$d\gamma_{\infty,\lambda}(y)=\left(\frac{\pi}{\lambda}\right)^{-d/2}e^{-\lambda|y|^{2}}dy$,
and

$\widetilde{h}^{Q,B}_{t}(x,y)=\det(Q_{\infty}Q^{-1}_{t})^{\frac{1}{2}}e^{-\frac{1}{2}\left[\langle
Q^{-1}_{t}(e^{tB}x-y),e^{tB}x-y\rangle-\langle
Q^{-1}_{\infty}y,y\rangle\right]},$

for $x$, $y\in\mathbb{R}^{d}$ and $t>0$.

By using the subordination formula, the Poisson semigroup
$\\{P^{Q,B}_{t}\\}_{t>0}$ is given by

$P^{Q,B}_{t}(f)=\frac{t}{2\sqrt{\pi}}\int_{0}^{\infty}\frac{e^{-\frac{t^{2}}{4u}}}{u^{\frac{3}{2}}}\mathcal{H}^{Q,B}_{u}(f)du,\quad
t>0.$

Let $k\in\mathbb{N}$ and $j=1,\dots,d.$ We consider the maximal operator
$P^{Q,B}_{*,k,j}$ defined by

$P^{Q,B}_{*,k,j}(f)=\sup_{t>0}|t^{k+1}\partial^{k}_{t}\partial_{x_{j}}P^{Q,B}_{t}(f)|.$

The Littlewood-Paley $g^{Q,B}_{k,j}$ is given by

$g^{Q,B}_{k,j}(f)=\left(\int_{0}^{\infty}\left|t^{k+1}\partial^{k}_{t}\partial_{x_{j}}P^{Q,B}_{t}(f)\right|^{2}\frac{dt}{t}\right)^{\frac{1}{2}}.$

Let $\rho>2$. If $g$ is a complex valued function defined in $(0,\infty)$, the
$\rho$-variation $V_{\rho}(g)$ of $g$ is defined by

(1.2) $V_{\rho}(g)=\sup_{\begin{subarray}{c}0<t_{\ell}<t_{\ell-1}<t_{1}\\\
\ell\in\mathbb{N}\end{subarray}}\left(\sum_{n=1}^{\ell-1}|g(t_{n+1})-g(t_{n})|^{\rho}\right)^{\frac{1}{\rho}}.$

Variation inequalities have been studied in probability, ergodic theory and
harmonic analysis in recent years, The first variation inequality was due to
Léplinge ([24]) in the martingales setting. Later, Bourgain ([2]) studied
variation operators associated with ergodic averages of dynamic systems. The
last paper has motivated a lot of researches in ergodic theory and harmonic
analysis. We recommend to the interested reader the following recent papers
and the reference therein: [3], [26], [27], [32], [33] and [42].

We consider the variation operator $V^{Q,B}_{\rho,k,j}$ given by

$V^{Q,B}_{\rho,k,j}(f)(x)=V_{\rho}\left(t^{k+1}\partial^{k}_{t}\partial_{x_{j}}P^{Q,B}_{t}(f)(x)\right),\quad
x\in\mathbb{R}^{d}.$

Note that, for every $f\in L^{p}(\mathbb R^{d},\gamma_{\infty,1})$, $1\leq
p<\infty$, the function $V^{Q,B}_{\rho,k,j}(f)$ is a Lebesgue measurable
function because, for almost everywhere $x\in\mathbb R^{d}$, the function
$F_{x}(t)=t^{k+1}\partial_{t}^{k}\partial_{x_{j}}P_{t}^{Q,B}(f)(x)$,
$t\in(0,\infty)$, is continuous (see the comments after [4, Theorem 1.2]).
This measurability property also holds for the other variation operators
considered in this paper.

$g$-Littlewood-Paley functions associated with the symmetric Ornstein-
Uhlenbeck operator $\mathcal{L}^{I,-I}$ were studied in [13], [36] and [38].
The $\rho$-variation operator for the Poisson semigroup $\\{P^{I,-I}_{t}\\}$
defined by the symmetric Ornstein-Uhlenbeck operator without any derivatives
was considered in [21].

Let $k\in\mathbb{N}$, $j=1,\dots,d$, and $M\geq 1$. We consider the following
maximal operators, Littlewood-Paley functions and variation operators
involving resolvent operators of $\mathcal{L}^{Q,B}$ defined by

$S^{Q,B}_{*,k,j,M}(f)=\sup_{t>0}|t^{k+\frac{1}{2}}\partial^{k}_{t}\partial_{x_{j}}(I+t\mathcal{L}^{Q,B})^{-M}(f)|,$

$G^{Q,B}_{k,j,M}(f)=\left(\int_{0}^{\infty}\left|t^{k+\frac{1}{2}}\partial^{k}_{t}\partial_{x_{j}}(I+t\mathcal{L}^{Q,B})^{-M}(f)\right|^{2}\frac{dt}{t}\right)^{\frac{1}{2}},$

and

$\mathcal{V}^{Q,B}_{\rho,k,j,M}(f)(x)=V_{\rho}\left(t^{k+\frac{1}{2}}\partial^{k}_{t}\partial_{x_{j}}(I+t\mathcal{L}^{Q,B})^{-M}(f)(x)\right),\quad
x\in\mathbb{R}^{d}.$

Square functions of $G$-type have been recently defined in other settings
([14]).

Our main result is the following one.

######  Theorem 1.1.

Let $k\in\mathbb{N}$, $M>(d+1)/2$, $j=1,\dots,d$ and $\rho>2$. Assume that
$Q=I$ and $B=-\lambda(I+R)$, where $\lambda>0$ and $R$ is a skew-adjoint
matrix that generates a periodic one parameter group
$\\{e^{tR}\\}_{t\in\mathbb{R}}$. Then, the operators $P^{Q,B}_{*,k,j}$,
$g^{Q,B}_{k,j}$, $V^{Q,B}_{\rho,k,j}$, $S^{Q,B}_{*,k,j,M}$, $G^{Q,B}_{k,j,M}$
and $\mathcal{V}^{Q,B}_{\rho,k,j,M}$ are bounded from
$L^{p}(\mathbb{R}^{d},\gamma_{\infty,\lambda})$ into
$L^{p}(\mathbb{R}^{d},\gamma_{\infty,\lambda})$ and from
$L^{1}(\mathbb{R}^{d},\gamma_{\infty,\lambda})$ into
$L^{1,\infty}(\mathbb{R}^{d},\gamma_{\infty,\lambda})$.

As it is usual since [34] was published, in the study of $L^{p}$-boundedness
properties of harmonic analysis operators in the Ornstein-Uhlenbeck setting,
in order to prove Theorem 1.1 we decompose the operator in two ones that are
named the local part and the global part of the operator under consideration.
The local part, as the original operator, is a singular integral, while the
global part is controlled by a positive operator.

In Section 2 we give the definitions and known results that will be useful in
the sequel. We also explain the method used in the proof of Theorem 1.1.
$L^{p}$-boundedness properties stated in Theorem 1.1 are proved in Sections 3
and 4.

Throughout this paper $C$ and $c$ always represent positive constants that can
change in each occurrence.

##  2\. Preliminaries

Assume that $Q=I$ and $B=-\lambda(I+R)$, where $\lambda>0$ and $R$ is skew-
adjoint. In this case we have that

$\displaystyle
d\gamma_{\infty}(x):=d\gamma_{\infty,\lambda}(x)=\left(\frac{\pi}{\lambda}\right)^{-d/2}e^{-\lambda|x|^{2}}dx.$

Actually we are going to work with $\lambda=1$. We define $B_{1}=-(I+R)$. We
have that

$\displaystyle\mathcal{H}_{t}^{Q,B}=\mathcal{U}_{\lambda}^{-1}\mathcal{H}_{\lambda
t}^{Q,B_{1}}\mathcal{U}_{\lambda},\;\;\;\;t>0,$

where $\mathcal{U}_{\lambda}(f)(x)=f(x/\sqrt{\lambda})$, $x\in\mathbb R^{d}$.
It is clear that $\mathcal{U}_{\lambda}$ is an isometry from $L^{p}(\mathbb
R^{d},\gamma_{\infty,\lambda})$ (respectively, $L^{p,\infty}(\mathbb
R^{d},\gamma_{\infty,\lambda})$) into $L^{p}(\mathbb R^{d},\gamma_{\infty,1})$
(respectively, $L^{p,\infty}(\mathbb R^{d},\gamma_{\infty,1})$), for every
$1\leq p<\infty$.

After a change of variable we can write

$\displaystyle P_{t}^{Q,B}(f)(x)$
$\displaystyle=\frac{t}{2\sqrt{\pi}}\int_{0}^{\infty}\frac{e^{-\frac{t^{2}}{4u}}}{u^{\frac{3}{2}}}U_{\lambda}^{-1}\mathcal{H}_{\lambda
u}^{Q,B_{1}}(U_{\lambda}f)(x)du$
$\displaystyle=\mathcal{U}_{\lambda}^{-1}\left[\frac{\sqrt{\lambda}t}{2\sqrt{\pi}}\int_{0}^{\infty}\frac{e^{-\frac{\lambda
t^{2}}{4v}}}{v^{\frac{3}{2}}}\mathcal{H}_{v}^{Q,B_{1}}(U_{\lambda}(f))(\cdot)dv\right](x)$
(2.1)
$\displaystyle=\mathcal{U}_{\lambda}^{-1}\left[P_{\sqrt{\lambda}t}^{Q,B_{1}}(U_{\lambda}(f))\right](x),\;\;\;\;x\in\mathbb
R^{d}\;\mbox{and}\;t>0.$

By using (2) we deduce that, by denoting $T^{Q,B}$ every of the operators
considered in Theorem 1.1, the following equality holds

$\displaystyle
T^{Q,B}=\mathcal{U}_{\lambda}^{-1}T^{Q,B_{1}}\mathcal{U}_{\lambda}.$

Thus, we show that it is sufficient to prove Theorem 1.1 when $\lambda=1$. In
the sequel we assume $\lambda=1$.

We can write

$\displaystyle\mathcal{H}_{t}^{I,B_{1}}(f)(x)=\int_{\mathbb
R^{d}}h_{t}^{I,B_{1}}(x,y)f(y)dy,\quad x\in\mathbb R^{d}\mbox{ and }t>0,$

where

$\displaystyle
h_{t}^{I,B_{1}}(x,y)=\frac{1}{(2\pi(1-e^{-2t}))^{d/2}}e^{-\frac{|e^{tB_{1}}x-y|^{2}}{1-e^{-2t}}},\quad
x,y\in\mathbb R^{d}\mbox{ and }t>0.$

As it was mentioned in the introduction in order to study $L^{p}$-boundedness
properties of the harmonic analysis operators in the Ornstein-Uhlenbeck
setting, those operators are decomposed in a local part and a global part.

Let $\delta>0$. We define the sets

$\displaystyle
L_{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\sigma}=\left\\{(x,y)\in\mathbb
R^{d}\times\mathbb
R^{d}:\;|x-y|\leq{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\sigma}\min\\{1,|x+y|^{-1}\\}\right\\},$

and
$G_{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\sigma}=(\mathbb
R^{d}\times\mathbb R^{d})\setminus
L_{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\sigma}$.
$L_{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\sigma}$
and
$G_{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\sigma}$
are named the
${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\sigma}$-local
and
${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\sigma}$-global
region respectively.

The integral kernel $h_{t}^{I,B_{1}}$ of $\mathcal{H}_{t}^{I,B_{1}}$, $t>0$,
can be estimated in different ways on
$L_{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\sigma}$
and
$G_{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\sigma}$.

######  Lemma 2.1.

([28, Lemma 3.3]). For every $\sigma>0$, there exists $C>0$ such that

$0\leq
h_{t}^{I,B_{1}}(x,y)\leq\frac{C}{(1-e^{-2t})^{d/2}}e^{-\frac{|x-y|^{2}}{2(1-e^{-2t})}},\quad(x,y)\in
L_{\sigma},\;x\neq y\mbox{ and }t>0.$

In order to obtain a more manageable form of the kernel $h_{t}^{I,B_{1}}$ in
the symmetric case, that is, when $R=0$, the following change of variable due
to S. Meda was introduced in [15]

$\tau(s)=\log\frac{1+s}{1-s},\quad s\in(0,1).$

We observe that $\tau$ maps $(0,1)$ onto $(0,\infty)$.

For every $s\in(0,1)$ we consider the quadratic form $Q_{s}$ defined by

$Q_{s}(x,y)=|(1+s)x-(1-s)y|^{2},\quad x,y\in\mathbb R^{d}.$

As in [29, p. 190] if $J$ is an interval in $(0,\infty)$ and $D>0$ we define
$J_{D}^{\\#}=\cup_{n\in\mathbb N}(J+nD)$.

After a careful reading of the proof of [29, Lemmas 5.5 and 5.6] we can see
that with minor modifications in those ones the following properties can be
proved.

######  Lemma 2.2.

Let $\delta\in(0,1)$.

  1. (i)

There exists $C$ and $t_{0}>0$ such that

(2.2) $\displaystyle h_{\tau(s)}^{I,B_{1}}(x,y)\leq
Cs^{-\frac{d}{2}}e^{|x|^{2}-|y|^{2}-\frac{\delta}{4s}Q_{s}(x,y)},\quad
x,y\in\mathbb R^{d}\mbox{ and }s\in\tau^{-1}((0,t_{0})).$

  2. (ii)

Suppose that the one-parameter group of rotations $\\{e^{tR}\\}_{t\in\mathbb
R}$ generated by the matrix $R$ is periodic of period $D$. Then, there exists
an interval $J=(a,b)$, with $0<a<b<\infty$, and $C>0$ such that

(2.3) $\displaystyle h_{\tau(s)}^{I,B_{1}}(x,y)\leq
Cs^{-\frac{d}{2}}e^{|x|^{2}-|y|^{2}-\frac{\delta}{4s}Q_{s}(x,y)},\quad
x,y\in\mathbb R^{d}\mbox{ and }s\in\tau^{-1}(J_{D}^{\\#}).$

Here $C$, $t_{0}$, $a$ and $b$ depend on $\delta$.

Note that (2.2) and (2.3) also hold when $t_{0}$ is replaced by
$t_{1}\in(0,t_{0})$ and $J$ is replaced by an interval $J_{1}\subset J$,
respectively. In the sequel we consider $t_{0}>0$ and an interval $J=(a,b)$,
with $0<a<b<\infty$, satisfying (2.2) and (2.3), respectively, and such that
there exist $n,m\in\mathbb N$ and $\beta>0$ for which

(2.4)
$(0,\infty)\setminus\mathcal{N}=\Big{[}\bigcup_{k=0}^{n}((0,t_{0})+kt_{0})\Big{]}\cup\Big{[}\bigcup_{\ell=0}^{m}(J_{D}^{\\#}+\ell\beta)\Big{]},$

for certain $\mathcal{N}\subset(0,\infty)$ of measure zero, and being a
disjoint union. Let $\sigma>0$. We choose an smooth function $\varphi$ in
$\mathbb R^{d}\times\mathbb R^{d}$ satisfying that

  1. (i)

$0\leq\varphi\leq 1$, $x,y\in\mathbb R^{d}$;

  2. (ii)

$\varphi(x,y)=1$, $(x,y)\in L_{\sigma}$, and $\varphi(x,y)=0$, $(x,y)\notin
L_{2\sigma}$;

  3. (iii)

$|\nabla_{x}\varphi(x,y)|+|\nabla_{y}\varphi(x,y)|\leq\frac{C}{|x-y|}$,
$x,y\in\mathbb R^{d}$, $x\neq y$.

By $L^{0}(\mathbb R^{d})$ we denote the space of Lebesgue measurable functions
in $\mathbb R^{d}$ and we represent by $C_{c}^{\infty}(\mathbb R^{d})$ the
space of smooth functions in $\mathbb R^{d}$ having compact support. Suppose
that $T$ is a linear or sublinear operator from $C_{c}^{\infty}(\mathbb
R^{d})$ into $L^{0}(\mathbb R^{d})$. We define the local part $T_{\rm loc}$ of
$T$ by

$T_{\rm loc}(f)(x)=T(f(\cdot)\varphi(x,\cdot))(x),\quad x\in\mathbb R^{d},$

and the global part $T_{\rm glob}$ of $T$ by

$T_{\rm glob}(f)(x)=T(f)(x)-T_{\rm loc}(f)(x),\quad x\in\mathbb R^{d}.$

The following results were proved in [16] and they will be useful to prove
that the global parts of the operators in Theorem 1.1 are bounded from
$L^{1}(\mathbb R^{d},\gamma_{\infty,1})$ into $L^{1,\infty}(\mathbb
R^{d},\gamma_{\infty,1})$. If $x,y\in\mathbb R^{d}\setminus\\{0\\}$,
$\theta(x,y)$ denotes the angle between $x$ and $y$.

######  Lemma 2.3.

([16, Lemma 4.1]) For every $\delta>0$ there exists $C>0$ such that

$\sup_{0<s<1}s^{-d/2}e^{(-\frac{\delta}{s})Q_{s}(x,y)}\leq
C\min\\{(1+|x|)^{d},(|x|\sin\theta(x,y))^{-d}\\},\;\;\;\;(x,y)\in
G_{1},\;x\neq 0\neq y.$

######  Lemma 2.4.

([16, Lemma 4.4]). The operator $T$ defined by

$T(f)(x)=e^{|x|^{2}}\int_{\mathbb
R^{d}}\min\\{(1+|x|)^{d},(|x|\sin\theta(x,y))^{-d}\\}f(y)e^{-|y|^{2}}dy,\quad
x\in\mathbb R^{d},$

is bounded from $L^{1}(\mathbb R^{d},\gamma_{\infty,1})$ into
$L^{1,\infty}(\mathbb R^{d},\gamma_{\infty,1})$.

In the study of the local parts of the operators in Theorem 1.1 we will use
the $L^{p}$-boundedness properties of the operator $S_{\sigma}$ defined by

$S_{\sigma}(f)(x)=\int_{\\{y\in\mathbb R^{d}:(x,y)\in
L_{\sigma}\\}}\frac{1+|x|}{|x-y|^{d-1}}f(y)dy,\quad x\in\mathbb R^{d},$

where $\delta>0$. Operators of this type appear also when the symmetric case
is considered (see, for instance, [21]).

######  Lemma 2.5.

Let $\sigma>0$. The operator $S_{\sigma}$ is bounded from $L^{p}(\mathbb
R^{d},dx)$ into itself and also from $L^{p}(\mathbb R^{d},\gamma_{\infty,1})$
into itself, for every $1\leq p\leq\infty$.

###### Proof.

We include a sketch of the proof of this property for the sake of
completeness.

We have that $|x+y|=|2x+y-x|\geq 2|x|-|x-y|\geq 2|x|-\sigma\geq|x|$, provided
that $|x-y|\leq\sigma\leq|x|$. It follows that

$\int_{\\{y\in\mathbb R^{d}:(x,y)\in
L_{\sigma}\\}}\frac{1+|x|}{|x-y|^{d-1}}dy\leq
C(1+|x|)\times\left\\{\begin{array}[]{ll}\displaystyle\int_{0}^{\sigma/|x|}dr,&|x|\geq\delta\\\
&\\\ \displaystyle\int_{0}^{\sigma}dr,&|x|\leq\delta\end{array}\right.\leq C.$

Hence, $\displaystyle\sup_{x\in\mathbb R^{d}}\int_{\\{y\in\mathbb
R^{d}:(x,y)\in L_{\sigma}\\}}\frac{1+|x|}{|x-y|^{d-1}}dy<\infty$ and in a
similar way $\displaystyle\sup_{y\in\mathbb R^{d}}\int_{\\{x\in\mathbb
R^{d}:(x,y)\in L_{\sigma}\\}}\frac{1+|x|}{|x-y|^{d-1}}dx<\infty.$

By using interpolation we deduce that the operator $S_{\sigma}$ is bounded
from $L^{p}(\mathbb R^{d},dx)$ into itself, for every $1\leq p\leq\infty$.
According to [18, Lemma 3.6] $S_{\sigma}$ is bounded from $L^{p}(\mathbb
R^{d},\gamma_{\infty,1})$ into itself, for every $1\leq p\leq\infty$. ∎

We are going to explain the method we use to prove Theorem 1.1. We extend the
procedure developed by Mauceri and Noselli ([28] and [29]).

Suppose that $X$ is a Banach space of complex functions defined in
$(0,\infty)$. Let $k\in\mathbb N$ and $j=1,...,d$. We consider the operator
$T_{k,j}^{X}$ defined by

$T_{k,j}^{X}(f)(x)=\big{\|}t\rightarrow
t^{k+1}\partial_{t}^{k}\partial_{x_{j}}P_{t}^{I,B_{1}}(f)(x)\big{\|}_{X},\quad
x\in\mathbb R^{d}.$

It is clear that $T_{k,j}^{X}$ reduces to $P_{*,k,j}^{I,B_{1}}$ and
$g_{k,j}^{I,B_{1}}$ when $X=L^{\infty}((0,\infty),dt)$ and
$X=L^{2}((0,\infty),\frac{dt}{t})$, respectively. Furthermore, let $\rho>2$.
We consider on the space $C(0,\infty)$ of continuous functions on $(0,\infty)$
and the seminorm $V_{\rho}$ defined in (1.2). By identifying those functions
in $C(0,\infty)$ that differ in a constant the space $V_{\rho}(0,\infty)$
consisting of all those $g\in C(0,\infty)$ such that $V_{\rho}(g)<\infty$
endowed with $V_{\rho}$ is a Banach space. We have that
$T_{k,j}^{V_{\rho}(0,\infty)}=V_{\rho,k,j}^{I,B_{1}}$.

Let $f\in C_{c}^{\infty}(\mathbb{R}^{d})$. We can write

(2.5)
$\displaystyle\partial_{t}^{k}\partial_{x_{j}}P_{t}^{I,B_{1}}(f)(x)=\int_{\mathbb
R^{d}}f(y)\partial_{t}^{k}\partial_{x_{j}}P_{t}^{I,B_{1}}(x,y)dy,\quad
x\in\mathbb R^{d}\mbox{ and }t>0,$

where $P_{t}^{I,B_{1}}(x,y)$, $x,y\in\mathbb R^{d}$ and $t>0$, denotes the
Poisson integral kernel and

$\displaystyle\partial_{t}^{k}\partial_{x_{j}}P_{t}^{I,B_{1}}(x,y)=\frac{1}{2\sqrt{\pi}}\int_{0}^{\infty}\partial_{t}^{k}[te^{-t^{2}/4u}]\partial_{x_{j}}h_{u}^{I,B_{1}}(x,y)\frac{du}{u^{\frac{3}{2}}},\quad
x,y\in\mathbb R^{d}\mbox{ and }t>0.$

Differentiations under the integral sign are justified. Indeed, we have

$\displaystyle|\partial_{x_{j}}h_{u}^{I,B_{1}}(x,y)|\leq
C\frac{|e^{uB_{1}^{*}}||e^{uB_{1}}x-y|}{(1-e^{-2u})^{d/2+1}}e^{-|e^{uB_{:}1}x-y|^{2}/(1-e^{-2u})},\quad
x,y\in\mathbb R^{d}\mbox{ and }u>0.$

Since $|e^{uB^{*}}|\leq e^{-u}$, $u>0$, we get

$|\partial_{x_{j}}h_{u}^{I,B_{1}}(x,y)|\leq
C\frac{e^{-u}}{(1-e^{-2u})^{(d+1)/2}},\quad x,y\in\mathbb{R}^{d}\mbox{ and
}t>0.$

By using [1, Lemma 4] we obtain, for each $x,y\in\mathbb{R}^{d}$ and $t>0$,

$\displaystyle\int_{0}^{\infty}|\partial_{t}^{k}[te^{-\frac{t^{2}}{4u}}]||\partial_{x_{j}}h_{u}^{I,B_{1}}(x,y)|\frac{du}{u^{\frac{3}{2}}}$
$\displaystyle\leq
C\int_{0}^{\infty}\frac{e^{-\frac{t^{2}}{8u}-u}}{u^{(k+2)/2}(1-e^{-2u})^{(d+1)/2}}du$
$\displaystyle\leq
C\left(\int_{1}^{\infty}e^{-u}du+\int_{0}^{1}\frac{e^{-\frac{t^{2}}{8u}}}{u^{(k+d+3)/2}}du\right)\leq
C(1+t^{-k-d-1}).$

Since $f\in C_{c}^{\infty}(\mathbb{R}^{d})$ (2.5) holds.

Suppose that $E\subset(0,\infty)$ and $h\geq 0$. We define the operator

$S_{k,j}^{E,h}(f)(x,t)=\int_{\mathbb{R}^{d}}\mathfrak{s}_{k,j}^{E,h}(x,y,t)f(y)dy,\quad
x\in\mathbb{R}^{d}\mbox{ and }t>0.$

where

$\mathfrak{s}_{k,j}^{E,h}(x,y,t)=\frac{t^{k+1}}{2\sqrt{\pi}}\int_{E}\partial_{t}^{k}[te^{-\frac{t^{2}}{u+h}}]\partial_{x_{j}}h_{u}^{I,B_{1}}(x,y)\frac{du}{(u+h)^{\frac{3}{2}}},\quad
x,y\in\mathbb{R}^{d}\mbox{ and }t>0.$

Assume that $X$ is one of the following Banach spaces:
$L^{\infty}((0,\infty),dt)$, $L^{2}((0,\infty),dt/t)$ and
$V_{\rho}(0,\infty)$. We consider the operator

$\widetilde{S}_{k,j}^{E,h}(f)(x)=\left\|S_{k,j}^{E,h}(f)(x,\cdot)\right\|_{X},\quad
x\in\mathbb{R}^{d}.$

Claim 1 Let $1\leq p<\infty$. We define $\delta_{p}$ as follows

$\delta_{p}=\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{10},&p=1,\\\\[8.5359pt]
\displaystyle\frac{1}{2}\Big{(}1+\max\big{\\{}1-\frac{1}{d},1-\frac{1}{p}\big{\\}}\Big{)},&1<p<\infty.\end{array}\right.$

We denote $E_{p}$ the sets $(0,t_{0})$ or $J_{D}^{\\#}$ in Lemma 2.2
associated to $t_{0}$ and satisfying the covering property (2.4).

Then, for every $h\geq 0$, the operator $\widetilde{S}_{k,j}^{E_{p},h}$ is
bounded from $L^{p}(\mathbb{R}^{d},\gamma_{\infty,1})$ into itself, when
$1<p<\infty$, and from $L^{1}(\mathbb{R}^{d},\gamma_{\infty,1})$ into
$L^{1,\infty}(\mathbb{R}^{d},\gamma_{\infty,1})$, when $p=1$.

Suppose that the claim has been proved. Since

$(0,\infty)\setminus{\mathcal{N}}=\Big{[}\bigcup_{\ell=0}^{n}((0,t_{0})+\ell
t_{0})\Big{]}\cup\Big{[}\bigcup_{\ell=0}^{m}(J_{P}^{\\#}+\ell\beta)\Big{]}$

we can write

$\displaystyle\int_{0}^{\infty}\partial_{t}^{k}[te^{-\frac{t^{2}}{4u}}]\partial_{x_{j}}h_{u}^{I,B_{1}}(x,y)\frac{du}{u^{\frac{3}{2}}}$
$\displaystyle=\left(\sum_{\ell=0}^{n}\int_{(0,t_{0})+\ell
t_{0}}+\sum_{\ell=0}^{m}\int_{J_{D}^{\\#}+\ell\beta}\right)\partial_{t}^{k}[te^{-\frac{t^{2}}{4u}}]\partial_{x_{j}}h_{u}^{I,B_{1}}(x,y)\frac{du}{u^{\frac{3}{2}}}$
$\displaystyle=\sum_{\ell=0}^{n}\int_{(0,t_{0})}\partial_{t}^{k}[te^{-\frac{t^{2}}{4(u+\ell
t_{0})}}]\partial_{x_{j}}h_{u+\ell t_{0}}^{I,B_{1}}(x,y)\frac{du}{(u+\ell
t_{0})^{\frac{3}{2}}}$
$\displaystyle\quad+\sum_{\ell=0}^{m}\int_{J_{D}^{\\#}}\partial_{t}^{k}[te^{-\frac{t^{2}}{2(u+\ell\beta)}}]\partial_{x_{j}}h_{u+\ell\beta}^{I,B_{1}}(x,y)\frac{du}{(u+\ell\beta)^{\frac{3}{2}}},\quad
x,y\in\mathbb{R}^{d}\mbox{ and }t>0.$

By using the semigroup property of $\\{H_{t}^{I,B_{1}}\\}_{t>0}$ we deduce
that

$t^{k+1}\partial_{t}^{k}\partial_{x_{j}}P_{t}^{I,B_{1}}(f)(x)=\sum_{\ell=0}^{n}S_{k,j}^{(0,t_{0}),\ell
t_{0}}(H_{\ell
t_{0}}^{I,B_{1}}(f))(x,t)+\sum_{\ell=0}^{m}S_{k,j}^{J_{D}^{\\#},\ell\beta}(H_{\ell\beta}^{I,B_{1}}(f))(x,t),\quad
x\in\mathbb{R}^{d}\mbox{ and }t>0.$

Since the semigroup $\\{H_{t}^{I,B_{1}}\\}_{t>0}$ is contractive in
$L^{p}(\mathbb{R}^{d},\gamma_{\infty,1})$, for every $1\leq p<\infty$, the
Claim 1 allows us to conclude that the operator $T_{k,j}^{X}$ is bounded from
$L^{p}(\mathbb{R}^{d},\gamma_{\infty,1})$ into itself, for every $1<p<\infty$,
and from $L^{1}(\mathbb{R}^{d},\gamma_{\infty,1})$ into
$L^{1,\infty}(\mathbb{R}^{d},\gamma_{\infty,1})$.

Our objective is to prove the Claim 1. In order to see the $L^{p}$-boundedness
properties of the operator $\widetilde{S}_{k,j}^{E,h}$ we study separately the
local part and the global part of $\widetilde{S}_{k,j}^{E,h}$. We analyze
firstly the local part $\widetilde{S}_{k,j,{\rm loc}}^{E,h}$ of
$\widetilde{S}_{k,j}^{E,h}$. We consider the operator

$U_{k,j}^{E,h}(f)(x,t)=\int_{\mathbb{R}^{d}}\mathfrak{U}_{k,j}^{E,h}(x-y,t)f(y)dy,\quad
x\in\mathbb{R}^{d}\mbox{ and }t>0,$

where

$\mathfrak{U}_{k,j}^{E,h}(z,t)=\frac{t^{k+1}}{2\sqrt{\pi}}\int_{E}\partial_{t}^{k}\big{[}te^{-\frac{t^{2}}{4(u+h)}}\big{]}\partial_{x_{j}}W_{u}(z)\frac{du}{(u+h)^{\frac{3}{2}}},\quad
z\in\mathbb{R}^{d}\mbox{ and }t>0.$

Here $W_{u}$, $u>0$ denotes the classical heat kernel given by

$W_{u}(z)=\frac{e^{-\frac{|z|^{2}}{2u}}}{(2\pi u)^{d/2}},\quad
z\in\mathbb{R}^{d}\mbox{ and }u>0.$

We also define

$\widetilde{U}_{k,j}^{E,h}(f)(x)=\left\|U_{k,j}^{E,h}(f)(x,\cdot)\right\|_{X},\quad
x\in\mathbb{R}^{d}.$

By using Minkowski inequality we deduce that

$\Big{|}\widetilde{S}_{k,j,{\rm loc}}^{E,h}(f)(x)-\widetilde{U}_{k,j,{\rm
loc}}^{E,h}(f)(x)\Big{|}\leq\int_{\mathbb{R}^{d}}\Big{\|}\mathfrak{s}_{k,j}^{E,h}(x,y,\cdot)-\mathfrak{U}_{k,j}^{E,h}(x-y,\cdot)\Big{\|}_{X}\varphi(x,y)|f(y)|dy,\quad
x\in\mathbb{R}^{d}.$

Claim 2. Let $E\subset(0,\infty)$ and $h\geq 0$. The operator $D_{k,j,{\rm
loc}}^{E,h}$ defined by

$D_{k,j,{\rm
loc}}^{E,h}(f)(x)=\int_{\mathbb{R}^{d}}\big{\|}\mathfrak{s}_{k,j}^{E,h}(x,y,\cdot)-\mathfrak{U}_{k,j}^{E,h}(x-y,\cdot)\Big{\|}_{X}\varphi(x,y)f(y)dy,\quad
x\in\mathbb{R}^{d},$

is bounded from $L^{p}(\mathbb{R}^{d},dx)$ into itself, and from
$L^{p}(\mathbb{R}^{d},\gamma_{\infty,1})$ into itself, for every $1\leq
p\leq\infty$.

Claim 3. Let $h\geq 0$ and $E\subset(0,\infty)$ such that $E\subset(0,\eta,)$
or $E\subset(\eta,\infty)$, for some $\eta>0$. The operator
$\widetilde{U}_{k,j,{\rm loc}}^{E,h}$ is bounded from
$L^{p}(\mathbb{R}^{d},dx)$ into itself, for every $1<p<\infty$, and from
$L^{1}(\mathbb{R}^{d},dx)$ into $L^{1,\infty}(\mathbb{R}^{d},dx)$.

By using [18, Lemma 3.6] from Claim 3 we deduce that the operator
$\widetilde{U}_{k,j,{\rm loc}}^{E,h}$ is bounded from
$L^{p}(\mathbb{R}^{d},\gamma_{\infty,1})$ into itself, for every $1<p<\infty$,
and from $L^{1}(\mathbb{R}^{d},\gamma_{\infty,1})$ into
$L^{1,\infty}(\mathbb{R}^{d},\gamma_{\infty,1})$. Then, according to Claim 2
it follows that the operator $\widetilde{S}_{k,j,{\rm loc}}^{E,h}(f)$ is
bounded from $L^{p}(\mathbb{R}^{d},\gamma_{\infty,1})$ into itself, for every
$1<p<\infty$, and from $L^{1}(\mathbb{R}^{d},\gamma_{\infty,1})$ into
$L^{1,\infty}(\mathbb{R}^{d},\gamma_{\infty,1})$.

Minkowski’s inequality leads to

$\widetilde{S}_{k,j,{\rm
glob}}^{E,h}(f)(x)\leq\int_{\mathbb{R}^{d}}\big{\|}\mathfrak{s}_{k,j}^{E,h}(x,y,\cdot)\big{\|}_{X}(1-\varphi(x,y))|f(y)|dy,\quad
x\in\mathbb{R}^{d}.$

Claim 4. Let $h\geq 0$. Assume that $1\leq p<\infty$ and that $E_{p}$ is the
set associated to $\delta_{p}$ as in Claim 1. Then, the operator
$\widetilde{V}_{k,j,{\rm glob}}^{E_{p},h}$ defined by

$\widetilde{V}_{k,j,{\rm
glob}}^{E_{p},h}(f)(x)=\int_{\mathbb{R}^{d}}\big{\|}\mathfrak{s}_{k,j}^{E_{p},h}(x,y,\cdot)\big{\|}_{X}(1-\varphi(x,y))|f(y)|dy,\quad
x\in\mathbb{R}^{d},$

is bounded from $L^{p}(\mathbb{R}^{d},\gamma_{\infty,1})$ into itself, when
$1<p<\infty$, and from $L^{1}(\mathbb{R}^{d},\gamma_{\infty,1})$ into
$L^{1,\infty}(\mathbb{R}^{d},\gamma_{\infty,1})$.

Thus, Claim 1 is proved when we establish Claims 2, 3 and 4.

Let $k\in\mathbb{N}$, $j=1,...,d$, and $M\geq 1$. We consider the operator
$\mathbb{T}_{k,j,M}^{X}$ defined by

$\mathbb{T}_{k,j,M}^{X}(f)(x)=\left\|t\rightarrow
t^{k+\frac{1}{2}}\partial_{t}^{k}\partial_{x_{j}}(I+t\mathcal{L}^{I,B_{1}})^{-M}(f)(x)\right\|_{X},\quad
x\in\mathbb{R}^{d}.$

$\mathbb{T}_{k,j,M}^{X}$ reduces to $S_{*,k,j,M}^{I,B_{1}}$,
$G_{k,j,M}^{I,B_{1}}$ and $\mathcal{V}_{\rho,k,j,M}^{I,B_{1}}$ when
$X=L^{\infty}((0,\infty),dt)$, $X=L^{2}((0,\infty),\frac{dt}{t})$ and
$X=V_{\rho}(0,\infty)$, respectively.

We have that

$(I+t\mathcal{L}^{I,B_{1}})^{-M}f=\frac{1}{\Gamma(M)}\int_{0}^{\infty}e^{-u}\mathcal{H}_{ut}^{I,B_{1}}(f)u^{M-1}du.$

It is clear that

$\sup_{t>0}|(I+t\mathcal{L}^{I,B_{1}})^{-M}f|\leq\sup_{t>0}|\mathcal{H}_{t}^{I,B_{1}}(f)|.$

According to [29, Corollary 4.3 and Theorem 5.1] we have that the maximal
operator $\sup_{t>0}|(I+t\mathcal{L}^{I,B_{1}})^{-M}|$ is bounded from
$L^{p}(\mathbb{R}^{d},\gamma_{\infty,1})$ into itself, for every $1<p<\infty$,
provided that the Ornstein-Uhlenbeck operator $\mathcal{L}^{I,B_{1}}$ is
normal, and from $L^{1}(\mathbb{R}^{d},\gamma_{\infty,1})$ into
$L^{1,\infty}(\mathbb{R}^{d},\gamma_{\infty,1})$ when the matrix $R$ generates
a periodic group $\\{e^{Rt}\\}_{t>0}$. In order to establish the
$L^{p}$-boundedness properties for the maximal operators
$S_{*,k,j,M}^{I,B_{1}}$ we need to work harder because, as far as we know, the
$L^{p}$-boundedness properties for the corresponding maximal operators
involving the heat semigroup $\\{\mathcal{H}_{t}^{I,B_{1}}\\}_{t>0}$ have not
been studied.

Let $f\in C_{c}^{\infty}(\mathbb{R}^{d})$. We can write

$\displaystyle(I+t\mathcal{L}^{I,B_{1}})^{-M}(f)(x)$
$\displaystyle=\frac{t^{-M}}{\Gamma(M)}\int_{0}^{\infty}e^{-\frac{u}{t}}\mathcal{H}_{u}^{I,B_{1}}(f)(x)u^{M-1}du$
$\displaystyle=\frac{t^{-M}}{\Gamma(M)}\int_{\mathbb{R}^{d}}f(y)\int_{0}^{\infty}e^{-\frac{u}{t}}h_{u}^{I,B_{1}}(x,y)u^{M-1}dudy,\quad
x\in\mathbb{R}^{d}\mbox{ and }t>0.$

Then, for every $x\in\mathbb{R}^{d}$ and $t>0$,

$t^{k+\frac{1}{2}}\partial_{t}^{k}\partial_{x_{j}}(I+t\mathcal{L}^{I,B_{1}})^{-M}(f)(x)=\frac{t^{k+\frac{1}{2}}}{\Gamma(M)}\int_{\mathbb{R}^{d}}f(y)\int_{0}^{\infty}\partial_{t}^{k}[t^{-M}e^{-\frac{u}{t}}]\partial_{x_{j}}h_{u}^{I,B_{1}}(x,y)u^{M-1}dudy.$

Differentiation under the integral can be justified as above by considering
that $M>(d+1)/2$.

We take $E\subset(0,\infty)$ and $h\geq 0$. As in the previous case, we define
the operator $\mathbb{S}_{k,j,M}^{E,h}$ by

$\mathbb{S}_{k,j,M}^{E,h}(f)(x,t)=\int_{\mathbb{R}^{d}}\mathfrak{s}_{k,j,M}^{E,h}(x,y,t)f(y)dy,\quad
x\in\mathbb{R}^{d}\mbox{ and }t>0,$

where

$\mathfrak{s}_{k,j,M}^{E,h}(x,y,t)=\frac{t^{k+\frac{1}{2}}}{\Gamma(M)}\int_{0}^{\infty}\partial_{t}^{k}[t^{-M}e^{-(u+h)/t}]\partial_{x_{j}}h_{u}^{I,B_{1}}(x,y)(u+h)^{M-1}du,\quad
x,y\in\mathbb{R}^{d}\mbox{ and }t>0.$

In order to prove the $L^{p}$-boundedness properties of the operator
$\mathbb{T}_{k,j,M}^{X}$ where $X=L^{\infty}((0,\infty),dt)$,
$X=L^{2}((0,\infty),\frac{dt}{t})$ and $X=V_{\rho}(0,\infty)$, we can proceed
by following the same steps than in the previous case by considering the
operator

$\widetilde{\mathbb{S}}_{k,j,M}^{E,h}(f)(x)=\big{\|}\mathbb{S}_{k,j,M}^{E,h}(f)(x,\cdot)\big{\|}_{X},\quad
x\in\mathbb{R}^{d}.$

######  Remark 2.6.

Let $k\in\mathbb{N}$, $\alpha=(\alpha_{1},...,\alpha_{d})\in\mathbb{N}^{d}$
such that $k+\widehat{\alpha}>0$ when
$\widehat{\alpha}=\alpha_{1}+...+\alpha_{d}$. Assume that
$X=L^{\infty}((0,\infty),dt)$, $X=L^{2}((0,\infty),\frac{dt}{t})$ or
$X=V_{\rho}(0,\infty)$. We define the operator $T_{k,\alpha}^{X}$ by

$T_{k,\alpha}^{X}(f)(x)=\left\|t\rightarrow
t^{k+\widehat{\alpha}}\partial_{t}^{k}\partial_{x}^{\alpha}P_{t}^{I,B_{1}}(f)(x)\right\|_{X},\quad
x\in\mathbb{R}^{d}.$

Here $\partial_{x}^{\alpha}=\frac{\partial^{\widehat{\alpha}}}{\partial
x_{1}^{\alpha_{1}}\cdots\partial x_{d}^{\alpha_{d}}}$.

It is natural to ask if $L^{p}$-boundedness properties of this operator when
$\widehat{\alpha}\not=1$ can be proved by using the procedure in this paper.
At this moment we can not apply our procedure because we do not know how to
deal with the global parts of the operators when $\widehat{\alpha}\not=1$.

We now comment about some special cases. We consider $\widehat{\alpha}=0$ and
$X=L^{\infty}((0,\infty),dt)$. By using the method in [25, §4] we can see that

$T_{k,0}^{L^{\infty}((0,\infty),dt)}(f)\leq
C\sup_{t>0}\frac{1}{t}\left|\int_{0}^{t}\mathcal{H}_{s}^{I,B_{1}}(f)ds\right|.$

Since $\\{\mathcal{H}_{t}^{I,B_{1}}\\}_{t>0}$ is contractive in
$L^{p}(\mathbb{R}^{d},\gamma_{\infty,1})$, $1\leq p\leq\infty$, the Hopf-
Dunford-Schwartz ergodic theorem ([9, Lemma VIII.7.6 and Theorem VIII.7.7])
leads to $T_{k,0}^{L^{\infty}((0,\infty),dt)}$ is bounded from
$L^{p}(\mathbb{R}^{d},\gamma_{\infty,1})$ into itself, for every $1<p<\infty$,
and from $L^{1}(\mathbb{R}^{d},\gamma_{\infty,1})$ into
$L^{1,\infty}(\mathbb{R}^{d},\gamma_{\infty,1})$.

We can write

$\partial_{t}^{k}P_{t}^{I,B_{1}}(x,y)=\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}\partial_{t}^{k-1}[te^{-\frac{t^{2}}{4u}}]\partial_{u}h_{u}^{I,B_{1}}(x,y)\frac{du}{\sqrt{u}},\quad
x,y\in\mathbb{R}^{d}\mbox{ and }t>0.$

We have that

$\partial_{u}h_{u}^{I,B_{1}}(x,y)=\frac{1}{2}\sum_{j=1}^{d}\partial_{x_{j}}^{2}h_{u}^{I,B_{1}}(x,y)+\langle
B_{1}x,\nabla_{x}h_{u}^{I,B_{1}}(x,y)\rangle,\quad x,y\in\mathbb{R}^{d}\mbox{
and }u>0.$

Then, the cases $\widehat{\alpha}=0$ and $\widehat{\alpha}=2$ are connected.

The arguments used in the symmetric Ornstein-Uhlenbeck setting in [21], [36],
[37] and [38] do not work for the global parts of the operator
$T_{k,\alpha}^{X}$ in the nonsymmetric context. Our objective in a next paper
is to establish $L^{p}$-boundedness properties of $T_{k,\alpha}^{X}$-type
operators for general nonsymmetric Ornstein-Uhlenbeck operators by using some
of the ideas developed by Casarino, Ciatti and Sjögren ([6], [7] and [5]).

##  3\. Proof of Claims 2 and 4

Our objective in this section is to prove Claims 2 and 4 stated in the
previous section for the operators in Theorem 1.1.

###  3.1. Proof of Claim 2

We consider firstly the operators $P_{*,k,j}^{I,B_{1}}$, $g_{k,j}^{I,B_{1}}$
and $V_{\rho,k,j}^{I,B_{1}}$. In the sequel $X$ represents one of the
following Banach spaces: $L^{\infty}((0,\infty),dt)$,
$L^{2}((0,\infty),\frac{dt}{t})$ and $V_{\rho}(0,\infty)$. We are going to
study the operator

$D_{k,j,{\rm
loc}}^{E,h}(f)(x)=\int_{\mathbb{R}^{d}}\Big{\|}\mathfrak{s}_{k,j}^{E,h}(x,y,\cdot)-\mathfrak{U}_{k,j}^{E,h}(x-y,\cdot)\Big{\|}_{X}\varphi(x,y)f(y)dy,\quad
x\in\mathbb{R}^{d},$

where $E\subset(0,\infty)$ and $h\geq 0$. The definitions can be found in
Section 2.

We can write, for each $x,y\in\mathbb{R}^{d}$ and $t>0$,

$\mathfrak{s}_{k,j}^{E,h}(x,y,t)-\mathfrak{U}_{k,j}^{E,h}(x-y,t)=\frac{t^{k+1}}{2\sqrt{\pi}}\int_{E}\frac{\partial_{t}^{k}[te^{-\frac{t^{2}}{4(u+h)}}]}{(u+h)^{\frac{3}{2}}}\big{(}\partial_{x_{j}}h_{u}^{I,B_{1}}(x,y)-\partial_{x_{j}}W_{u}(x-y)\big{)}du.$

By using Minkowski inequality we get

(3.1)
$\big{\|}\mathfrak{s}_{k,j}^{E,h}(x,y,\cdot)-\mathfrak{U}_{k,j}^{E,h}(x-y,\cdot)\big{\|}_{X}\leq
C\int_{E}\frac{\|t^{k+1}\partial_{t}^{k}[te^{-\frac{t^{2}}{4(u+h)}}]\|_{X}}{(u+h)^{\frac{3}{2}}}\big{|}\partial_{x_{j}}h_{u}^{I,B_{1}}(x,y)-\partial_{x_{j}}W_{u}(x-y)\big{|}du,\quad
x,y\in\mathbb{R}^{d}.$

According to [1, Lemma 4] it follows that

(3.2) $\big{|}\partial_{t}^{k}[te^{-\frac{t^{2}}{4(u+h)}}]\big{|}\leq
C\frac{e^{-\frac{t^{2}}{8(u+h)}}}{(u+h)^{(k-1)/2}},\quad t,u\in(0,\infty).$

We have that

(3.3) $\big{|}t^{k+1}\partial_{t}^{k}[te^{-\frac{t^{2}}{4(u+h)}}]\big{|}\leq
C\frac{t^{k+1}e^{-\frac{t^{2}}{8(u+h)}}}{(u+h)^{(k-1)/2}}\leq C(u+h),\quad
t,u\in(0,\infty).$

By using again (3.2) we obtain

(3.4)
$\left\|t^{k+1}\partial_{t}^{k}[te^{-\frac{t^{2}}{4(u+h)}}]\right\|_{L^{2}((0,\infty),\frac{dt}{t})}\leq\frac{C}{(u+h)^{(k-1)/2}}\left(\int_{0}^{\infty}|t^{k+1}e^{-\frac{t^{2}}{8(u+h)}}|^{2}\frac{dt}{t}\right)^{\frac{1}{2}}\leq
C(u+h),\quad u>0.$

Suppose that $g:(0,\infty)\longrightarrow\mathbb{C}$ is a derivable function.
If $t_{1}>t_{2}>\cdots>t_{k}>0$ we have that

$\left(\sum_{j=1}^{k-1}|g(t_{j+1})-g(t_{j})|^{\rho}\right)^{1/\rho}=\left(\sum_{j=1}^{k-1}\Big{|}\int_{t_{j}}^{t_{j+1}}g^{\prime}(t)dt\Big{|}^{\rho}\right)^{1/\rho}\leq\sum_{j=1}^{k-1}\int_{t_{j}}^{t_{j+1}}|g^{\prime}(t)|dt\leq\int_{0}^{\infty}|g^{\prime}(t)|dt.$

Then,

$V_{\rho}(g)\leq\int_{0}^{\infty}|g^{\prime}(t)|dt.$

From (3.2) we deduce that

$\displaystyle\left\|t^{k+1}\partial_{t}^{k}[te^{-\frac{t^{2}}{4(u+h)}}]\right\|_{V_{\rho}}$
$\displaystyle\leq\int_{0}^{\infty}\Big{|}\partial_{t}(t^{k+1}\partial_{t}^{k}(te^{-\frac{t^{2}}{4(u+h)}}))\Big{|}dt\leq
C\left(\int_{0}^{\infty}|t^{k}\partial_{t}^{k}(te^{-\frac{t^{2}}{4(u+h)}})|dt\right.$
$\displaystyle\left.\quad+\int_{0}^{\infty}|t^{k+1}\partial_{t}^{k+1}(te^{-\frac{t^{2}}{4(u+h)}})|dt\right)$
$\displaystyle\leq
C\left((u+h)^{-(k-1)/2}\int_{0}^{\infty}t^{k}e^{-\frac{t^{2}}{8(u+h)}}dt+(u+h)^{-k/2}\int_{0}^{\infty}t^{k+1}e^{-\frac{t^{2}}{4(u+h)}}dt\right)$
(3.5) $\displaystyle\leq C(u+h),\quad u>0.$

By (3.1) we get

(3.6)
$\displaystyle\Big{\|}\mathfrak{s}_{k,j}^{E,h}(x,y,\cdot)-\mathfrak{U}_{k,j}^{E,h}(x-y,\cdot)\Big{\|}_{X}\leq
C\int_{0}^{\infty}\frac{1}{\sqrt{u}}|\partial_{x_{j}}h_{u}^{I,B_{1}}(x,y)-\partial_{x_{j}}W_{u}(x-y)|du,\quad
x,y\in\mathbb{R}^{d}.$

We are going to see that

(3.7)
$\int_{0}^{\infty}\frac{1}{\sqrt{u}}|\partial_{x_{j}}h_{u}^{I,B_{1}}(x,y)-\partial_{x_{j}}W_{u}(x-y)|du\leq
C\frac{1+|x|}{|x-y|^{d-1}},\quad(x,y)\in L_{2}.$

When $B_{1}=-I$ the operator $\mathcal{L}^{I,B_{1}}$ is the symmetric
Ornstein-Uhlenbeck and we have that

$h_{u}^{I,-I}(x,y)=\frac{e^{-\frac{|e^{-u}x-y|^{2}}{1-e^{-2u}}}}{(2\pi(1-e^{-2u}))^{d/2}},\quad
x,y\in\mathbb{R}^{d}\mbox{ and }u>0.$

By using [22, Lemma 3.4] we obtain

$\int_{0}^{\infty}\frac{1}{\sqrt{u}}|\partial_{x_{j}}h_{u}^{I,-I}(x,y)-\partial_{x_{j}}W_{u}(x-y)|du\leq
C\frac{1+|x|}{|x-y|^{d-1}},\quad(x,y)\in L_{2}.$

Then, (3.7) will be proved when we see that

(3.8)
$\int_{0}^{\infty}\frac{1}{\sqrt{u}}|\partial_{x_{j}}h_{u}^{I,B_{1}}(x,y)-\partial_{x_{j}}h_{u}^{I,-I}(x,y)|du\leq
C\frac{1+|x|}{|x-y|^{d-1}},\quad(x,y)\in L_{2}.$

Since $R+R^{*}=0$ we get

$|e^{uB_{1}}x-y|^{2}=\langle
e^{-uR}(e^{-u}x-e^{uR}y),e^{-uR}(e^{-u}x-e^{uR}y)\rangle=|e^{-u}x-e^{uR}y|^{2},\quad
x,y\in\mathbb{R}^{d}\mbox{ and }u>0.$

We have that

(3.9)
$\displaystyle\partial_{x_{j}}h_{u}^{I,B_{1}}(x,y)=-\frac{2e^{-u}}{(2\pi)^{\frac{d}{2}}}(e^{-u}x_{j}-(e^{uR}y)_{j})\frac{e^{-\frac{|e^{-u}x-e^{uR}y|^{2}}{1-e^{-2u}}}}{(1-e^{-2u})^{\frac{d}{2}+1}},\quad
x,y\in\mathbb{R}^{d}\mbox{ and }u>0,$

and

(3.10)
$\partial_{x_{j}}h_{u}^{I,-I}(x,y)=-\frac{2e^{-u}}{(2\pi)^{\frac{d}{2}}}(e^{-u}x_{j}-y_{j})\frac{e^{-\frac{|e^{-u}x-y|^{2}}{1-e^{-2u}}}}{(1-e^{-2u})^{\frac{d}{2}+1}},\quad
x,y\in\mathbb{R}^{d}\mbox{ and }u>0.$

We can write, for every $x,y\in\mathbb{R}^{d}$ and $u>0$,

$\displaystyle\partial_{x_{j}}h_{u}^{I,B_{1}}(x,y)-\partial_{x_{j}}h_{u}^{I,-I}(x,y)$
$\displaystyle=-\frac{2e^{-u}}{(2\pi)^{\frac{d}{2}}}\frac{1}{(1-e^{-2u})^{\frac{d}{2}+1}}$
$\displaystyle\times\left[\Big{(}e^{-\frac{|e^{-u}x-e^{uR}y|^{2}}{1-e^{-2u}}}-e^{-\frac{|e^{-u}x-y|^{2}}{1-e^{-2u}}}\Big{)}(e^{-u}x_{j}-(e^{uR}y)_{j})+e^{-\frac{|e^{-u}x-y|^{2}}{1-e^{-2u}}}(y_{j}-(e^{uR}y)_{j})\right].$

We need to establish some estimations. We have that

$|e^{-a}-e^{-b}|\leq e^{-\min\\{a,b\\}}|a-b|,\quad a,b>0.$

Then, according to [28, Lemma 3.3 (i)] we obtain

$\left|e^{-\frac{|e^{-u}x-e^{uR}y|^{2}}{1-e^{-2u}}}-e^{-\frac{|e^{-u}x-y|^{2}}{1-e^{-2u}}}\right|\leq
C\frac{e^{-c\frac{|x-y|^{2}}{1-e^{-2u}}}}{1-e^{-2u}}\big{|}|e^{-u}x-y|^{2}-|e^{-u}x-e^{uR}y|^{2}\big{|},\quad(x,y)\in
L_{2}\mbox{ and }u>0.$

We manipulate to get, for every $x,y\in\mathbb{R}^{d}$ and $u>0$,

$\displaystyle|e^{-u}x-y|^{2}-|e^{-u}x-e^{uR}y|^{2}$
$\displaystyle=|e^{-u}x-y|^{2}-|e^{-u}x-y-(e^{uR}-I)y|^{2}$
$\displaystyle=-|(e^{uR}-I)y|^{2}+2\langle e^{-u}x-y,(e^{uR}-I)y\rangle$
$\displaystyle=-|(e^{uR}-I)y|^{2}+2\Big{[}(e^{-u}-1)\langle
x,(e^{uR}-I)y\rangle+\langle x-y,(e^{uR}-I)y\rangle\Big{]}.$

It follows that

$\left||e^{-u}x-y|^{2}-|e^{-u}x-e^{uR}y|^{2}\right|\leq
C(u^{2}|y|^{2}+u^{2}|x||y|+u|x-y||y|),\quad x,\;y\in\mathbb{R}^{d}\text{ and
}u>0.$

We conclude that, for $x$, $y\in\mathbb{R}^{d}$ and $u\in(0,1)$,

$\left|e^{-\frac{|e^{-u}x-e^{uR}y|^{2}}{1-e^{-2u}}}-e^{-\frac{|e^{-u}x-y|^{2}}{1-e^{-2u}}}\right|\leq
Ce^{-c\frac{|x-y|^{2}}{u}}\big{[}u|y|(|x|+|y|)+|x-y||y|\big{]}.$

On the other hand we have that, for $x$, $y\in\mathbb{R}^{d}$ and $u>0$,

$|y_{j}-(e^{uR}y)_{j}|\leq|(e^{uR}-I)y|\leq Cu|y|,$

and

$|e^{-u}x_{j}-(e^{-uR}y)_{j}|\leq|e^{-u}x_{j}-x_{j}|+|x_{j}-y_{j}|+|y_{j}-(e^{uR}y)_{j}|\leq
C\big{[}u(|x|+|y|)+|x-y|\big{]}.$

We get, for $x$, $y\in\mathbb{R}^{d}$ and $u\in(0,1)$,

$\displaystyle|\partial_{x_{j}}h^{I,B_{1}}_{u}(x,y)-\partial_{x_{j}}h^{I,-I}_{u}(x,y)|$
$\displaystyle\leq
C\frac{e^{-c\frac{|x-y|^{2}}{u}}}{u^{\frac{d}{2}+1}}\left(\big{[}u|y|(|x|+|y|)+|x-y||y|\big{]}\big{[}u(|x|+|y|)+|x-y|\big{]}+u|y|\right)$
$\displaystyle\leq
C\frac{e^{-c\frac{|x-y|^{2}}{u}}}{u^{\frac{d}{2}+1}}\left(\big{[}u|y|(|x|+|y|)+\sqrt{u}|y|\big{]}\big{[}u(|x|+|y|)+\sqrt{u}\big{]}+u|y|\right)$
$\displaystyle=C\frac{e^{-c\frac{|x-y|^{2}}{u}}}{u^{\frac{d}{2}}}\left(\big{[}\sqrt{u}|y|(|x|+|y|)+|y|\big{]}\big{[}\sqrt{u}(|x|+|y|)+1\big{]}+|y|\right).$

Let us define the function $m$ by $m(x)=\min\left\\{1,|x|^{-2}\right\\}$,
$x\in\mathbb{R}^{d}\setminus\\{0\\}$, and $m(0)=1$. When $(x,y)\in L_{2}$ and
$0<u<m(x)$, we have that $|y|\leq C(1+|x|)$ and $\sqrt{u}(|x|+|y|)\leq C$.
Then, it follows that

$\big{[}\sqrt{u}|y|(|x|+|y|)+|y|\big{]}\big{[}\sqrt{u}(|x|+|y|)+1\big{]}+|y|\leq
C(1+|x|),\quad(x,y)\in L_{2}\mbox{ and }0<u<m(x).$

Thus we obtain that

$|\partial_{x_{j}}h^{I,B_{1}}_{u}(x,y)-\partial_{x_{j}}h^{I,-I}_{u}(x,y)|\leq
C\frac{e^{-c\frac{|x-y|^{2}}{u}}}{u^{\frac{d}{2}}}(1+|x|),\quad(x,y)\in
L_{2}\mbox{ and }0<u<m(x).$

By using this estimation, we get, when $(x,y)\in L_{2}$,

$\begin{split}\int_{0}^{m(x)}\frac{1}{\sqrt{u}}|\partial_{x_{j}}h^{I,B_{1}}_{u}(x,y)-\partial_{x_{j}}h^{I,-I}_{u}(x,y)|du&\leq
C(1+|x|)\int_{0}^{m(x)}\frac{e^{-c\frac{|x-y|^{2}}{u}}}{u^{\frac{d+1}{2}}}du\leq
C\frac{1+|x|}{|x-y|^{d-1}}.\end{split}$

On the other hand, by (3.9) and (3.10) and using Lemma 2.1 we deduce that

$\begin{split}\int_{m(x)}^{\infty}\frac{1}{\sqrt{u}}|\partial_{x_{j}}h^{I,B_{1}}_{u}(x,y)-\partial_{x_{j}}h^{I,-I}_{u}(x,y)|du&\leq\int_{m(x)}^{\infty}\frac{1}{\sqrt{u}}|\partial_{x_{j}}h^{I,B_{1}}_{u}(x,y)|du+\int_{m(x)}^{\infty}\frac{1}{\sqrt{u}}|\partial_{x_{j}}h^{I,-I}_{u}(x,y)|du\\\
&\leq
C\left(\int_{m(x)}^{\infty}\frac{e^{-\frac{|e^{-u}x-e^{uR}y|^{2}}{1-e^{-2u}}}}{(1-e^{-2u})^{\frac{d}{2}+1}\sqrt{u}}|e^{-u}x-e^{uR}y|e^{-u}du+\int_{m(x)}^{\infty}\frac{e^{-\frac{|e^{-u}x-y|^{2}}{1-e^{-2u}}}}{(1-e^{-2u})^{\frac{d}{2}+1}\sqrt{u}}|e^{-u}x-y|e^{-u}du\right)\\\
&\leq
C\int_{m(x)}^{\infty}\frac{e^{-c\frac{|x-y|^{2}}{1-e^{-2u}}}e^{-u}}{(1-e^{-2u})^{\frac{d+1}{2}}\sqrt{u}}du\leq
C\int_{m(x)}^{\infty}\frac{e^{-\frac{|x-y|^{2}}{u}}}{u^{\frac{d}{2}+1}}du\leq\frac{C}{\sqrt{m(x)}}\int_{m(x)}^{\infty}\frac{e^{-\frac{|x-y|^{2}}{u}}}{u^{\frac{d+1}{2}}}du\\\
&\leq\frac{C}{\sqrt{m(x)}|x-y|^{d-1}}\leq
C\frac{1+|x|}{|x-y|^{d-1}},\quad(x,y)\in L_{2}.\end{split}$

We conclude that (3.8) holds. Thus we proved (3.7).

According to (3.6) and (3.7), by using Lemma 2.5 we conclude that the operator
$D^{E,h}_{k,j,\text{\rm loc}}$ is bounded from
$L^{p}(\mathbb{R}^{d},\gamma_{\infty,1})$ into itself, for every $1\leq
p\leq\infty$.

We now consider the operators $S^{I,B_{1}}_{*,k,j,M}$, $G^{I,B_{1}}_{k,j,M}$
and $\mathcal{V}^{I,B_{1}}_{\rho,k,j,M}$. Let $E\subset(0,\infty)$ and $h\geq
0$. We define the operator

$\mathcal{H}^{E,h}_{k,j,M}(f)(x,t)=\int_{\mathbb{R}^{d}}H^{E,h}_{k,j,M}(x-y,t)f(y)dy,\quad
x\in\mathbb{R}^{d}\text{ and }t>0,$

where

$H^{E,h}_{k,j,M}(z,t)=\frac{t^{k+\frac{1}{2}}}{\Gamma(M)}\int_{E}\partial^{k}_{t}\left[t^{-M}e^{-\frac{u+h}{t}}\right]\partial_{x_{j}}W_{u}(z)(u+h)^{M-1}du,\quad
z\in\mathbb{R}^{n}\mbox{ and }t>0.$

Minkowski inequality leads to

$\left\|\mathbb{S}^{E,h}_{k,j,M}(f)(x,\cdot)-\mathcal{H}^{E,h}_{k,j,M}(f)(x,\cdot)\right\|_{X}\leq\int_{\mathbb{R}^{d}}\left\|\mathfrak{s}^{E,h}_{k,j,M}(x,y,\cdot)-H^{E,h}_{k,j,M}(f)(x-y,\cdot)\right\|_{X}|f(y)|dy,\quad
x\in\mathbb{R}^{d}.$

Our objective is to see that the operator $Z^{E,h}_{k,j,M,\text{loc}}$ defined
by

$Z^{E,h}_{k,j,M,\text{loc}}(f)(x)=\int_{\mathbb{R}^{d}}\left\|\mathfrak{s}^{E,h}_{k,j,M}(x,y,\cdot)-H^{E,h}_{k,j,M}(f)(x-y,\cdot)\right\|_{X}\varphi(x,y)|f(y)|dy,\quad
x\in\mathbb{R}^{d},$

is bounded from $L^{p}(\mathbb{R}^{d},\gamma_{\infty,1})$ into itself, for
every $1\leq p\leq\infty$.

By using again Minkowski inequality we get

$\begin{split}\left\|\mathfrak{s}^{E,h}_{k,j,M}(x,y,\cdot)\right.&\left.-H^{E,h}_{k,j,M}(f)(x-y,\cdot)\right\|_{X}\\\
&\leq\frac{1}{\Gamma(M)}\int_{E}\left\|t^{k+\frac{1}{2}}\partial^{k}_{t}\left[t^{-M}e^{-\frac{u+h}{t}}\right]\right\|_{X}|\partial_{x_{j}}h^{I,B_{1}}_{u}(x,y)-\partial_{x_{j}}W_{u}(x-y)|(u+h)^{M-1}du,\quad
x,y\in\mathbb{R}^{d}.\end{split}$

We are going to see that

(3.11)
$\left\|t^{k+\frac{1}{2}}\partial_{t}^{k}\big{[}t^{-M}e^{-\frac{u+h}{t}}\big{]}\right\|_{X}\leq
C(u+h)^{\frac{1}{2}-M},\quad u>0.$

We firstly consider $k=0$. We have that

$\left\|t^{\frac{1}{2}-M}e^{-\frac{u+h}{t}}\right\|_{L^{\infty}((0,\infty),dt)}\leq
C(u+h)^{\frac{1}{2}-M},\quad u>0,$

and

$\left\|t^{\frac{1}{2}-M}e^{-\frac{u+h}{t}}\right\|_{L^{2}((0,\infty),\frac{dt}{t})}=\left(\int_{0}^{\infty}t^{-2M}e^{-2\frac{u+h}{t}}dt\right)^{\frac{1}{2}}\leq
C(u+h)^{\frac{1}{2}-M},\quad u>0,$

provided that $M>\frac{1}{2}$.

We also get

$\left\|t^{\frac{1}{2}-M}e^{-\frac{u+h}{t}}\right\|_{V_{\rho}(0,\infty)}\leq\int_{0}^{\infty}\Big{|}\partial_{t}\big{[}t^{\frac{1}{2}-M}e^{-\frac{u+h}{t}}\big{]}\Big{|}dt\leq
C\int_{0}^{\infty}t^{-M-\frac{1}{2}}e^{-c\frac{u+h}{t}}dt\leq
C(u+h)^{\frac{1}{2}-M},\quad u>0,$

when $M>\frac{1}{2}$.

Suppose now $k\geq 1$. We can write

$\partial_{t}^{k}=\sum_{i=0}^{k-1}c_{i}w^{2k-i}\partial_{w}^{k-i},\quad
w=\frac{1}{t}\in(0,\infty),$

where $c_{i}\in\mathbb{R}$, $i=0,...,k-1$. It follows that

(3.12)
$\partial_{t}^{k}\big{[}t^{-M}e^{-\frac{u+h}{t}}\big{]}=\sum_{i=0}^{k-1}\sum_{\ell=0}^{k-i}c_{i,\ell}w^{2k+M-\ell-i}e^{-(u+h)w}(u+h)^{k-i-\ell},\quad
w=\frac{1}{t},\quad u\in(0,\infty),$

being $c_{i,\ell}\in\mathbb{R}$, $i=0,...,k-1$, $\ell=0,...,k-i$. Here
$c_{i,\ell}=0$, when $M<\ell\leq k-i$, $i=0,...,k-1$.

We have that

$\displaystyle\left\|t^{k+\frac{1}{2}}\partial_{t}^{k}\big{[}t^{-M}e^{-\frac{u+h}{t}}\big{]}\right\|_{L^{\infty}((0,\infty),dt)}$
$\displaystyle\leq
C\sum_{i=0}^{k-1}\sum_{\ell=0}^{k-i}(u+h)^{k-i-\ell}\sup_{w\in(0,\infty)}w^{k+M-\ell-i-\frac{1}{2}}e^{-(u+h)w}$
$\displaystyle\leq C(u+h)^{\frac{1}{2}-M},\quad u>0,$

when $M>\frac{1}{2}$.

We also obtain

$\displaystyle\left\|t^{k+\frac{1}{2}}\partial_{t}^{k}\big{[}t^{-M}e^{-\frac{u+h}{t}}\big{]}\right\|_{L^{2}((0,\infty),\frac{dt}{t})}$
$\displaystyle\leq
C\sum_{i=0}^{k-1}\sum_{\ell=0}^{k-i}(u+h)^{k-i-\ell}\left(\int_{0}^{\infty}w^{2k+2M-2\ell-2i-2}e^{-(u+h)w}dw\right)^{\frac{1}{2}}$
$\displaystyle\leq C(u+h)^{\frac{1}{2}-M},\quad u>0,$

and

$\displaystyle\left\|t^{k+\frac{1}{2}}\partial_{t}^{k}\big{[}t^{-M}e^{-\frac{u+h}{t}}\big{]}\right\|_{V_{\rho}(0,\infty)}$
$\displaystyle\leq\int_{0}^{\infty}\Big{|}\partial_{t}\Big{[}t^{k+\frac{1}{2}}\partial_{t}^{k}\big{[}t^{-M}e^{-\frac{u+h}{t}}\big{]}\Big{]}\Big{|}dt$
$\displaystyle\leq
C\left(\int_{0}^{\infty}\Big{|}t^{k-\frac{1}{2}}\partial_{t}^{k}\big{[}t^{-M}e^{-\frac{u+h}{t}}\big{]}\Big{|}dt+\int_{0}^{\infty}\Big{|}t^{k+\frac{1}{2}}\partial_{t}^{k+1}\big{[}t^{-M}e^{-\frac{u+h}{t}}\big{]}\Big{|}dt\right)$
$\displaystyle\leq
C\left(\sum_{i=0}^{k-1}\sum_{\ell=0}^{k-i}(u+h)^{k-i-\ell}\int_{0}^{\infty}w^{k+M-\ell-i-\frac{3}{2}}e^{-(u+h)w}dw\right.$
$\displaystyle\left.\quad+\sum_{i=0}^{k}\sum_{\ell=0}^{k+1-i}(u+h)^{k+1-i-\ell}\int_{0}^{\infty}w^{k+M-\ell-i-\frac{1}{2}}e^{-(u+h)w}dw\right)$
$\displaystyle\leq C(u+h)^{\frac{1}{2}-M},\quad u>0,$

provided that $M>\frac{1}{2}$. Thus (3.11) is established.

From (3.11) we deduce that

$\left\|\mathfrak{s}_{k,j,M}^{E,h}(x,y,\cdot)-H_{k,j,M}^{E,h}(x-y,\cdot)\right\|_{X}\leq
C\int_{0}^{\infty}\frac{1}{\sqrt{u}}|\partial_{x_{j}}h_{u}^{I,B_{1}}(x,y)-\partial_{x_{j}}W_{u}(x-y)|du,\quad
x,y\in\mathbb{R}^{d}.$

By using (3.7) we get

$\left\|\mathfrak{s}_{k,j,M}^{E,h}(x,y,\cdot)-H_{k,j,M}^{E,h}(x-y,\cdot)\right\|_{X}\leq
C\frac{1+|x|}{|x-y|^{d-1}},\quad(x,y)\in L_{2},$

and then, by virtue of Lemma 2.5, the operator $Z^{E,h}_{k,j,M,{\rm loc}}$ is
bounded from $L^{p}(\mathbb{R}^{d},\gamma_{\infty,1})$ into itself, for every
$1\leq p\leq\infty$.

###  3.2. Proof of Claim 4

Let $E\subset(0,\infty)$ and $h\geq 0$. By using Minkowski inequality we
obtain

$\left\|\mathfrak{s}_{k,j}^{E,h}(x,y,\cdot)\right\|_{X}\leq
C\int_{E}\Big{\|}t^{k+1}\partial_{t}^{k}\big{[}te^{-\frac{t^{2}}{4(u+h}}\big{]}\Big{\|}_{X}|\partial_{x_{j}}h_{u}^{I,B_{1}}(x,y)|\frac{du}{(u+h)^{\frac{3}{2}}},\quad
x,y\in\mathbb{R}^{d}.$

It was proved (see (3.3), (3.4) and (3.1)) that

$\|t^{k+1}\partial_{t}^{k}\big{[}te^{-\frac{t^{2}}{4(u+h}}\big{]}\Big{\|}_{X}\leq
C(u+h),\quad u>0.$

Then,

$\left\|\mathfrak{s}_{k,j}^{E,h}(x,y,\cdot)\right\|_{X}\leq
C\int_{E}|\partial_{x_{j}}h_{u}^{I,B_{1}}(x,y)|\frac{du}{\sqrt{u+h}},\quad
x,y\in\mathbb{R}^{d}.$

In a similar way by using (3.11) we can see that

$\left\|\mathfrak{s}_{k,j,M}^{E,h}(x,y,\cdot)\right\|_{X}\leq
C\int_{E}|\partial_{x_{j}}h_{u}^{I,B_{1}}(x,y)|\frac{du}{\sqrt{u+h}},\quad
x,y\in\mathbb{R}^{d}.$

We denote now by $E_{1}$ the sets $(0,t_{0})$ and $J_{D}^{\\#}$ associated
with $\delta_{1}=1/10$ given in Lemma 2.2. The arguments developed in the
proof of [28, Proposition 4.7] allow us to prove that

$\int_{E}|\partial_{x_{j}}h_{u}^{I,B_{1}}(x,y)|\frac{du}{\sqrt{u+h}}\leq
Ce^{|x|^{2}-|y|^{2}}\min\big{\\{}(1+|x|)^{d},(|x|\sin\theta(x,y))^{-d}\big{\\}},\quad(x,y)\in
G_{1}.$

We define

$\widetilde{\mathbb{V}}_{k,j,M}^{E,h}(f)(x)=\int_{\mathbb{R}^{d}}\Big{\|}\mathfrak{s}_{k,j,M}^{E,h}(x,y,\cdot)\Big{\|}_{X}f(y)dy,\quad
x\in\mathbb{R}^{d}.$

From Lemma 2.4 we deduce that the operators $\widetilde{V}_{k,j,{\rm
glob}}^{E,h}$ and $\widetilde{\mathbb{V}}_{k,j,{\rm glob}}^{E,h}$ are bounded
from $L^{1}(\mathbb{R}^{d},\gamma_{\infty,1})$ into
$L^{1,\infty}(\mathbb{R}^{d},\gamma_{\infty,1})$.

Let now $1<p<\infty$ and $\delta\in(\max\\{1-\frac{1}{d},1-\frac{1}{p}\\},1)$.
Observe that $\delta_{p}$ in Claim 4 is a particular case of such $\delta$.
Consider $E_{p}$ one of the sets $(0,t_{0})$ or $J_{D}^{\\#}$ given in Lemma
2.2 which is associated to $\delta$. By considering the above estimations it
is sufficient to see that the operator $Z_{\rm glob}^{E_{p},h}$ is bounded
from $L^{p}(\mathbb{R}^{d},\gamma_{\infty,1})$ into itself, where

$Z^{E_{p},h}(f)(x)=\int_{\mathbb{R}^{d}}Z^{E_{p},h}(x,y)f(y)dy,\quad
x\in\mathbb{R}^{d},$

being

$Z^{E_{p},h}(x,y)=\int_{E_{p}}|\partial_{x_{j}}h_{u}^{I,B_{1}}(x,y)|\frac{du}{\sqrt{u+h}},\quad
x,y\in\mathbb{R}^{d}.$

We observe that

$\displaystyle\Big{\|}Z^{E_{p},h}_{\rm
glob}(f)\Big{\|}_{L^{p}(\mathbb{R}^{d},\gamma_{\infty,1})}^{p}$
$\displaystyle=\pi^{-\frac{d}{2}}\int_{\mathbb R^{d}}|Z^{E_{p},h}_{\rm
glob}(f)(x)|^{p}e^{-|x|^{2}}dx$ $\displaystyle=\pi^{-\frac{d}{2}}\int_{\mathbb
R^{d}}\left(\int_{\mathbb
R^{d}}Z^{E_{p},h}(x,y)(1-\varphi(x,y))e^{-\frac{1}{p}(|x|^{2}-|y|^{2})}f(y)e^{-\frac{|y|^{2}}{p}}dy\right)^{p}dx$
$\displaystyle=\Big{\|}\widetilde{Z}^{E_{p},h}_{\rm
glob}(\pi^{-\frac{d}{2}}e^{-\frac{|y|^{2}}{p}}f)\Big{\|}_{L^{p}(\mathbb{R}^{d},dx)}^{p},$

where

$\widetilde{Z}^{E_{p},h}(g)(x)=\int_{\mathbb
R^{d}}Z^{E_{p},h}(x,y)e^{-\frac{1}{p}(|x|^{2}-|y|^{2})}g(y)dy,\quad
x\in\mathbb R^{d}.$

Then, we only have to establish that the operator $\widetilde{Z}_{\rm
glob}^{E_{p},h}$ is bounded from $L^{p}(\mathbb R^{d},dx)$ into itself. For
that, we first prove the following estimations.

(a) For every $\eta\in(0,1)$,

(3.13) $Z^{E_{p},h}(x,y)\leq
Ce^{\eta(1-\delta)|x|^{2}-\eta|y|^{2}},\quad(x,y)\in G_{1},\;\langle
x,y\rangle<0.$

(b) For every $\eta\in(0,1)$ such that $\eta\delta>1-\frac{1}{d}$,

(3.14) $Z^{E_{p},h}(x,y)\leq
C|x+y|^{d}e^{\eta(1-\frac{\delta}{2})(|x|^{2}-|y|^{2})-\frac{\eta\delta}{2}|x+y||x-y|},\quad(x,y)\in
G_{1},\;\langle x,y\rangle\geq 0.$

Let $\eta\in(0,1)$. First we observe that from (3.9) we deduce that

$|\partial_{x_{j}}h_{u}^{I,B_{1}}(x,y)|\leq
Ce^{-u}\frac{|e^{-u}x-e^{uR}y|}{(1-e^{-2u})^{\frac{d}{2}+1}}e^{-\frac{|e^{-u}x-e^{uR}y|^{2}}{1-e^{-2u}}},\quad
x,y\in\mathbb{R}^{d}\mbox{ and }u>0.$

Then,

$|\partial_{x_{j}}h_{u}^{I,B_{1}}(x,y)|\leq
Ce^{-u}\frac{e^{-\eta\frac{|e^{-u}x-e^{uR}y|^{2}}{1-e^{-2u}}}}{(1-e^{-2u})^{\frac{d+1}{2}}}\leq
C\frac{e^{-u}}{\sqrt{1-e^{-2u}}}h_{u}^{I,B_{1}}(\sqrt{\eta}x,\sqrt{\eta}y),\quad
x,y\in\mathbb{R}^{d}\mbox{ and }u>0.$

By using Lemma 2.2 we have that

$|\partial_{x_{j}}h_{\tau(s)}^{I,B_{1}}(x,y)|\leq
C\frac{1-s}{s^{\frac{d+1}{2}}}e^{\eta(|x|^{2}-|y|^{2}-\frac{\delta}{4s}Q_{s}(x,y))}\quad
x,y\in\mathbb{R}^{d}\mbox{ and }s\in\tau^{-1}(E_{p}).$

Since

$\frac{1}{4s}Q_{s}(x,y)=\frac{s}{4}|x+y|^{2}+\frac{1}{4s}|x-y|^{2}+\frac{|x|^{2}-|y|^{2}}{2}=\frac{1}{4s}Q_{s}(y,x)+|x|^{2}-|y|^{2},\quad
x,y\in\mathbb R^{d}\mbox{ and }s\in(0,1),$

and

$\frac{|e^{-u}x-y|^{2}}{1-e^{-2u}}=\frac{1}{4s}Q_{s}(y,x),\quad
u=\log\frac{1+s}{1-s},\;x,y\in\mathbb R^{d}\mbox{ and }s\in(0,1),$

we get

$|x|^{2}-|y|^{2}-\frac{\delta}{4s}Q_{s}(x,y)=(1-\delta)(|x|^{2}-|y|^{2})-\delta\frac{|e^{-u}x-y|^{2}}{1-e^{-2u}},\quad
u=\log\frac{1+s}{1-s}\in(0,\infty),\;x,y\in\mathbb R^{d}.$

Thus we obtain

$|\partial_{x_{j}}h_{u}^{I,B_{1}}(x,y)|\leq
C\frac{e^{-u}}{(1-e^{-2u})^{\frac{d+1}{2}}}e^{\eta(1-\delta)(|x|^{2}-|y|^{2})}e^{-\eta\delta\frac{|e^{-u}x-y|^{2}}{1-e^{-2u}}}\quad
x,y\in\mathbb{R}^{d}\;\mbox{and}\;u\in E_{p}.$

By making the change of variables $t=1-e^{-2u}$, $u\in(0,\infty)$, it follows
that

$\displaystyle Z^{E_{p},h}(x,y)$ $\displaystyle\leq
Ce^{\eta(1-\delta)(|x|^{2}-|y|^{2})}\int_{0}^{1}\frac{e^{-\eta\delta
u(t)}}{t^{\frac{d+1}{2}}\sqrt{|\log(1-t)|}}\frac{dt}{\sqrt{1-t}}$
$\displaystyle\leq
Ce^{\eta(1-\delta)(|x|^{2}-|y|^{2})}\int_{0}^{1}\frac{e^{-\eta\delta
u(t)}}{t^{\frac{d}{2}+1}}\frac{dt}{\sqrt{1-t}},\quad x,y\in\mathbb R^{d}.$

Here $u(t)=|\sqrt{1-t}x-y|^{2}/t$, $x,y\in\mathbb R^{d}$ and $t\in(0,1)$.

Then, as in the proof of [37, Theorem 4.2] we deduce that (3.13) is satisfied.

Assume now that $\langle x,y\rangle\geq 0$ and $\eta\delta>1-1/d$. As in [37]
we define $t_{0}=2\frac{\sqrt{a^{2}-b^{2}}}{a+\sqrt{a^{2}-b^{2}}}$, being
$a=|x|^{2}+|y|^{2}$ and $b=2\langle x,y\rangle$, and
$u_{0}=\frac{|y|^{2}-|x|^{2}+|x+y||x-y|}{2}$. By using [37, Lemma 4.1] we
obtain

$\displaystyle\int_{0}^{1}\frac{e^{-\eta\delta
u(t)}}{t^{\frac{d}{2}+1}\sqrt{1-t}}dt$ $\displaystyle\leq
C\left(\frac{e^{-u_{0}}}{t_{0}^{d/2}}\right)^{1-\frac{1}{d}}\int_{0}^{1}\frac{e^{-(\eta\delta-1+\frac{1}{d})u(t)}}{t^{\frac{3}{2}}\sqrt{1-t}}dt$
$\displaystyle\leq
C\left(\frac{e^{-u_{0}}}{t_{0}^{d/2}}\right)^{1-\frac{1}{d}}\frac{e^{-(\eta\delta-1+\frac{1}{d})u_{0}}}{t_{0}^{\frac{1}{2}}}\leq
C\frac{e^{-\eta\delta u_{0}}}{t_{0}^{d/2}},\quad(x,y)\in G_{1}.$

Then,

$Z^{E_{p},h}(x,y)\leq C\frac{e^{\eta(1-\delta)(|x|^{2}-|y|^{2})-\eta\delta
u_{0}}}{t_{0}^{d/2}},\quad(x,y)\in G_{1}\mbox{.}$

Since $t_{0}\sim\frac{\sqrt{a^{2}-b^{2}}}{a}=\frac{|x+y||x-y|}{a}$,
$|x-y||x+y|\geq 1$, $(x,y)\in G_{1}$ we obtain

$Z^{E_{p},h}(x,y)\leq
C|x+y|^{d}e^{\eta(1-\frac{\delta}{2})(|x|^{2}-|y|^{2})-\frac{\eta\delta}{2}|x-y||x+y|},\quad(x,y)\in
G_{1},$

and (3.14) is proved.

We now choose $\eta$ such that

$\max\Big{\\{}\frac{1}{p},\frac{1}{\delta}\big{(}1-\frac{1}{d}\big{)}\Big{\\}}<\eta<1.$

Note that our hypothesis on $\delta$ leads to
$0<\frac{1}{\delta}(1-\frac{1}{d})<1$ and also $\eta(1-\delta)<\frac{1}{p}$.

By using (3.13) and (3.14) we can deduce that $\widetilde{Z}^{Ep,h}_{\rm
glob}$ is a bounded operator from $L^{q}(\mathbb{R}^{d},dx)$ into itself, for
every $1\leq q\leq\infty$. Indeed, from (3.13) we have that

$\int_{\\{y\in\mathbb R^{d}:\langle
x,y\rangle<0\\}}Z^{E_{p},h}(x,y)e^{-\frac{1}{p}(|x|^{2}-|y|^{2})}(1-\varphi(x,y))dy\leq
Ce^{-(\frac{1}{p}-\eta(1-\delta))|x|^{2}}\int_{\mathbb
R^{d}}e^{-(\eta-\frac{1}{p})|y|^{2}}dy,\quad x\in\mathbb R^{d}.$

Since $\frac{1}{p}<\eta<1$ and $\eta(1-\delta)<\frac{1}{p}$, it follows that

$\sup_{x\in\mathbb R^{d}}\int_{\\{y\in\mathbb R^{d}:\langle
x,y\rangle<0\\}}Z^{E_{p},h}(x,y)e^{-\frac{1}{p}(|x|^{2}-|y|^{2})}(1-\varphi(x,y))dy<\infty.$

On the other hand, from (3.14) we can write

$Z^{E_{p},h}(x,y)e^{-\frac{1}{p}(|x|^{2}-|y|^{2})}\leq
C|x+y|^{d}e^{-\left(\frac{\eta\delta}{2}-\left|\frac{1}{p}-\eta(1-\frac{\delta}{2})\right|\right)|x+y||x-y|},\quad(x,y)\in
G_{1}\mbox{ and }\langle x,y\rangle\geq 0.$

By taking into account that $\frac{1}{p}<\eta<1$ and
$\eta(1-\delta)<\frac{1}{p}$, we get that
$\frac{\eta\delta}{2}-\left|\frac{1}{p}-\eta(1-\frac{\delta}{2})\right|>0$. By
proceeding as in [37, p. 501] we obtain

$\sup_{x\in\mathbb R^{d}}\int_{\\{y\in\mathbb R^{d}:\langle x,y\rangle\geq
0\\}}Z^{E_{p},h}(x,y)e^{-\frac{1}{p}(|x|^{2}-|y|^{2})}(1-\varphi(x,y))dy<\infty.$

We conclude that

$\sup_{x\in\mathbb R^{d}}\int_{\mathbb
R^{d}}Z^{E_{p},h}(x,y)e^{-\frac{1}{p}(|x|^{2}-|y|^{2})}(1-\varphi(x,y))dy<\infty.$

In a similar way we get

$\sup_{y\in\mathbb R^{d}}\int_{\mathbb
R^{d}}Z^{E_{p},h}(x,y)e^{-\frac{1}{p}(|x|^{2}-|y|^{2})}(1-\varphi(x,y))dx<\infty.$

We deduce that the operator $\widetilde{Z}_{\rm glob}^{E_{p},h}$ is bounded
from $L^{q}(\mathbb R^{d},dx)$ into itself, for every $1\leq q\leq\infty$.
Thus we prove that the operator $Z_{\rm glob}^{E_{p},h}$ is bounded from
$L^{p}(\mathbb R^{d},\gamma_{\infty,1})$ into itself, and the proof of the
Claim 4 is finished.

##  4\. Proof of Claim 3

In this section we prove the Claim 3. Assume that $E\subset(0,\infty)$ and
$h\geq 0$.

###  4.1. Maximal operators

#### 4.1.1.

We consider firstly the local maximal operator
$\widetilde{U}^{E,h}_{*,k,j,{\rm loc}}$ defined by

$\widetilde{U}^{E,h}_{*,k,j,{\rm
loc}}(f)(x)=\sup_{t>0}\left|\int_{\mathbb{R}^{d}}f(y)\mathfrak{U}_{k,j}^{E,h}(x-y,t)\varphi(x,y)dy\right|,\quad
x\in\mathbb{R}^{d},$

where

$\mathfrak{U}_{k,j}^{E,h}(z,t)=\frac{t^{k+1}}{2\sqrt{\pi}}\int_{E}\partial_{t}^{k}\big{[}te^{-\frac{t^{2}}{4(u+h)}}\big{]}\partial_{x_{j}}W_{u}(z)\frac{du}{(u+h)^{\frac{3}{2}}},\quad
z\in\mathbb{R}^{d}\mbox{ and }t>0.$

We recall that

$W_{u}(z)=\frac{e^{-\frac{|z|^{2}}{2u}}}{(2\pi u)^{\frac{d}{2}}},\quad
z\in\mathbb{R}^{d}\mbox{ and }u>0.$

Assume that $f\in C_{c}^{\infty}(\mathbb{R}^{d})$. We can write, for each
$x,y\in\mathbb{R}^{d}$ and $t>0$,

$\displaystyle\int_{\mathbb{R}^{d}}f(y)\mathfrak{U}_{k,j}^{E,h}(x-y,t)\varphi(x,y)dy$
$\displaystyle=\frac{t^{k+1}}{2\sqrt{\pi}}\int_{E}\partial_{t}^{k}\big{[}te^{-\frac{t^{2}}{4(u+h)}}\big{]}\int_{\mathbb{R}^{d}}\partial_{x_{j}}W_{u}(x-y)f(y)\varphi(x,y)\frac{dydu}{(u+h)^{\frac{3}{2}}}$
$\displaystyle=-\frac{t^{k+1}}{2\sqrt{\pi}}\int_{E}\partial_{t}^{k}\big{[}te^{-\frac{t^{2}}{4(u+h)}}\big{]}\int_{\mathbb{R}^{d}}\frac{x_{j}-y_{j}}{u}W_{u}(x-y)f(y)\varphi(x,y)\frac{dydu}{(u+h)^{\frac{3}{2}}}.$

Then, by using Minkowski inequality and (3.3) we get that

$\displaystyle\widetilde{U}_{*,k,j,{\rm loc}}^{E,h}(f)(x)$ $\displaystyle\leq
C\int_{E}\|t^{k+1}\partial_{t}^{k}[te^{-\frac{t^{2}}{4(u+h)}}]\|_{L^{\infty}((0,\infty),dt)}\int_{\mathbb{R}^{d}}|x-y|W_{u}(x-y)|f(y)|\varphi(x,y)\frac{dydu}{u(u+h)^{\frac{3}{2}}}$
$\displaystyle\leq
C\int_{E}\frac{1}{u\sqrt{u+h}}\int_{\mathbb{R}^{d}}|x-y|W_{u}(x-y)|f(y)|\varphi(x,y)dydu,\quad
x\in\mathbb{R}^{d}.$

If $h>0$, by taking into account that $|x-y|\leq C$, when $(x,y)\in L_{2}$,
and that $\frac{|z|}{\sqrt{u}}W_{u}(z)\leq CW_{2u}(z)$, $z\in\mathbb{R}^{d}$,
it follows that

$\displaystyle\widetilde{U}_{*,k,j,{\rm loc}}^{E,h}(f)(x)$ $\displaystyle\leq
C\left(\int_{1}^{\infty}\frac{1}{u\sqrt{u+h}}\int_{\mathbb{R}^{d}}W_{u}(x-y)|f(y)|dydu\right.$
$\displaystyle\left.\quad+\int_{0}^{1}\frac{1}{\sqrt{u(u+h)}}\int_{\mathbb{R}^{d}}W_{2u}(x-y)|f(y)|dydu\right)$
$\displaystyle\leq
C\left(\int_{1}^{\infty}u^{-\frac{3}{2}}du+\int_{0}^{1}u^{-\frac{1}{2}}du\right)\sup_{v>0}W_{v}(|f|)(x),\quad
x\in\mathbb{R}^{d}.$

On the other hand, we have that, for every $x\in\mathbb{R}^{d}$ and $t>0$,

$\widetilde{U}_{*,k,j,{\rm loc}}^{E,0}(f)(x)\leq
C\sup_{t>0}\left(t^{k+1}\int_{0}^{\infty}\frac{e^{-\frac{t^{2}}{8u}}}{u^{\frac{k+3}{2}}}du\right)\sup_{v>0}W_{v}(|f|)(x)\leq
C\sup_{v>0}W_{v}(|f|)(x).$

We conclude that

$\widetilde{U}_{*,k,j,{\rm loc}}^{E,h}(f)(x)\leq
C\sup_{t>0}W_{t}(|f|)(x),\quad x\in\mathbb{R}^{n}.$

From wellknown $L^{p}$-boundedness properties of the maximal operator defined
by the classical heat semigroup we deduce that the local maximal operator
$\widetilde{U}_{*,k,j,{\rm loc}}^{E,h}$ is bounded from
$L^{p}(\mathbb{R}^{d},dx)$ into itself, for every $1<p<\infty$ and from
$L^{1}(\mathbb{R}^{d},dx)$ into $L^{1,\infty}(\mathbb{R}^{d},dx)$.

#### 4.1.2.

We now study the local maximal operator

$\widetilde{\mathbb{U}}_{*,k,j,M,{\rm
loc}}^{E,h}(f)(x)=\sup_{t>0}\left|\int_{\mathbb{R}^{d}}\mathfrak{U}_{k,j,M}^{E,h}(x-y,t)f(y)\varphi(x,y)dy\right|,\quad
x\in\mathbb{R}^{d},$

where

$\mathfrak{U}_{k,j,M}^{E,h}(z,t)=\frac{t^{k+\frac{1}{2}}}{\Gamma(M)}\int_{E}\partial_{t}^{k}\big{[}t^{-M}e^{-\frac{u+h}{t}}\big{]}\partial_{x_{j}}W_{u}(z)(u+h)^{M-1}du,\quad
z\in\mathbb{R}^{d}\mbox{ and }t>0.$

According to (3.11) and Minkowski inequality we can write

$\displaystyle\widetilde{\mathbb{U}}_{*,k,j,M,{\rm loc}}^{E,h}(f)(x)$
$\displaystyle\leq
C\int_{E}\|t^{k+\frac{1}{2}}\partial_{t}^{k}[t^{-M}e^{-\frac{u+h}{t}}]\|_{L^{\infty}((0,\infty),dt)}\frac{|x-y|}{u}W_{u}(x-y)|f(y)|\varphi(x,y)(u+h)^{M-1}dydu$
$\displaystyle\leq
C\int_{0}^{\infty}\frac{1}{u\sqrt{u+h}}\int_{\mathbb{R}^{d}}|x-y|W_{u}(x-y)|f(y)|\varphi(x,y)dydu,\quad
x\in\mathbb{R}^{d}.$

By proceeding as in section 4.1.1 we get that, if $h>0$,

$\widetilde{\mathbb{U}}_{*,k,j,M,{\rm loc}}^{E,h}(f)(x)\leq
C\sup_{v>0}W_{v}(|f|)(x),\quad x\in\mathbb{R}^{d}.$

On the other hand, by using (3.12), if $k\geq 1$ we get

$t^{k+\frac{1}{2}}\int_{0}^{\infty}\Big{|}\partial_{t}^{k}\big{[}t^{-M}e^{-\frac{u}{t}}\big{]}\Big{|}u^{M-\frac{3}{2}}du\leq
Ct^{k+\frac{1}{2}}\sum_{i=0}^{k-1}\sum_{\ell=0}^{k-i}t^{-2k-M+\ell+i}\int_{0}^{\infty}e^{-\frac{u}{t}}u^{k-i-\ell+M-\frac{3}{2}}du\leq
C,\quad t>0.$

Furthermore, we obtain

(4.1)
$t^{\frac{1}{2}-M}\int_{0}^{\infty}e^{-\frac{u}{t}}u^{M-\frac{3}{2}}du\leq
C,\quad t>0.$

Then,

$\displaystyle\widetilde{\mathbb{U}}_{*,k,j,M,{\rm loc}}^{E,0}(f)(x)$
$\displaystyle\leq
Ct^{k+\frac{1}{2}}\int_{0}^{\infty}\partial_{t}^{k}\big{[}t^{-M}e^{-\frac{u}{t}}\big{]}\Big{|}u^{M-\frac{3}{2}}\int_{\mathbb{R}^{d}}\frac{|x-y|}{\sqrt{u}}W_{u}(x-y)|f(y)|dydu$
$\displaystyle\leq C\sup_{v>0}W_{v}(|f|)(x),\quad x\in\mathbb{R}^{d}.$

We conclude that

$\widetilde{\mathbb{U}}_{*,k,j,M,{\rm loc}}^{E,h}(f)(x)\leq
C\sup_{t>0}W_{t}(|f|)(x),\quad x\in\mathbb{R}^{d},$

and thus, we establish that $\widetilde{\mathbb{U}}_{*,k,j,M,{\rm loc}}^{E,h}$
is bounded from $L^{p}(\mathbb{R}^{d},dx)$ into itself, for every
$1<p<\infty$, and from $L^{1}(\mathbb{R}^{d},dx)$ into
$L^{1,\infty}(\mathbb{R}^{d},dx)$.

###  4.2. Littlewood-Paley functions

#### 4.2.1.

We consider the local Littlewood-Paley function $\mathfrak{g}_{k,j,{\rm
loc}}^{E,h}$ defined by

$\mathfrak{g}_{k,j,{\rm
loc}}^{E,h}(f)(x)=\left(\int_{0}^{\infty}\left|\int_{\mathbb{R}^{d}}f(y)\mathfrak{U}_{k,j}^{E,h}(x-y,t)\varphi(x,y)dy\right|^{2}\frac{dt}{t}\right)^{\frac{1}{2}},\quad
x\in\mathbb{R}^{d}.$

By using Minkowski inequality and (3.4) we get

$\displaystyle\mathfrak{g}_{k,j,{\rm loc}}^{E,h}(f)(x)$ $\displaystyle\leq
C\int_{E}\Big{\|}t^{k+1}\partial_{t}^{k}\big{[}te^{-\frac{t^{2}}{4(u+h)}}\big{]}\Big{\|}_{L^{2}((0,\infty),\frac{dt}{t})}\int_{\mathbb{R}^{d}}|x_{j}-y_{j}|W_{u}(x-y)|f(y)|\varphi(x,y)\frac{dydu}{u(u+h)^{\frac{3}{2}}}$
$\displaystyle\leq
C\int_{0}^{\infty}\frac{1}{u\sqrt{u+h}}\int_{\mathbb{R}^{d}}|x-y|W_{u}(x-y)|f(y)|\varphi(x,y)dydu\leq
C\sup_{v>0}W_{v}(|f|)(x),\quad x\in\mathbb{R}^{d}.$

As in section 4.1.1 we deduce that if $h>0$ the operator
$\mathfrak{g}_{k,j,{\rm loc}}^{E,h}$ is bounded from
$L^{p}(\mathbb{R}^{d},dx)$ into itself, for every $1<p<\infty$, and from
$L^{1}(\mathbb{R}^{d},dx)$ into $L^{1,\infty}(\mathbb{R}^{d},dx)$.

We now study the operator $\mathfrak{g}_{k,j,{\rm loc}}^{E,0}$. We consider
the Littlewood-Paley function $\mathfrak{g}_{k,j}^{E,0}$ defined by

$\mathfrak{g}_{k,j}^{E,0}(f)(x)=\left(\int_{0}^{\infty}\left|\int_{\mathbb{R}^{d}}f(y)\mathfrak{U}_{k,j}^{E,0}(x-y,t)dy\right|^{2}\frac{dt}{t}\right)^{\frac{1}{2}},\quad
x\in\mathbb{R}^{d}.$

We are going to see that $\mathfrak{g}_{k,j}^{E,0}$ is bounded from
$L^{2}(\mathbb{R}^{d},dx)$ into itself. If $\mathcal{F}$ denotes the Fourier
transform defined in $L^{1}(\mathbb{R}^{d})$ by

$\mathcal{F}(f)(z)=\int_{\mathbb{R}^{d}}g(x)e^{-i\langle x,z\rangle}dx,\quad
z\in\mathbb{R}^{d},$

we have that (see for instance, [10, p. 15 (11)])

(4.2) $\mathcal{F}(W_{t})(x)=e^{-\frac{t|x|^{2}}{2}},\quad
x\in\mathbb{R}^{d}\mbox{ and }t>0.$

By using Plancherel equality and (4.2) we get

$\displaystyle\int_{\mathbb{R}^{d}}|\mathfrak{g}_{k,j}^{E,0}(f)(x)|^{2}dx$
$\displaystyle=\int_{0}^{\infty}\int_{\mathbb{R}^{d}}\left|\int_{\mathbb{R}^{d}}f(y)\mathfrak{U}_{k,j}^{E,0}(x-y,t)dy\right|^{2}dx\frac{dt}{t}$
$\displaystyle=\int_{0}^{\infty}\int_{\mathbb{R}^{d}}\left|\mathcal{F}\left(\int_{\mathbb{R}^{d}}f(y)\mathfrak{U}_{k,j}^{E,0}(\cdot-y,t)dy\right)(z)\right|^{2}dz\frac{dt}{t}$
$\displaystyle=\int_{0}^{\infty}\int_{\mathbb{R}^{d}}\Big{|}\int_{E}\frac{t^{k+1}}{2\sqrt{\pi}}\frac{\partial_{t}^{k}\big{[}te^{-\frac{t^{2}}{4u}}\big{]}}{u^{\frac{3}{2}}}\int_{\mathbb
R^{d}}f(y)\mathcal{F}(\partial_{x_{j}}W_{u}(\cdot-y))(z)dydu\Big{|}^{2}dz\frac{dt}{t}$
$\displaystyle=\int_{0}^{\infty}\int_{\mathbb{R}^{d}}\Big{|}\mathcal{F}(f)(z)\frac{t^{k+1}}{2\sqrt{\pi}}\int_{E}\frac{\partial_{t}^{k}\big{[}te^{-\frac{t^{2}}{4u}}\big{]}}{u^{\frac{3}{2}}}\mathcal{F}(\partial_{x_{j}}W_{u})(z)du\Big{|}^{2}dz\frac{dt}{t}$
$\displaystyle=\int_{\mathbb{R}^{d}}|\mathcal{F}(f)(z)|^{2}\int_{0}^{\infty}\Big{|}\frac{t^{k+1}}{2\sqrt{\pi}}\int_{E}\frac{\partial_{t}^{k}\big{[}te^{-\frac{t^{2}}{4u}}\big{]}}{u^{\frac{3}{2}}}z_{j}e^{-\frac{u|z|^{2}}{2}}du\Big{|}^{2}\frac{dt}{t}dz,\quad
f\in C_{c}^{\infty}(\mathbb{R}^{d}).$

Minkowski inequality and (3.4) leads to

$\displaystyle\int_{\mathbb{R}^{d}}|\mathfrak{g}_{k,j}^{E,0}(f)(x)|^{2}dx$
$\displaystyle\leq
C\int_{\mathbb{R}^{d}}|\mathcal{F}(f)(z)|^{2}\left(\int_{0}^{\infty}|z|e^{-\frac{u|z|^{2}}{2}}\big{\|}t^{k+1}\partial_{t}^{k}\big{[}te^{-\frac{t^{2}}{4u}}\big{]}\big{\|}_{L^{2}(0,\infty),\frac{dt}{t})}\frac{du}{u^{\frac{3}{2}}}\right)^{2}dz$
$\displaystyle\leq
C\int_{\mathbb{R}^{d}}|\mathcal{F}(f)(z)|^{2}\left(\int_{0}^{\infty}\frac{|z|}{\sqrt{u}}e^{-\frac{u|z|^{2}}{2}}du\right)^{2}dz$
$\displaystyle\leq
C\int_{\mathbb{R}^{d}}|\mathcal{F}(f)(z)|^{2}dz=C\|f\|_{L^{2}(\mathbb{R}^{d},dx)}^{2},\quad
f\in C_{c}^{\infty}(\mathbb{R}^{d}).$

We now use the Banach-valued Calderón-Zygmund theory for singular integrals
(see [39]). We recall that the operator $U_{k,j}^{E,0}$ is defined by

$U_{k,j}^{E,0}(f)(x,t)=\int_{\mathbb{R}^{d}}\mathfrak{U}_{k,j}^{E,0}(x-y,t)f(y)dy,\quad
x\in\mathbb{R}^{d}\mbox{ and }t>0.$

It is clear that

$\mathfrak{g}_{k,j}^{E,0}(f)(x)=\left\|U_{k,j}^{E,0}(f)(x,\cdot)\right\|_{L^{2}((0,\infty),\frac{dt}{t})},\quad
x\in\mathbb{R}^{d}.$

By using again Minkowski inequality and (3.4)

$\displaystyle\left\|\mathfrak{U}_{k,j}^{E,0}(x-y,\cdot)\right\|_{L^{2}((0,\infty),\frac{dt}{t})}$
$\displaystyle\leq
C\int_{0}^{\infty}\Big{\|}t^{k+1}\partial_{t}^{k}[te^{-\frac{t^{2}}{4u}]}\Big{\|}_{L^{2}((0,\infty),\frac{dt}{t})}\frac{|x-y|}{u}\frac{e^{-\frac{|x-y|^{2}}{2u}}}{u^{\frac{d+3}{2}}}du$
(4.3) $\displaystyle\leq
C\int_{0}^{\infty}\frac{e^{-c\frac{|x-y|^{2}}{u}}}{u^{\frac{d}{2}+1}}du\leq\frac{C}{|x-y|^{d}},\quad
x,y\in\mathbb{R}^{d},\;x\not=y.$

In a similar way we obtain, for every $i=1,...,d$,

(4.4)
$\left\|\partial_{x_{i}}\mathfrak{U}_{k,j}^{E,0}(x-y,\cdot)\right\|_{L^{2}((0,\infty),\frac{dt}{t})}+\left\|\partial_{y_{i}}\mathfrak{U}_{k,j}^{E,0}(x-y,\cdot)\right\|_{L^{2}((0,\infty),\frac{dt}{t})}\leq\frac{C}{|x-y|^{d+1}},\quad
x,y\in\mathbb{R}^{d},\;x\not=y.$

Let $N\in\mathbb{N}$, $N\geq 2$. The space
$\mathbb{F}_{N}=L^{2}((1/N,N),dt/t)$ is a Banach and separable space.

Assume that $f\in C_{c}^{\infty}(\mathbb{R}^{d})$. Let $x\in\mathbb{R}^{d}$.
We consider the mapping $F_{x}:\mathbb{R}^{d}\longrightarrow\mathbb{F}_{N}$
defined, for every $y\in\mathbb{R}^{d}$, by
$F_{x}(y):[\frac{1}{N},N]\longrightarrow\mathbb{C}$ such that

$[F_{x}(y)](t)=f(y)\mathfrak{U}_{k,j}^{E,0}(x-y,t),\quad
t\in\Big{[}\frac{1}{N},N\Big{]}.$

We observe that $F_{x}(y)$, $y\in\mathbb{R}^{d}$, is continuous in $[1/N,N]$.
Thus, $F_{x}$ is continuous in $\mathbb{R}^{d}$. Indeed, let
$y_{0}\in\mathbb{R}^{d}$. We can write, by (3.2) and Minkowski inequality

$\displaystyle\|F_{x}(y)-F_{x}(y_{0})\|_{\mathbb{F}_{N}}$ $\displaystyle\leq
C\Big{\|}\int_{0}^{\infty}\frac{t^{k+1}e^{-\frac{t^{2}}{8u}}}{u^{(k+2)/2}}|f(y)\partial_{x_{j}}W_{u}(x-y)-f(y_{0})\partial_{x_{j}}W_{u}(x-y_{0})|du\Big{\|}_{\mathbb{F}_{N}}$
$\displaystyle\leq
C\Big{\|}\int_{0}^{\infty}te^{-v}v^{k/2-1}|f(y)\partial_{x_{j}}W_{\frac{t^{2}}{8v}}(x-y)-f(y_{0})\partial_{x_{j}}W_{\frac{t^{2}}{8v}}(x-y_{0})|dv\Big{\|}_{\mathbb{F}_{N}}$
$\displaystyle\leq
C\int_{0}^{\infty}e^{-v}v^{k/2-1}\big{\|}t(f(y)\partial_{x_{j}}W_{\frac{t^{2}}{8v}}(x-y)-f(y_{0})\partial_{x_{j}}W_{\frac{t^{2}}{8v}}(x-y_{0}))\big{\|}_{\mathbb{F}_{N}}dv,\quad
y\in\mathbb{R}^{d}.$

Since

$\displaystyle\big{\|}t(f(y)\partial_{x_{j}}W_{\frac{t^{2}}{8v}}(x-y)-f(y_{0})\partial_{x_{j}}W_{\frac{t^{2}}{8v}}(x-y_{0}))\big{\|}_{\mathbb{F}_{N}}$
$\displaystyle=(2\pi)^{-d/2}\left(\int_{\frac{1}{N}}^{N}t\Big{(}|f(y)(x_{j}-y_{j})e^{-4\frac{|x-y|^{2}v}{t^{2}}}-f(y_{0})(x_{j}-(y_{0})_{j})e^{-4\frac{|x-y_{0}|^{2}v}{t^{2}}}|\Big{(}\frac{8v}{t^{2}}\Big{)}^{{\frac{d}{2}}+1}\Big{)}^{2}dt\right)^{\frac{1}{2}}$
$\displaystyle\leq
C\left(\int_{\frac{1}{N}}^{N}\frac{v^{d+2}}{t^{2d+3}}\big{(}|x-y|^{2}e^{-8\frac{|x-y|^{2}v}{t^{2}}}+|x-y_{0}|^{2}e^{-8\frac{|x-y_{0}|^{2}v}{t^{2}}}\big{)}dt\right)^{2}$
$\displaystyle\leq
Cv^{\frac{d+1}{2}}\int_{\frac{1}{N}}^{N}\frac{dt}{t^{2d+1}}\leq
Cv^{\frac{d+1}{2}},\quad y\in\mathbb{R}^{d}\mbox{ and }v>0,$

by using the dominated convergence theorem we deduce that

$\lim_{y\rightarrow y_{0}}\|F_{x}(y)-F_{x}(y_{0})\|_{\mathbb{F}_{N}}=0.$

Since $\mathbb{F}_{N}$ is a separable Banach space, Pettis’ Theorem ([41,
Theorem p. 131]) implies that $F_{x}$ is $\mathbb{F}_{N}$ -strongly
measurable.

By (4.2.1) we get

$\int_{\mathbb{R}^{d}}\|\mathfrak{U}_{k,j}^{E,0}(x-y,\cdot)\|_{\mathbb{F}_{N}}|f(y)|dy<\infty,\quad
x\not\in{\rm supp}f.$

We define

$\widetilde{U}_{k,j}^{E,0}(f)(x)=\int_{\mathbb{R}^{d}}\mathfrak{U}_{k,j}^{E,0}(x-y,\cdot)f(y)dy,\quad
x\not\in{\rm supp}f,$

where the integral is understood in the $\mathbb{F}_{N}$-Bochner sense.
Suppose that $g\in\mathbb{F}_{N}$. We have that

$\int_{\frac{1}{N}}^{N}\int_{\mathbb{R}^{d}}|\mathfrak{U}_{k,j}^{E,0}(x-y,t)f(y)g(t)|dydt\leq
C\|g\|_{\mathbb{F}_{N}}\int_{\mathbb{R}^{d}}\frac{|f(y)|}{|x-y|^{d}}dy<\infty,\quad
x\not\in{\rm supp}f.$

We can write

$\int_{\frac{1}{N}}^{N}g(t)\Big{[}\widetilde{U}_{k,j}^{E,0}(f)(x)\Big{]}(t)\frac{dt}{t}=\int_{\mathbb{R}^{d}}f(y)\int_{\frac{1}{N}}^{N}\mathfrak{U}_{k,j}^{E,0}(x-y,t)g(t)\frac{dt}{t}dy=\int_{\frac{1}{N}}^{N}g(t)U_{k,j}^{E,0}(f)(x,t)\frac{dt}{t},\quad
x\not\in{\rm supp}f.$

Hence, for every $x\not\in{\rm supp}f$,
$\widetilde{U}_{k,j}^{E,0}(f)(x)=U_{k,j}^{E,0}(f)(x,\cdot)$, in
$\mathbb{F}_{N}$.

We also have that

(4.5)
$\int_{\mathbb{R}^{d}}\Big{\|}\widetilde{U}_{k,j}^{E,0}(f)(x)\Big{\|}_{\mathbb{F}_{N}}^{2}dx\leq\int_{\mathbb{R}^{d}}|\mathfrak{g}_{k,j}^{E,0}(f)(x)|^{2}dx\leq
C\|f\|_{L^{2}(\mathbb{R}^{d},dx)}^{2}.$

According to (4.2.1), (4.4) and (4.5) and by taking into account that the
constant $C$ in (4.5) does not depend on $N$, the vector-valued Calderón-
Zygmund theory ([39]) allows us to conclude that $\widetilde{U}_{k,j}^{E,0}$
defines a bounded operator from

(i) $L^{p}(\mathbb{R}^{d},dx)$ into
$L_{\mathbb{F}_{N}}^{p}(\mathbb{R}^{d},dx)$ and

$\sup_{\begin{subarray}{c}N\in\mathbb{N}\\\ N\geq
2\end{subarray}}\Big{\|}\widetilde{U}_{k,j}^{E,0}\Big{\|}_{L^{p}(\mathbb{R}^{d},dx)\longrightarrow
L^{p}_{\mathbb{F}_{N}}(\mathbb{R}^{d},dx)}<\infty,$

for every $1<p<\infty$;

(ii) $L^{1}(\mathbb{R}^{d},dx)$ into
$L_{\mathbb{F}_{N}}^{1,\infty}(\mathbb{R}^{d},dx)$ and

$\sup_{\begin{subarray}{c}N\in\mathbb{N}\\\ N\geq
2\end{subarray}}\Big{\|}\widetilde{U}_{k,j}^{E,0}\Big{\|}_{L^{1}(\mathbb{R}^{d},dx)\longrightarrow
L_{\mathbb{F}_{N}}^{1,\infty}(\mathbb{R}^{d},dx)}<\infty.$

By using monotone convergence theorem we deduce that
$\mathfrak{g}_{k,j}^{E,0}$ is bounded from $L^{p}(\mathbb{R}^{d},dx)$ into
itself, for every $1<p<\infty$, and from $L^{1}(\mathbb{R}^{d},dx)$ into
$L^{1,\infty}(\mathbb{R}^{d},dx)$.

#### 4.2.2.

We are going to study the local Littlewood-Paley function
$\mathbb{G}_{k,j,M,{\rm loc}}^{E,h}$ defined by

$\mathbb{G}_{k,j,M,{\rm
loc}}^{E,h}(f)(x)=\left(\int_{0}^{\infty}\Big{|}\int_{\mathbb{R}^{d}}\mathfrak{U}_{k,j,M}^{E,h}(x-y,t)f(y)\varphi(x,y)dy\Big{|}^{2}\frac{dt}{t}\right)^{\frac{1}{2}},\quad
x\in\mathbb{R}^{d}.$

Suppose that $h>0$. Minkowski inequality and (3.11) lead to

$\displaystyle\mathbb{G}_{k,j,M,{\rm loc}}^{E,h}(f)(x)$ $\displaystyle\leq
C\int_{E}\Big{\|}t^{k+\frac{1}{2}}\partial_{t}^{k}\big{[}t^{-M}e^{-\frac{u+h}{t}}\big{]}\Big{\|}_{L^{2}((0,\infty),\frac{dt}{t})}\frac{(u+h)^{M-1}}{u}\int_{\mathbb{R}^{d}}|x_{j}-y_{j}|W_{u}(x-y)|f(y)|\varphi(x,y)dydu$
$\displaystyle\leq
C\int_{E}\frac{1}{u\sqrt{u+h}}\int_{\mathbb{R}^{d}}|x_{j}-y_{j}|W_{u}(x-y)|f(y)|\varphi(x,y)dydu\leq
C\sup_{v>0}|W_{v}(|f|)(x)|,\quad x\in\mathbb{R}^{d}.$

It follows that $\mathbb{G}_{k,j,M,{\rm loc}}^{E,h}$ is bounded from
$L^{p}(\mathbb{R}^{d},dx)$ into itself, for every $1<p<\infty$, and from
$L^{1}(\mathbb{R}^{d},dx)$ into $L^{1,\infty}(\mathbb{R}^{d},dx)$.

In order to study the operator $\mathbb{G}_{k,j,M,{\rm loc}}^{E,0}$ we use the
vector-valued Calderón-Zygmund theory ([39]).

We define

$\mathbb{G}_{k,j,M}^{E,0}(f)(x)=\left(\int_{0}^{\infty}\left|\int_{\mathbb{R}^{d}}\mathfrak{U}_{k,j,M}^{E,0}(x-y,t)f(y)dy\right|^{2}\frac{dt}{t}\right)^{\frac{1}{2}},\quad
x\in\mathbb{R}^{d}.$

By using Plancherel equality, Minkowski inequality and (3.11) we obtain

$\displaystyle\int_{\mathbb{R}^{d}}|\mathbb{G}_{k,j,M}^{E,0}(f)(x)|^{2}dx$
$\displaystyle=\int_{0}^{\infty}\int_{\mathbb{R}^{d}}\left|\int_{\mathbb{R}^{d}}\mathfrak{U}_{k,j,M}^{E,0}(x-y,t)f(y)dy\right|^{2}dx\frac{dt}{t}$
$\displaystyle=\int_{0}^{\infty}\int_{\mathbb{R}^{d}}\left|\mathcal{F}(f)(z)\frac{t^{k+\frac{1}{2}}}{\Gamma(M)}\int_{E}\partial_{t}^{k}\big{[}t^{-M}e^{-\frac{u}{t}}\big{]}\mathcal{F}(\partial_{x_{j}}W_{u})(z)u^{M-1}dy\right|^{2}dz\frac{dt}{t}$
$\displaystyle\leq
C\int_{\mathbb{R}^{d}}|\mathcal{F}(f)(z)|^{2}\left(\int_{0}^{\infty}|z|e^{-u|z|^{2}}\frac{du}{\sqrt{u}}\right)^{2}dz\leq
C\|f\|_{L^{2}(\mathbb{R}^{d},dx)}^{2}.$

We consider the operator $\mathbb{U}_{k,j,M}^{E,0}$ defined by

$\mathbb{U}_{k,j,M}^{E,0}(f)(x,t)=\int_{\mathbb{R}^{d}}\mathfrak{U}_{k,j,M}^{E,0}(x-y,t)f(y)dy,\quad
x\in\mathbb{R}^{d}\mbox{ and }t>0.$

We have that

$\mathbb{G}_{k,j,M}^{E,0}(f)(x)=\left\|\mathbb{U}_{k,j,M}^{E,0}(f)(x,\cdot)\right\|_{L^{2}((0,\infty),\frac{dt}{t})},\quad
x\in\mathbb{R}^{d}.$

Minkowski inequality and (3.11) lead to

$\Big{\|}\mathfrak{U}_{k,j,M}^{E,0}(x-y,\cdot)\Big{\|}_{L^{2}((0,\infty),\frac{dt}{t})}\leq
C\int_{0}^{\infty}\frac{e^{-\frac{|x-y|^{2}}{u}}}{u^{\frac{d}{2}+1}}du\leq\frac{C}{|x-y|^{d}},\quad
x,y\in\mathbb{R}^{d},\;x\not=y.$

We also obtain, for every $i=1,..,d$,

$\Big{\|}\partial_{x_{i}}\mathfrak{U}_{k,j,M}^{E,0}(x-y,\cdot)\Big{\|}_{L^{2}((0,\infty),\frac{dt}{t})}+\Big{\|}\partial_{y_{i}}\mathfrak{U}_{k,j,M}^{E,0}(x-y,\cdot)\Big{\|}_{L^{2}((0,\infty),\frac{dt}{t})}\leq\frac{C}{|x-y|^{d+1}},\quad
x,y\in\mathbb{R}^{d},\;x\not=y.$

By proceeding as in the previous case we can conclude that the operator
$\mathbb{G}_{k,j,M,{\rm loc}}^{E,0}$ is bounded from
$L^{p}(\mathbb{R}^{d},dx)$ into itself, for every $1<p<\infty$, and from
$L^{1}(\mathbb{R}^{d},dx)$ into $L^{1,\infty}(\mathbb{R}^{d},dx)$.

###  4.3. Variation operators

#### 4.3.1.

We study the local variation operator defined by

$\mathfrak{v}^{E,h}_{\rho,k,j,\text{loc}}(f)(x)=V_{\rho}\left(t\rightarrow
U^{E,h}_{k,j,\text{loc}}(f)(x,t)\right),\quad x\in\mathbb{R}^{d},$

where

$U^{E,h}_{k,j}(f)(x,t)=\int_{\mathbb{R}^{d}}\mathfrak{U}^{E,h}_{k,j}(x-y,t)f(y)dy,\quad
x\in\mathbb{R}^{d}\text{ and }t>0.$

By using (3.1) we can proceed as in section 4.1.1 to prove that, when $h>0$,
$\mathfrak{v}^{E,h}_{\rho,k,j,\text{loc}}$ is bounded from
$L^{p}(\mathbb{R}^{d},dx)$ into itself, for every, $1<p<\infty$, and from
$L^{1}(\mathbb{R}^{d},dx)$ into $L^{1,\infty}(\mathbb{R}^{d},dx)$.

We now study the variation operator
$\mathfrak{v}^{E,0}_{\rho,k,j,\text{loc}}$. We consider first, $E=(0,\infty)$.
The classical Poisson kernel is given by

$\mathbb{P}_{t}(z)=\frac{\Gamma(\frac{d+1}{2})}{\pi^{\frac{d+1}{2}}}\frac{t}{(t^{2}+|z|^{2})^{\frac{d+1}{2}}},\quad
z\in\mathbb{R}^{d}\text{ and }t>0.$

We have that

$\partial^{k+1}_{t}\left[\mathbb{P}_{t}(z)\right]=\partial^{k+2}_{t}\left[\frac{\Gamma(\frac{d+1}{2})}{\pi^{\frac{d+1}{2}}(1-d)}\frac{1}{(t^{2}+|z|^{2})^{\frac{d-1}{2}}}\right],\quad
z\in\mathbb{R}^{d}\text{ and }t>0.$

According to Faà di Bruno’s formula

$\partial^{k+2}_{t}\left[\frac{1}{(t^{2}+|z|^{2})^{\frac{d-1}{2}}}\right]=\sum_{0\leq\ell\leq\frac{k+2}{2}}a_{\ell}t^{k+2-2\ell}\frac{1}{(t^{2}+|z|^{2})^{\frac{d-1}{2}+k+2-\ell}},\quad
z\in\mathbb{R}^{d}\text{ and }t>0,$

for certain $a_{\ell}\in\mathbb{R}$, $0\leq\ell\leq\frac{k+1}{2}$,
$\ell\in\mathbb{N}$. Then,

$t^{k+1}\partial^{k+1}_{t}\mathbb{P}_{t}(z)=\phi_{t}(z),\quad
z\in\mathbb{R}^{d}\text{ and }t>0,$

where $\phi_{t}(z)=t^{-d}\phi(z/t)$, $z\in\mathbb{R}^{d}$ and $t>0$, and

$\phi(z)=\sum_{0\leq\ell\leq\frac{k+2}{2}}\frac{a_{\ell}}{(1+|z|^{2})^{\frac{d-1}{2}+k+2-\ell}},\quad
z\in\mathbb{R}^{d}.$

On the other hand, given that $\mathbb{P}_{t}$ is a radial function, the
relation between the Fourier and the Hankel Transform, leads to (see [11, p. 7
(4) and p. 24 (18)]),

$\mathcal{F}(\partial_{x_{j}}\partial^{k}_{t}\mathbb{P}_{t})(z)=iz_{j}|z|^{k}e^{-t|z|}=i\frac{z_{j}}{|z|}|z|^{k+1}e^{-t|z|},\quad
z\in\mathbb{R}^{d}\setminus\\{0\\}\text{ and }t>0.$

We get

$t^{k+1}\partial_{x_{j}}\partial^{k}_{t}\mathbb{P}_{t}(f)(x)=Ct^{k+1}\partial^{k+1}_{t}\mathbb{P}_{t}(R_{j}f)(x)=C(\phi_{t}*R_{j}f)(x),\quad
x\in\mathbb{R}^{d}\text{ and }t>0,$

for a certain $C\in\mathbb{R}$. Here, ${R}_{j}$ denotes the $j$-th Euclidean
Riesz transform.

We define

$\psi(u)=\sum_{0\leq\ell\leq\frac{k+2}{2}}\frac{a_{\ell}}{(1+u^{2})^{\frac{d+1}{2}+k+2-\ell}},\quad
u\in(0,\infty).$

It is clear that $\phi(z)=\psi(|z|)$, $z\in\mathbb{R}^{d}$. We have that
$\psi(u)\rightarrow 0$ as $u\rightarrow\infty$, and
$\int_{0}^{\infty}|\psi^{\prime}(u)|u^{d}du<\infty$. According to Lemma 2.4 in
[4], the variation operator associated with $\\{T_{t}\\}_{t>0}$ where
$T_{t}f=\phi_{t}*f$, $t>0$, is bounded from $L^{p}(\mathbb{R}^{d},dx)$ into
itself, for every $1<p<\infty$, and from $L^{1}(\mathbb{R}^{d},dx)$ into
$L^{1,\infty}(\mathbb{R}^{d},dx)$. Since $R_{j}$ is bounded from
$L^{p}(\mathbb{R}^{d},dx)$ into itself, for every $1<p<\infty$ we derive the
same boundedness property for $\mathfrak{v}^{(0,\infty),0}_{\rho,k,j}$.

In order to see that $\mathfrak{v}^{(0,\infty),0}_{\rho,k,j}$ is bounded from
$L^{1}(\mathbb{R}^{d},dx)$ into $L^{1,\infty}(\mathbb{R}^{d},dx)$, we use
vector valued Calderón-Zygmund theory.

According to (3.1) and Minkoswki inequality we obtain

(4.6)
$\begin{split}\Big{\|}\mathfrak{U}^{(0,\infty),0}_{k,j}(x-y,\cdot)\Big{\|}_{V_{\rho}}&\leq
C\int_{0}^{\infty}\frac{\|t^{k+1}\partial^{k}_{t}[te^{\frac{-t^{2}}{4u}}]\|_{V_{\rho}}}{u^{\frac{3}{2}}}\frac{|x_{j}-y_{j}|}{u^{\frac{d}{2}+1}}e^{\frac{-|x-y|^{2}}{4u}}du\\\
&\leq
C\int_{0}^{\infty}\frac{e^{\frac{-c|x-y|^{2}}{u}}}{u^{\frac{d}{2}+1}}du\leq\frac{C}{|x-y|^{d}},\quad
x,y\in\mathbb{R}^{d},\;x\neq y,\end{split}$

and, in a similar way, for every $i=1,...,d$,

(4.7)
$\Big{\|}\partial_{x_{i}}\mathfrak{U}^{(0,\infty),0}_{k,j}(x-y,\cdot)\Big{\|}_{V_{\rho}}+\Big{\|}\partial_{y_{i}}\mathfrak{U}^{(0,\infty),0}_{k,j}(x-y,\cdot)\Big{\|}_{V_{\rho}}\leq\frac{C}{|x-y|^{d+1}},\quad
x,y\in\mathbb{R}^{d},\;x\neq y.$

Let $N\in\mathbb{N}$, $N\geq 2$. We consider the space
$V_{\rho}\left(\left[\frac{1}{N},N\right]\right)$ consisting in all $g\in
C\left(\left[\frac{1}{N},N\right]\right)$ such that

$V^{N}_{\rho}(g)=\sup_{\begin{subarray}{c}\frac{1}{N}<t_{\ell}<t_{\ell-1}<\dots<t_{1}<N\\\
\ell\in\mathbb{N}\end{subarray}}\left(\sum_{i=1}^{\ell-1}|g(t_{i+1})-g(t_{i})|^{\rho}\right)^{\frac{1}{\rho}}<\infty.$

By identifying those functions that differ in a constant
$\left(V_{\rho}\left(\left[\frac{1}{N},N\right]\right),V^{N}_{\rho}\right)$ is
a Banach space. Assume that $f\in C_{c}^{\infty}(\mathbb{R}^{d})$. Let
$x\in\mathbb{R}^{d}$. We define, for every $y\in\mathbb{R}^{d}$, the function
$F_{x}(y)\in C\left(\left[\frac{1}{N},N\right]\right)$ defined by

$[F_{x}(y)](t)=f(y)\mathfrak{U}^{(0,\infty),0}_{k,j}(x-y,t),\quad
t\in\left[\tfrac{1}{N},N\right].$

By [1, Lemma 4] we get

$\begin{split}V^{N}_{\rho}(F_{x}(y))&\leq
C\int_{\frac{1}{N}}^{N}\int_{0}^{\infty}\left|\partial_{t}\left[t^{k+1}\partial^{k}_{t}\left(te^{-\frac{t^{2}}{4u}}\right)\right]\right|\frac{|x_{j}-y_{j}|}{u^{\frac{d}{2}+\frac{5}{2}}}e^{-\frac{|x-y|^{2}}{2u}}dudt\\\
&\leq
C\int_{\frac{1}{N}}^{N}\int_{0}^{\infty}\left(\frac{t^{k+1}}{u^{\frac{k}{2}}}+\frac{t^{k}}{u^{\frac{k-1}{2}}}\right)\frac{e^{-\frac{t^{2}}{8u}}}{u^{\frac{d}{2}+2}}e^{-c\frac{|x-y|^{2}}{u}}dudt\\\
&\leq
C\int_{0}^{\infty}\frac{e^{-c\frac{N^{-2}+|x-y|^{2}}{u}}}{u^{\frac{d+3}{2}}}du\\\
&\leq\frac{C}{(N^{-2}+|x-y|^{2})^{d+1}},\quad y\in\mathbb{R}^{d}.\end{split}$

We define the mapping

$\begin{split}F_{x}:\mathbb{R}^{d}&\longrightarrow
V_{\rho}\left(\left[\tfrac{1}{N},N\right]\right)\\\ y&\longrightarrow
F_{x}(y).\end{split}$

$F_{x}$ is continuous. Indeed, let $y_{0}\in\mathbb{R}^{d}$. By Lemma 4 in [1]
we obtain

$\begin{split}V^{N}_{\rho}&(F_{x}(y)-F_{x}(y_{0}))\\\ &\leq
C\int_{\frac{1}{N}}^{N}\int_{0}^{\infty}\frac{\left|\partial_{t}\left[t^{k+1}\partial^{k}_{t}\left(te^{-\frac{t^{2}}{4u}}\right)\right]\right|}{u^{\frac{3}{2}}}\left|\frac{x_{j}-y_{j}}{u}W_{u}(x-y)f(y)-\frac{x_{j}-(y_{0})_{j}}{u}W_{u}(x-y_{0})f(y_{0})\right|dudt\\\
&\leq
C\int_{\frac{1}{N}}^{N}\int_{0}^{\infty}\frac{t^{k}e^{-c\frac{t^{2}}{u}}}{u^{\frac{k+2}{2}}}\left|\frac{x_{j}-y_{j}}{u}W_{u}(x-y)f(y)-\frac{x_{j}-(y_{0})_{j}}{u}W_{u}(x-y_{0})f(y_{0})\right|dudt\\\
&\leq
C\int_{\frac{1}{N}}^{N}\int_{0}^{\infty}\frac{e^{-cv}v^{\frac{k}{2}}}{t^{2}}\left|(x_{j}-y_{j})W_{\frac{t^{2}}{v}}(x-y)f(y)-(x_{j}-(y_{0})_{j})W_{\frac{t^{2}}{v}}(x-y_{0})f(y_{0})\right|dvdt.\end{split}$

Since

$\int_{\frac{1}{N}}^{N}\frac{1}{t^{2}}\left|(x_{j}-y_{j})W_{\frac{t^{2}}{v}}(x-y)f(y)-(x_{j}-y_{0,j})W_{\frac{t^{2}}{v}}(x-y_{0})f(y_{0})\right|dt\leq
Cv^{{\frac{d-1}{2}}},\quad y\in\mathbb{R}^{d}\text{ and }v>0,$

the dominated convergence theorem leads to

$\lim_{y\rightarrow y_{0}}V_{\rho}^{N}(F_{x}(y)-F_{x}(y_{0}))=0.$

Since $F_{x}$ is continuous, it is
$V_{\rho}\left(\left[\tfrac{1}{N},N\right]\right)$-strongly measurable by
Pettis’ Theorem. Indeed, $F_{x}$ is weakly measurable. Furthermore, if
$\mathbb{Q}$ represents the set of rational numbers, we have that
$\displaystyle
F_{x}(\mathbb{R}^{d})=\overline{F_{x}(\mathbb{Q}^{d})}^{V_{\rho}\left(\left[\tfrac{1}{N},N\right]\right)}$.

By using (4.6) we get

$\int_{\mathbb{R}^{d}}\left\|\mathfrak{U}^{(0,\infty),0}_{k,j}(x-y,\cdot)\right\|_{V^{N}_{\rho}}|f(y)|dy\leq\int_{\mathbb{R}^{d}}\frac{|f(y)|}{|x-y|^{d}}dy<\infty,\quad
x\notin\text{supp}(f).$

We define

$\widetilde{M}^{(0,\infty),0}_{k,j}(f)(x)=\int_{\mathbb{R}^{d}}\mathfrak{U}^{(0,\infty),0}_{k,j}(x-y,\cdot)f(y)dy,\quad
x\notin\text{supp}(f),$

where the integral is understood in the $V_{\rho}$\- Bochner sense.

Let $0<a\neq 1$. We define the functional $T_{a}$ in
$V_{\rho}\left(\left[\tfrac{1}{N},N\right]\right)$ as follows

$T_{a}(g)=g(a)-g(1).$

It is clear that $T_{a}\in
V_{\rho}\left(\left[\tfrac{1}{N},N\right]\right)^{\prime}$, the dual space of
$V_{\rho}\left(\left[\tfrac{1}{N},N\right]\right)$. We have that

$\begin{split}T_{a}\left(\widetilde{M}^{(0,\infty),0}_{k,j}(f)(x)\right)&=\int_{\mathbb{R}^{d}}T_{a}\left(\mathfrak{U}^{(0,\infty),0}_{k,j}(x-y,\cdot)\right)f(y)dy\\\
&=U^{(0,\infty),0}_{k,j}(f)(x,a)-U^{(0,\infty),0}_{k,j}(f)(x,1),\quad
x\notin\text{supp}(f).\end{split}$

We deduce that, for every $x\notin\text{supp}(f)$, there exists
$c_{x}\in\mathbb{R}$ such that

$\left[\widetilde{M}^{(0,\infty),0}_{k,j}(f)(x)\right](t)=U^{(0,\infty),0}_{k,j}(f)(x,t)+c_{x},\quad
t\in(0,\infty),$

and

$\widetilde{M}^{(0,\infty),0}_{k,j}(f)(x)=U^{(0,\infty),0}_{k,j}(f)(x,\cdot)$

as elements of $V_{\rho}\left(\left[\tfrac{1}{N},N\right]\right)$.

Since $\mathfrak{v}^{(0,\infty),0}_{\rho,k,j}$ is bounded from
$L^{p}(\mathbb{R}^{d},dx)$ into itself, for every $1<p<\infty$, from (4.6) and
(4.7)we deduce that $\widetilde{M}^{(0,\infty),0}_{k,j}$ is bounded from
$L^{1}(\mathbb{R}^{d},dx)$ into
$L_{V_{\rho}([\tfrac{1}{N},N])}^{1,\infty}\left(\mathbb{R}^{d},dx\right)$.
Furthermore

$\sup_{N\in\mathbb{N},N\geq
2}\|\widetilde{M}^{(0,\infty),0}_{k,j}\|_{L^{1}(\mathbb{R}^{d},dx)\hookrightarrow
L_{V_{\rho}([1/N,N])}^{1,\infty}(\mathbb{R}^{d},dx)}<\infty.$

By using the monotone convergence theorem we conclude that
$\mathfrak{v}^{(0,\infty),0}_{\rho,k,j}$ is bounded from
$L^{1}(\mathbb{R}^{d},dx)$ into $L^{1,\infty}(\mathbb{R}^{d},dx)$.

Suppose now that $E\subset(\eta,\infty)$ for some $\eta>0$. By using Lemma 4
in [1] we get

$\begin{split}\|\mathfrak{U}^{E,0}_{k,j}(x-y,\cdot)\|_{V_{\rho}}&\leq
C\int_{E}\int_{0}^{\infty}\frac{\left|\partial_{t}\left[t^{k+1}\partial^{k}_{t}\left(te^{-\frac{t^{2}}{4u}}\right)\right]\right|}{u^{\frac{3}{2}}}\frac{|x_{j}-y_{j}|}{u}W_{u}(x-y)dtdu\\\
&\leq
C\int_{\eta}^{\infty}\int_{0}^{\infty}\frac{t^{k}e^{-c\frac{t^{2}}{u}}}{u^{\frac{k+4}{2}}}dt\,\frac{e^{-c\frac{|x-y|^{2}}{u}}}{u^{\frac{d-1}{2}}}du\\\
&\leq\frac{C}{|x-y|^{d-1}}\int_{\eta}^{\infty}\frac{du}{u^{\frac{3}{2}}}\int_{0}^{\infty}v^{\frac{k-1}{2}}e^{-v}dv\\\
&\leq C\frac{1+|x|}{|x-y|^{d-1}},\quad x\neq y.\end{split}$

According to Lemma 2.5, the operator
$\mathfrak{v}^{E,0}_{\rho,k,j,\text{loc}}$ is bounded from
$L^{p}(\mathbb{R}^{d},dx)$ into itself, for every $1<p<\infty$, and from
$L^{1}(\mathbb{R}^{d},dx)$ into $L^{1,\infty}(\mathbb{R}^{d},dx)$.

By combining the above results we deduce that if $E\subset(0,\eta)$, for some
$\eta>0$, the operator $\mathfrak{v}^{E,0}_{\rho,k,j,\text{loc}}$ is bounded
from $L^{p}(\mathbb{R}^{d},dx)$ into itself, for every $1<p<\infty$, and from
$L^{1}(\mathbb{R}^{d},dx)$ into $L^{1,\infty}(\mathbb{R}^{d},dx)$.

#### 4.3.2.

We consider the local variation operator defined by

$\mathfrak{v}^{E,h}_{\rho,k,j,M,\text{loc}}f(x)=V_{\rho}\left(t\rightarrow
U^{E,h}_{k,j,M,\text{loc}}(f)(x,t)\right),\quad x\in\mathbb{R}^{d},$

where

$U^{E,h}_{k,j,M}(f)(x,t)=\int_{\mathbb{R}^{d}}f(y)\mathfrak{U}^{E,h}_{k,j,M}(x-y,t)dy,\quad
x\in\mathbb{R}^{d}\text{ and }t>0.$

Minkowski inequality and (3.11) allow us, as in Section 4.1.2, to see that the
operator $\mathfrak{v}^{E,h}_{\rho,k,j,M,\text{loc}}$, when $h>0$, is bounded
from $L^{p}(\mathbb{R}^{d},dx)$ into itself, for every $1<p<\infty$, and from
$L^{1}(\mathbb{R}^{d},dx)$ into $L^{1,\infty}(\mathbb{R}^{d},dx)$.

We are going to study the variation operator
$\mathfrak{v}^{E,0}_{\rho,k,j,M,\text{loc}}$. In order to do this we consider
firstly the operator $\mathfrak{v}^{(0,\infty),0}_{\rho,k,j,M}$. We recall
that

$\mathfrak{U}_{k,j,M}^{(0,\infty),0}(z,t)=\frac{t^{k+1/2}}{\Gamma(M)}\partial^{k}_{t}\partial_{x_{j}}\int_{0}^{\infty}t^{-M}e^{-u/t}W_{u}(z)u^{M-1}du,\quad
z\in\mathbb R^{d}\mbox{ and }t>0.$

We can write

$\mathfrak{U}_{k,j,M}^{(0,\infty),0}(z,t)=L_{k,j,M}(z,t),\quad z\in\mathbb
R^{d}\mbox{ and }t>0,$

where

$L_{k,j,M}(z,t)=\frac{t^{k+1/2}}{\Gamma(M)}\partial^{k}_{t}\partial_{x_{j}}\int_{0}^{\infty}e^{-v}W_{vt}(z)v^{M-1}dv,\quad
z\in\mathbb R^{d}\mbox{ and }t>0.$

We have that
$\mathcal{F}(\partial_{z_{j}}W_{vt}(z))(w)=iw_{j}e^{-\frac{|w|^{2}tv}{2}}$,
$w\in\mathbb R^{d}$ and $t,v>0$. Then,

$\mathcal{F}(t^{k+1/2}\partial^{k}_{t}\partial_{z_{j}}W_{vt}(z))(w)=\frac{(-1)^{k}}{2^{k}}iw_{j}t^{k+1/2}v^{k}|w|^{2k}e^{-\frac{tv|w|^{2}}{2}},\quad
w\in\mathbb R^{d}\mbox{ and }t,v>0,$

and we get

$\displaystyle\mathcal{F}(U_{k,j,M}^{(0,\infty),0}(\cdot,t))(w)$
$\displaystyle=\mathcal{F}(f*L_{k,j,M}(\cdot,t))(w)=-i\frac{w_{j}}{|w|}\mathcal{F}(f)(w)\Psi_{k,t}(w)$
$\displaystyle=\mathcal{F}(R_{j}f)(w)\mathcal{F}(\mathcal{F}^{-1}(\Psi_{k,t}))(w),\quad
w\in\mathbb{R}^{d}\mbox{ and }t>0,$

where $R_{j}$ denotes the $j$-th Euclidean Riesz transformation and

$\Psi_{k,t}(w)=\frac{(-1)^{k+1}}{2^{k}\Gamma(M)}t^{k+\frac{1}{2}}\int_{0}^{\infty}e^{-v}v^{k+M-1}|w|^{2k+1}e^{-\frac{tv|w|^{2}}{2}}dv,\quad
w\in\mathbb{R}^{d}\mbox{ and }t>0.$

We consider the function $\psi_{k}(w)=|w|^{2k+1}e^{-a|w|^{2}}$, $w\in\mathbb
R^{d}$, where $a>0$. We define the Hankel transform $h_{\nu}$ by

$h_{\nu}(g)(s)=\int_{0}^{\infty}(sr)^{-\nu}J_{\nu}(sr)g(r)r^{2\nu+1}d\nu,\quad
s\in(0,\infty),$

where $J_{\nu}$ denotes the Bessel function of the first kind and order
$\nu>-1$. Since $\psi_{k}$ is a radial function we obtain

$\mathcal{F}^{-1}(\psi_{k})(x)=\frac{1}{(2\pi)^{d}}h_{\frac{d-2}{2}}(\widetilde{\psi}_{k})(|x|),\quad
x\in\mathbb R^{d},$

where $\widetilde{\psi}_{k}(r)=r^{2k+1}e^{-ar^{2}}$, $r\in(0,\infty)$.
According to [11, (14), p. 30] we get

$\mathcal{F}^{-1}(\psi_{k})(s)=c_{k}a^{-\frac{d+1}{2}-k}\;_{1}F_{1}\Big{(}\frac{d+1}{2}+k,\frac{d}{2};-\frac{s^{2}}{4a}\Big{)},\quad
s\in(0,\infty),$

for certain $c_{k}>0$. Here ${}_{1}F_{1}$ represents the Kummer’s confluent
hypergeometric function (see, for example, [23, §9]).

Thus,

$\displaystyle\mathcal{F}^{-1}(\Psi_{k,t})(x)$
$\displaystyle=\frac{(-1)^{k+1}2^{\frac{d+1}{2}}c_{k}}{\Gamma(M)}t^{-\frac{d}{2}}\int_{0}^{\infty}e^{-v}v^{M-1-\frac{d+1}{2}}\;_{1}F_{1}\Big{(}\frac{d+1}{2}+k,\frac{d}{2};-\frac{|x|^{2}}{2tv}\Big{)}dv,\quad
x\in\mathbb{R}^{d},$

and we can write

$U_{k,j,M}^{(0,\infty),0}(f)(x,t)=\big{(}(\phi_{k})_{\sqrt{t}}*R_{j}f\big{)}(x),\quad
x\in\mathbb{R}^{d}\mbox{ and }t>0,$

where

$\phi_{k}(x)=\frac{(-1)^{k+1}2^{\frac{d+1}{2}}c_{k}}{\Gamma(M)}\int_{0}^{\infty}e^{-v}v^{M-1-\frac{d+1}{2}}\;_{1}F_{1}\Big{(}\frac{d+1}{2}+k,\frac{d}{2};-\frac{|x|^{2}}{2v}\Big{)}dv,\quad
x\in\mathbb{R}^{d}.$

By taking into account [23, (9.11.2)] with the adequate correct form (see also
[35, Theorem 1]), for every $\alpha,\gamma\in\mathbb{R}$, with
$\gamma\not=0,-1,-2,...$, we have that
$\;{}_{1}F_{1}(\alpha,\gamma;z)=e^{z}\;_{1}F_{1}(\gamma-\alpha,\gamma;-z)$,
$z\in\mathbb{R}$. Then,

$\lim_{z\rightarrow+\infty}\;{}_{1}F_{1}\Big{(}\frac{d+1}{2}+k,\frac{d}{2};-z\Big{)}=0.$

Since $M>\frac{d+1}{2}$, we infer by the dominated convergence Theorem that
$\lim_{|x|\rightarrow+\infty}\phi_{k}(x)=0$.

On the other hand let us consider the function

$\widetilde{\phi}_{k}(r)=\frac{(-1)^{k+1}2^{\frac{d+1}{2}}c_{k}}{\Gamma(M)}\int_{0}^{\infty}e^{-v}v^{M-1-\frac{d+1}{2}}\;_{1}F_{1}\Big{(}\frac{d+1}{2}+k,\frac{d}{2};-\frac{r^{2}}{2v}\Big{)}dv,\quad
r\in(0,\infty).$

It is clear that $\phi_{k}(x)=\widetilde{\phi}_{k}(|x|)$,
$x\in\mathbb{R}^{d}$.

Again by [23, (9.11.2)] we deduce that

$\displaystyle\int_{0}^{\infty}\big{|}\frac{d}{dr}\widetilde{\phi}_{k}(r)\big{|}r^{d}dr$
$\displaystyle\leq
C\int_{0}^{\infty}e^{-v}v^{M-2-\frac{d+1}{2}}\int_{0}^{\infty}r^{d+1}\Big{|}\;_{1}F_{1}\Big{(}\frac{d+1}{2}+k+1,\frac{d}{2}+1;-\frac{r^{2}}{2v}\Big{)}\Big{|}drdv$
$\displaystyle\leq
C\int_{0}^{\infty}e^{-v}v^{M-2-\frac{d+1}{2}}\left(\int_{0}^{\sqrt{v}}+\int_{\sqrt{v}}^{\infty}\right)r^{d+1}\Big{|}\;_{1}F_{1}\Big{(}\frac{d+1}{2}+k+1,\frac{d}{2}+1;-\frac{r^{2}}{2v}\Big{)}\Big{|}drdv$
$\displaystyle\leq
C\left(\int_{0}^{\infty}e^{-v}v^{M-\frac{3}{2}}dv+\int_{0}^{\infty}e^{-v}v^{M-2-\frac{d+1}{2}}\int_{\sqrt{v}}^{\infty}r^{-2k-2}v^{\frac{d+1}{2}+k+1}drdv\right)$
$\displaystyle=C\int_{0}^{\infty}e^{-v}v^{M-\frac{3}{2}}dv<\infty.$

From [4, Lemma 2.4] we deduce that the variation operator
$V_{\rho}(t\rightarrow(\phi_{k})_{\sqrt{t}}*f)$ is bounded from
$L^{p}(\mathbb{R}^{d},dx)$ into itself, for every $1<p<\infty$, and from
$L^{1}(\mathbb{R}^{d},dx)$ into $L^{1,\infty}(\mathbb{R}^{d},dx)$. Since
$R_{j}$ is bounded from $L^{p}(\mathbb{R}^{d},dx)$ into itself, for every
$1<p<\infty$, we deduce that $U_{\rho,k,j,M}^{(0,\infty),0}$ is bounded from
$L^{p}(\mathbb{R}^{d},dx)$ into itself, for every $1<p<\infty$.

In order to prove that $\mathfrak{v}_{\rho,k,j,M}^{(0,\infty),0}$ is bounded
from $L^{1}(\mathbb{R}^{d},dx)$ into $L^{1,\infty}(\mathbb{R}^{d},dx)$ we can
use the vector-valued Calderón-Zygmund theory by using the techniques
developed to prove the corresponding property for the operator
$\mathfrak{v}_{\rho,k,j,{\rm loc}}^{(0,\infty),0}$ in section 4.3.1.

Suppose that $E\subset(\eta,\infty)$ for some $\eta>0$. From (3.11) it follows
that

$\left\|\mathfrak{U}_{k,j,M}^{E,0}(x,y,\cdot)\right\|_{V_{\rho}}\leq
C\int_{E}|\partial_{x_{j}}W_{u}(x-y)|\frac{du}{\sqrt{u}}\leq\frac{C}{|x-y|^{d-1}}\int_{\eta}^{\infty}\frac{du}{u^{\frac{3}{2}}}\leq
C\frac{1+|x|}{|x-y|^{d-1}},\quad x,y\in\mathbb R^{d},\;x\neq y.$

By using Lemma 2.5 we prove that $\mathfrak{v}_{\rho,k,j,M,{\rm loc}}^{E,0}$
is bounded from $L^{p}(\mathbb{R}^{d},dx)$ into itself, for every
$1<p<\infty$, and from $L^{1}(\mathbb{R}^{d},dx)$ into
$L^{1,\infty}(\mathbb{R}^{d},dx)$.

The above properties allow us to conclude that $\mathfrak{v}_{\rho,k,j,M,{\rm
loc}}^{E,0}$ is bounded from $L^{p}(\mathbb{R}^{d},dx)$ into itself, for every
$1<p<\infty$, and from $L^{1}(\mathbb{R}^{d},dx)$ into
$L^{1,\infty}(\mathbb{R}^{d},dx)$ provided that $E\subset(0,\eta)$, for some
$\eta>0$.

######  Remark 4.1.

Note that the properties proved in this section for maximal operators and
Littlewood-Paley functions hold for every subset $E$ of $(0,\infty)$. However,
we prove the properties for variation operators when $E\subset(0,\eta)$ or
$E\subset(\eta,\infty)$, for some $\eta>0$.

## References

  * [1] Betancor, J. J., Castro, A. J., Curbelo, J., Fariña, J. C., and Rodríguez-Mesa, L. Square functions in the Hermite setting for functions with values in UMD spaces.  Ann. Mat. Pura Appl. (4) 193, 5 (2014), 1397–1430.
  * [2] Bourgain, J. Pointwise ergodic theorems for arithmetic sets.  Inst. Hautes Études Sci. Publ. Math., 69 (1989), 5–45.  With an appendix by the author, Harry Furstenberg, Yitzhak Katznelson and Donald S. Ornstein.
  * [3] Bourgain, J., Mirek, M., Stein, E. M., and Wróbel, B. a. On dimension-free variational inequalities for averaging operators in $\mathbb R^{d}$.  Geom. Funct. Anal. 28, 1 (2018), 58–99.
  * [4] Campbell, J. T., Jones, R. L., Reinhold, K., and Wierdl, M. Oscillation and variation for the Hilbert transform.  Duke Math. J. 105, 1 (2000), 59–83.
  * [5] Casarino, V., Ciatti, P., and Sjögren, P. On the maximal operator of a general Ornstein-Uhlenbeck semigroup, 2020.
  * [6] Casarino, V., Ciatti, P., and Sjögren, P. The maximal operator of a normal Ornstein-Uhlenbeck semigroup is of weak type $(1,1)$.  Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 21 (2020), 385–410.
  * [7] Casarino, V., Ciatti, P., and Sjögren, P. Riesz transforms of a general Ornstein-Uhlenbeck semigroup.  Calc. Var. Partial Differential Equations 60, 4 (2021), Paper No. 135, 32.
  * [8] Da Prato, G., and Zabczyk, J. Stochastic equations in infinite dimensions, vol. 44 of Encyclopedia of Mathematics and its Applications.  Cambridge University Press, Cambridge, 1992.
  * [9] Dunford, N., and Schwartz, J. T. Linear operators. Part I.  Wiley Classics Library. John Wiley & Sons, Inc., New York, 1988.  General theory, With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1958 original, A Wiley-Interscience Publication.
  * [10] Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G. Tables of integral transforms. Vol. I.  McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954.  Based, in part, on notes left by Harry Bateman.
  * [11] Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G. Tables of integral transforms. Vol. II.  McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954.  Based, in part, on notes left by Harry Bateman.
  * [12] Forzani, L., Harboure, E., and Scotto, R. Weak type inequality for a family of singular integral operators related with the Gaussian measure.  Potential Anal. 31, 2 (2009), 103–116.
  * [13] Forzani, L., Scotto, R., and Urbina, W. Riesz and Bessel potentials, the $g^{k}$ functions and an area function for the Gaussian measure $\gamma$.  Rev. Un. Mat. Argentina 42, 1 (2000), 17–37 (2001).
  * [14] Frey, D., McIntosh, A., and Portal, P. Conical square function estimates and functional calculi for perturbed Hodge-Dirac operators in $L^{P}$.  J. Anal. Math. 134, 2 (2018), 399–453.
  * [15] García-Cuerva, J., Mauceri, G., Meda, S., Sjögren, P., and Torrea, J. L. Functional calculus for the Ornstein-Uhlenbeck operator.  J. Funct. Anal. 183, 2 (2001), 413–450.
  * [16] García-Cuerva, J., Mauceri, G., Meda, S., Sjögren, P., and Torrea, J. L. Maximal operators for the holomorphic Ornstein-Uhlenbeck semigroup.  J. London Math. Soc. (2) 67, 1 (2003), 219–234.
  * [17] García-Cuerva, J., Mauceri, G., Sjögren, P., and Torrea, J.-L. Higher-order Riesz operators for the Ornstein-Uhlenbeck semigroup.  Potential Anal. 10, 4 (1999), 379–407.
  * [18] García-Cuerva, J., Mauceri, G., Sjögren, P., and Torrea, J. L. Spectral multipliers for the Ornstein-Uhlenbeck semigroup.  J. Anal. Math. 78 (1999), 281–305.
  * [19] Gutiérrez, C. E. On the Riesz transforms for Gaussian measures.  J. Funct. Anal. 120, 1 (1994), 107–134.
  * [20] Gutiérrez, C. E., Segovia, C., and Torrea, J. L. On higher Riesz transforms for Gaussian measures.  J. Fourier Anal. Appl. 2, 6 (1996), 583–596.
  * [21] Harboure, E., Macías, R. A., Menárguez, M. T., and Torrea, J. L. Oscillation and variation for the Gaussian Riesz transforms and Poisson integral.  Proc. Roy. Soc. Edinburgh Sect. A 135, 1 (2005), 85–104.
  * [22] Harboure, E., Torrea, J. L., and Viviani, B. Vector-valued extensions of operators related to the Ornstein-Uhlenbeck semigroup.  J. Anal. Math. 91 (2003), 1–29.
  * [23] Lebedev, N. N. Special functions and their applications.  Dover Publications, Inc., New York, 1972.  Revised edition, translated from the Russian and edited by Richard A. Silverman, Unabridged and corrected republication.
  * [24] Lépingle, D. La variation d’ordre $p$ des semi-martingales.  Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 36, 4 (1976), 295–316.
  * [25] Li, H.-Q., and Sjögren, P. Weak type $(1,1)$ bounds for some operators related to the Laplacian with drift on real hyperbolic spaces.  Potential Anal. 46, 3 (2017), 463–484.
  * [26] Ma, T., Torrea, J. L., and Xu, Q. Weighted variation inequalities for differential operators and singular integrals.  J. Funct. Anal. 268, 2 (2015), 376–416.
  * [27] Ma, T., Torrea, J. L., and Xu, Q. Weighted variation inequalities for differential operators and singular integrals in higher dimensions.  Sci. China Math. 60, 8 (2017), 1419–1442.
  * [28] Mauceri, G., and Noselli, L. Riesz transforms for a non-symmetric Ornstein-Uhlenbeck semigroup.  Semigroup Forum 77, 3 (2008), 380–398.
  * [29] Mauceri, G., and Noselli, L. The maximal operator associated to a nonsymmetric Ornstein-Uhlenbeck semigroup.  J. Fourier Anal. Appl. 15, 2 (2009), 179–200.
  * [30] Menárguez, T., Pérez, S., and Soria, F. Pointwise and norm estimates for operators associated with the Ornstein-Uhlenbeck semigroup.  C. R. Acad. Sci. Paris Sér. I Math. 326, 1 (1998), 25–30.
  * [31] Metafune, G., Prüss, J., Rhandi, A., and Schnaubelt, R. The domain of the Ornstein-Uhlenbeck operator on an $L^{p}$-space with invariant measure.  Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1, 2 (2002), 471–485.
  * [32] Mirek, M., Stein, E. M., and Zorin-Kranich, P. A bootstrapping approach to jump inequalities and their applications.  Anal. PDE 13, 2 (2020), 527–558.
  * [33] Mirek, M., Trojan, B., and Zorin-Kranich, P. Variational estimates for averages and truncated singular integrals along the prime numbers.  Trans. Amer. Math. Soc. 369, 8 (2017), 5403–5423.
  * [34] Muckenhoupt, B. Hermite conjugate expansions.  Trans. Amer. Math. Soc. 139 (1969), 243–260.
  * [35] Paris, R. B. Exponentially small expansions of the confluent hypergeometric functions.  Appl. Math. Sci. (Ruse) 7, 133-136 (2013), 6601–6609.
  * [36] Pérez, S. Boundedness of Littlewood-Paley $g$-functions of higher order associated with the Ornstein-Uhlenbeck semigroup.  Indiana Univ. Math. J. 50, 2 (2001), 1003–1014.
  * [37] Pérez, S. The local part and the strong type for operators related to the Gaussian measure.  J. Geom. Anal. 11, 3 (2001), 491–507.
  * [38] Pérez, S., and Soria, F. Operators associated with the Ornstein-Uhlenbeck semigroup.  J. London Math. Soc. (2) 61, 3 (2000), 857–871.
  * [39] Rubio de Francia, J. L., Ruiz, F. J., and Torrea, J. L. Calderón-Zygmund theory for operator-valued kernels.  Adv. in Math. 62, 1 (1986), 7–48.
  * [40] Sjögren, P. On the maximal function for the Mehler kernel.  In Harmonic analysis (Cortona, 1982), vol. 992 of Lecture Notes in Math. Springer, Berlin, 1983, pp. 73–82.
  * [41] Yosida, K. Functional analysis.  Classics in Mathematics. Springer-Verlag, Berlin, 1995.  Reprint of the sixth (1980) edition.
  * [42] Zorin-Kranich, P. Variation estimates for averages along primes and polynomials.  J. Funct. Anal. 268, 1 (2015), 210–238.