Maximal Operators, Littlewood–Paley Functions and Variation Operators Associated with Nonsymmetric Ornstein–Uhlenbeck Operators

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 semigroups associated with these Ornstein–Uhlenbeck operators are the basic building blocks of all the normal Ornstein–Uhlenbeck semigroups.

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 $d\in \mathbb {N}$, $d\ge 1$. 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

$$\begin{aligned} Q_t = \int _0^t e^{sB} Q e^{sB^*}ds, \end{aligned}$$

and the Gaussian measure $\gamma _t$ on ${\mathbb {R}}^d$ having mean zero and covariance matrix $Q_t$, defined by

$$\begin{aligned} d\gamma _t(x) = (2\pi )^{-\frac{d}{2}}(\det Q_t)^{-\frac{1}{2}} e^{-\frac{1}{2}\langle Q^{-1}_t x, x\rangle } dx. \end{aligned}$$

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

$$\begin{aligned} \mathcal {H}^{Q,B}_t(f)(x) = \int _{{\mathbb {R}}^d} f\left( e^{tB}x-y\right) d\gamma _t(y), \;\;\; x\in {\mathbb {R}}^d, \end{aligned}$$

(1.1)

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 $1\le p <\infty$.

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

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

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 $1\le p < \infty$, 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 [29], 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 ([30]) studied maximal operator and Riesz transforms in the one dimensional $\mathcal {L}^{I,-I}$-setting. Sjögren ([36]) 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 [15] and [28] (maximal operators); in [13, 32] and [34] (Littlewood–Paley functions); in [12, 16, 33] and [34] (Riesz transforms); in [14] and [17] (spectral multipliers) and in [20] (variation and oscillation operators). Gutiérrez, Segovia and Torrea ([19]) and Gutiérrez ([18]) 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 ([27]) and Riesz transforms ([26]) 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 ([4,5,6] and [7]) have extended the results about maximal operators and Riesz transforms due to Mauceri and Noselli ([26] and [27]).

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 ([26] and [27]).

Assume that $Q = I$ and $B = -\lambda (I + R)$ where $\lambda > 0$ and R is a skew-adjoint matrix as in [26] and [27]. After making a change of variables in (1.1) we get

$$\begin{aligned} \mathcal {H}^{Q,B}_t (f)(x) = \int _{\mathbb {R}^d} \widetilde{h}^{Q,B}_t (x,y) f(y) d\gamma _{\infty ,\lambda } (y), \quad x\in \mathbb {R}^d\text { and }t>0, \end{aligned}$$

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

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

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

$$\begin{aligned} P^{Q,B}_t(f) = \frac{t}{2\sqrt{\pi }} \int _0^{\infty } \frac{e^{-\frac{t^2}{4u}}}{u^{\frac{3}{2}}} \mathcal {H}^{Q,B}_u (f)du, \quad t>0. \end{aligned}$$

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

$$\begin{aligned} P^{Q,B}_{*,k,j} (f) = \sup _{t>0} |t^{k+1} \partial ^k_t \partial _{x_j} P^{Q,B}_t (f)|. \end{aligned}$$

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

$$\begin{aligned} g^{Q,B}_{k,j} (f) = \left( \int _0^\infty \left| t^{k+1} \partial ^k_t \partial _{x_j} P^{Q,B}_t (f)\right| ^2 \frac{dt}{t} \right) ^{\frac{1}{2}}. \end{aligned}$$

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

$$\begin{aligned} V_\rho (g) = \sup _{\begin{array}{c} 0<t_\ell<t_{\ell -1}<t_1\\ \ell \in \mathbb {N} \end{array}} \left( \sum _{n=1}^{\ell -1} |g(t_{n+1}) - g(t_n)|^\rho \right) ^{\frac{1}{\rho }}. \end{aligned}$$

(1.2)

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, a paper which has motivated a lot of researches in ergodic theory and harmonic analysis.

We consider the variation operator $V^{Q,B}_{\rho ,k,j}$, $k\in {\mathbb {N}}$, $j=1,...,d$, given by

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

Note that, for every $f\in L^p({\mathbb {R}}^d,\gamma _{\infty ,1})$, $1\le p<\infty$, 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 [3, 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, 32] and [34]. 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 [20].

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

$$\begin{aligned} S^{Q,B}_{*,k,j,M} (f)&= \sup _{t>0} |t^{k+\frac{1}{2}} \partial ^k_t \partial _{x_j} (I + t\mathcal {L}^{Q,B})^{-M}(f)|, \\ G^{Q,B}_{k,j,M} (f)&= \left( \int _0^\infty \left| t^{k+\frac{1}{2}} \partial ^k_t \partial _{x_j} (I + t\mathcal {L}^{Q,B})^{-M} (f)\right| ^2 \frac{dt}{t} \right) ^{\frac{1}{2}}, \end{aligned}$$

and

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

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 [30] 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 Sect. 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 Sects. 3 and 4.

Throughout this paper C and c always represent positive constants that can change at 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

$$\begin{aligned} d\gamma _\infty (x):=d\gamma _{\infty ,\lambda }(x)=\left( \frac{\pi }{\lambda }\right) ^{-d/2}e^{-\lambda |x|^2}dx. \end{aligned}$$

Actually we are going to work with $\lambda =1$. We define $B_1=-(I+R)$. We have that

$$\begin{aligned} {\mathcal {H}}_t^{Q,B}={\mathcal {U}}_\lambda ^{-1}{\mathcal {H}}_{\lambda t}^{Q,B_1}{\mathcal {U}}_\lambda ,\;\;\;\;t>0, \end{aligned}$$

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 $1\le p <\infty$.

After a change of variable we can write

$$\begin{aligned} P_t^{Q,B}(f)(x)= & {} \frac{t}{2\sqrt{\pi }}\int _0^\infty \frac{e^{-\frac{t^2}{4u}}}{u^{\frac{3}{2}}}U_\lambda ^{-1}{\mathcal {H}}_{\lambda u}^{Q,B_1}(U_\lambda f)(x)du \nonumber \\= & {} {\mathcal {U}}_\lambda ^{-1}\left[ \frac{\sqrt{\lambda }t}{2\sqrt{\pi }}\int _0^\infty \frac{e^{-\frac{\lambda t^2}{4v}}}{v^{\frac{3}{2}}}{\mathcal {H}}_{v}^{Q,B_1}(U_\lambda (f))(\cdot )dv\right] (x) \nonumber \\= & {} \mathcal U_\lambda ^{-1}\left[ P_{\sqrt{\lambda }t}^{Q,B_1}(U_\lambda (f))\right] (x),\;\;\;\;x\in \mathbb R^d\;\text{ and }\;t>0. \end{aligned}$$

(2.1)

By using (2.1) we deduce that, by denoting $T^{Q,B}$ every of the operators considered in Theorem 1.1, the following equality holds

$$\begin{aligned} T^{Q,B}={\mathcal {U}}_\lambda ^{-1}T^{Q,B_1}{\mathcal {U}}_\lambda . \end{aligned}$$

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

$$\begin{aligned} {\mathcal {H}}_t^{I,B_1}(f)(x)=\int _{\mathbb R^d}h_t^{I,B_1}(x,y)f(y)dy,\quad x\in {\mathbb {R}}^d \text{ and } t>0, \end{aligned}$$

where

$$\begin{aligned} h_t^{I,B_1}(x,y)=\frac{1}{(2\pi (1-e^{-2t}))^{d/2}}e^{-\frac{|e^{tB_1}x-y|^2}{1-e^{-2t}}},\quad x,y\in {\mathbb {R}}^d \text{ and } t>0. \end{aligned}$$

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

$$\begin{aligned} L_\sigma =\left\{ (x,y)\in {\mathbb {R}}^d\times {\mathbb {R}}^d:\;|x-y|\le \sigma \min \{1,|x+y|^{-1}\}\right\} , \end{aligned}$$

and $G_\sigma =({\mathbb {R}}^d\times {\mathbb {R}}^d){\setminus } L_\sigma$. $L_\sigma$ and $G_\sigma$ are named the $\sigma$-local and $\sigma$-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 $L_\sigma$ and $G_\sigma$.

Lemma 2.1

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

$$\begin{aligned} 0\le h_t^{I,B_1}(x,y)\le \frac{C}{(1-e^{-2t})^{d/2}}e^{-\frac{|x-y|^2}{2(1-e^{-2t})}},\quad (x,y)\in L_\sigma ,\;x\ne y \text{ and } t>0. \end{aligned}$$

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 [14]

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

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

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

As in [27, 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 [27, 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 $\delta \in (0,1)$.

• (i) There exists C and $t_0>0$ such that

  $$\begin{aligned} h_{\tau (s)}^{I,B_1}(x,y)\le Cs^{-\frac{d}{2}}e^{|x|^2-|y|^2- \frac{\delta }{4s}Q_s(x,y)},\quad x,y\in {\mathbb {R}}^d \text{ and } s\in \tau ^{-1}((0,t_0)). \end{aligned}$$

  (2.2)

• (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

  $$\begin{aligned} h_{\tau (s)}^{I,B_1}(x,y)\le Cs^{-\frac{d}{2}}e^{|x|^2-|y|^2-\frac{\delta }{4s}Q_s(x,y)},\quad x,y\in {\mathbb {R}}^d \text{ and } s\in \tau ^{-1}(J_D^\#).\nonumber \\ \end{aligned}$$

  (2.3)

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

$$\begin{aligned} (0,\infty ){\setminus } \mathcal {N}=\Big [\bigcup _{k=0}^n((0,t_0)+kt_0)\Big ]\cup \Big [\bigcup _{\ell =0}^m(J_D^\# +\ell \beta )\Big ], \end{aligned}$$

(2.4)

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

• (i) $0\le \varphi \le 1$, $x,y\in {\mathbb {R}}^d$;

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

• (iii) $|\nabla _x\varphi (x,y)|+|\nabla _y\varphi (x,y)|\le \frac{C}{|x-y|}$, $x,y\in {\mathbb {R}}^d$, $x\ne y$.

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 $T_{\textrm{loc}}$ of T by

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

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

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

The following results were proved in [15] 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

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

$$\begin{aligned}{} & {} \sup _{0<s<1}s^{-d/2}e^{ (-\frac{\delta }{s})Q_s(x,y)}\le C\min \{(1+|x|)^d, (|x|\sin \theta (x,y))^{-d}\},\;\;\;\;\\{} & {} \quad (x,y)\in G_1,\;x\ne 0\ne y. \end{aligned}$$

Lemma 2.4

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

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

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

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

where $\delta >0$. Operators of this type appear also when the symmetric case is considered (see, for instance, [20]).

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 $1\le p\le \infty$.

Proof

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

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

$$\begin{aligned} \int _{\{y\in {\mathbb {R}}^d:(x,y)\in L_\sigma \}}\frac{1+|x|}{|x-y|^{d-1}}dy\le C(1+|x|)\times \left\{ \begin{array}{ll} \displaystyle \int _0^{\sigma /|x|}dr, &{} |x|\ge \delta \\ &{} \\ \displaystyle \int _0^\sigma dr, &{} |x|\le \delta \end{array}\right. \le C. \end{aligned}$$

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 $1\le p\le \infty$. According to [17, Lemma 3.6] $S_\sigma$ is bounded from $L^p({\mathbb {R}}^d,\gamma _{\infty ,1})$ into itself, for every $1\le p\le \infty$. $\square$

We are going to explain the method we use to prove Theorem 1.1. We extend the procedure developed by Mauceri and Noselli ([26] and [27]).

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

$$\begin{aligned} T_{k,j}^X(f)(x)=\big \Vert t\rightarrow t^{k+1}\partial _t^k\partial _{x_j}P_t^{I,B_1}(f)(x)\big \Vert _X,\quad x\in {\mathbb {R}}^d. \end{aligned}$$

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 by 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

$$\begin{aligned} \partial _t^k\partial _{x_j}P_t^{I,B_1}(f)(x)=\int _{{\mathbb {R}}^d}f(y) \partial _t^k\partial _{x_j}P_t^{I,B_1}(x,y)dy,\quad x\in {\mathbb {R}}^d \text{ and } t>0,\nonumber \\ \end{aligned}$$

(2.5)

where $P_t^{I,B_1}(x,y)$, $x,y\in {\mathbb {R}}^d$ and $t>0$, denotes the Poisson integral kernel and

$$\begin{aligned}&\partial _t^k\partial _{x_j}P_t^{I,B_1}(x,y)=\frac{1}{2\sqrt{\pi }}\int _0^\infty \partial _t^k[te^{-\frac{t^2}{4u}}]\partial _{x_j}h_u^{I,B_1}(x,y)\frac{du}{u^{\frac{3}{2}}},\quad \\ {}&\quad x,y\in {\mathbb {R}}^d \text { and } t>0. \end{aligned}$$

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

$$\begin{aligned}&|\partial _{x_j}h_u^{I,B_1}(x,y)|\le C\frac{|e^{uB_1^*}||e^{uB_1}x-y|}{(1-e^{-2u})^{d/2+1}}e^{-|e^{uB_1}x-y|^2/(1-e^{-2u})},\quad \\&\quad x,y\in {\mathbb {R}}^d \text{ and } u>0. \end{aligned}$$

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

$$\begin{aligned} |\partial _{x_j}h_u^{I,B_1}(x,y)|\le C\frac{e^{-cu}}{(1-e^{-2u})^{(d+1)/2}},\quad x,y\in \mathbb {R}^d \text{ and } t>0. \end{aligned}$$

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

$$\begin{aligned}&\int _0^\infty |\partial _t^k[te^{-\frac{t^2}{4u}}]||\partial _{x_j}h_u^{I,B_1}(x,y)|\frac{du}{u^{\frac{3}{2}}}\le C\int _0^\infty \frac{e^{-\frac{t^2}{8u}-cu}}{u^{(k+2)/2}(1-e^{-2u})^{(d+1)/2}}du\\&\quad \le C\left( \int _1^\infty e^{-cu}du+\int _0^1\frac{e^{-\frac{t^2}{8u}}}{u^{(k+d+3)/2}}du\right) \le C(1+t^{-k-d-1}). \end{aligned}$$

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

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

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

where

$$\begin{aligned}{} & {} \mathfrak {s}_{k,j}^{E,h}(x,y,t)=\frac{t^{k+1}}{2\sqrt{\pi }}\int _E\partial _t^k[te^{-\frac{t^2}{u+h}}]\partial _{x_j}h_u^{I,B_1}(x,y)\frac{du}{(u+h)^{\frac{3}{2}}},\quad \\{} & {} \quad x,y\in \mathbb {R}^d \text{ and } t>0. \end{aligned}$$

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

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

Claim 1

Let $1\le p<\infty$. We define $\delta _p$ as follows

$$\begin{aligned} \delta _p=\left\{ \begin{array}{ll} \displaystyle \frac{1}{10}, &{} p=1, \\[0.3cm] \displaystyle \frac{1}{2}\left( 1+\max \left\{ 1-\frac{1}{d},1-\frac{1}{p}\right\} \right) , &{} 1<p<\infty . \end{array}\right. \end{aligned}$$

We denote by $E_p$ 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 $h\ge 0$, 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

$$\begin{aligned} (0,\infty ){\setminus }{\mathcal {N}}=\left[ \bigcup _{\ell =0}^n((0,t_0)+\ell t_0)\right] \cup \left[ \bigcup _{\ell =0}^m(J_P^\#+\ell \beta )\right] \end{aligned}$$

we can write

$$\begin{aligned} \int _0^\infty&\partial _t^k[te^{-\frac{t^2}{4u}}]\partial _{x_j}h_u^{I,B_1}(x,y)\frac{du}{u^{\frac{3}{2}}}\\&\quad =\left( \sum _{\ell =0}^n\int _{(0,t_0)+\ell t_0}+\sum _{\ell =0}^m\int _{J_D^\#+\ell \beta }\right) \partial _t^k[te^{-\frac{t^2}{4u}}]\partial _{x_j}h_u^{I,B_1}(x,y)\frac{du}{u^{\frac{3}{2}}}\\&\quad =\sum _{\ell =0}^n\int _{(0,t_0)}\partial _t^k[te^{-\frac{t^2}{4(u+\ell t_0)}}]\partial _{x_j}h_{u+\ell t_0}^{I,B_1}(x,y)\frac{du}{(u+\ell t_0)^{\frac{3}{2}}}\\&\qquad +\sum _{\ell =0}^m\int _{J_D^\#}\partial _t^k[te^{-\frac{t^2}{2(u+\ell \beta )}}]\partial _{x_j}h_{u+\ell \beta }^{I,B_1}(x,y)\frac{du}{(u+\ell \beta )^{\frac{3}{2}}},\quad \\&\qquad x,y\in \mathbb {R}^d \text{ and } t>0. \end{aligned}$$

By using the semigroup property of $\{H_t^{I,B_1}\}_{t>0}$ we deduce that

$$\begin{aligned}{} & {} t^{k+1}\partial _t^k\partial _{x_j}P_t^{I,B_1}(f)(x)=\sum _{\ell =0}^nS_{k,j}^{(0,t_0),\ell t_0}(H_{\ell t_0}^{I,B_1}(f))(x,t)\\{} & {} \quad +\sum _{\ell =0}^mS_{k,j}^{J_D^\#,\ell \beta }(H_{\ell \beta }^{I,B_1}(f))(x,t),\quad x\in \mathbb {R}^d \text{ and } t>0. \end{aligned}$$

Since the semigroup $\{H_t^{I,B_1}\}_{t>0}$ is contractive in $L^p(\mathbb {R}^d,\gamma _{\infty ,1})$, for every $1\le p<\infty$, 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 $\widetilde{S}_{k,j,\textrm{loc}}^{E,h}$ of $\widetilde{S}_{k,j}^{E,h}$. We consider the operator

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

where

$$\begin{aligned} \mathfrak {U}_{k,j}^{E,h}(z,t)=\frac{t^{k+1}}{2\sqrt{\pi }}\int _E\partial _t^k\big [te^{-\frac{t^2}{4(u+h)}}\big ]\partial _{x_j}W_u(z)\frac{du}{(u+h)^{\frac{3}{2}}},\quad z\in \mathbb {R}^d \text{ and } t>0. \end{aligned}$$

Here $W_u$, $u>0$ denotes the classical heat kernel given by

$$\begin{aligned} W_u(z)=\frac{e^{-\frac{|z|^2}{2u}}}{(2\pi u)^{d/2}},\quad z\in \mathbb {R}^d \text{ and } u>0. \end{aligned}$$

We also define

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

By using Minkowski inequality we deduce that

$$\begin{aligned}{} & {} \Big |\widetilde{S}_{k,j,{\textrm{loc}}}^{E,h}(f)(x)-\widetilde{U}_{k,j,\textrm{loc}}^{E,h}(f)(x)\Big |\le \int _{\mathbb {R}^d}\Big \Vert \mathfrak {s}_{k,j}^{E,h}(x,y,\cdot )\\{} & {} \quad -\mathfrak {U}_{k,j}^{E,h}(x-y,\cdot )\Big \Vert _X\varphi (x,y)|f(y)|dy,\quad x\in \mathbb {R}^d. \end{aligned}$$

Claim 2

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

$$\begin{aligned} D_{k,j,\textrm{loc}}^{E,h}(f)(x)=\int _{\mathbb {R}^d}\big \Vert \mathfrak {s}_{k,j}^{E,h}(x,y,\cdot )-\mathfrak {U}_{k,j}^{E,h}(x-y,\cdot )\Big \Vert _X\varphi (x,y)f(y)dy,\quad x\in \mathbb {R}^d, \end{aligned}$$

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 $1\le p\le \infty$.

Claim 3

Let $h\ge 0$ and $E\subset (0,\infty )$ such that $E\subset (0,\eta ,)$ or $E\subset (\eta ,\infty )$, for some $\eta >0$. The operator $\widetilde{U}_{k,j,\textrm{loc}}^{E,h}$ 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 [17, Lemma 3.6] from Claim 3 we deduce that the operator $\widetilde{U}_{k,j,\textrm{loc}}^{E,h}$ 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 $\widetilde{S}_{k,j,\textrm{loc}}^{E,h}(f)$ 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

$$\begin{aligned} \widetilde{S}_{k,j,\textrm{glob}}^{E,h}(f)(x)\le \int _{\mathbb {R}^d}\big \Vert \mathfrak {s}_{k,j}^{E,h}(x,y,\cdot )\big \Vert _X(1-\varphi (x,y))|f(y)|dy,\quad x\in \mathbb {R}^d. \end{aligned}$$

Claim 4

Let $h\ge 0$. Assume that $1\le p <\infty$ and that $E_p$ is the set associated to $\delta _p$ as in Claim 1. Then, the operator $\widetilde{V}_{k,j,\textrm{glob}}^{E_p,h}$ defined by

$$\begin{aligned} \widetilde{V}_{k,j,\textrm{glob}}^{E_p,h}(f)(x)=\int _{\mathbb {R}^d}\big \Vert \mathfrak {s}_{k,j}^{E_p,h}(x,y,\cdot )\big \Vert _X(1-\varphi (x,y))|f(y)|dy,\quad x\in \mathbb {R}^d, \end{aligned}$$

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 $M\ge 1$. We consider the operator $\mathbb {T}_{k,j,M}^X$ defined by

$$\begin{aligned} \mathbb {T}_{k,j,M}^X(f)(x)=\left\| t\rightarrow t^{k+\frac{1}{2}}\partial _t^k\partial _{x_j}(I+t\mathcal {L}^{I,B_1})^{-M}(f)(x)\right\| _X,\quad x\in \mathbb {R}^d. \end{aligned}$$

$\mathbb {T}_{k,j,M}^X$ 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

$$\begin{aligned} (I+t\mathcal {L}^{I,B_1})^{-M}f=\frac{1}{\Gamma (M)}\int _0^\infty e^{-u}\mathcal {H}_{ut}^{I,B_1}(f)u^{M-1}du. \end{aligned}$$

It is clear that

$$\begin{aligned} \sup _{t>0}|(I+t\mathcal {L}^{I,B_1})^{-M}f|\le \sup _{t>0}|\mathcal {H}_t^{I,B_1}(f)|. \end{aligned}$$

According to [27, 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

$$\begin{aligned}&(I+t\mathcal {L}^{I,B_1})^{-M}(f)(x)=\frac{t^{-M}}{\Gamma (M)}\int _0^\infty e^{-\frac{u}{t}}\mathcal {H}_u^{I,B_1}(f)(x)u^{M-1}du\\&\quad =\frac{t^{-M}}{\Gamma (M)}\int _{\mathbb {R}^d}f(y)\int _0^\infty e^{-\frac{u}{t}}h_u^{I,B_1}(x,y)u^{M-1}dudy,\quad \\&\qquad x\in \mathbb {R}^d \text{ and } t>0. \end{aligned}$$

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

$$\begin{aligned}{} & {} t^{k+\frac{1}{2}}\partial _t^k\partial _{x_j}(I+t\mathcal {L}^{I,B_1})^{-M}(f)(x)=\frac{t^{k+\frac{1}{2}}}{\Gamma (M)}\\{} & {} \quad \int _{\mathbb {R}^d}f(y)\int _0^\infty \partial _t^k[t^{-M}e^{-\frac{u}{t}}]\partial _{x_j}h_u^{I,B_1}(x,y)u^{M-1}dudy. \end{aligned}$$

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

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

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

where

$$\begin{aligned}{} & {} \mathfrak {s}_{k,j,M}^{E,h}(x,y,t)=\frac{t^{k+\frac{1}{2}}}{\Gamma (M)}\int _0^\infty \partial _t^k[t^{-M}e^{-(u+h)/t}]\partial _{x_j}h_u^{I,B_1}(x,y)(u+h)^{M-1}du,\quad \\{} & {} \quad x,y\in \mathbb {R}^d \text{ and } t>0. \end{aligned}$$

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

$$\begin{aligned} \widetilde{\mathbb {S}}_{k,j,M}^{E,h}(f)(x)=\big \Vert \mathbb {S}_{k,j,M}^{E,h}(f)(x,\cdot )\big \Vert _X, \quad x\in \mathbb {R}^d. \end{aligned}$$

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

$$\begin{aligned} T_{k,\alpha }^X(f)(x)=\left\| t\rightarrow t^{k+\widehat{\alpha }}\partial _t^k\partial _x^\alpha P_t^{I,B_1}(f)(x)\right\| _X,\quad x\in \mathbb {R}^d. \end{aligned}$$

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

$$\begin{aligned} T_{k,0}^{L^\infty ((0,\infty ),dt)}(f)\le C\sup _{t>0}\frac{1}{t}\left| \int _0^t\mathcal {H}_s^{I,B_1}(f)ds\right| . \end{aligned}$$

Since $\{\mathcal {H}_t^{I,B_1}\}_{t>0}$ is contractive in $L^p(\mathbb {R}^d,\gamma _{\infty ,1})$, $1\le p\le \infty$, 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

$$\begin{aligned} \partial _t^kP_t^{I,B_1}(x,y)=\frac{1}{\sqrt{\pi }}\int _0^\infty \partial _t^{k-1}[t e^{-\frac{t^2}{4u}}]\partial _uh_u^{I,B_1}(x,y)\frac{du}{\sqrt{u}},\quad x,y\in \mathbb {R}^d \text{ and } t>0. \end{aligned}$$

We have that

$$\begin{aligned}{} & {} \partial _uh_u^{I,B_1}(x,y)=\frac{1}{2}\sum _{j=1}^d\partial _{x_j}^2h_u^{I,B_1}(x,y)\\{} & {} \quad +\langle B_1x, \nabla _x h_u^{I,B_1}(x,y)\rangle ,\quad x,y \in \mathbb {R}^d \text{ and } u>0. \end{aligned}$$

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

The arguments used in the symmetric Ornstein–Uhlenbeck setting in [20, 32, 33] and [34] 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 ([5, 6] and [4]).

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

$$\begin{aligned} D_{k,j,\textrm{loc}}^{E,h}(f)(x)=\int _{\mathbb {R}^d}\Big \Vert \mathfrak {s}_{k,j}^{E,h}(x,y,\cdot )-\mathfrak {U}_{k,j}^{E,h}(x-y,\cdot )\Big \Vert _X\varphi (x,y)f(y)dy,\quad x\in \mathbb {R}^d, \end{aligned}$$

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

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

$$\begin{aligned}{} & {} \mathfrak {s}_{k,j}^{E,h}(x,y,t)-\mathfrak {U}_{k,j}^{E,h}(x-y,t)=\frac{t^{k+1}}{2\sqrt{\pi }}\int _E\frac{\partial _t^k[te^{-\frac{t^2}{4(u+h)}}]}{(u+h)^{\frac{3}{2}}}\big (\partial _{x_j}h_u^{I,B_1}(x,y)\\{} & {} \quad -\partial _{x_j}W_u(x-y)\big )du. \end{aligned}$$

By using Minkowski inequality we get

$$\begin{aligned}{} & {} \big \Vert \mathfrak {s}_{k,j}^{E,h}(x,y,\cdot )-\mathfrak {U}_{k,j}^{E,h}(x-y,\cdot )\big \Vert _X\le C\int _E \frac{\Vert t^{k+1}\partial _t^k[te^{-\frac{t^2}{4(u+h)}}]\Vert _X}{(u+h)^{\frac{3}{2}}}\big |\partial _{x_j}h_u^{I,B_1}(x,y)\nonumber \\{} & {} \quad -\partial _{x_j}W_u(x-y)\big |du,\quad x,y\in \mathbb {R}^d. \end{aligned}$$

(3.1)

According to [1, Lemma 4] it follows that

$$\begin{aligned} \big |\partial _t^k[te^{-\frac{t^2}{4(u+h)}}]\big |\le C\frac{e^{-\frac{t^2}{8(u+h)}}}{(u+h)^{(k-1)/2}},\quad t,u\in (0,\infty ). \end{aligned}$$

(3.2)

We have that

$$\begin{aligned} \big |t^{k+1}\partial _t^k[te^{-\frac{t^2}{4(u+h)}}]\big |\le C\frac{t^{k+1}e^{-\frac{t^2}{8(u+h)}}}{(u+h)^{(k-1)/2}}\le C(u+h),\quad t,u\in (0,\infty ). \end{aligned}$$

(3.3)

By using again (3.2) we obtain

$$\begin{aligned}{} & {} \left\| t^{k+1}\partial _t^k[te^{-\frac{t^2}{4(u+h)}}]\right\| _{L^2((0,\infty ),\frac{dt}{t})}\le \frac{C}{(u+h)^{(k-1)/2}}\left( \int _0^\infty |t^{k+1}e^{-\frac{t^2}{8(u+h)}}|^2\frac{dt}{t}\right) ^{\frac{1}{2}}\nonumber \\{} & {} \quad \le C(u+h),\quad u>0. \end{aligned}$$

(3.4)

It is well-known that if $g:(0,\infty )\longrightarrow \mathbb {C}$ is a differentiable function and $g'\in L^1(0,\infty )$, then $V_\rho (g)\le \int _0^\infty |g'(t)|dt$ (see, for example, [22, Lemma 5.2.9]).

From (3.2) we deduce that

$$\begin{aligned}{} & {} {} \left\| t^{k+1}\partial _t^k[te^{-\frac{t^2}{4(u+h)}}]\right\| _{V_\rho }\le \int _0^\infty \Big |\partial _t(t^{k+1}\partial _t^k(te^{-\frac{t^2}{4(u+h)}}))\Big |dt \nonumber \\{}{} & {} {} \quad \le C\left( \int _0^\infty |t^k\partial _t^k(te^{-\frac{t^2}{4(u+h)}})|dt + \int _0^\infty |t^{k+1}\partial _t^{k+1}(te^{-\frac{t^2}{4(u+h)}})|dt\right) \nonumber \\{}{} & {} {} \quad \le C\left( (u+h)^{-(k-1)/2}\int _0^\infty t^ke^{-\frac{t^2}{8(u+h)}}dt+(u+h)^{-k/2}\int _0^\infty t^{k+1}e^{-\frac{t^2}{4(u+h)}}dt\right) \nonumber \\{}{} & {} {} \quad \le C(u+h),\quad u>0.\end{aligned}$$

(3.5)

By (3.1) we get

$$\begin{aligned}{} & {} \Big \Vert \mathfrak {s}_{k,j}^{E,h}(x,y,\cdot )-\mathfrak {U}_{k,j}^{E,h}(x-y,\cdot )\Big \Vert _X\le C\int _0^\infty \frac{1}{\sqrt{u}}|\partial _{x_j}h_u^{I,B_1}(x,y)\nonumber \\{} & {} \quad -\partial _{x_j}W_u(x-y)|du,\quad x,y\in \mathbb {R}^d. \end{aligned}$$

(3.6)

We first prove a technical result.

Lemma 3.1

If $(x,y)\in L_2$ then

$$\begin{aligned} \int _0^\infty \frac{1}{\sqrt{u}}|\partial _{x_j}h_u^{I,B_1}(x,y)-\partial _{x_j}W_u(x-y)|du\le C\frac{1+|x|}{|x-y|^{d-1}}. \end{aligned}$$

(3.7)

Proof

When $B_1=-I$ the operator $\mathcal {L}^{I,B_1}$ is the symmetric Ornstein–Uhlenbeck and we have that

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

By using [21, Lemma 3.4] we obtain

$$\begin{aligned} \int _0^\infty \frac{1}{\sqrt{u}}|\partial _{x_j}h_u^{I,-I}(x,y)-\partial _{x_j}W_u(x-y)|du\le C\frac{1+|x|}{|x-y|^{d-1}},\quad (x,y)\in L_2. \end{aligned}$$

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

$$\begin{aligned} \int _0^\infty \frac{1}{\sqrt{u}}|\partial _{x_j}h_u^{I,B_1}(x,y)-\partial _{x_j}h_u^{I,-I}(x,y)|du\le C\frac{1+|x|}{|x-y|^{d-1}},\quad (x,y)\in L_2.\nonumber \\ \end{aligned}$$

(3.8)

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

$$\begin{aligned}{} & {} |e^{uB_1}x-y|^2=\langle e^{-uR}(e^{-u}x-e^{uR}y),e^{-uR}(e^{-u}x-e^{uR}y)\rangle \\{} & {} \quad =|e^{-u}x-e^{uR}y|^2,\quad x,y\in \mathbb {R}^d \text{ and } u>0. \end{aligned}$$

We have that

$$\begin{aligned}{} & {} \partial _{x_j}h_u^{I,B_1}(x,y)=-\frac{2e^{-u}}{(2\pi )^{\frac{d}{2}}}(e^{-u}x_j-(e^{uR}y)_j)\frac{e^{-\frac{|e^{-u}x-e^{uR}y|^2}{1-e^{-2u}}}}{(1-e^{-2u})^{\frac{d}{2}+1}},\quad \nonumber \\{} & {} \quad x,y\in \mathbb {R}^d \text{ and } u>0, \end{aligned}$$

(3.9)

and

$$\begin{aligned}{} & {} \partial _{x_j}h_u^{I,-I}(x,y)=-\frac{2e^{-u}}{(2\pi )^{\frac{d}{2}}}(e^{-u}x_j-y_j)\frac{e^{-\frac{|e^{-u}x-y|^2}{1-e^{-2u}}}}{(1-e^{-2u})^{\frac{d}{2}+1}},\quad \nonumber \\{} & {} \quad x,y\in \mathbb {R}^d \text{ and } u>0. \end{aligned}$$

(3.10)

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

$$\begin{aligned}&\partial _{x_j}h_u^{I,B_1}(x,y)-\partial _{x_j}h_u^{I,-I}(x,y)=-\frac{2e^{-u}}{(2\pi )^{\frac{d}{2}}}\frac{1}{(1-e^{-2u})^{\frac{d}{2}+1}}\\&\quad \times \left[ \Big (e^{-\frac{|e^{-u}x-e^{uR}y|^2}{1-e^{-2u}}}-e^{-\frac{|e^{-u}x-y|^2}{1-e^{-2u}}}\Big )(e^{-u}x_j-(e^{uR}y)_j)\right. \\&\quad \left. +e^{-\frac{|e^{-u}x-y|^2}{1-e^{-2u}}}(y_j-(e^{uR}y)_j)\right] . \end{aligned}$$

We need to establish some estimations. We have that

$$\begin{aligned} |e^{-a}-e^{-b}|\le e^{-\min \{a,b\}}|a-b|,\quad a,b>0. \end{aligned}$$

Then, according to [26, Lemma 3.3 (i)] we obtain

$$\begin{aligned}{} & {} \left| e^{-\frac{|e^{-u}x-e^{uR}y|^2}{1-e^{-2u}}}-e^{-\frac{|e^{-u}x-y|^2}{1-e^{-2u}}}\right| \le C \frac{e^{-c\frac{|x-y|^2}{1-e^{-2u}}}}{1-e^{-2u}} \big ||e^{-u}x-y|^2-|e^{-u}x-e^{uR}y|^2\big |,\quad \\{} & {} \quad (x,y)\in L_2 \text{ and } u>0. \end{aligned}$$

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

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

It follows that

$$\begin{aligned}{} & {} \left| |e^{-u}x - y|^2 - |e^{-u}x - e^{uR}y|^2 \right| \le C (u^2|y|^2 + u^2|x||y| + u|x-y||y|), \quad \\{} & {} \quad x,\; y\in \mathbb {R}^d \text { and }u\in (0,1). \end{aligned}$$

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

$$\begin{aligned} \left| e^{-\frac{|e^{-u}x - e^{uR}y|^2}{1-e^{-2u}}} - e^{-\frac{|e^{-u}x - y|^2}{1-e^{-2u}}} \right| \le C e^{-c \frac{|x-y|^2}{u}} \big [u|y|(|x| + |y|) + |x-y||y|\big ]. \end{aligned}$$

On the other hand we have that, for x, $y\in \mathbb {R}^d$ and $u\in (0,1)$,

$$\begin{aligned} |y_j - (e^{uR}y)_j|\le |(e^{uR}-I)y|\le Cu|y|, \end{aligned}$$

and

$$\begin{aligned}{} & {} |e^{-u}x_j - (e^{-uR}y)_j| \le |e^{-u}x_j - x_j| + | x_j - y_j | + |y_j - (e^{uR}y)_j| \\{} & {} \quad \le C\big [ u(|x| + |y|) + |x-y|\big ]. \end{aligned}$$

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

$$\begin{aligned}&|\partial _{x_j} h^{I, B_1}_u (x,y) - \partial _{x_j} h^{I, -I}_u (x,y)|\\&\quad \le C \frac{e^{-c\frac{|x-y|^2}{u}}}{u^{\frac{d}{2}+1}} \left( \big [u|y|(|x|+ |y|) + |x-y||y|\big ] \big [u(|x| + |y|) + |x-y|\big ] + u|y| \right) \\&\quad \le C \frac{e^{-c\frac{|x-y|^2}{u}}}{u^{\frac{d}{2}+1}} \left( \big [u|y|(|x| + |y|) + \sqrt{u}|y|\big ] \big [u(|x| + |y|) + \sqrt{u}\big ] + u|y| \right) \\&\quad = C \frac{e^{-c\frac{|x-y|^2}{u}}}{u^{\frac{d}{2}}} \left( \big [\sqrt{u}|y|(|x| +|y|)+ |y|\big ] \big [\sqrt{u}(|x| + |y|) + 1\big ] + |y| \right) . \end{aligned}$$

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 $|y|\le C(1+|x|)$ and $\sqrt{u}(|x|+|y|)\le C$. Then, it follows that

$$\begin{aligned}{} & {} \big [\sqrt{u}|y|(|x|+|y|) + |y|\big ] \big [\sqrt{u}(|x| + |y|) + 1\big ] + |y|\le C(1+|x|), \quad \\{} & {} \quad (x,y)\in L_2 \text{ and } 0<u<m(x). \end{aligned}$$

Thus we obtain that

$$\begin{aligned}{} & {} |\partial _{x_j} h^{I, B_1}_u (x,y) - \partial _{x_j} h^{I, -I}_u (x,y)| \le C \frac{e^{-c\frac{|x-y|^2}{u}}}{u^{\frac{d}{2}}} (1+ |x|),\quad \\{} & {} \quad (x,y)\in L_2 \text{ and } 0<u<m(x). \end{aligned}$$

By using this estimation, we get, when $(x,y)\in L_2$,

$$\begin{aligned} \begin{aligned}&\int _0^{m(x)} \frac{1}{\sqrt{u}}|\partial _{x_j} h^{I, B_1}_u (x,y) - \partial _{x_j} h^{I, -I}_u (x,y)| du \\&\quad \le C (1+ |x|)\int _0^{m(x)} \frac{e^{-c\frac{|x-y|^2}{u}}}{u^{\frac{d + 1}{2}}} du \le C \frac{1+|x|}{|x-y|^{d-1}}. \end{aligned} \end{aligned}$$

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

$$\begin{aligned} \begin{aligned}&\int _{m(x)}^\infty \frac{1}{\sqrt{u}} |\partial _{x_j} h^{I, B_1}_u (x,y) - \partial _{x_j} h^{I, -I}_u (x,y)| du \\&\quad \le \int _{m(x)}^\infty \frac{1}{\sqrt{u}} |\partial _{x_j} h^{I, B_1}_u (x,y)| du+ \int _{m(x)}^\infty \frac{1}{\sqrt{u}} |\partial _{x_j} h^{I, -I}_u (x,y)| du \\ {}&\quad \le C\left( \int _{m(x)}^\infty \frac{e^{-\frac{|e^{-u}x-e^{uR}y|^2}{1-e^{-2u}}}}{(1-e^{-2u})^{\frac{d}{2}+1}\sqrt{u}} |e^{-u}x-e^{uR}y| e^{-u} du\right. \\&\qquad \left. + \int _{m(x)}^\infty \frac{e^{-\frac{|e^{-u}x-y|^2}{1-e^{-2u}}}}{(1-e^{-2u})^{\frac{d}{2}+1}\sqrt{u}} |e^{-u}x-y| e^{-u} du\right) \\ {}&\quad \le C \int _{m(x)}^\infty \frac{e^{-c\frac{|x-y|^2}{1-e^{-2u}}}e^{-u}}{(1-e^{-2u})^{\frac{d+1}{2}}\sqrt{u}} du \le C \int _{m(x)}^\infty \frac{e^{-\frac{|x-y|^2}{u}}}{u^{\frac{d}{2}+1}} du\\&\quad \le \frac{C}{\sqrt{m(x)}} \int _{m(x)}^\infty \frac{e^{-\frac{|x-y|^2}{u}}}{u^{\frac{d+1}{2}}} du \\ {}&\quad \le \frac{C}{\sqrt{m(x)}|x-y|^{d-1}} \le C \frac{1+|x|}{|x-y|^{d-1}},\quad (x,y)\in L_2. \end{aligned} \end{aligned}$$

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

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

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 $h\ge 0$. We define the operator

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

where

$$\begin{aligned}{} & {} H^{E,h}_{k,j,M}(z,t) = \frac{t^{k+\frac{1}{2}}}{\Gamma (M)} \int _E \partial ^k_t \left[ t^{-M}e^{-\frac{u+h}{t}} \right] \partial _{x_j} W_u(z) (u+h)^{M-1}du,\quad \\{} & {} \quad z\in \mathbb {R}^n \text{ and } t>0. \end{aligned}$$

Minkowski inequality leads to

$$\begin{aligned}{} & {} \left\| \mathbb {S}^{E,h}_{k,j,M} (f)(x,\cdot ) - \mathcal {H}^{E,h}_{k,j,M} (f)(x,\cdot ) \right\| _X \le \int _{\mathbb {R}^d} \left\| \mathfrak {s}^{E,h}_{k,j,M} (x, y, \cdot )\right. \\{} & {} \quad \left. - H^{E,h}_{k,j,M} (f)(x-y,\cdot ) \right\| _X |f(y)|dy,\quad x\in \mathbb {R}^d. \end{aligned}$$

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

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

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

By using again Minkowski inequality we get

$$\begin{aligned} \begin{aligned}&\left\| \mathfrak {s}^{E,h}_{k,j,M} (x, y, \cdot ) \right. \left. - H^{E,h}_{k,j,M} (f)(x- y,\cdot ) \right\| _X \\&\quad \le \frac{1}{\Gamma (M)} \int _E \left\| t^{k+\frac{1}{2}} \partial ^k_t \left[ t^{-M} e^{-\frac{u+h}{t}} \right] \right\| _X |\partial _{x_j}h^{I,B_1}_u(x,y) \\&\qquad - \partial _{x_j}W_u(x-y)|(u+h)^{M-1}du,\quad x,y\in \mathbb {R}^d. \end{aligned} \end{aligned}$$

We are going to prove a preliminary bound.

Lemma 3.2

If $u>0$,

$$\begin{aligned} \left\| t^{k+\frac{1}{2}}\partial _t^k\big [t^{-M}e^{-\frac{u+h}{t}}\big ]\right\| _X\le C(u+h)^{\frac{1}{2}-M}. \end{aligned}$$

(3.11)

Proof

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

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

and

$$\begin{aligned}{} & {} \left\| t^{\frac{1}{2}-M}e^{-\frac{u+h}{t}}\right\| _{L^2 ((0,\infty ),\frac{dt}{t})}=\left( \int _0^\infty t^{-2M}e^{-2\frac{u+h}{t}}dt\right) ^{\frac{1}{2}}\\{} & {} \quad \le C(u+h)^{\frac{1}{2}-M},\quad u>0, \end{aligned}$$

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

We also get

$$\begin{aligned}{} & {} \left\| t^{\frac{1}{2}-M}e^{-\frac{u+h}{t}}\right\| _{V_\rho (0,\infty )}\le \int _0^\infty \Big |\partial _t\big [t^{\frac{1}{2}-M}e^{-\frac{u+h}{t}}\big ]\Big |dt\\{} & {} \quad \le C\int _0^\infty t^{-M-\frac{1}{2}}e^{-c\frac{u+h}{t}}dt\le C(u+h)^{\frac{1}{2}-M},\quad u>0, \end{aligned}$$

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

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

$$\begin{aligned} \partial _t^k=\sum _{i=0}^{k-1}c_iw^{2k-i}\partial _w^{k-i},\quad w=\frac{1}{t}\in (0,\infty ), \end{aligned}$$

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

$$\begin{aligned}{} & {} \partial _t^k\big [t^{-M}e^{-\frac{u+h}{t}}\big ]=\sum _{i=0}^{k-1}\sum _{\ell =0}^{k-i}c_{i,\ell }w^{2k+M-\ell -i}e^{-(u+h)w}(u+h)^{k-i-\ell }, \quad \nonumber \\{} & {} \quad w=\frac{1}{t},\quad u\in (0,\infty ), \end{aligned}$$

(3.12)

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

We have that

$$\begin{aligned}&\left\| t^{k+\frac{1}{2}}\partial _t^k\big [t^{-M}e^{-\frac{u+h}{t}}\big ]\right\| _{L^\infty ((0,\infty ),dt)}\\&\quad \le C\sum _{i=0}^{k-1}\sum _{\ell =0}^{k-i}(u+h)^{k-i-\ell }\sup _{w\in (0,\infty )}w^{k+M-\ell -i-\frac{1}{2}}e^{-(u+h)w}\\&\quad \le C(u+h)^{\frac{1}{2}-M},\quad u>0, \end{aligned}$$

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

We also obtain

$$\begin{aligned}&\left\| t^{k+\frac{1}{2}}\partial _t^k\big [t^{-M}e^{-\frac{u+h}{t}}\big ]\right\| _{L^2((0,\infty ),\frac{dt}{t})}\\&\quad \le C\sum _{i=0}^{k-1}\sum _{\ell =0}^{k-i}(u+h)^{k-i-\ell }\left( \int _0^\infty w^{2k+2M-2\ell -2i-2}e^{-(u+h)w}dw\right) ^{\frac{1}{2}}\\&\quad \le C(u+h)^{\frac{1}{2}-M},\quad u>0, \end{aligned}$$

and

$$\begin{aligned}&\left\| t^{k+\frac{1}{2}}\partial _t^k\big [t^{-M}e^{-\frac{u+h}{t}}\big ]\right\| _{V_\rho (0,\infty )}\le \int _0^\infty \Big |\partial _t\Big [t^{k+\frac{1}{2}}\partial _t^k\big [t^{-M}e^{-\frac{u+h}{t}}\big ]\Big ]\Big |dt\\&\quad \le C\left( \int _0^\infty \Big |t^{k-\frac{1}{2}}\partial _t^k\big [t^{-M}e^{-\frac{u+h}{t}}\big ]\Big |dt+\int _0^\infty \Big |t^{k+\frac{1}{2}}\partial _t^{k+1}\big [t^{-M}e^{-\frac{u+h}{t}}\big ]\Big |dt\right) \\&\quad \le C\left( \sum _{i=0}^{k-1}\sum _{\ell =0}^{k-i}(u+h)^{k-i-\ell }\int _0^\infty w^{k+M-\ell -i-\frac{3}{2}}e^{-(u+h)w}dw\right. \\&\left. \qquad +\sum _{i=0}^k\sum _{\ell =0}^{k+1-i}(u+h)^{k+1-i-\ell }\int _0^\infty w^{k+M-\ell -i-\frac{1}{2}}e^{-(u+h)w}dw\right) \\&\quad \le C(u+h)^{\frac{1}{2}-M},\quad u>0, \end{aligned}$$

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

From (3.11) we deduce that

$$\begin{aligned}{} & {} \left\| \mathfrak {s}_{k,j,M}^{E,h}(x,y,\cdot )-H_{k,j,M}^{E,h}(x-y,\cdot )\right\| _X\le C\int _0^\infty \frac{1}{\sqrt{u}}|\partial _{x_j}h_u^{I,B_1}(x,y)\\{} & {} \quad -\partial _{x_j}W_u(x-y)|du,\quad x,y\in \mathbb {R}^d. \end{aligned}$$

By using (3.7) we get

$$\begin{aligned} \left\| \mathfrak {s}_{k,j,M}^{E,h}(x,y,\cdot )-H_{k,j,M}^{E,h}(x-y,\cdot )\right\| _X\le C\frac{1+|x|}{|x-y|^{d-1}},\quad (x,y)\in L_2, \end{aligned}$$

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

3.2 Proof of Claim 4

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

$$\begin{aligned}{} & {} {} \left\| \mathfrak {s}_{k,j}^{E,h}(x,y,\cdot )\right\| _X\le C\int _E\Big \Vert t^{k+1}\partial _t^k\big [te^{-\frac{t^2}{4(u+h)}}\big ]\Big \Vert _X|\partial _{x_j}h_u^{I,B_1}(x,y)|\frac{du}{(u+h)^{\frac{3}{2}}},\quad \\{}{} & {} {} \quad x,y\in \mathbb {R}^d. \end{aligned}$$

It was proved (see (3.3), (3.4) and (3.5)) that

$$\begin{aligned} \Vert t^{k+1}\partial _t^k\big [te^{-\frac{t^2}{4(u+h)}}\big ]\Big \Vert _X\le C(u+h),\quad u>0. \end{aligned}$$

Then,

$$\begin{aligned} \left\| \mathfrak {s}_{k,j}^{E,h}(x,y,\cdot )\right\| _X\le C\int _E|\partial _{x_j}h_u^{I,B_1}(x,y)|\frac{du}{\sqrt{u+h}},\quad x,y\in \mathbb {R}^d. \end{aligned}$$

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

$$\begin{aligned} \left\| \mathfrak {s}_{k,j,M}^{E,h}(x,y,\cdot )\right\| _X\le C\int _E|\partial _{x_j}h_u^{I,B_1}(x,y)|\frac{du}{\sqrt{u+h}},\quad x,y\in \mathbb {R}^d. \end{aligned}$$

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 [26, Proposition 4.7] allow us to prove that

$$\begin{aligned}{} & {} \int _E|\partial _{x_j}h_u^{I,B_1}(x,y)|\frac{du}{\sqrt{u+h}}\le Ce^{|x|^2-|y|^2}\min \big \{(1+|x|)^d,(|x|\sin \theta (x,y))^{-d}\big \},\quad \\{} & {} \quad (x,y)\in G_1. \end{aligned}$$

We define

$$\begin{aligned} \widetilde{\mathbb {V}}_{k,j,M}^{E,h}(f)(x)=\int _{\mathbb {R}^d}\Big \Vert \mathfrak {s}_{k,j,M}^{E,h}(x,y,\cdot )\Big \Vert _Xf(y)dy,\quad x\in \mathbb {R}^d. \end{aligned}$$

From Lemma 2.4 we deduce that the operators $\widetilde{V}_{k,j,\textrm{glob}}^{E,h}$ and $\widetilde{\mathbb {V}}_{k,j,\textrm{glob}}^{E,h}$ 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 estimates it is sufficient to see that the operator $Z_{\textrm{glob}}^{E_p,h}$ is bounded from $L^p(\mathbb {R}^d,\gamma _{\infty ,1})$ into itself, where

$$\begin{aligned} Z^{E_p,h}(f)(x)=\int _{\mathbb {R}^d}Z^{E_p,h}(x,y)f(y)dy,\quad x\in \mathbb {R}^d, \end{aligned}$$

with

$$\begin{aligned} Z^{E_p,h}(x,y)=\int _{E_p}|\partial _{x_j}h_u^{I,B_1}(x,y)|\frac{du}{\sqrt{u+h}},\quad x,y\in \mathbb {R}^d. \end{aligned}$$

We observe that

$$\begin{aligned}&\Big \Vert Z^{E_p,h}_{\textrm{glob}}(f)\Big \Vert _{L^p(\mathbb {R}^d,\gamma _{\infty ,1})}^p=\pi ^{-\frac{d}{2}}\int _{{\mathbb {R}}^d}|Z^{E_p,h}_{\textrm{glob}}(f)(x)|^pe^{-|x|^2}dx \\&\quad =\pi ^{-\frac{d}{2}}\int _{{\mathbb {R}}^d}\left( \int _{{\mathbb {R}}^d} Z^{E_p,h}(x,y)(1-\varphi (x,y))e^{-\frac{1}{p}(|x|^2-|y|^2)}f(y)e^{-\frac{|y|^2}{p}}dy\right) ^p dx\\&\quad =\Big \Vert \widetilde{Z}^{E_p,h}_{\textrm{glob}}(\pi ^{-\frac{d}{2}}e^{-\frac{|y|^2}{p}}f)\Big \Vert _{L^p(\mathbb {R}^d,dx)}^p, \end{aligned}$$

where

$$\begin{aligned} \widetilde{Z}^{E_p,h}(g)(x)=\int _{\mathbb R^d}Z^{E_p,h}(x,y)e^{-\frac{1}{p}(|x|^2-|y|^2)}g(y)dy,\quad x\in {\mathbb {R}}^d. \end{aligned}$$

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

Lemma 3.3

• (a) For every $\eta \in (0,1)$,

  $$\begin{aligned} Z^{E_p,h}(x,y)\le Ce^{\eta (1-\delta )|x|^2-\eta |y|^2},\quad (x,y)\in G_1,\;\langle x,y\rangle <0. \end{aligned}$$

  (3.13)

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

  $$\begin{aligned}{} & {} Z^{E_p,h}(x,y)\le C|x+y|^de^{\eta (1-\frac{\delta }{2})(|x|^2-|y|^2)-\frac{\eta \delta }{2}|x+y||x-y|},\nonumber \\{} & {} \quad \quad (x,y)\in G_1,\;\langle x,y\rangle \ge 0. \end{aligned}$$

  (3.14)

Proof

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

$$\begin{aligned} |\partial _{x_j}h_u^{I,B_1}(x,y)|\le Ce^{-u}\frac{|e^{-u}x-e^{uR}y|}{(1-e^{-2u})^{\frac{d}{2}+1}}e^{-\frac{|e^{-u}x-e^{uR}y|^2}{1-e^{-2u}}},\quad x,y\in \mathbb {R}^d \text{ and } u>0. \end{aligned}$$

Then,

$$\begin{aligned}{} & {} |\partial _{x_j}h_u^{I,B_1}(x,y)| \le Ce^{-u}\frac{e^{-\eta \frac{|e^{-u}x-e^{uR}y|^2}{1-e^{-2u}}}}{(1-e^{-2u})^{\frac{d+1}{2}}} \le C\frac{e^{-u}}{\sqrt{1-e^{-2u}}}h_u^{I,B_1}(\sqrt{\eta }x,\sqrt{\eta }y),\quad \\{} & {} \quad x,y\in \mathbb {R}^d \text{ and } u>0. \end{aligned}$$

By using Lemma 2.2 we have that

$$\begin{aligned} |\partial _{x_j}h_{\tau (s)}^{I,B_1}(x,y)|\le C\frac{1-s}{s^{\frac{d+1}{2}}}e^{\eta (|x|^2-|y|^2-\frac{\delta }{4s}Q_s(x,y))}\quad x,y\in \mathbb {R}^d \text{ and } s\in \tau ^{-1}(E_p). \end{aligned}$$

Since

$$\begin{aligned}{} & {} \frac{1}{4s}Q_s(x,y)=\frac{s}{4}|x+y|^2+\frac{1}{4s}|x-y|^2+\frac{|x|^2-|y|^2}{2}=\frac{1}{4s}Q_s(y,x)+|x|^2-|y|^2, \quad \\{} & {} \quad x,y\in {\mathbb {R}}^d \text{ and } s\in (0,1), \end{aligned}$$

and

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

we get

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

Thus we obtain

$$\begin{aligned}{} & {} |\partial _{x_j}h_u^{I,B_1}(x,y)|\le C\frac{e^{-u}}{(1-e^{-2u})^{\frac{d+1}{2}}}e^{\eta (1-\delta )(|x|^2-|y|^2)}e^{-\eta \delta \frac{|e^{-u}x-y|^2}{1-e^{-2u}}}\quad \\{} & {} \quad x,y\in \mathbb {R}^d\;\text{ and }\;u\in E_p. \end{aligned}$$

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

$$\begin{aligned} Z^{E_p,h}(x,y)&\le Ce^{\eta (1-\delta )(|x|^2-|y|^2)} \int _0^1\frac{e^{-\eta \delta u(t)}}{t^{\frac{d+1}{2}}\sqrt{|\log (1-t)|}}\frac{dt}{\sqrt{1-t}}\\&\le Ce^{\eta (1-\delta )(|x|^2-|y|^2)}\int _0^1\frac{e^{-\eta \delta u(t)}}{t^{ \frac{d}{2}+1}}\frac{dt}{\sqrt{1-t}},\quad x,y\in \mathbb R^d. \end{aligned}$$

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 [33, Theorem 4.2] we deduce that (3.13) is satisfied.

Assume now that $\langle x,y\rangle \ge 0$ and $\eta \delta >1-1/d$. As in [33] 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 [33, Lemma 4.1] we obtain

$$\begin{aligned} \int _0^1\frac{e^{-\eta \delta u(t)}}{t^{\frac{d}{2}+1}\sqrt{1-t}}dt&\le C\left( \frac{e^{-u_0}}{t_0^{d/2}}\right) ^{1-\frac{1}{d}}\int _0^1\frac{e^{-(\eta \delta -1+\frac{1}{d})u(t)}}{t^{ \frac{3}{2}}\sqrt{1-t}}dt \\&\le C\left( \frac{e^{-u_0}}{t_0^{d/2}}\right) ^{1-\frac{1}{d}}\frac{e^{-(\eta \delta -1+\frac{1}{d})u_0}}{t_0^{ \frac{1}{2}}} \le C \frac{e^{-\eta \delta u_0}}{t_0^{d/2}},\quad (x,y)\in G_1. \end{aligned}$$

Then,

$$\begin{aligned} Z^{E_p,h}(x,y)\le C\frac{e^{\eta (1-\delta )(|x|^2-|y|^2)-\eta \delta u_0}}{t_0^{ d/2}},\quad (x,y)\in G_1. \end{aligned}$$

Since $t_0\sim \frac{\sqrt{a^2-b^2}}{a}=\frac{|x+y||x-y|}{a}$, $|x-y||x+y|\ge 1$, $(x,y)\in G_1$, we obtain

$$\begin{aligned} Z^{E_p,h}(x,y)\le C|x+y|^de^{\eta (1-\frac{\delta }{2})(|x|^2-|y|^2)-\frac{\eta \delta }{2}|x-y||x+y|},\quad (x,y)\in G_1, \end{aligned}$$

and (3.14) is proved. $\square$

We now choose $\eta$ such that

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

Note that our hypothesis on $\delta$ (recall that $\delta \in (\max \{1-\frac{1}{d},1-\frac{1}{p}\},1)$) implies $0<\frac{1}{\delta }(1-\frac{1}{d})<1$ and also $\eta (1-\delta )<\frac{1}{p}$.

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

$$\begin{aligned}{} & {} {} \int _{\{y\in {\mathbb {R}}^d:\langle x,y\rangle <0\}}Z^{E_p,h}(x,y)e^{-\frac{1}{p}(|x|^2-|y|^2)}(1-\varphi (x,y))dy \\{}{} & {} {}\quad \le C e^{-(\frac{1}{p}-\eta (1-\delta ))|x|^2} \int _{\mathbb R^d}e^{-(\eta -\frac{1}{p})|y|^2}dy,\quad x\in {\mathbb {R}}^d. \end{aligned}$$

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

$$\begin{aligned} \sup _{x\in {\mathbb {R}}^d}\int _{\{y\in {\mathbb {R}}^d:\langle x,y\rangle<0\}}Z^{E_p,h}(x,y)e^{-\frac{1}{p}(|x|^2-|y|^2)}(1-\varphi (x,y))dy<\infty . \end{aligned}$$

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

$$\begin{aligned}{} & {} Z^{E_p,h}(x,y)e^{-\frac{1}{p}(|x|^2-|y|^2)}\le C |x+y|^de^{-\left( \frac{\eta \delta }{2}-\left| \frac{1}{p}-\eta (1-\frac{\delta }{2})\right| \right) |x+y||x-y|},\quad \\{} & {} \qquad (x,y)\in G_1 \text{ and } \langle x,y\rangle \ge 0. \end{aligned}$$

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 [33, p. 501] we obtain

$$\begin{aligned} \sup _{x\in {\mathbb {R}}^d}\int _{\{y\in {\mathbb {R}}^d:\langle x,y\rangle \ge 0\}}Z^{E_p,h}(x,y)e^{-\frac{1}{p}(|x|^2-|y|^2)}(1-\varphi (x,y))dy<\infty . \end{aligned}$$

We conclude that

$$\begin{aligned} \sup _{x\in {\mathbb {R}}^d}\int _{\mathbb R^d}Z^{E_p,h}(x,y)e^{-\frac{1}{p}(|x|^2-|y|^2)}(1-\varphi (x,y))dy<\infty . \end{aligned}$$

In a similar way we get

$$\begin{aligned} \sup _{y\in {\mathbb {R}}^d}\int _{\mathbb R^d}Z^{E_p,h}(x,y)e^{-\frac{1}{p}(|x|^2-|y|^2)}(1-\varphi (x,y))dx<\infty . \end{aligned}$$

We deduce that the operator $\widetilde{Z}_{\textrm{glob}}^{E_p,h}$ is bounded from $L^q({\mathbb {R}}^d,dx)$ into itself, for every $1\le q\le \infty$. Thus we prove that the operator $Z_\textrm{glob}^{E_p,h}$ 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 $h\ge 0$.

4.1 Maximal operators

4.1.1 4.1.1.

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

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

where

$$\begin{aligned} \mathfrak {U}_{k,j}^{E,h}(z,t)=\frac{t^{k+1}}{2\sqrt{\pi }}\int _E \partial _t^k\big [te^{-\frac{t^2}{4(u+h)}}\big ]\partial _{x_j}W_u(z)\frac{du}{(u+h)^{\frac{3}{2}}},\quad z\in \mathbb {R}^d \text{ and } t>0. \end{aligned}$$

We recall that

$$\begin{aligned} W_u(z)=\frac{e^{-\frac{|z|^2}{2u}}}{(2\pi u)^{\frac{d}{2}}},\quad z\in \mathbb {R}^d \text{ and } u>0. \end{aligned}$$

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

$$\begin{aligned}&\int _{\mathbb {R}^d}f(y)\mathfrak {U}_{k,j}^{E,h}(x-y,t)\varphi (x,y)dy\\&\quad =\frac{t^{k+1}}{2\sqrt{\pi }}\int _E \partial _t^k\big [te^{-\frac{t^2}{4(u+h)}}\big ]\int _{\mathbb {R}^d}\partial _{x_j}W_u(x-y)f(y)\varphi (x,y)\frac{dydu}{(u+h)^{\frac{3}{2}}}\\&\quad =-\frac{t^{k+1}}{2\sqrt{\pi }}\int _E \partial _t^k\big [te^{-\frac{t^2}{4(u+h)}}\big ]\int _{\mathbb {R}^d}\frac{x_j-y_j}{u}W_u(x-y)f(y)\varphi (x,y)\frac{dydu}{(u+h)^{\frac{3}{2}}}. \end{aligned}$$

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

$$\begin{aligned}&\widetilde{U}_{*,k,j,\textrm{loc}}^{E,h}(f)(x)\le C\int _E\Vert t^{k+1}\partial _t^k[te^{-\frac{t^2}{4(u+h)}}]\Vert _{L^\infty ((0,\infty ),dt)}\\&\qquad \int _{\mathbb {R}^d}|x-y|W_u(x-y)|f(y)|\varphi (x,y)\frac{dydu}{u(u+h)^{\frac{3}{2}}}\\&\quad \le C\int _E \frac{1}{u\sqrt{u+h}}\int _{\mathbb {R}^d}|x-y|W_u(x-y)|f(y)|\varphi (x,y)dydu,\quad x\in \mathbb {R}^d. \end{aligned}$$

If $h>0$, by taking into account that $|x-y|\le C$, when $(x,y)\in L_2$, and that $\frac{|z|}{\sqrt{u}}W_u(z)\le CW_{2u}(z)$, $z\in \mathbb {R}^d$, it follows that

$$\begin{aligned} \widetilde{U}_{*,k,j,\textrm{loc}}^{E,h}(f)(x)&\le C\left( \int _1^\infty \frac{1}{u\sqrt{u+h}}\int _{\mathbb {R}^d}W_u(x-y)|f(y)|dydu\right. \\&\left. \quad +\int _0^1\frac{1}{\sqrt{u(u+h)}}\int _{\mathbb {R}^d}W_{2u}(x-y)|f(y)|dydu\right) \\&\le C\left( \int _1^\infty u^{-\frac{3}{2}}du+\int _0^1u^{-\frac{1}{2}}du\right) \sup _{v>0}W_v(|f|)(x),\quad x\in \mathbb {R}^d. \end{aligned}$$

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

$$\begin{aligned}{} & {} \widetilde{U}_{*,k,j,\textrm{loc}}^{E,0}(f)(x)\\{} & {} \le C\sup _{t>0}\left( t^{k+1}\int _0^\infty \frac{e^{-\frac{t^2}{8u}}}{u^{\frac{k+3}{2}}}du\right) \sup _{v>0}W_v(|f|)(x)\le C\sup _{v>0}W_v(|f|)(x). \end{aligned}$$

We conclude that

$$\begin{aligned} \widetilde{U}_{*,k,j,\textrm{loc}}^{E,h}(f)(x)\le C\sup _{t>0}W_t(|f|)(x),\quad x\in \mathbb {R}^n. \end{aligned}$$

From well-known $L^p$-boundedness properties of the maximal operator defined by the classical heat semigroup we deduce that the local maximal operator $\widetilde{U}_{*,k,j,\textrm{loc}}^{E,h}$ 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 4.1.2.

We now study the local maximal operator

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

where

$$\begin{aligned}{} & {} \mathfrak {U}_{k,j,M}^{E,h}(z,t)=\frac{t^{k+\frac{1}{2}}}{\Gamma (M)}\int _E \partial _t^k\big [t^{-M}e^{-\frac{u+h}{t}}\big ]\\{} & {} \quad \partial _{x_j}W_u(z)(u+h)^{M-1}du,\quad z\in \mathbb {R}^d \text{ and } t>0. \end{aligned}$$

According to (3.11) and Minkowski inequality we can write

$$\begin{aligned}&\widetilde{\mathbb {U}}_{*,k,j,M, \text {loc}}^{E,h}(f)(x)\le C\int _E\Vert t^{k+\frac{1}{2}}\partial _t^k[t^{-M}e^{-\frac{u+h}{t}}]\Vert _{L^\infty ((0,\infty ),dt)}\\&\qquad \frac{|x-y|}{u}W_u(x-y)|f(y)|\varphi (x,y)(u+h)^{M-1}dydu\\&\qquad \le C\int _0^\infty \frac{1}{u\sqrt{u+h}}\int _{\mathbb {R}^d}|x-y|W_u(x-y)|f(y)|\varphi (x,y)dydu,\quad x\in \mathbb {R}^d. \end{aligned}$$

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

$$\begin{aligned} \widetilde{\mathbb {U}}_{*,k,j,M, \textrm{loc}}^{E,h}(f)(x)\le C\sup _{v>0}W_v(|f|)(x),\quad x\in \mathbb {R}^d. \end{aligned}$$

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

$$\begin{aligned}{} & {} t^{k+\frac{1}{2}}\int _0^\infty \Big |\partial _t^k\big [t^{-M}e^{-\frac{u}{t}}\big ]\Big |u^{M-\frac{3}{2}}du \le Ct^{k+\frac{1}{2}}\sum _{i=0}^{k-1}\sum _{\ell =0}^{k-i}t^{-2k-M+\ell +i}\\{} & {} \quad \int _0^\infty e^{-\frac{u}{t}}u^{k-i-\ell +M-\frac{3}{2}}du\le C,\quad t>0. \end{aligned}$$

Furthermore, we obtain

$$\begin{aligned} t^{\frac{1}{2}-M}\int _0^\infty e^{-\frac{u}{t}}u^{M-\frac{3}{2}}du\le C,\quad t>0. \end{aligned}$$

(4.1)

Then,

$$\begin{aligned}&\widetilde{\mathbb {U}}_{*,k,j,M, \textrm{loc}}^{E,0}(f)(x)\\&\quad \le Ct^{k+\frac{1}{2}} \int _0^\infty \partial _t^k\big [t^{-M}e^{-\frac{u}{t}}\big ]\Big |u^{M-\frac{3}{2}}\int _{\mathbb {R}^d}\frac{|x-y|}{\sqrt{u}}W_u(x-y)|f(y)|dydu\\&\quad \le C\sup _{v>0}W_v(|f|)(x),\quad x\in \mathbb {R}^d. \end{aligned}$$

We conclude that

$$\begin{aligned} \widetilde{\mathbb {U}}_{*,k,j,M,\textrm{loc}}^{E,h}(f)(x)\le C\sup _{t>0}W_t(|f|)(x),\quad x\in \mathbb {R}^d, \end{aligned}$$

and thus, we establish that $\widetilde{\mathbb {U}}_{*,k,j,M,\textrm{loc}}^{E,h}$ 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 4.2.1.

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

$$\begin{aligned} \mathfrak {g}_{k,j,\textrm{loc}}^{E, h}(f)(x)=\left( \int _0^\infty \left| \int _{\mathbb {R}^d}f(y)\mathfrak {U}_{k,j}^{E,h}(x-y,t)\varphi (x,y)dy\right| ^2\frac{dt}{t}\right) ^{\frac{1}{2}},\quad x\in \mathbb {R}^d. \end{aligned}$$

By using Minkowski inequality and (3.4) we get

$$\begin{aligned}&\mathfrak {g}_{k,j, \textrm{loc}}^{E, h}(f)(x)\le C\int _E \Big \Vert t^{k+1}\partial _t^k\big [te^{-\frac{t^2}{4(u+h)}}\big ]\Big \Vert _{L^2((0,\infty ),\frac{dt}{t})}\\&\qquad \int _{\mathbb {R}^d}|x_j-y_j|W_u(x-y)|f(y)|\varphi (x,y)\frac{dydu}{u(u+h)^{\frac{3}{2}}}\\&\quad \le C \int _0^\infty \frac{1}{u\sqrt{u+h}}\int _{\mathbb {R}^d}|x-y|W_u(x-y)|f(y)|\varphi (x,y)dydu\\&\quad \le C\sup _{v>0}W_v(|f|)(x),\quad x\in \mathbb {R}^d. \end{aligned}$$

As in section 4.1.1 we deduce that if $h>0$ the operator $\mathfrak {g}_{k,j,\textrm{loc}}^{E, h}$ 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 $\mathfrak {g}_{k,j,\textrm{loc}}^{E, 0}$. We consider the Littlewood–Paley function $\mathfrak {g}_{k,j}^{E, 0}$ defined by

$$\begin{aligned} \mathfrak {g}_{k,j}^{E, 0}(f)(x)=\left( \int _0^\infty \left| \int _{\mathbb {R}^d}f(y)\mathfrak {U}_{k,j}^{E,0}(x-y,t)dy\right| ^2\frac{dt}{t}\right) ^{\frac{1}{2}},\quad x\in \mathbb {R}^d. \end{aligned}$$

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

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

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

$$\begin{aligned} \mathcal {F}(W_t)(x)= e^{-\frac{t|x|^2}{2}},\quad x\in \mathbb {R}^d \text{ and } t>0. \end{aligned}$$

(4.2)

By using Plancherel equality and (4.2) we get

$$\begin{aligned}&\int _{\mathbb {R}^d}|\mathfrak {g}_{k,j}^{E ,0}(f)(x)|^2dx\\&\quad =\int _0^\infty \int _{\mathbb {R}^d}\left| \int _{\mathbb {R}^d}f(y)\mathfrak {U}_{k,j}^{E, 0}(x-y,t)dy\right| ^2dx\frac{dt}{t}\\&\quad =\int _0^\infty \int _{\mathbb {R}^d}\left| {\mathcal {F}}\left( \int _{\mathbb {R}^d}f(y)\mathfrak {U}_{k,j}^{E, 0}(\cdot -y,t)dy\right) (z)\right| ^2dz\frac{dt}{t}\\&\quad =\int _0^\infty \int _{\mathbb {R}^d}\Big |\int _E\frac{t^{k+1}}{2\sqrt{\pi }} \frac{\partial _t^k\big [te^{-\frac{t^2}{4u}}\big ]}{u^{\frac{3}{2}}}\int _{{\mathbb {R}}^d} f(y)\mathcal {F}(\partial _{x_j}W_u(\cdot -y))(z)dydu\Big |^2dz\frac{dt}{t}\\&\quad =\int _0^\infty \int _{\mathbb {R}^d}\Big |\mathcal {F}(f)(z)\frac{t^{k+1}}{2\sqrt{\pi }}\int _E \frac{\partial _t^k\big [te^{-\frac{t^2}{4u}}\big ]}{u^{\frac{3}{2}}}\mathcal {F}(\partial _{x_j}W_u)(z)du\Big |^2dz\frac{dt}{t}\\&\quad = \int _{\mathbb {R}^d}|\mathcal {F}(f)(z)|^2\int _0^\infty \Big |\frac{t^{k+1}}{2\sqrt{\pi }}\int _E \frac{\partial _t^k\big [te^{-\frac{t^2}{4u}}\big ]}{u^{\frac{3}{2}}}z_je^{-\frac{u|z|^2}{2}}du\Big |^2\frac{dt}{t}dz,\\&\qquad f\in C_c^\infty (\mathbb {R}^d). \end{aligned}$$

Minkowski inequality and (3.4) lead to

$$\begin{aligned}&\int _{\mathbb {R}^d}|\mathfrak {g}_{k,j}^{E ,0}(f)(x)|^2dx\\&\quad \le C\int _{\mathbb {R}^d}|\mathcal {F}(f)(z)|^2\left( \int _0^\infty |z|e^{-\frac{u|z|^2}{2}}\big \Vert t^{k+1}\partial _t^k\big [te^{-\frac{t^2}{4u}}\big ]\big \Vert _{L^2(0,\infty ),\frac{dt}{t})}\frac{du}{u^{\frac{3}{2}}}\right) ^2dz\\&\quad \le C\int _{\mathbb {R}^d}|\mathcal {F}(f)(z)|^2\left( \int _0^\infty \frac{|z|}{\sqrt{u}}e^{-\frac{u|z|^2}{2}}du\right) ^2dz\\&\quad \le C\int _{\mathbb {R}^d}|\mathcal {F}(f)(z)|^2dz=C\Vert f\Vert _{L^2(\mathbb {R}^d,dx)}^2,\quad f\in C_c^\infty (\mathbb {R}^d). \end{aligned}$$

We now use the Banach-valued Calderón-Zygmund theory for singular integrals (see [35]). We recall that the operator $U_{k,j}^{E,0}$ is defined by

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

It is clear that

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

By using again Minkowski inequality and (3.4)

$$\begin{aligned}{} & {} \left\| \mathfrak {U}_{k,j}^{E, 0}(x-y,\cdot )\right\| _{L^2((0,\infty ),\frac{dt}{t})}\nonumber \\{} & {} \quad \le C\int _0^\infty \Big \Vert t^{k+1}\partial _t^k[te^{-\frac{t^2}{4u}]}\Big \Vert _{L^2((0,\infty ),\frac{dt}{t})}\frac{|x-y|}{u}\frac{e^{-\frac{|x-y|^2}{2u}}}{u^{\frac{d+3}{2}}}du\nonumber \\{} & {} \quad \le C\int _0^\infty \frac{e^{-c\frac{|x-y|^2}{u}}}{u^{\frac{d}{2}+1}}du\le \frac{C}{|x-y|^d},\quad x,y\in \mathbb {R}^d,\;x\not =y. \end{aligned}$$

(4.3)

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

$$\begin{aligned}{} & {} \left\| \partial _{x_i}\mathfrak {U}_{k,j}^{E, 0}(x-y,\cdot )\right\| _{L^2((0,\infty ),\frac{dt}{t})} + \left\| \partial _{y_i}\mathfrak {U}_{k,j}^{E, 0}(x-y,\cdot )\right\| _{L^2((0,\infty ),\frac{dt}{t})}\nonumber \\{} & {} \quad \le \frac{C}{|x-y|^{d+1}},\quad x,y\in \mathbb {R}^d,\;x\not =y. \end{aligned}$$

(4.4)

Let $N\in \mathbb {N}$, $N\ge 2$. 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

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

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

$$\begin{aligned}&\Vert F_x(y)-F_x(y_0)\Vert _{\mathbb {F}_N}\\&\quad \le C\Big \Vert \int _0^\infty \frac{t^{k+1}e^{-\frac{t^2}{8u}}}{u^{(k+2)/2}}|f(y)\partial _{x_j}W_u(x-y)-f(y_0)\partial _{x_j}W_u(x-y_0)|du\Big \Vert _{\mathbb {F}_N}\\&\quad \le C\Big \Vert \int _0^\infty te^{-v}v^{k/2-1}|f(y)\partial _{x_j}W_{\frac{t^2}{8v}}(x-y)-f(y_0)\partial _{x_j}W_{\frac{t^2}{8v}}(x-y_0)|dv\Big \Vert _{\mathbb {F}_N}\\&\quad \le C\int _0^\infty e^{-v}v^{k/2-1}\big \Vert t(f(y)\partial _{x_j}W_{\frac{t^2}{8v}}(x-y)\\&\qquad -f(y_0)\partial _{x_j}W_{\frac{t^2}{8v}}(x-y_0))\big \Vert _{\mathbb {F}_N}dv,\quad y\in \mathbb {R}^d. \end{aligned}$$

Since

$$\begin{aligned}&\big \Vert t(f(y)\partial _{x_j}W_{\frac{t^2}{8v}}(x-y)-f(y_0)\partial _{x_j}W_{\frac{t^2}{8v}}(x-y_0))\big \Vert _{\mathbb {F}_N}\\&\quad = (2\pi )^{-d/2}\left( \int _{\frac{1}{N}}^N t\Big (|f(y)(x_j-y_j)e^{-4\frac{|x-y|^2v}{t^2}}\right. \\&\qquad \left. -f(y_0)(x_j-(y_0)_j)e^{-4\frac{|x-y_0|^2v}{t^2}}|\Big (\frac{8v}{t^2}\Big )^{{\frac{d}{2}}+1}\Big )^2dt\right) ^{\frac{1}{2}}\\&\quad \le C\left( \int _{\frac{1}{N}}^N\frac{v^{d+2}}{t^{2d+3}}\big (|x-y|^2e^{-8\frac{|x-y|^2v}{t^2}}+|x-y_0|^2e^{-8\frac{|x-y_0|^2v}{t^2}}\big )dt\right) ^2\\&\quad \le Cv^{\frac{d+1}{2}}\int _{\frac{1}{N}}^N\frac{dt}{t^{2d+1}}\le Cv^{\frac{d+1}{2}},\quad y\in \mathbb {R}^d \text{ and } v>0, \end{aligned}$$

by using the dominated convergence theorem we deduce that

$$\begin{aligned} \lim _{y\rightarrow y_0}\Vert F_x(y)-F_x(y_0)\Vert _{\mathbb {F}_N}=0. \end{aligned}$$

Since $\mathbb {F}_N$ is a separable Banach space, Pettis’ Theorem ([37, Theorem p. 131]) implies that $F_x$ is $\mathbb {F}_N$ -strongly measurable.

By (4.3) we get

$$\begin{aligned} \int _{\mathbb {R}^d}\Vert \mathfrak {U}_{k,j}^{E, 0}(x-y,\cdot )\Vert _{\mathbb {F}_N}|f(y)|dy<\infty ,\quad x\not \in \textrm{supp }f. \end{aligned}$$

We define

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

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

$$\begin{aligned}{} & {} \int _{\frac{1}{N}}^N\int _{\mathbb {R}^d}|\mathfrak {U}_{k,j}^{E,0}(x-y,t)f(y)g(t)|dydt\\{} & {} \le C\Vert g\Vert _{\mathbb {F}_N}\int _{\mathbb {R}^d}\frac{|f(y)|}{|x-y|^d}dy<\infty ,\quad x\not \in \textrm{supp }f. \end{aligned}$$

We can write

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

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

We also have that

$$\begin{aligned} \int _{\mathbb {R}^d}\Big \Vert \widetilde{U}_{k,j}^{E,0}(f)(x)\Big \Vert _{\mathbb {F}_N}^2dx\le \int _{\mathbb {R}^d}|\mathfrak {g}_{k,j}^{E, 0}(f)(x)|^2dx\le C\Vert f\Vert _{L^2(\mathbb {R}^d,dx)}^2. \end{aligned}$$

(4.5)

According to (4.3), (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 ([35]) 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

  $$\begin{aligned} \sup _{\begin{array}{c} N\in \mathbb {N}\\ N\ge 2 \end{array}}\Big \Vert \widetilde{U}_{k,j}^{E, 0}\Big \Vert _{L^p(\mathbb {R}^d,dx)\longrightarrow L^p_{\mathbb {F}_N}(\mathbb {R}^d,dx)}<\infty , \end{aligned}$$

  for every $1<p<\infty$;

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

  $$\begin{aligned} \sup _{\begin{array}{c} N\in \mathbb {N}\\ N\ge 2 \end{array}}\Big \Vert \widetilde{U}_{k,j}^{E, 0}\Big \Vert _{L^1(\mathbb {R}^d,dx)\longrightarrow L_{\mathbb {F}_N}^{1,\infty }(\mathbb {R}^d,dx)}<\infty . \end{aligned}$$

  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 4.2.2.

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

$$\begin{aligned} \mathbb {G}_{k,j,M,\textrm{loc}}^{E, h}(f)(x)=\left( \int _0^\infty \Big |\int _{\mathbb {R}^d}\mathfrak {U}_{k,j,M}^{E, h}(x-y,t)f(y)\varphi (x,y)dy\Big |^2\frac{dt}{t}\right) ^{\frac{1}{2}},\quad x\in \mathbb {R}^d. \end{aligned}$$

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

$$\begin{aligned}&\mathbb {G}_{k,j,M,\textrm{loc}}^{E, h}(f)(x)\le C\int _{E}\Big \Vert t^{k+\frac{1}{2}}\partial _t^k\big [t^{-M}e^{-\frac{u+h}{t}}\big ]\Big \Vert _{L^2((0,\infty ),\frac{dt}{t})}\frac{(u+h)^{M-1}}{u}\\&\qquad \int _{\mathbb {R}^d}|x_j-y_j|W_u(x-y)|f(y)|\varphi (x,y)dydu\\&\quad \le C\int _E\frac{1}{u\sqrt{u+h}}\int _{\mathbb {R}^d}|x_j-y_j|W_u(x-y)|f(y)|\varphi (x,y)dydu\\&\quad \le C\sup _{v>0}|W_v(|f|)(x)|,\quad x\in \mathbb {R}^d. \end{aligned}$$

It follows that $\mathbb {G}_{k,j,M,\textrm{loc}}^{E, h}$ 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 $\mathbb {G}_{k,j,M,\textrm{loc}}^{E, 0}$ we use the vector-valued Calderón-Zygmund theory ([35]).

We define

$$\begin{aligned} \mathbb {G}_{k,j,M}^{E, 0}(f)(x)=\left( \int _0^\infty \left| \int _{\mathbb {R}^d}\mathfrak {U}_{k,j,M}^{E,0}(x-y,t)f(y)dy\right| ^2\frac{dt}{t}\right) ^{\frac{1}{2}},\quad x\in \mathbb {R}^d. \end{aligned}$$

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

$$\begin{aligned}&\int _{\mathbb {R}^d}|\mathbb {G}_{k,j,M}^{E, 0}(f)(x)|^2dx=\int _0^\infty \int _{\mathbb {R}^d}\left| \int _{\mathbb {R}^d}\mathfrak {U}_{k,j,M}^{E ,0}(x-y,t)f(y)dy\right| ^2dx\frac{dt}{t}\\&\quad =\int _0^\infty \int _{\mathbb {R}^d}\left| \mathcal {F}(f)(z)\frac{t^{k+\frac{1}{2}}}{\Gamma (M)}\int _E \partial _t^k\big [t^{-M}e^{-\frac{u}{t}}\big ]\mathcal {F}(\partial _{x_j}W_u)(z)u^{M-1}dy\right| ^2dz\frac{dt}{t}\\&\quad \le C\int _{\mathbb {R}^d}|\mathcal {F}(f)(z)|^2\left( \int _0^\infty |z|e^{-u|z|^2}\frac{du}{\sqrt{u}}\right) ^2dz\le C\Vert f\Vert _{L^2(\mathbb {R}^d,dx)}^2. \end{aligned}$$

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

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

We have that

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

Minkowski inequality and (3.11) lead to

$$\begin{aligned}{} & {} \Big \Vert \mathfrak {U}_{k,j,M}^{E,0}(x-y,\cdot )\Big \Vert _{L^2((0,\infty ),\frac{dt}{t})}\le C\int _0^\infty \frac{e^{-\frac{|x-y|^2}{u}}}{u^{\frac{d}{2}+1}}du\le \frac{C}{|x-y|^d},\quad \\{} & {} \quad x,y\in \mathbb {R}^d,\;x\not =y. \end{aligned}$$

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

$$\begin{aligned}{} & {} \Big \Vert \partial _{x_i}\mathfrak {U}_{k,j,M}^{E,0}(x-y,\cdot )\Big \Vert _{L^2((0,\infty ),\frac{dt}{t})}+\Big \Vert \partial _{y_i}\mathfrak {U}_{k,j,M}^{E,0}(x-y,\cdot )\Big \Vert _{L^2((0,\infty ),\frac{dt}{t})}\\{} & {} \quad \le \frac{C}{|x-y|^{d+1}},\quad x,y\in \mathbb {R}^d,\;x\not =y. \end{aligned}$$

By proceeding as in the previous case we can conclude that the operator $\mathbb {G}_{k,j,M,\textrm{loc}}^{E, 0}$ 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 4.3.1.

We study the local variation operator defined by

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

where

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

By using (3.5) 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

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

We have that

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

According to Faà di Bruno’s formula

$$\begin{aligned}{} & {} \partial ^{k+2}_t \left[ \frac{1}{(t^2 + |z|^2)^{\frac{d-1}{2}}} \right] = \sum _{0\le \ell \le \frac{k+2}{2}} a_\ell t^{k+2-2\ell } \frac{1}{(t^2 + |z|^2)^{\frac{d-1}{2}+k+2-\ell }}, \quad \\{} & {} \quad z\in \mathbb {R}^d \text { and } t>0, \end{aligned}$$

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

$$\begin{aligned} t^{k+1}\partial ^{k+1}_t \mathbb {P}_t(z) = \phi _t(z), \quad z\in \mathbb {R}^d \text { and } t>0, \end{aligned}$$

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

$$\begin{aligned} \phi (z) = \sum _{0\le \ell \le \frac{k+2}{2}} \frac{a_\ell }{(1 + |z|^2)^{\frac{d-1}{2}+k+2-\ell }}, \quad z\in \mathbb {R}^d. \end{aligned}$$

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)]),

$$\begin{aligned} \mathcal {F}(\partial _{x_j}\partial ^k_t \mathbb {P}_t)(z) = i z_j |z|^k e^{-t|z|} = i \frac{z_j}{|z|}|z|^{k+1}e^{-t|z|}, \quad z\in \mathbb {R}^d{\setminus }\{0\} \text { and } t>0. \end{aligned}$$

We get

$$\begin{aligned} t^{k+1}\partial _{x_j}\partial ^k_t \mathbb {P}_t (f)(x) = C t^{k+1} \partial ^{k+1}_t \mathbb {P}_t (R_jf)(x) \\ = C(\phi _t * R_j f)(x), \quad x\in \mathbb {R}^d \text { and } t>0, \end{aligned}$$

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

We define

$$\begin{aligned} \psi (u) = \sum _{0\le \ell \le \frac{k+2}{2}} \frac{a_\ell }{(1+u^2)^{\frac{d+1}{2}+k+2-\ell }},\quad u\in (0,\infty ). \end{aligned}$$

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 $\int _0^\infty |\psi '(u)|u^d du <\infty$. According to Lemma 2.4 in [3], 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 (3.5) and Minkoswki inequality we obtain

$$\begin{aligned} \begin{aligned} \Big \Vert \mathfrak {U}^{(0,\infty ),0}_{k,j}(x-y,\cdot )\Big \Vert _{V_\rho }&\le C \int _0^\infty \frac{\Vert t^{k+1}\partial ^k_t [te^{\frac{-t^2}{4u}}]\Vert _{V_\rho }}{u^{\frac{3}{2}}} \frac{|x_j - y_j|}{u^{\frac{d}{2} +1}} e^{\frac{-|x-y|^2}{4u}} du \\ {}&\le C \int _0^\infty \frac{e^{\frac{-c|x-y|^2}{u}}}{u^{\frac{d}{2} +1}} du \le \frac{C}{|x-y|^d},\quad x, y\in \mathbb {R}^d,\;x\ne y, \end{aligned}\nonumber \\ \end{aligned}$$

(4.6)

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

$$\begin{aligned}{} & {} \Big \Vert \partial _{x_i}\mathfrak {U}^{(0,\infty ),0}_{k,j}(x-y,\cdot )\Big \Vert _{V_\rho }+\Big \Vert \partial _{y_i}\mathfrak {U}^{(0,\infty ),0}_{k,j}(x-y,\cdot )\Big \Vert _{V_\rho } \le \frac{C}{|x-y|^{d+1}},\quad \nonumber \\{} & {} \quad x, y\in \mathbb {R}^d,\;x\ne y. \end{aligned}$$

(4.7)

Let $N\in \mathbb {N}$, $N\ge 2$. 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

$$\begin{aligned} V^N_\rho (g) = \sup _{\begin{array}{c} \frac{1}{N}< t_\ell< t_{\ell -1}< \dots< t_1<N\\ \ell \in \mathbb {N} \end{array}} \left( \sum _{i = 1}^{\ell -1} |g(t_{i+1}) - g(t_i)|^\rho \right) ^{\frac{1}{\rho }} < \infty . \end{aligned}$$

By identifying those functions that differ by 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

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

By [1, Lemma 4] we get

$$\begin{aligned} \begin{aligned} V^N_\rho (F_x(y))&\le C \int _{\frac{1}{N}}^N \int _0^\infty \left| \partial _t \left[ t^{k+1} \partial ^k_t\left( t e^{-\frac{t^2}{4u}} \right) \right] \right| \frac{|x_j - y_j|}{u^{\frac{d}{2}+ \frac{5}{2}}} e^{-\frac{|x-y|^2}{2u}} du dt \\ {}&\le C \int _{\frac{1}{N}}^N \int _0^\infty \left( \frac{t^{k+1}}{u^{\frac{k}{2}}} + \frac{t^{k}}{u^{\frac{k-1}{2}}} \right) \frac{e^{-\frac{t^2}{8u}}}{u^{\frac{d}{2}+ 2}} e^{-c\frac{|x-y|^2}{u}} du dt \\ {}&\le C \int _0^\infty \frac{e^{-c\frac{N^{-2}+|x-y|^2}{u}}}{u^{\frac{d+3}{2}}} du \\ {}&\le \frac{C}{(N^{-2}+|x-y|^2)^{d+1}}, \quad y\in \mathbb {R}^d. \end{aligned} \end{aligned}$$

We define the mapping

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

$F_x$ is continuous. Indeed, let $y_0\in \mathbb {R}^d$. By Lemma 4 in [1] we obtain

$$\begin{aligned} \begin{aligned} V^N_\rho&(F_x(y) - F_x(y_0)) \\ {}&\le C \int _{\frac{1}{N}}^N \int _0^\infty \frac{\left| \partial _t \left[ t^{k+1} \partial ^k_t\left( t e^{-\frac{t^2}{4u}} \right) \right] \right| }{u^{\frac{3}{2}}} \left| \frac{x_j-y_j}{u} W_u(x-y) f(y)\right. \\&\quad \left. - \frac{x_j - (y_0)_j}{u} W_u(x-y_0) f(y_0) \right| du dt \\ {}&\le C \int _{\frac{1}{N}}^N \int _0^\infty \frac{t^k e^{-c\frac{t^2}{u}}}{u^{\frac{k+2}{2}}} \left| \frac{x_j-y_j}{u} W_u(x-y) f(y) \right. \\&\quad \left. - \frac{x_j - (y_0)_j}{u} W_u(x-y_0) f(y_0) \right| du dt \\ {}&\le C \int _{\frac{1}{N}}^N \int _0^\infty \frac{e^{-cv}v^{\frac{k}{2}}}{t^2} \left| (x_j-y_j) W_{\frac{t^2}{v}}(x-y) f(y) \right. \\&\quad \left. - (x_j - (y_0)_j) W_{\frac{t^2}{v}}(x-y_0) f(y_0) \right| dv dt. \end{aligned} \end{aligned}$$

Since

$$\begin{aligned}{} & {} \int _{\frac{1}{N}}^N \frac{1}{t^2} \left| (x_j-y_j) W_{\frac{t^2}{v}}(x-y) f(y) - (x_j - y_{0,j}) W_{\frac{t^2}{v}}(x-y_0) f(y_0) \right| dt \\{} & {} \quad \le C v^{{\frac{d-1}{2}}}, \quad y\in \mathbb {R}^d\text { and } v>0, \end{aligned}$$

the dominated convergence theorem leads to

$$\begin{aligned} \lim _{y\rightarrow y_0} V_{\rho }^N (F_x(y) - F_x(y_0)) = 0. \end{aligned}$$

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

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

We define

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

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

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

$$\begin{aligned} T_a(g) = g(a) - g(1). \end{aligned}$$

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

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

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

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

and

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

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

$$\begin{aligned} \sup _{N\in \mathbb {N},N\ge 2} \Vert \widetilde{M}^{(0,\infty ),0}_{k,j}\Vert _{ L^1(\mathbb {R}^d,dx)\hookrightarrow L_{V_\rho ([1/N, N])}^{1,\infty }(\mathbb {R}^d,dx) } <\infty . \end{aligned}$$

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

$$\begin{aligned} \begin{aligned}&\Vert \mathfrak {U}^{E,0}_{k,j} (x-y,\cdot ) \Vert _{V_\rho }\\&\le C \int _{E} \int _0^{\infty } \frac{\left| \partial _t \left[ t^{k+1} \partial ^k_t\left( t e^{-\frac{t^2}{4u}} \right) \right] \right| }{u^{\frac{3}{2}}} \frac{|x_j-y_j|}{u} W_u(x-y) dt du \\ {}&\le C \int _\eta ^\infty \int _0^{\infty } \frac{t^k e^{-c\frac{t^2}{u}}}{u^{\frac{k+4}{2}}} dt \, \frac{e^{-c\frac{|x-y|^2}{u}}}{u^{\frac{d-1}{2}}} du \\ {}&\le \frac{C}{|x-y|^{d-1}} \int _\eta ^\infty \frac{du}{u^{\frac{3}{2}}} \int _0^{\infty } v^{\frac{k-1}{2}}e^{-v}dv \\ {}&\le C \frac{1 + |x|}{|x-y|^{d-1}}, \quad x\ne y. \end{aligned} \end{aligned}$$

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 4.3.2.

We consider the local variation operator defined by

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

where

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

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

$$\begin{aligned} \mathfrak U_{k,j,M}^{(0,\infty ),0}(z,t)=\frac{t^{k+1/2}}{\Gamma (M)}\partial ^k_t\partial _{x_j}\int _0^\infty t^{-M}e^{-u/t}W_u(z)u^{M-1}du,\quad z\in {\mathbb {R}}^d \text{ and } t>0. \end{aligned}$$

We can write

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

where

$$\begin{aligned} L_{k,j,M}(z,t)=\frac{t^{k+1/2}}{\Gamma (M)}\partial ^k_t\partial _{x_j}\int _0^\infty e^{-v}W_{vt}(z)v^{M-1}dv,\quad z\in {\mathbb {R}}^d \text{ and } t>0. \end{aligned}$$

We have that $\mathcal F(\partial _{z_j}W_{vt}(z))(w)=iw_je^{-\frac{|w|^2tv}{2}}$, $w\in {\mathbb {R}}^d$ and $t, v>0$. Then,

$$\begin{aligned}{} & {} \mathcal {F}(t^{k+1/2}\partial ^k_t\partial _{z_j}W_{vt}(z))(w)=\frac{(-1)^k}{2^k}iw_jt^{k+1/2}v^k|w|^{2k}e^{-\frac{tv|w|^2}{2}}, \quad \\{} & {} \quad w\in {\mathbb {R}}^d \text{ and } t,v>0, \end{aligned}$$

and we get

$$\begin{aligned}&\mathcal {F}(U_{k,j,M}^{(0,\infty ),0}(\cdot ,t))(w)=\mathcal {F}(f*L_{k,j,M}(\cdot ,t))(w)=-i\frac{w_j}{|w|}\mathcal {F}(f)(w)\Psi _{k,t}(w)\\&\quad =\mathcal {F}(R_jf)(w)\mathcal {F}(\mathcal {F}^{-1}(\Psi _{k,t}))(w),\quad w\in \mathbb {R}^d \text{ and } t>0, \end{aligned}$$

where $R_j$ denotes the j-th Euclidean Riesz transformation and

$$\begin{aligned}{} & {} \Psi _{k,t}(w)=\frac{(-1)^{k+1}}{2^k\Gamma (M)}t^{k+\frac{1}{2}}\int _0^\infty e^{-v} v^{k+M-1}|w|^{2k+1}e^{-\frac{tv|w|^2}{2}}dv,\quad \\{} & {} \quad w\in \mathbb {R}^d \text{ and } t>0. \end{aligned}$$

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

$$\begin{aligned} h_\nu (g)(s)=\int _0^\infty (sr)^{-\nu }J_\nu (sr)g(r)r^{2\nu +1}d\nu ,\quad s\in (0,\infty ), \end{aligned}$$

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

$$\begin{aligned} {\mathcal {F}}^{-1}(\psi _k)(x)=\frac{1}{(2\pi )^d}h_{\frac{d-2}{2}}(\widetilde{\psi }_k)(|x|),\quad x\in \mathbb R^d, \end{aligned}$$

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

$$\begin{aligned} \mathcal {F}^{-1}(\psi _k)(s)=c_ka^{-\frac{d+1}{2}-k}\;_1F_1\Big (\frac{d+1}{2}+k,\frac{d}{2};-\frac{s^2}{4a}\Big ),\quad s\in (0,\infty ), \end{aligned}$$

for certain $c_k>0$. Here $_1F_1$ represents the Kummer’s confluent hypergeometric function (see, for example, [23, §9]).

Thus,

$$\begin{aligned}&\mathcal {F}^{-1}(\Psi _{k,t})(x)=\frac{(-1)^{k+1}2^{\frac{d+1}{2}}c_k}{\Gamma (M)}t^{-\frac{d}{2}}\\&\quad \int _0^\infty e^{-v} v^{M-1-\frac{d+1}{2}}\;_1F_1\Big (\frac{d+1}{2}+k,\frac{d}{2};-\frac{|x|^2}{2tv}\Big )dv,\quad x\in \mathbb {R}^d, \end{aligned}$$

and we can write

$$\begin{aligned} U_{k,j,M}^{(0,\infty ),0}(f)(x,t)=\big ((\phi _k)_{\sqrt{t}}*R_jf\big )(x),\quad x\in \mathbb {R}^d \text{ and } t>0, \end{aligned}$$

where

$$\begin{aligned}{} & {} \phi _k(x)=\frac{(-1)^{k+1}2^{\frac{d+1}{2}}c_k}{\Gamma (M)}\int _0^\infty e^{-v}v^{M-1-\frac{d+1}{2}}\;_1F_1\Big (\frac{d+1}{2}+k,\frac{d}{2};-\frac{|x|^2}{2v}\Big )dv,\quad \\{} & {} \quad x\in \mathbb {R}^d. \end{aligned}$$

By taking into account [23, (9.11.2)] with the adequate correct form (see also [31, Theorem 1]), for every $\alpha ,\gamma \in \mathbb {R}$, with $\gamma \not =0,-1,-2,...$, we have that $\;_1F_1(\alpha ,\gamma ;z)=e^z\;_1F_1(\gamma -\alpha ,\gamma ; -z)$, $z\in \mathbb {R}$. Then,

$$\begin{aligned} \lim _{z\rightarrow +\infty }\;_1F_1\Big (\frac{d+1}{2}+k,\frac{d}{2};-z\Big )=0. \end{aligned}$$

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

$$\begin{aligned}{} & {} \widetilde{\phi }_k(r)=\frac{(-1)^{k+1}2^{\frac{d+1}{2}}c_k}{\Gamma (M)}\int _0^\infty e^{-v}v^{M-1-\frac{d+1}{2}}\;_1F_1\Big (\frac{d+1}{2}+k,\frac{d}{2};-\frac{r^2}{2v}\Big )dv,\quad \\{} & {} \quad r\in (0,\infty ). \end{aligned}$$

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

$$\begin{aligned}&\int _0^\infty \big |\frac{d}{dr}\widetilde{\phi }_k(r)\big |r^ddr\\&\quad \le C\int _0^\infty e^{-v}v^{M-2-\frac{d+1}{2}}\int _0^\infty r^{d+1}\Big |\;_1F_1\Big (\frac{d+1}{2}+k+1,\frac{d}{2}+1;-\frac{r ^2}{2v}\Big )\Big |drdv\\&\quad \le C\int _0^\infty e^{-v}v^{M-2-\frac{d+1}{2}}\left( \int _0^{\sqrt{v}}+\int _{\sqrt{v}}^\infty \right) r^{d+1} \Big |\;_1F_1\\&\qquad \times \Big (\frac{d+1}{2}+k+1,\frac{d}{2}+1;-\frac{r^2}{2v}\Big )\Big |drdv\\&\quad \le C\left( \int _0^\infty e^{-v}v^{M-\frac{3}{2}}dv+\int _0^\infty e^{-v}v^{M-2-\frac{d+1}{2}}\int _{\sqrt{v}}^\infty r^{-2k-2}v^{\frac{d+1}{2}+k+1}dr dv\right) \\&\quad =C\int _0^\infty e^{-v}v^{M-\frac{3}{2}}dv<\infty . \end{aligned}$$

From [3, 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 ${\mathfrak {v}}_{\rho ,k,j,\textrm{loc}}^{(0,\infty ),0}$ in section 4.3.1.

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

$$\begin{aligned}{} & {} \left\| \mathfrak {U}_{k,j,M}^{E, 0}(x,y,\cdot )\right\| _{V_\rho }\le C\int _E |\partial _{x_j}W_u(x-y)|\frac{du}{\sqrt{u}}\le \frac{C}{|x-y|^{d-1}}\int _\eta ^\infty \frac{du}{u^{\frac{3}{2}}}\\{} & {} \quad \le C\frac{1+|x|}{|x-y|^{d-1}},\quad x,y\in {\mathbb {R}}^d,\;x\ne y. \end{aligned}$$

By using Lemma 2.5 we prove that ${\mathfrak {v}}_{\rho ,k,j,M,\textrm{loc}}^{E, 0}$ 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 ${\mathfrak {v}}_{\rho ,k,j,M,\textrm{loc}}^{E, 0}$ 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 $\eta >0$.