# Harmonic analysis operators associated with Laguerre polynomial expansions
on variable Lebesgue spaces

Jorge J. Betancor  Jorge J. Betancor
Departamento de Análisis Matemático, Universidad de La Laguna,
Campus de Anchieta, Avda. Astrofísico Sánchez, s/n,
38721 La Laguna (Sta. Cruz de Tenerife), Spain<EMAIL_ADDRESS>,  Estefanía
Dalmasso  Estefanía Dalmasso, Pablo Quijano
Instituto de Matemática Aplicada del Litoral, UNL, CONICET, FIQ.
Colectora Ruta Nac. Nº 168, Paraje El Pozo,
S3007ABA, Santa Fe, Argentina<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Pablo Quijano  and  Roberto Scotto  Roberto
Scotto
Universidad Nacional del Litoral, FIQ.
Santiago del Estero 2829,
S3000AOM, Santa Fe, Argentina<EMAIL_ADDRESS>
###### Abstract.

In this paper we give sufficient conditions on a measurable function
$p:(0,\infty)^{n}\rightarrow[1,\infty)$ in order that harmonic analysis
operators (maximal operators, Riesz transforms, Littlewood–Paley functions and
multipliers) associated with $\alpha$-Laguerre polynomial expansions are
bounded on the variable Lebesgue space
$L^{p(\cdot)}((0,\infty)^{n},\mu_{\alpha})$, where
$d\mu_{\alpha}(x)=2^{n}\prod_{j=1}^{n}\frac{x_{j}^{2\alpha_{j}+1}e^{-x_{j}^{2}}}{\Gamma(\alpha_{j}+1)}dx$,
being $\alpha=(\alpha_{1},\dots,\alpha_{n})\in[0,\infty)^{n}$ and
$x=(x_{1},\dots,x_{n})\in(0,\infty)^{n}$.

###### Key words and phrases:

Variable exponent $L^{p}$-spaces, Laguerre polynomials, diffusion semigroups,
maximal operators, Riesz transforms, Littlewood-Paley functions.

###### 2020 Mathematics Subject Classification:

42B15, 42B20, 42B25, 42B35

The first author is partially supported by grant PID2019-106093GB-I00 from the
Spanish Government. The second and fourth authors are partially supported by
grants PICT-2019-2019-00389 (ANPCyT) and CAI+D 2019-015 (UNL)

##  1\. Introduction and main results

In this article we establish $L^{p(\cdot)}-$boundedness properties of harmonic
analysis operators appearing in the context of Laguerre polynomials.

For every $\alpha>-1$ and $k\in\mathbb N:=\\{0,1,2,\dots\\}$, the normalized
Laguerre polynomial of type $\alpha$ and degree $k$ is defined by the formula
(c.f. [24], [43])

$L_{k}^{\alpha}(x)=\sqrt{\frac{\Gamma(\alpha+1)}{\Gamma(\alpha+k+1)k!}}e^{x}x^{-\alpha}\frac{d^{k}}{dx^{k}}\left(e^{-x}x^{k+\alpha}\right),\quad
x\in(0,\infty).$

Let $\alpha=(\alpha_{1},\dots,\alpha_{n})\in\left(-1,\infty\right)^{n}$. For
every $k=(k_{1},\dots,k_{n})\in\mathbb N^{n}$, the $k$–th Laguerre polynomial
of type $\alpha$ and degree $\widehat{k}:=k_{1}+\cdots+k_{n}$ is defined by

$L_{k}^{\alpha}(x)=\prod_{i=1}^{n}L_{k_{i}}^{\alpha_{i}}(x_{i}),\quad
x=(x_{1},\dots,x_{n})\in(0,\infty)^{n}:=\mathbb{R}_{+}^{n}.$

The sequence of polynomials $\\{L_{k}^{\alpha}\\}_{k\in\mathbb N^{n}}$ is an
orthonormal basis for $L^{2}(\mathbb{R}_{+}^{n},\nu_{\alpha})$ being
$d\nu_{\alpha}(x)=\prod_{j=1}^{n}\frac{x_{j}^{\alpha_{j}}e^{-x_{j}}}{\Gamma(\alpha_{j}+1)}dx$
a non-doubling measure defined on $\mathbb{R}_{+}^{n}$, see [24, §4.21] for
the orthonormality of the family.

We define, for each $k\in\mathbb N^{n}$,
$\mathcal{L}_{k}^{\alpha}(x)=\prod_{i=1}^{n}L_{k_{i}}^{\alpha_{i}}(x_{i}^{2})$,
$x=(x_{1},\dots,x_{n})\in\mathbb{R}_{+}^{n}$. The sequence
$\\{\mathcal{L}_{k}^{\alpha}\\}_{k\in\mathbb N^{n}}$ is an orthonormal basis
for $L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ where

$d\mu_{\alpha}(x)=2^{n}\prod_{j=1}^{n}\frac{x_{j}^{2\alpha_{j}+1}e^{-x_{j}^{2}}}{\Gamma(\alpha_{j}+1)}dx$

is the pull-back measure from $d\nu_{\alpha}$ on $\mathbb{R}_{+}^{n}$ through
the one-to-one and onto change of variables
$\Psi:\mathbb{R}_{+}^{n}\to\mathbb{R}_{+}^{n}$ defined as
$\Psi(x)=x^{2}:=(x_{1}^{2},\cdots,x_{n}^{2}),$ for
$x=(x_{1},\cdots,x_{n})\in\mathbb{R}_{+}^{n}.$

We consider the differential Laguerre operator defined on $\mathbb{R}_{+}^{n}$
as follows

$\amsbb{\Delta}_{\alpha}=-\frac{1}{4}\sum_{j=1}^{n}\left(\frac{d^{2}}{dx_{j}^{2}}+\left(\frac{2\alpha_{j}+1}{x_{j}}-2x_{j}\right)\frac{d}{dx_{j}}\right).$

It turns out that the polynomials $\mathcal{L}_{k}^{\alpha}$ are
eigenfunctions of the operator $\amsbb{\Delta}_{\alpha}$, with
$\amsbb{\Delta}_{\alpha}\mathcal{L}_{k}^{\alpha}=\widehat{k}\mathcal{L}_{k}^{\alpha}$
for every $k\in\mathbb N^{n}.$

For every $f\in L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ and $k\in\mathbb
N^{n}$, we denote

$c_{k}^{\alpha}(f)=\int_{\mathbb{R}_{+}^{n}}\mathcal{L}_{k}^{\alpha}(x)f(x)d\mu_{\alpha}(x).$

We define the operator $\Delta_{\alpha}$ by

$\Delta_{\alpha}f=\sum_{k\in\mathbb
N^{n}}\lambda_{k}c_{k}^{\alpha}(f)\mathcal{L}_{k}^{\alpha},\quad f\in
D(\Delta_{\alpha}),$

where $\lambda_{k}=\widehat{k}$ for every $k=(k_{1},\dots,k_{n})\in\mathbb
N^{n}$, and

$D(\Delta_{\alpha})=\left\\{f\in
L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha}):\sum_{k\in\mathbb
N^{n}}\left|\lambda_{k}c_{k}^{\alpha}(f)\right|^{2}<\infty\right\\},$

is the domain of $\Delta_{\alpha}$ on
$L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha}).$ Note that
$\Delta_{\alpha}f=\amsbb{\Delta}_{\alpha}f$ for every $f\in
C_{c}^{\infty}(\mathbb{R}_{+}^{n})$, the space of smooth and compactly
supported functions on $\mathbb{R}_{+}^{n}$.

The operator $\Delta_{\alpha}$ is symmetric and positive, and
$-\Delta_{\alpha}$ generates a semigroup of operators
$\\{W_{t}^{\alpha}\\}_{t>0}$ in $L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$
where, for every $t>0$,

$W_{t}^{\alpha}(f)=\sum_{k\in\mathbb
N^{n}}e^{-\lambda_{k}t}c_{k}^{\alpha}(f)\mathcal{L}_{k}^{\alpha},\quad f\in
L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha}).$

According to the Hille-Hardy formula ([24, (4.17.6)] with $x$ and $y$ replaced
by $x^{2}$ and $y^{2}$ respectively, and $t$ by $e^{-t}$), we have that

$\displaystyle\sum_{k\in\mathbb
N^{n}}e^{-\lambda_{k}t}\mathcal{L}_{k}^{\alpha}(x)\mathcal{L}_{k}^{\alpha}(y)$
$\displaystyle=\prod_{j=1}^{n}\frac{\Gamma(\alpha_{j}+1)}{1-e^{-t}}\left(e^{-t/2}x_{j}y_{j}\right)^{-\alpha_{j}}I_{\alpha_{j}}\left(\frac{2e^{-t/2}x_{j}y_{j}}{1-e^{-t}}\right)$
$\displaystyle\qquad\times\exp\left(-\frac{e^{-t}}{1-e^{-t}}(x_{j}^{2}+y_{j}^{2})\right),$

for $x=(x_{1},\dots,x_{n}),y=(y_{1},\dots,y_{n})\in\mathbb{R}_{+}^{n}$ and
$t>0$. Here $I_{\nu}$ is the modified Bessel function of the first kind and
order $\nu>-1$. We can write, for every $f\in
L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ and $t>0$,

(1.1)
$W_{t}^{\alpha}(f)(x)=\int_{\mathbb{R}_{+}^{n}}W_{t}^{\alpha}(x,y)f(y)d\mu_{\alpha}(y),\quad
x\in\mathbb{R}_{+}^{n},$

being

$W_{t}^{\alpha}(x,y)=\prod_{j=1}^{n}\frac{\Gamma(\alpha_{j}+1)}{1-e^{-t}}\left(e^{-t/2}x_{j}y_{j}\right)^{-\alpha_{j}}I_{\alpha_{j}}\left(\frac{2e^{-t/2}x_{j}y_{j}}{1-e^{-t}}\right)e^{-\frac{e^{-t}}{1-e^{-t}}(x_{j}^{2}+y_{j}^{2})},$

for $x=(x_{1},\dots,x_{n}),y=(y_{1},\dots,y_{n})\in\mathbb{R}_{+}^{n}$ and
$t>0.$ The integral in (1.1) defines, for every $t>0$, a contraction on
$L^{p}(\mathbb{R}_{+}^{n},\mu_{\alpha})$, for every $1\leq p\leq\infty$. By
defining, for each $t>0$, $W_{t}^{\alpha}$ by (1.1), the family
$\\{W_{t}^{\alpha}\\}_{t>0}$ is a symmetric diffusion semigroup in Stein’s
sense in $(\mathbb{R}_{+}^{n},\mu_{\alpha})$ (see [42, p. 65]).

The Poisson semigroup $\\{P_{t}^{\alpha}\\}_{t>0}$ associated with the
operators $-\sqrt{\Delta_{\alpha}}$ is defined by

$P_{t}^{\alpha}(f)=\sum_{k\in\mathbb
N^{n}}e^{-t\sqrt{\lambda_{k}}}c_{k}^{\alpha}(f)\mathcal{L}_{k}^{\alpha},\quad
f\in L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha}),t>0.$

By using the subordination formula, we have that

(1.2)
$P_{t}^{\alpha}(f)=\frac{t}{2\sqrt{\pi}}\int_{0}^{\infty}\frac{e^{-\frac{t^{2}}{4u}}}{u^{\frac{3}{2}}}W_{u}^{\alpha}(f)du,\quad
f\in L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha}),t>0.$

We can write, for every $t>0$ and $f\in
L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$,

(1.3)
$P_{t}^{\alpha}(f)(x)=\int_{\mathbb{R}_{+}^{n}}P_{t}^{\alpha}(x,y)f(y)d\mu_{\alpha}(y),\quad
x\in\mathbb{R}_{+}^{n},$

where

$P_{t}^{\alpha}(x,y)=\frac{t}{2\sqrt{\pi}}\int_{0}^{\infty}\frac{e^{-\frac{t^{2}}{4u}}}{u^{\frac{3}{2}}}W_{u}^{\alpha}(x,y)du,\quad
x,y\in\mathbb{R}_{+}^{n},t>0.$

For each $t>0$, the integral in (1.3) defines a contraction on
$L^{p}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ when $1\leq p\leq\infty$. By defining
$P_{t}^{\alpha}$ as in (1.2), $\\{P_{t}^{\alpha}\\}_{t>0}$ is a Stein
symmetric diffusion semigroup in $(\mathbb{R}_{+}^{n},\mu_{\alpha})$.

The study of harmonic analysis in the Laguerre setting was initiated by
Muckenhoupt ([29, 30]). Muckenhoupt’s context is transferred to ours by
applying the transform mapping $\Psi$ mentioned above (see, for instance,
[41]).

The maximal operators $W_{*}^{\alpha}$ and $P_{*}^{\alpha}$ are defined by

$W_{*}^{\alpha}(f)=\sup_{t>0}|W_{t}^{\alpha}(f)|,\quad
P_{*}^{\alpha}(f)=\sup_{t>0}|P_{t}^{\alpha}(f)|.$

From [42, p. 73], it follows that both $W_{*}^{\alpha}$ and $P_{*}^{\alpha}$
are bounded on $L^{p}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ for every
$1<p\leq\infty$. Muckenhoupt ([29]) proved that $W_{*}^{\alpha}$ is bounded
from $L^{1}(\mathbb{R}_{+},\mu_{\alpha})$ into
$L^{1,\infty}(\mathbb{R}_{+},\mu_{\alpha})$. He considered the one-dimensional
case. This result was extended to higher dimensions by Dinger ([20]). Note
that the subordination formula (1.2) allows us to deduce the
$L^{p}$-boundedness properties for $P_{*}^{\alpha}$ from the corresponding
ones of $W_{*}^{\alpha}$. The holomorphic Laguerre semigroups and the maximal
operators associated with them where studied in [40].

Taking into account the spectral decomposition of $\Delta_{\alpha}$ and [33,
§7.2] we define the first order Riesz-Laguerre transform associated to
$\Delta_{\alpha}$ as

$R_{\alpha}^{i}f=\sum_{k\in\mathbb
N^{n}\setminus\\{(0,\dots,0)\\}}\frac{1}{\sqrt{\lambda_{k}}}\partial_{x_{i}}\mathcal{L}_{k}^{\alpha}(x)c_{k}^{\alpha}(f),\qquad
f\in L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha}).$

Thus the operator $R_{\alpha}^{i}$ turns out to be bounded on
$L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha}).$

Moreover, we can also define the higher order Riesz-Laguerre transforms as an
extension of the first order ones in the following way

$R_{\alpha}^{\beta}f=\sum_{k\in\mathbb
N^{n}\setminus\\{(0,\dots,0)\\}}\frac{1}{\lambda_{k}^{\widehat{\beta}/2}}D^{\beta}_{x}\mathcal{L}_{k}^{\alpha}(x)c_{k}^{\alpha}(f),$

with $\beta\in\mathbb N^{n}\setminus\\{(0,\dots,0)\\}$ and
$D^{\beta}_{x}=\frac{\partial^{\widehat{\beta}}}{\partial
x_{1}^{\beta_{1}}\dots\ \partial x_{n}^{\beta_{n}}}.$ They are also bounded on
$L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha}),$ see [33]. Let us remark that
$R_{\alpha}^{i}f=R_{\alpha}^{e_{i}}f$ with $e_{i}$ the $i$-th unit vector and
$f\in L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha}).$

For every $b>0$ we define the fractional integral $\Delta_{\alpha}^{-b}$ as
the $-b$ power of $\Delta_{\alpha}$, given, for every $f\in
L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$, by

$\Delta_{\alpha}^{-b}f=\sum_{k\in\mathbb{N}^{n}\setminus\\{0\\}}\lambda_{k}^{-b}c_{k}^{\alpha}(f)\mathcal{L}_{k}^{\alpha}.$

Let us notice that for any number $b>0$ in [41] it was proved that the
integral kernel for $\Delta_{\alpha}^{-b}$ is given by

$\displaystyle K_{b}(x,y)$
$\displaystyle=\frac{1}{\Gamma(b)}\int_{0}^{\infty}t^{b-1}\left(W_{t}^{\alpha}(x,y)-1\right)dt$
$\displaystyle=\frac{1}{\Gamma(b)}\int_{0}^{1}(-\log
r)^{b-1}\left(W^{\alpha}_{-\log r}(x,y)-1\right)\,\frac{dr}{r}.$

If $f\in C_{c}^{\infty}(\mathbb{R}_{+}^{n})$ we have that, for every
$\beta\in\mathbb N^{n}\setminus\\{(0,\dots,0)\\}$,

$R_{\alpha}^{\beta}f=D^{\alpha}\Delta_{\alpha}^{-\widehat{\beta}/2}f.$

According to what was done in [41] we can also conclude that the operator
$R_{\alpha}^{\beta}$, off the diagonal, is given by the smooth kernel
$D_{x}^{\beta}K_{\frac{\widehat{\beta}}{2}}(x,y),$ i.e.

$R_{\alpha}^{\beta}f(x)=\int_{\mathbb{R}_{+}^{n}}D_{x}^{\beta}K_{\frac{\widehat{\beta}}{2}}(x,y)f(y)\,d\mu_{\alpha}(y),$

for all $x\notin\text{supp}(f)$ when $f\in
C_{c}^{\infty}(\mathbb{R}_{+}^{n})$.

From [8, Theorem 1.1], [41, Theorem 1.1] and [32, Theorem 13], we deduce that
$R_{\alpha}^{\beta}$ can be extended from
$L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})\cap
L^{p}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ to
$L^{p}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ as a bounded operator on
$L^{p}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ when $1<p<\infty$. It can also be
extended from $L^{1}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ into
$L^{1,\infty}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ for $\widehat{\beta}\leq 2$
and from $L^{1}(\mathbb{R}_{+}^{n},w\mu_{\alpha})$ into
$L^{1,\infty}(\mathbb{R}_{+}^{n},\mu_{\alpha})$, with
$w(y)=(1+\sqrt{|y|})^{\widehat{\beta}-2}$, for $\widehat{\beta}>2$ (see [21]).
We continue denoting by $R_{\alpha}^{\beta}$ to those extensions. Furthermore,
there exists a constant $c_{\beta}$ such that, for every $f\in
L^{p}(\mathbb{R}_{+}^{n},\mu_{\alpha})$, $1\leq p<\infty$,

$R_{\alpha}^{\beta}(f)(x)=c_{\beta}f(x)+\lim_{\epsilon\to
0^{+}}\int_{y\in\mathbb{R}_{+}^{n},\,|x-y|>\epsilon}R_{\alpha}^{\beta}(x,y)f(y)d\mu_{\alpha}(y),\qquad\text{a.e.
}x\in\mathbb{R}_{+}^{n},$

where

(1.4)
$\begin{split}R_{\alpha}^{\beta}(x,y)&=\frac{1}{\Gamma\left(\frac{\widehat{\beta}}{2}\right)}\int_{0}^{\infty}t^{\frac{\widehat{\beta}}{2}-1}D_{x}^{\beta}W_{t}^{\alpha}(x,y)dt,\\\
&=\frac{1}{\Gamma\left(\frac{\widehat{\beta}}{2}\right)}\int_{0}^{1}(-\log
r)^{\frac{\widehat{\beta}}{2}-1}D_{x}^{\beta}W_{-\log
r}^{\alpha}(x,y)\,\frac{dr}{r},\end{split}$

for $x,y\in\mathbb{R}_{+}^{n}$, $x\neq y.$

We consider the Littlewood–Paley functions $g_{\alpha}^{\beta,k}$ defined for
Poisson semigroups $\\{P_{t}^{\alpha}\\}_{t>0}$ for $k\in\mathbb N$ and
$\beta\in\mathbb{N}^{n}$ such that $k+\widehat{\beta}>0$, as follows

$g^{\beta,k}_{\alpha}(f)(x)=\left(\int_{0}^{\infty}\left|t^{k+\widehat{\beta}}\partial^{k}_{t}D^{\beta}_{x}P_{t}^{\alpha}(f)(x)\right|^{2}\frac{dt}{t}\right)^{1/2},\;x\in\mathbb{R}_{+}^{n}.$

For simplicity, when $\beta=\boldsymbol{0}=(0,\dots,0)$, we shall write
$g_{\alpha}^{k}=g_{\alpha}^{\boldsymbol{0},k}$. According to [42, Corollary
1], $g_{\alpha}^{k}$ is bounded on $L^{p}(\mathbb{R}_{+}^{n},\mu_{\alpha})$
for every $k\in\mathbb N$, $k\geq 1$ and $1<p<\infty$. In [9, Theorem 1.2] it
was recently proved that $g_{\alpha}^{k}$ is bounded from
$L^{1}(\mathbb{R}_{+},\mu_{\alpha})$ into
$L^{1,\infty}(\mathbb{R}_{+},\mu_{\alpha})$. Nowak in [32, Theorems 6 and 7]
proved $L^{p}$-boundedness properties for $1<p<\infty$ for Littlewood-Paley
functions associated with Laguerre polynomial expansions in the
$\nu_{\alpha}$-context including one spatial derivative.

We say that a function $m$ is of Laplace transform type when

$m(x)=x\int_{0}^{\infty}\phi(y)e^{-xy}dy,\quad x\in\mathbb{R}_{+},$

being $\phi\in L^{\infty}(\mathbb{R}_{+})$. Given $m$ of Laplace transform
type, we define the spectral multiplier for $\Delta_{\alpha}$,
$T_{m}^{\alpha}$, associated with $m$ by

$T_{m}^{\alpha}(f)=\sum_{k\in\mathbb
N^{n}}m(\lambda_{k})c_{k}^{\alpha}(f)\mathcal{L}_{\alpha}^{k},\quad f\in
L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha}).$

Since $m$ is bounded, $T_{m}^{\alpha}$ is bounded on
$L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$. According to [42, Corollary 3, p.
121], $T_{m}^{\alpha}$ can be extended from
$L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})\cap
L^{p}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ to
$L^{p}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ as a bounded operator on
$L^{p}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ when $1<p<\infty$. In [38] it was
established that $T_{m}^{\alpha}$ can be extended from
$L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})\cap
L^{1}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ to
$L^{1}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ as a bounded operator from
$L^{1}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ into
$L^{1,\infty}(\mathbb{R}_{+}^{n},\mu_{\alpha})$. From a higher dimension
version of [6, Theorem 1.1] we deduce that, for every $f\in
L^{p}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ with $1\leq p<\infty$,

$T_{m}^{\alpha}(f)(x)=\lim_{\epsilon\rightarrow
0^{+}}\left(\Lambda(\epsilon)f(x)+\int_{\begin{subarray}{c}|x-y|>\epsilon,\\\
y\in\mathbb{R}_{+}^{n}\end{subarray}}K_{\phi}^{\alpha}(x,y)f(y)d\mu_{\alpha}(y)\right),\quad\text{a.e.
}x\in\mathbb{R}_{+}^{n},$

where $\Lambda\in L^{\infty}(\mathbb{R}_{+})$ and

$K_{\phi}^{\alpha}(x,y)=\int_{0}^{\infty}\phi(t)\left(-\frac{\partial}{\partial
t}\right)W_{t}^{\alpha}(x,y)dt,\quad x,y\in\mathbb{R}_{+}^{n},x\neq y.$

A special case of multiplier of Laplace transform type is the imaginary power
$\Delta_{\alpha}^{i\beta}$ of $\Delta_{\alpha}$ that appears when
$m(x)=x^{i\beta}$, for $x\in\mathbb{R}_{+}$ and $\beta\in\mathbb R$.

Our objective is to give conditions on a function
$p:\mathbb{R}_{+}^{n}\rightarrow[1,\infty)$ in order that the operators we
have just defined (maximal operators, Riesz transforms, Littlewood–Paley
functions and multipliers of Laplace transform type) are bounded on
$L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$.

Exhaustive studies about Lebesgue spaces with variable exponent (also called
generalized Lebesgue spaces or variable Lebesgue spaces) can be found in the
monographs [14] and [18].

Assume that $p:\mathbb{R}_{+}^{n}\rightarrow[1,\infty)$ is measurable. We say
that a measurable function $f$ on $\mathbb{R}_{+}^{n}$ belongs to
$L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ if the modular
$\varrho_{p(\cdot),\mu_{\alpha}}(f/\lambda)$ is finite for some $\lambda>0$,
where

$\varrho_{p(\cdot),\mu_{\alpha}}(g)=\int_{\mathbb{R}_{+}^{n}}|g(x)|^{p(x)}d\mu_{\alpha}(x).$

We define on $L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ the Luxemburg
norm $\|\cdot\|_{L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})}$ associated
with $\varrho_{p(\cdot),\mu_{\alpha}}$, that is,

$\|f\|_{L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})}=\inf\left\\{\lambda>0:\varrho_{p(\cdot),\mu_{\alpha}}\left(\frac{f}{\lambda}\right)\leq
1\right\\}.$

The space
$\left(L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha}),\|\cdot\|_{L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})}\right)$
is a Banach function space. The variable Lebesgue space
$L^{p(\cdot)}(\mathbb{R}_{+}^{n}):=L^{p(\cdot)}(\mathbb{R}_{+}^{n},dx)$ and
its norm
$\|\cdot\|_{L^{p(\cdot)}(\mathbb{R}_{+}^{n})}:=\|\cdot\|_{L^{p(\cdot)}(\mathbb{R}_{+}^{n},dx)}$
are defined in the obvious way.

Lebesgue spaces with variable exponents appear associated to physics problems,
image processing and modeling of electrorheological fluids (see, for instance,
[1], [10] and [37]).

As it is well-known, the Hardy–Littlewood maximal function $M_{\textup{HL}}$
plays a central role in the study of $L^{p}$-boundedness properties of
harmonic analysis operators. The following conditions on the exponent
$p(\cdot)$ arise related with the boundedness of $M_{\textup{HL}}$ on
$L^{p(\cdot)}(\mathbb R^{n})$ ([13] and [17]):

  1. (a)

Local log-Hölder condition: a measurable function $p:\Omega\subset\mathbb
R^{n}\rightarrow[1,\infty)$ is said to be in $\textup{LH}_{0}(\Omega)$ if
there exists $C>0$ such that

$|p(x)-p(y)|\leq\frac{C}{-\log|x-y|},\quad x,y\in\Omega,0<|x-y|<\tfrac{1}{2}.$

  2. (b)

Decay log-Hölder condition: a measurable function ${p:\Omega\subset\mathbb
R^{n}\rightarrow[1,\infty)}$ is said to be in $\textup{LH}_{\infty}(\Omega)$
when there exists $C>0$ and $p_{\infty}\geq 1$ such that

$|p(x)-p_{\infty}|\leq\frac{C}{\log(e+|x|)},\quad x\in\Omega.$

We define
$\textup{LH}(\Omega)=\textup{LH}_{0}(\Omega)\cap\textup{LH}_{\infty}(\Omega)$,
where $\Omega\subset\mathbb R^{n}.$

If $p:\Omega\subset\mathbb R^{n}\rightarrow[1,\infty)$ is measurable, we
denote by $p^{-}=\operatorname*{ess\ inf}_{\Omega}p$ and
${p^{+}=\operatorname*{ess\ sup}_{\Omega}p}$ the essential infimum and
supremum of $p$ on $\Omega$, respectively.

If $1<p^{-}\leq p^{+}<\infty$ and $p\in\textup{LH}(\mathbb R^{n})$, then the
Hardy–Littlewood maximal function is bounded on $L^{p(\cdot)}(\mathbb R^{n})$
([13]). However, $p\in\textup{LH}(\mathbb R^{n})$ is not necessary for this
boundedness ([14, Examples 4.1 and 4.43]). The same conditions on $p$,
$1<p^{-}\leq p^{+}<\infty$ and $p\in\textup{LH}(\mathbb R^{n})$, assure that
the Calderón–Zygmund singular integrals are bounded on $L^{p(\cdot)}(\mathbb
R^{n})$ ([14, Theorem 5.39]).

In [15], Dalmasso and Scotto studied Riesz transforms in the Gaussian setting
on variable Lebesgue spaces. In order to do this, they introduced a new class
of exponents which is contained in $\textup{LH}_{\infty}(\mathbb R^{n})$. A
measurable function $p:\Omega\subset\mathbb R^{n}\rightarrow[1,\infty)$ is
said to be in $\mathcal{P}^{\infty}_{e}(\Omega)$ when there exists $C>0$ and
$p_{\infty}\geq 1$ such that

$|p(x)-p_{\infty}|\leq\frac{C}{|x|^{2}},\quad
x\in\Omega\setminus\\{(0,\dots,0)\\}.$

If $p_{\infty}\geq 1$, $A>0$ and $q\geq 2$ are given, the functions
$p(x)=p_{\infty}+\frac{A}{(e+|x|)^{q}}$, for $x\in\mathbb R^{n}$, are in
$\mathcal{P}^{\infty}_{e}(\mathbb R^{n})$. Main properties of the functions in
$\mathcal{P}^{\infty}_{e}(\mathbb R^{n})$ were established in [15]. Maximal
operators defined by the heat semigroup ([28]) and Riesz type singular
integrals ([16] and [31]) associated with the Ornstein-Uhlenbeck differential
operator were studied on $L^{p(\cdot)}(\mathbb R^{n},\gamma_{n})$ with
$p\in\textup{LH}_{0}(\mathbb R^{n})\cap\mathcal{P}^{\infty}_{e}(\mathbb
R^{n})$, where $d\gamma_{n}$ denotes the Gaussian measure.

We now state the main results of this article concerning
$L^{p(\cdot)}$-boundedness properties of harmonic analysis operators in the
Laguerre setting.

######  Theorem 1.1.

Let $\alpha\in[0,\infty)^{n}$. Assume that
$p\in\textup{LH}_{0}(\mathbb{R}_{+}^{n})\cap\mathcal{P}^{\infty}_{e}(\mathbb{R}_{+}^{n})$
with $1<p^{-}\leq p^{+}<\infty$. We denote by $T_{\alpha}$ one of the
following operators:

  1. (a)

The maximal operators $W_{*}^{\alpha}$ and $P_{*}^{\alpha}$;

  2. (b)

The Laguerre-Riesz transformation $R^{\beta}_{\alpha}$, $\beta\in\mathbb
N^{n}\setminus\\{(0,\dots,0)\\}$;

  3. (c)

The Littlewood–Paley functions $g_{\alpha}^{\beta,k}$ associated with the
Poisson semigroup $\\{P_{t}^{\alpha}\\}_{t>0}$, where $\beta\in\mathbb{N}^{n}$
and $k\in\mathbb N$, such that $k+\widehat{\beta}>0$;

  4. (d)

The Laguerre spectral multipliers $T_{m}^{\alpha}$, where $m$ is a Laplace
transform type function.

Then, $T_{\alpha}$ is bounded on
$L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$.

Hereinafter, we prove Theorem 1.1. In Section 2, we explain the method we
develop in order to prove that the operators given in (a)–(d) are bounded on
$L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$. In Section 3, we introduce a
global operator that will be a key ingredient for proving our main theorem. In
the following sections, we establish the $L^{p(\cdot)}$-boundedness for each
class of operators. Our method exploits the decomposition of the operators
into a local part and a global part, which is usual in the study of harmonic
analysis in the Laguerre setting, but we need a careful adaptation to the
variable exponent context.

Throughout this paper, $C$ and $c$ will always denote positive constants that
may change in each occurrence.

##  2\. The method for proving our results

In this section we describe the method we apply to prove the boundedness
results.

The polynomial measure $\mathfrak{m}_{\alpha}$ on $\mathbb{R}_{+}^{n}$ defined
by $d\mathfrak{m}_{\alpha}(x)=\prod_{i=1}^{n}x_{i}^{2\alpha_{i}+1}dx_{i}$ is
doubling on $\mathbb{R}_{+}^{n}$. Thus, the triple
$(\mathbb{R}_{+}^{n},|\cdot|,\mathfrak{m}_{\alpha})$ is a homogeneous space in
the sense of Coifman and Weiss ([11]).

Let $X$ be a Banach space. Suppose that
$K:\mathbb{R}_{+}^{n}\times\mathbb{R}_{+}^{n}\setminus D\rightarrow X$ is a
strongly measurable function, where $D=\\{(x,x):x\in\mathbb{R}_{+}^{n}\\}$,
satisfying the following two conditions:

  1. (i)

Size condition: there exists $C>0$ such that

$\|K(x,y)\|_{X}\leq\frac{C}{\mathfrak{m}_{\alpha}(B(x,|x-y|))},\quad
x,y\in\mathbb{R}_{+}^{n},x\neq y;$

  2. (ii)

Regularity condition: there exists $C>0$ such that

$\|K(x,y)-K(z,y)\|_{X}\leq\frac{C|x-z|}{|x-y|\
\mathfrak{m}_{\alpha}(B(x,|x-y|))}$

and

$\|K(x,y)-K(x,z)\|_{X}\leq\frac{C|y-z|}{|x-y|\
\mathfrak{m}_{\alpha}(B(x,|x-y|))}$

for every $x,y,z\in\mathbb{R}_{+}^{n}$ with $|x-z|\leq\frac{1}{2}|x-y|$.

When the function $K$ verifies (i) and (ii), we say that $K$ is an $X$-valued
Calderón–Zygmund kernel with respect to the homogeneous space
$(\mathbb{R}_{+}^{n},|\cdot|,\mathfrak{m}_{\alpha})$ in the Banach space $X$.

For every exponent $q:\mathbb{R}_{+}^{n}\rightarrow[1,\infty)$, we denote by
$L^{q(\cdot)}_{X}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})$ the $X$-Bochner
Lebesgue space with variable exponent $q$, defined in the natural way.

Assume $T$ is a bounded operator from
$L^{2}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})$ into
$L^{2}_{X}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})$. We say that $T$ is an
$X$-valued Calderón–Zygmund operator associated with the Calderón–Zygmund
kernel $K$ when, for every $f\in C_{c}^{\infty}(\mathbb{R}_{+}^{n})$,

$Tf(x)=\int_{\mathbb{R}_{+}^{n}}K(x,y)f(y)d\mathfrak{m}_{\alpha}(y),\quad\text{
a.e. }x\notin\operatorname{supp}(f).$

Here, the integral is understood in the $X$-Bochner sense.

According to [23, Theorem 1.1] (see also [36]), if $T$ is an $X$-valued
Calderón–Zygmund operator on
$(\mathbb{R}_{+}^{n},|\cdot|,\mathfrak{m}_{\alpha})$, $T$ can be extended, for
every $1\leq p<\infty$, from
$L^{2}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})\cap
L^{p}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})$ to
$L^{p}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})$ as a bounded operator from
$L^{p}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})$ into
$L^{p}_{X}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})$ when $1<p<\infty$, and
from $L^{1}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})$ into
$L^{1,\infty}_{X}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})$ when $p=1$.

Any non-negative measurable function $w$ on $\mathbb{R}_{+}^{n}$ is named a
weight. For every $1<p<\infty$, we say that a weight $w$ on
$\mathbb{R}_{+}^{n}$ is in the Muckenhoupt class
$A_{p}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})$ when

$\sup_{B}\left(\frac{1}{\mathfrak{m}_{\alpha}(B)}\int_{B}w(x)d\mathfrak{m}_{\alpha}(x)\right)\left(\frac{1}{\mathfrak{m}_{\alpha}(B)}\int_{B}w(x)^{-\frac{1}{p-1}}d\mathfrak{m}_{\alpha}(x)\right)^{p-1}<\infty,$

where the supremum is taken over all the balls $B$ in $\mathbb{R}_{+}^{n}$.

A weight $w$ on $\mathbb{R}_{+}^{n}$ is said to be in the Muckenhoupt class
$A_{1}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})$ when there exists $C>0$ such
that, for every ball $B\subset\mathbb{R}_{+}^{n}$,

$\frac{1}{\mathfrak{m}_{\alpha}(B)}\int_{B}w(x)d\mathfrak{m}_{\alpha}(x)\leq
C\operatorname*{ess\ inf}_{y\in B}w(y).$

We also define
$A_{\infty}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})=\bigcup_{p\geq
1}A_{p}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})$.

If $T$ is an $X$-valued Calderón–Zygmund operator on
$(\mathbb{R}_{+}^{n},|\cdot|,\mathfrak{m}_{\alpha})$, for every $w\in
A_{p}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})$ and $1<p<\infty$, the
operator $T$ can be extended from
$L^{2}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})\cap
L^{p}(\mathbb{R}_{+}^{n},w,\mathfrak{m}_{\alpha})$ to
$L^{p}(\mathbb{R}_{+}^{n},w,\mathfrak{m}_{\alpha})$ as a bounded operator from
$L^{p}(\mathbb{R}_{+}^{n},w,\mathfrak{m}_{\alpha})$ into
$L^{p}_{X}(\mathbb{R}_{+}^{n},w,\mathfrak{m}_{\alpha})$ (see, for instance,
[25, Theorem 1.1]).

Rubio de Francia’s extrapolation theorem works for spaces of homogeneous type
([3, Theorem 3.5]). The arguments in the proof of [12, Theorem 1.3] allow us
to deduce that if $T$ is an $X$-valued Calderón–Zygmund operator on
$(\mathbb{R}_{+}^{n},|\cdot|,\mathfrak{m}_{\alpha})$, $T$ defines a bounded
operator from $L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})$ into
$L_{X}^{p(\cdot)}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})$, provided that
$1<p^{-}\leq p^{+}<\infty$ and the $\mathfrak{m}_{\alpha}$-Hardy–Littlewood
maximal function is bounded on
$L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})$ (see also [19,
Theorem 4.8]). We recall that according to [2, Theorems 1.4 and 1.7], the
Hardy–Littlewood maximal operator defined by the measure
$\mathfrak{m}_{\alpha}$ is bounded on
$L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})$ provided $1<p^{-}\leq
p^{+}<\infty$ and $p\in\textup{LH}(\mathbb{R}_{+}^{n})$ (see also [16, Theorem
5.2]). We also notice that $T$ is well-defined for $f\in
L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})$ thanks to the
embedding
$L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})\hookrightarrow
L^{p^{-}}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})+L^{p^{+}}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})$
([18, Theorem 3.3.11]).

The maximal operators and the Littlewood–Paley function can be studied by
using Banach valued operators. Indeed, we can write

$P_{*}^{\alpha}(f)=\|P_{t}^{\alpha}(f)\|_{L^{\infty}(\mathbb{R}_{+})},\qquad
W_{*}^{\alpha}(f)=\|W_{t}^{\alpha}(f)\|_{L^{\infty}(\mathbb{R}_{+})}$

and

$g_{\alpha}^{\beta,k}(f)=\left\|t^{k+\widehat{\beta}}\partial_{t}^{k}D_{x}^{\beta}P_{t}^{\alpha}(f)\right\|_{L^{2}\left(\mathbb{R}_{+},\frac{dt}{t}\right)}.$

We define

$q_{\pm}(x,y,s)=\sum_{i=1}^{n}(x_{i}^{2}+y_{i}^{2}\pm 2x_{i}y_{i}s_{i}),$

with $x=(x_{1},\dots,x_{n}),y=(y_{1},\dots,y_{n})\in\mathbb{R}_{+}^{n}$ and
$s=(s_{1},\dots,s_{n})\in(-1,1)^{n}$. We split
$\mathbb{R}_{+}^{n}{\times\mathbb{R}_{+}^{n}\times(-1,1)^{n}}$ into two parts.
Let $\tau>0$ and let us fix $C_{0}>0$ whose exact value will be specified
later. The local part $L_{\tau}$ is defined by

$L_{\tau}=\left\\{(x,y,s)\in\mathbb{R}_{+}^{n}\times\mathbb{R}_{+}^{n}\times(-1,1)^{n}:\sqrt{q_{-}(x,y,s)}\leq\frac{C_{0}\tau}{1+|x|+|y|}\right\\}$

and the global part $G_{\tau}$ is given by

$G_{\tau}=\mathbb{R}_{+}^{n}{\times\mathbb{R}_{+}^{n}\times(-1,1)^{n}}\setminus
L_{\tau}.$

By taking into account the integral representation for the modified Bessel
function $I_{\nu}$, $\nu>-\frac{1}{2}$ ([24, (5.10.22)]), for every $t>0$, the
integral kernel of $W_{t}^{\alpha}$ can be written as

$W_{t}^{\alpha}(x,y)=\frac{1}{(1-e^{-t})^{n+\widehat{\alpha}}}\int_{(-1,1)^{n}}\exp\left(-\frac{q_{-}\left(e^{-t/2}x,y,s\right)}{1-e^{-t}}+|y|^{2}\right)\Pi_{\alpha}(s)ds,$

for $x,y\in\mathbb{R}_{+}^{n}$, where
$\widehat{\alpha}=\sum_{i=1}^{n}\alpha_{i}$ and
$\Pi_{\alpha}(s)=\prod_{i=1}^{n}\frac{\Gamma(\alpha_{i}+1)}{\Gamma(\alpha_{i}+1/2)\sqrt{\pi}}(1-{s_{i}^{2}})^{\alpha_{i}-1/2}$
for $s=(s_{1},\dots,s_{n})\in(-1,1)^{n}$.

As in [41], we consider a smooth function $\varphi$ on
$\mathbb{R}_{+}^{n}\times\mathbb{R}_{+}^{n}\times(-1,1)^{n}$ such that
$0\leq\varphi\leq 1$,

$\varphi(x,y,s)=\begin{cases}1,&(x,y,s)\in L_{1},\\\ 0,&(x,y,s)\notin
L_{2},\end{cases}$

and

$|\nabla_{x}\varphi(x,y,s)|+|\nabla_{y}\varphi(x,y,s)|\leq\frac{C}{q_{-}(x,y,s)^{1/2}},\quad
x,y\in\mathbb{R}_{+}^{n},s\in(-1,1)^{n}.$

We also define, for $x,y\in\mathbb{R}_{+}^{n}$ and $t>0$,

$W_{t,\text{loc}}^{\alpha}(x,y)=\int_{(-1,1)^{n}}\frac{\exp\left(-\frac{q_{-}\left(e^{-t/2}x,y,s\right)}{1-e^{-t}}+|y|^{2}\right)}{(1-e^{-t})^{n+\widehat{\alpha}}}\Pi_{\alpha}(s)\varphi(x,y,s)ds$

and

$W_{t,\text{glob}}^{\alpha}(x,y)=W_{t}^{\alpha}(x,y)-W_{t,\text{loc}}^{\alpha}(x,y).$

Suppose that $T_{\alpha}$ is one of the operators considered in Theorem 1.1.
This operator is defined by using the heat integral kernel
$W_{t}^{\alpha}(x,y)$. We decompose the operator $T_{\alpha}$ as

$|T_{\alpha}|\leq|T_{\alpha,\text{loc}}|+|T_{\alpha,\text{glob}}|,$

where $T_{\alpha,\text{loc}}$ is defined as $T_{\alpha}$ but replacing
$W_{t}^{\alpha}(x,y)$ by $W_{t,\text{loc}}^{\alpha}(x,y)$, and in
$T_{\alpha,\text{glob}}$ the kernel $W_{t}^{\alpha}(x,y)$ is replaced by
$W_{t,\text{glob}}^{\alpha}(x,y)$.

We shall prove that both $T_{\alpha,\text{loc}}$ and $T_{\alpha,\text{glob}}$
are bounded on $L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ provided that
$p$ satisfies the hypotheses imposed on Theorem 1.1.

In order to prove the $L^{p(\cdot)}$-boundedness of $T_{\alpha,\text{glob}}$,
we introduce, for every $\varepsilon\in[0,1)$, a positive measurable function
$H_{\alpha,\varepsilon}$ defined on
$\mathbb{R}_{+}^{n}\times\mathbb{R}_{+}^{n}$ verifying that the operator
$\mathcal{H}_{\alpha,\varepsilon}$ given by

$\mathcal{H}_{\alpha,\varepsilon}(f)(x)=\int_{\mathbb{R}_{+}^{n}}H_{\alpha,\varepsilon}(x,y)f(y)d\mu_{\alpha}(y),\quad
x\in\mathbb{R}_{+}^{n}$

is bounded on $L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$. Then, we prove
that there exists $\varepsilon\in[0,1)$ for which

$\left|T_{\alpha,\text{glob}}f(x)\right|\leq\mathcal{H}_{\alpha,\varepsilon}(|f|)(x),\quad
x\in\mathbb{R}_{+}^{n}.$

Secondly, we prove that $T_{\alpha,\text{loc}}$ is bounded on
$L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$. We consider the following
Banach spaces

$X(W_{*}^{\alpha})=X(P_{*}^{\alpha})=L^{\infty}(\mathbb{R}_{+}),$

for every $k\in\mathbb{N}$ and $\beta\in\mathbb{N}^{n}$ such that
$k+\widehat{\beta}>0$,

$X\left(g_{\alpha}^{\beta,k}\right)=L^{2}\left(\mathbb{R}_{+},\frac{dt}{t}\right),$

and for every $\beta\in\mathbb{N}\setminus\\{0\\}$ and every multiplier $m$ of
Laplace transform type,

$X\left(R_{\alpha}^{\beta}\right)=X(T_{m})=\mathbb C.$

We can write

$\left|T_{\alpha,\text{loc}}(f)\right|=\left\|\mathbb
T_{\alpha}(f)\right\|_{X(T_{\alpha})}$

where, for $x\in\mathbb{R}_{+}^{n}$,

$\mathbb
T_{\alpha}(f)(x)=\int_{\mathbb{R}_{+}^{n}}\int_{(-1,1)^{n}}\mathcal{M}_{\alpha}(x,y,s)\varphi(x,y,s)\Pi_{\alpha}(s)ds\
f(y)d\mathfrak{m}_{\alpha}(y).$

Here, the function
$\mathcal{M}_{\alpha}:\mathbb{R}_{+}^{n}\times\mathbb{R}_{+}^{n}\times(-1,1)^{n}\rightarrow
X(T_{\alpha})$ is strongly measurable and the integral is understood in the
$X(T_{\alpha})$-Bochner sense. We write

$\mathbb{M}_{\alpha}(x,y)=\int_{(-1,1)^{n}}\mathcal{M}_{\alpha}(x,y,s)\varphi(x,y,s)\Pi_{\alpha}(s)ds,\quad
x,y\in\mathbb{R}_{+}^{n}.$

Thus, $\mathbb{M}_{\alpha}:\mathbb{R}_{+}^{n}\times\mathbb{R}_{+}^{n}\setminus
D\rightarrow X(T_{\alpha})$ is strongly measurable.

The operator $\mathbb T_{\alpha}$ is bounded from
$L^{2}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})$ into
$L^{2}_{X(T_{\alpha})}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})$. We prove
that $\mathbb T_{\alpha}$ is an $X(T_{\alpha})$-valued Calderón–Zygmund
operator associated with $\mathbb{M}_{\alpha}$. Then, according to the above-
mentioned arguments, $\mathbb T_{\alpha}$ defines a bounded operator from
$L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})$ into
$L^{p(\cdot)}_{X(T_{\alpha})}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})$. We
are going to see that $\mathbb T_{\alpha}$ is also bounded from
$L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ into
$L^{p(\cdot)}_{X(T_{\alpha})}(\mathbb{R}_{+}^{n},\mu_{\alpha})$. Note that the
measure $\mu_{\alpha}$ is not doubling on $(\mathbb{R}_{+}^{n},|\cdot|)$.

As stated in [39, Lemma 4], there exists a sequence
$\\{x(\ell)\\}_{\ell\in\mathbb N}\subset\mathbb{R}_{+}^{n}$ such that, if we
set

$B_{\ell}=\left\\{x\in\mathbb{R}_{+}^{n}:|x-x(\ell)|\leq\frac{1}{2(1+|x(\ell)|)}\right\\},\quad\ell\in\mathbb
N,$

the following properties hold

  1. (i)

$\mathbb{R}_{+}^{n}=\bigcup\limits_{\ell\in\mathbb N}B_{\ell}$;

  2. (ii)

for every $\delta>1$, the family $\\{\delta B_{\ell}\\}_{\ell\in\mathbb N}$
has bounded overlap;

  3. (iii)

there exists $C>1$ such that, for every $\ell\in\mathbb N$ and every
measurable subset $E$ of $B_{\ell}$,

$\frac{1}{C}e^{-|x(\ell)|^{2}}\mathfrak{m}_{\alpha}(E)\leq\mu_{\alpha}(E)\leq
Ce^{-|x(\ell)|^{2}}{\mathfrak{m}_{\alpha}(E)}.$

Furthermore, for every $\eta>0$, there exists $\delta>1$ such that, if
$\ell\in\mathbb N,\,x\in B_{\ell}$ and $y\notin\delta B_{\ell}$, then
$(x,y,s)\notin L_{\eta}$ for each $s\in(-1,1)^{n}$ (see [39, Remark 5]).

We have that

$\|\mathbb
T_{\alpha}f\|_{L^{p(\cdot)}_{X(T_{\alpha})}(\mathbb{R}_{+}^{n},\mu_{\alpha})}=\left\|\|\mathbb
T_{\alpha}f\|_{X(T_{\alpha})}\right\|_{L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})},$

so, according to [18, Corollary 3.2.14],

$\|\mathbb
T_{\alpha}f\|_{L^{p(\cdot)}_{X(T_{\alpha})}(\mathbb{R}_{+}^{n},\mu_{\alpha})}\leq
2\sup_{\|F\|_{L^{p^{\prime}(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})}\leq
1}\int_{\mathbb{R}_{+}^{n}}\|\mathbb
T_{\alpha}f(x)\|_{X(T_{\alpha})}|F(x)|d\mu_{\alpha}(x).$

Here, $p^{\prime}$ denotes the Hölder conjugate exponent of $p$, i.e.,
$\frac{1}{p(x)}+\frac{1}{p^{\prime}(x)}=1$ for every $x\in\mathbb{R}_{+}^{n}$.

Fix $F\in L^{p^{\prime}(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ with
$\|F\|_{L^{p^{\prime}(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})}\leq 1$. By
virtue of the properties (i), (ii) and (iii), for certain $\delta>1$ we get

$\displaystyle\int_{\mathbb{R}_{+}^{n}}$ $\displaystyle\|\mathbb
T_{\alpha}f(x)\|_{X(T_{\alpha})}|F(x)|d\mu_{\alpha}(x)$
$\displaystyle\leq\sum_{\ell\in\mathbb N}\int_{B_{\ell}}\|\mathbb
T_{\alpha}f(x)\|_{X(T_{\alpha})}|F(x)|d\mu_{\alpha}(x)$
$\displaystyle=\sum_{\ell\in\mathbb N}\int_{B_{\ell}}\|\mathbb
T_{\alpha}\left(f\chi_{\delta
B_{\ell}}\right)(x)\|_{X(T_{\alpha})}|F(x)|d\mu_{\alpha}(x)$
$\displaystyle\leq C\sum_{\ell\in\mathbb
N}e^{-|x(\ell)|^{2}}\int_{B_{\ell}}\|\mathbb T_{\alpha}\left(f\chi_{\delta
B_{\ell}}\right)(x)\|_{X(T_{\alpha})}|F(x)|d\mathfrak{m}_{\alpha}(x)$
$\displaystyle\leq C\sum_{\ell\in\mathbb N}e^{-|x(\ell)|^{2}}\left\|\mathbb
T_{\alpha}(f\chi_{\delta
B_{\ell}})\right\|_{L^{p(\cdot)}_{X(T_{\alpha})}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})}\|F\chi_{B_{\ell}}\|_{L^{p^{\prime}(\cdot)}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})}$
$\displaystyle\leq C\sum_{\ell\in\mathbb N}e^{-|x(\ell)|^{2}}\|f\chi_{\delta
B_{\ell}}\|_{L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})}\|F\chi_{B_{\ell}}\|_{L^{p^{\prime}(\cdot)}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})}.$

We have used Hölder’s inequality with variable exponents (see, for instance,
[18, Lemma 3.2.20]).

Since $p\in\mathcal{P}^{\infty}_{e}(\mathbb{R}_{+}^{n})$ and $1<p^{-}\leq
p^{+}<\infty$, we also have
$p^{\prime}\in\mathcal{P}^{\infty}_{e}(\mathbb{R}_{+}^{n})$ with
$1<(p^{\prime})^{-}\leq(p^{\prime})^{+}<\infty$. From [15, Lemma 2.5], by
proceeding as in [15, (3.12)] and the following lines, we get

$e^{-|x(\ell)|^{2}/p_{\infty}}\|f\chi_{\delta
B_{\ell}}\|_{L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})}\leq\|f\chi_{\delta
B_{\ell}}\|_{L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})}$

and

$e^{-|x(\ell)|^{2}/p^{\prime}_{\infty}}\|F\chi_{B_{\ell}}\|_{L^{p^{\prime}(\cdot)}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})}\leq\|F\chi_{B_{\ell}}\|_{L^{p^{\prime}(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})},$

where $p^{\prime}_{\infty}$ is the conjugate exponent of $p_{\infty}$.

By means of [15, Corollary 2.8], we obtain

$\displaystyle\int_{\mathbb{R}_{+}^{n}}\|\mathbb
T_{\alpha}f(x)\|_{X(T_{\alpha})}$ $\displaystyle|F(x)|d\mu_{\alpha}(x)\leq
C\sum_{\ell\in\mathbb N}\|f\chi_{\delta
B_{\ell}}\|_{L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})}\|F\chi_{B_{\ell}}\|_{L^{p^{\prime}(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})}$
$\displaystyle\leq C\sum_{\ell\in\mathbb N}\left\|f\chi_{\delta
B_{\ell}}e^{-|\cdot|^{2}/p(\cdot)}\prod_{i=1}^{n}\frac{x_{i}^{(2\alpha_{i}+1)/p(\cdot)}}{\Gamma(\alpha_{i}+1/2)^{1/p(\cdot)}}\right\|_{L^{p(\cdot)}(\mathbb{R}_{+}^{n})}$
$\displaystyle\qquad\qquad\times\left\|F\chi_{B_{\ell}}e^{-|\cdot|^{2}/p^{\prime}(\cdot)}\prod_{i=1}^{n}\frac{x_{i}^{(2\alpha_{i}+1)/p^{\prime}(\cdot)}}{\Gamma(\alpha_{i}+1/2)^{1/p^{\prime}(\cdot)}}\right\|_{L^{p^{\prime}(\cdot)}(\mathbb{R}_{+}^{n})}$
$\displaystyle\leq
C\|f\|_{L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})}\|F\|_{L^{p^{\prime}(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})}.$

Hence, we conclude that

$\|\mathbb
T_{\alpha}f\|_{L^{p(\cdot)}_{X(T_{\alpha})}(\mathbb{R}_{+}^{n},\mu_{\alpha})}\leq
C\|f\|_{L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})}.$

We have thus proved that the operator $T_{\alpha,\text{loc}}$ is bounded on
$L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ provided that the exponent
function $p$ satisfies the conditions of Theorem 1.1.

##  3\. An auxiliary result

In this section we establish a result that will be useful to prove
$L^{p(\cdot)}$-boundedness for the global parts of the operators considered in
Theorem 1.1.

Given $\alpha\in[0,\infty)^{n}$ and $\varepsilon\in[0,1)$, we define the
global operator

$\mathcal{H}_{\alpha,\varepsilon}(f)(x)=\int_{\mathbb{R}_{+}^{n}}H_{\alpha,\varepsilon}(x,y)f(y)d\mathfrak{m}_{\alpha}(y),\quad
x\in\mathbb{R}_{+}^{n},$

where

$H_{\alpha,\epsilon}(x,y)=\int_{(-1,1)^{n}}H_{\alpha,\varepsilon}(x,y,s)(1-\varphi(x,y,s))\Pi_{\alpha}(s)ds$

and

(3.1)
$H_{\alpha,\varepsilon}(x,y,s)=\begin{cases}e^{-(1-\varepsilon)|y|^{2}},&\sum\limits_{i=1}^{n}x_{i}y_{i}s_{i}\leq
0,\\\
q_{+}(x,y,s)^{n+\widehat{\alpha}}e^{-\frac{(1-\varepsilon)}{2}\left(|y|^{2}-|x|^{2}+\sqrt{q_{+}(x,y,s)q_{-}(x,y,s)}\right)},&\sum\limits_{i=1}^{n}x_{i}y_{i}s_{i}>0.\end{cases}$

######  Proposition 3.1.

Let $\alpha\in[0,\infty)^{n}$. Suppose that
$p\in\textup{LH}_{0}(\mathbb{R}_{+}^{n})\cap\mathcal{P}_{e}^{\infty}(\mathbb{R}_{+}^{n})$
with $1<p^{-}\leq p^{+}<\infty$ and let
$0<\varepsilon<\frac{1}{(p^{-})^{\prime}}\wedge\frac{1}{n+\widehat{\alpha}}$.
Then, the operator $\mathcal{H}_{\alpha,\varepsilon}$ is bounded on
$L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$.

###### Proof.

We decompose
$\mathcal{H}_{\alpha,\varepsilon}(f)=\mathcal{H}_{\alpha,\varepsilon}^{(1)}(f)+\mathcal{H}_{\alpha,\varepsilon}^{(2)}(f)$,
where

$\mathcal{H}_{\alpha,\varepsilon}^{(1)}(f)(x)=\int_{E_{x}}H_{\alpha,\varepsilon}(x,y,s)(1-\varphi(x,y,s))\Pi_{\alpha}(s)dsf(y)d\mathfrak{m}_{\alpha}(y),$

and

$\mathcal{H}_{\alpha,\varepsilon}^{(2)}(f)(x)=\int_{F_{x}}H_{\alpha,\varepsilon}(x,y,s)(1-\varphi(x,y,s))\Pi_{\alpha}(s)dsf(y)d\mathfrak{m}_{\alpha}(y),$

being

$\displaystyle E_{x}$
$\displaystyle=\left\\{(y,s)\in\mathbb{R}_{+}^{n}\times(-1,1)^{n}:\sum_{i=1}^{n}x_{i}y_{i}s_{i}\leq
0\right\\},$ $\displaystyle F_{x}$
$\displaystyle=\left\\{(y,s)\in\mathbb{R}_{+}^{n}\times(-1,1)^{n}:\sum_{i=1}^{n}x_{i}y_{i}s_{i}>0\right\\}.$

Let $f\in L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ be given such that
$\|f\|_{L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})}\leq 1$. For
$x\in\mathbb{R}_{+}^{n}$, we have that

$\displaystyle\left|\mathcal{H}_{\alpha,\varepsilon}^{(1)}(f)(x)\right|$
$\displaystyle\leq\int_{\mathbb{R}_{+}^{n}}e^{-(1-\varepsilon)|y|^{2}}|f(y)|\int_{(-1,1)^{n}}|1-\varphi(x,y,s)|\Pi_{\alpha}(s)dsd\mathfrak{m}_{\alpha}(y)$
$\displaystyle\leq
C\int_{\mathbb{R}_{+}^{n}}e^{-(1-\varepsilon)|y|^{2}}|f(y)|d\mathfrak{m}_{\alpha}(y).$

Since $\varepsilon<1/(p^{-})^{\prime}$, we can write
$1-\varepsilon=\tilde{\varepsilon}+1/p^{-}$ with $\tilde{\varepsilon}>0$.
Thus, by Hölder’s inequality with $p^{-}>1$ we have

$\displaystyle\left|\mathcal{H}_{\alpha,\varepsilon}^{(1)}(f)(x)\right|$
$\displaystyle\quad\leq
C\int_{\mathbb{R}_{+}^{n}}e^{-\left(\tilde{\varepsilon}+\frac{1}{p^{-}}\right)|y|^{2}}|f(y)|d\mathfrak{m}_{\alpha}(y)$
$\displaystyle\quad\leq
C\left(\int_{\mathbb{R}_{+}^{n}}e^{-|y|^{2}}|f(y)|^{p^{-}}d\mathfrak{m}_{\alpha}(y)\right)^{1/p^{-}}\left(\int_{\mathbb{R}_{+}^{n}}e^{-\tilde{\varepsilon}(p^{-})^{\prime}|y|^{2}}d\mathfrak{m}_{\alpha}(y)\right)^{1/(p^{-})^{\prime}}$
$\displaystyle\quad\leq
C\left(\int_{\mathbb{R}_{+}^{n}\cap\\{|f|>1\\}}|f(y)|^{p(y)}d\mu_{\alpha}(y)+\int_{\mathbb{R}_{+}^{n}\cap\\{|f|\leq
1\\}}d\mu_{\alpha}(y)\right)^{1/p^{-}}\leq C,$

since $\int_{\mathbb{R}_{+}^{n}}|f(y)|^{p(y)}d\mu_{\alpha}(y)\leq 1$ and
$\mu_{\alpha}$ is a probability measure on $\mathbb{R}_{+}^{n}$.

Therefore, by the homogeneity of the norm,

$\left\|\mathcal{H}_{\alpha,\varepsilon}^{(1)}(f)\right\|_{L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})}\leq
C\|f\|_{L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})}.$

for any $f\in L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$.

We now study $\mathcal{H}_{\alpha,\varepsilon}^{(2)}$. We have that

$\begin{split}&\int_{\mathbb{R}_{+}^{n}}|\mathcal{H}_{\alpha,\varepsilon}^{(2)}(f)(x)|^{p(x)}d\mu_{\alpha}(x)\leq
C\int_{\mathbb{R}_{+}^{n}}\left(\int_{F_{x}}|f(y)|e^{\frac{-|y|^{2}}{p(y)}}e^{\frac{|y|^{2}}{p(y)}-\frac{|x|^{2}}{p(x)}}\right.\\\
&\;\times\left.q_{+}(x,y,s)^{n+\widehat{\alpha}}e^{-\frac{(1-\varepsilon)}{2}(|y|^{2}-|x|^{2}+\sqrt{q_{+}(x,y,s)q_{-}(x,y,s)})}\Pi_{\alpha}(s)dsd\mathfrak{m}_{\alpha}(y)\vphantom{\int}\right)^{p(x)}d\mathfrak{m}_{\alpha}(x).\end{split}$

Note that we can write

$\displaystyle q_{+}(x,y,s)$ $\displaystyle q_{-}(x,y,s)$
$\displaystyle=\left(|x|^{2}+|y|^{2}+2\sum_{i=1}^{n}x_{i}y_{i}s_{i}\right)\left(|x|^{2}+|y|^{2}-2\sum_{i=1}^{n}x_{i}y_{i}s_{i}\right)$
$\displaystyle=\left(|x|^{2}+|y|^{2}\right)^{2}-4\left(\sum_{i=1}^{n}x_{i}y_{i}s_{i}\right)^{2}$
$\displaystyle=|x|^{4}+|y|^{4}+2|x|^{2}|y|^{2}-4\left(\sum_{i=1}^{n}x_{i}y_{i}s_{i}\right)^{2}$
$\displaystyle=\left(|x|^{2}-|y|^{2}\right)^{2}+4\left(|x|^{2}|y|^{2}-\left(\sum_{i=1}^{n}x_{i}y_{i}s_{i}\right)^{2}\right)$
$\displaystyle\geq\left(|x|^{2}-|y|^{2}\right)^{2}+4\left(|x|^{2}|y|^{2}-\left|\langle(x_{1},\dots,x_{n}),(s_{1}y_{1},\dots,s_{n}y_{n})\rangle\right|^{2}\right)$
$\displaystyle\geq\left(|x|^{2}-|y|^{2}\right)^{2}+4\left(|x|^{2}|y|^{2}-|x|^{2}|(s_{1}y_{1},\dots,s_{n}y_{n})|^{2}\right)$
$\displaystyle\geq\left(|x|^{2}-|y|^{2}\right)^{2},$

for each
$x=(x_{1},\ldots,x_{n}),\,y=(y_{1},\ldots,y_{n})\in\mathbb{R}_{+}^{n}$ and
$s=(s_{1},\ldots,s_{n})\in(-1,1)^{n}$.

On the other hand, according to [15, Lemma 2.5], since
$p\in\mathcal{P}_{e}^{\infty}(\mathbb{R}_{+}^{n})$ then

$e^{\frac{|y|^{2}}{p(y)}-\frac{|x|^{2}}{p(x)}}\sim
e^{\frac{|y|^{2}-|x|^{2}}{p_{\infty}}},\quad x,y\in\mathbb{R}_{+}^{n}.$

Here $p_{\infty}>1$. Whence, it follows that

$\displaystyle
q_{+}(x,y,s)^{n+\widehat{\alpha}}\exp\left(-\tfrac{(1-\varepsilon)}{2}(|y|^{2}-|x|^{2}+\sqrt{q_{+}(x,y,s)q_{-}(x,y,s)}\right)e^{\frac{|y|^{2}}{p(y)}-\frac{|x|^{2}}{p(x)}}$
$\displaystyle\;\leq
Cq_{+}(x,y,s)^{n+\widehat{\alpha}}\exp\left(\left(\tfrac{1}{p_{\infty}}-\tfrac{1-\varepsilon}{2}\right)\left(|y|^{2}-|x|^{2}\right)-\tfrac{(1-\varepsilon)}{2}\sqrt{q_{+}(x,y,s)q_{-}(x,y,s)}\right)$
$\displaystyle\;\leq
C\left(q_{+}(x,y,s)\right)^{n+\widehat{\alpha}}\exp\left(-a_{\varepsilon}\sqrt{q_{+}(x,y,s)q_{-}(x,y,s)}\right),$

for every
$x=(x_{1},\ldots,x_{n}),\,y=(y_{1},\ldots,y_{n})\in\mathbb{R}_{+}^{n}$ such
that $(x,y,s)\in G_{1}$ and $\sum_{i=1}^{n}x_{i}y_{i}s_{i}\geq 0$. We recall
that

$G_{1}=\left\\{(x,y,s)\in\mathbb{R}_{+}^{n}\times\mathbb{R}_{+}^{n}\times(-1,1)^{n}:\sqrt{q_{-}(x,y,s)}\geq\frac{C_{0}}{1+|x|+|y|}\right\\}.$

Above we have set
$a_{\varepsilon}=\frac{1-\varepsilon}{2}-\left|\frac{1}{p_{\infty}}-\frac{1-\varepsilon}{2}\right|$.
Note that $a_{\varepsilon}>0$ because $\varepsilon<1/(p^{-})^{\prime}$ and
$(p^{-})^{\prime}=(p^{\prime})^{+}\geq p^{\prime}_{\infty}$.

We get

$\displaystyle\int_{\mathbb{R}_{+}^{n}}$
$\displaystyle\left|\mathcal{H}_{\alpha,\varepsilon}^{(2)}(f)(x)\right|^{p(x)}d\mu_{\alpha}(x)$
$\displaystyle\leq
C\int_{\mathbb{R}_{+}^{n}}\left(\int_{F_{x}}|f(y)|e^{\frac{-|y|^{2}}{p(y)}}|1-\varphi(x,y,s)|q_{+}(x,y,s)^{n+\widehat{\alpha}}\right.$
$\displaystyle\left.\quad\times\exp\left(-a_{\varepsilon}\sqrt{q_{+}(x,y,s)q_{-}(x,y,s)}\right)\Pi_{\alpha}(s)dsd\mathfrak{m}_{\alpha}(y)\right)^{p(x)}d\mathfrak{m}_{\alpha}(x).$

In order to complete the study of $\mathcal{H}_{\alpha,\varepsilon}^{(2)}$ we
use Stein complex interpolation. We consider firstly $n=1$. For every
$z\in\mathbb{C}$ with $\operatorname{Re}(z)>-\frac{1}{2}$, we define the
operator $\mathbb{H}_{z,\varepsilon}^{(2)}$ by

$\displaystyle\mathbb{H}_{z,\varepsilon}^{(2)}(h)(x)$
$\displaystyle=\int_{0}^{\infty}K_{z,\varepsilon}^{(2)}(x,y)h(y)y^{2z+1}dy\
x^{\frac{2z+1}{p(x)}}$
$\displaystyle=\widehat{\mathcal{H}}_{z,\varepsilon}^{(2)}(h)(x)\
x^{\frac{2z+1}{p(x)}},\quad x\in\mathbb{R}_{+},$

where

$\displaystyle K_{z,\varepsilon}^{(2)}(x,y)$
$\displaystyle=\int_{-1}^{1}\chi_{F_{x}}(y,s)(1-\varphi(x,y,s))(q_{+}(x,y,s))^{z+1}$
$\displaystyle\quad\times\exp\left(-a_{\varepsilon}\sqrt{q_{+}(x,y,s)q_{-}(x,y,s)}\right)(1-s^{2})^{z-\frac{1}{2}}ds,\quad
x,y\in\mathbb{R}_{+},$

and $a_{\varepsilon}$ is as above.

For every $z\in\mathbb{C}$ with $\operatorname{Re}(z)>-\frac{1}{2}$ and every
simple function $h$ defined on $(\mathbb{R}_{+},dx)$,
$\mathbb{H}_{z,\varepsilon}^{(2)}(h)$ is a measurable function on
$(\mathbb{R}_{+},dx)$.

Assume that $r$, $y$, $c_{1}$, $c_{2}>0$, $b_{1}$, $b_{2}$, $m_{1}$ and
$m_{2}$ are positive bounded measurable functions on $\mathbb{R}_{+}$, and
$A_{1}$ and $A_{2}$ are two measurable subsets of $\mathbb{R}_{+}$ with finite
Lebesgue measure. We define

$F_{y,r}(z)=\int_{B(y,r)}{\mathbb{H}_{z,\varepsilon}^{(2)}}\left(c_{1}^{m_{1}(\cdot)z+b_{1}(\cdot)}\chi_{A_{1}}(\cdot)\right)(x)c_{2}^{m_{2}(x)z+b_{2}(x)}\chi_{A_{2}}(x)dx,$

for $z\in\mathbb{C}$, $\operatorname{Re}(z)>-\frac{1}{2}$. The function
$F_{y,r}$ is analytic on $\Omega=\left\\{z\in\mathbb
C:\operatorname{Re}(z)>-\frac{1}{2}\right\\}$. Furthermore, for every
$-\frac{1}{2}<c<d<\infty$,

$\sup_{c\leq\operatorname{Re}(z)\leq d}|F_{y,r}(z)|<\infty.$

Thus, the family
$\left\\{\mathbb{H}_{z,\varepsilon}^{(2)}\right\\}_{z\in\Omega}$ is an
analytic family of admissible growth in every strip
$\\{z\in\mathbb{C}:c<\operatorname{Re}(z)<d\\}$, with
$-\frac{1}{2}<c<d<\infty$ (see [27, §3]).

Let $k\in\mathbb{N}$, $k>1$. We take $\alpha=\frac{k}{2}-1$. For every
$\overline{x}\in\mathbb{R}^{k}$ we write $x=|\overline{x}|$. If
$\overline{x}$, $\overline{y}\in\mathbb{R}^{k}$ and $\theta$ is the angle
between $\overline{x}$ and $\overline{y}$, we have that

$|\overline{x}\pm\overline{y}|^{2}=q_{\pm}(x,y,\cos(\theta)),$

and also that $(x,y,\cos(\theta))\in L_{1}$ if and only if
$|\overline{x}-\overline{y}|<C_{0}/(1+x+y)$. By integrating in spherical
coordinates on $\mathbb{R}^{k}$ and by performing the change of variable
$s=\cos(\theta)$ we obtain

$\displaystyle\left|\mathbb{H}_{\alpha,\varepsilon}^{(2)}(h)(x)\right|$
$\displaystyle\leq
Cx^{\frac{k-1}{p(x)}}\,\int_{|\overline{x}-\overline{y}|>\frac{C_{0}}{1+x+y}}|\overline{x}+\overline{y}|^{k}e^{-a_{\varepsilon}|\overline{x}-\overline{y}||\overline{x}+\overline{y}|}|{h}(y)|d\overline{y},$

for $x=|\overline{x}|\in\mathbb{R}_{+}$. We consider the operators

$\begin{split}T_{1}(h)(\overline{x})&=\int_{\begin{subarray}{c}|\overline{x}-\overline{y}|>\frac{C_{0}}{1+x+y}\\\
2|\overline{x}-\overline{y}|\geq|\overline{x}+\overline{y}|\end{subarray}}|\overline{x}+\overline{y}|^{k}e^{-a_{\varepsilon}|\overline{x}-\overline{y}||\overline{x}+\overline{y}|}h(\overline{y})d\overline{y},\end{split}$

and

$\begin{split}T_{2}(h)(\overline{x})&=\int_{\begin{subarray}{c}|\overline{x}-\overline{y}|>\frac{C_{0}}{1+x+y}\\\
2|\overline{x}-\overline{y}|<|\overline{x}+\overline{y}|\end{subarray}}|\overline{x}+\overline{y}|^{k}e^{-a_{\varepsilon}|\overline{x}-\overline{y}||\overline{x}+\overline{y}|}h(\overline{y})d\overline{y},\end{split}$

for $\overline{x}\in\mathbb{R}^{k}$. We are going to see that $T_{1}$ and
$T_{2}$ are bounded on $L^{\overline{p}(\cdot)}(\mathbb{R}^{k},dx)$, where
$\overline{p}(\overline{x})=p(|\overline{x}|)$,
$\overline{x}\in\mathbb{R}^{k}$.

Note firstly that

$\begin{split}|T_{1}(h)(\overline{x})|&\leq
C\left(\int_{B({-\overline{x}},1)}|h(\overline{y})|d\overline{y}+\sum_{\ell=1}^{\infty}\int_{\ell\leq|\overline{x}{+}\overline{y}|<\ell+1}e^{-c|\overline{x}{+}\overline{y}|^{2}}|h(\overline{y})|d\overline{y}\right)\\\
&\leq
C\sum_{\ell=0}^{\infty}e^{-c\ell^{2}}\int_{B(-\overline{x},\ell+1)}|h(\overline{y})|d\overline{y}\\\
&\leq
CM_{\textup{HL}}(h)({-\overline{x}}),\quad\overline{x}\in\mathbb{R}^{k}.\end{split}$

Here, $M_{\textup{HL}}$ represents the Hardy–Littlewood maximal function in
$\mathbb{R}^{k}$.

On the other hand, according to [22, (16) and (17)], if
$2|\overline{x}-\overline{y}|<|\overline{x}+\overline{y}|$, then
$|\overline{y}|\leq 3|\overline{x}|$ and
$\frac{4}{3}|\overline{x}|\leq|\overline{x}+\overline{y}|\leq
4|\overline{x}|$. We obtain

$\begin{split}|T_{2}(h)(\overline{x})|&\leq
C\int_{|\overline{x}-\overline{y}|>\frac{C_{0}}{1+4x}}|\overline{x}|^{k}e^{-c|\overline{x}||\overline{x}-\overline{y}|}|h(\overline{y})|d\overline{y}\\\
&\leq\begin{cases}\int_{|\overline{x}-\overline{y}|\leq
4}|h(\overline{y})|d\overline{y}\leq
CM_{\textup{HL}}(h)({\overline{x}})&\text{if}\;\;|\overline{x}|\leq 1,\\\
\int_{|\overline{x}-\overline{y}|>C_{0}/(5|\overline{x}|)}|\overline{x}|^{k}e^{-c|\overline{x}||\overline{x}-\overline{y}|}|h(\overline{y})|d\overline{y}&\text{if}\;\;|\overline{x}|>1.\end{cases}\end{split}$

Since $\overline{p}(\overline{x})=\overline{p}(-\overline{x})$, and under the
imposed conditions for $p(\cdot)$, $M_{\textup{HL}}$ is bounded on
$L^{\overline{p}(\cdot)}(\mathbb{R}^{k},dx)$ (see Lemma A.2 for $n=1$), the
arguments developed in [15, pp. 417 and 418] allow us to conclude that $T_{1}$
and $T_{2}$ are bounded on $L^{\overline{p}(\cdot)}(\mathbb{R}^{k},dx)$.

We have, therefore, that the operator $T{:=T_{1}+T_{2}}$ is bounded on
$L^{\overline{p}(\cdot)}(\mathbb{R}^{k},dx)$.

Since

$\left|\mathbb{H}_{\alpha,\varepsilon}^{(2)}(h)(x)\right|\leq
Cx^{\frac{k-1}{p(x)}}T(|\widetilde{h}|)(\overline{x}),\quad
x=|\overline{x}|,\quad x\in\mathbb{R}^{k},$

we get

$\begin{split}\int_{0}^{\infty}\left|\mathbb{H}_{\alpha,\varepsilon}^{(2)}(h)(x)\right|^{p(x)}dx&=\int_{0}^{\infty}\left|\widehat{\mathcal{H}}_{\alpha,\varepsilon}^{(2)}(h)(x)\right|^{p(x)}x^{k-1}dx\leq
C\int_{\mathbb{R}^{k}}\left|T({|\widetilde{h}|})\left(|\overline{x}|\right)\right|^{\overline{p}(\overline{x})}d\overline{x},\end{split}$

where ${\widetilde{h}}(\overline{y})={h}(|\overline{y}|)$,
$\overline{y}\in\mathbb{R}^{k}$. Hence

$\left\|\mathbb{H}_{\alpha,\varepsilon}^{(2)}(h)\right\|_{L^{p(\cdot)}(\mathbb{R}_{+},dx)}\leq
C\left\|T\left(|\widetilde{h}|\right)\right\|_{L^{\overline{p}(\cdot)}(\mathbb{R}^{k},dx)}\leq
C\left\|{\widetilde{h}}\right\|_{L^{\overline{p}(\cdot)}(\mathbb{R}^{k},dx)}.$

Naming $h_{k}(u)=h(u)u^{\frac{k-1}{p(u)}}$, $u\in\mathbb{R}_{+}$, we also have

$\begin{split}\int_{\mathbb{R}^{k}}\left|{\widetilde{h}}(\overline{x})\right|^{\overline{p}(x)}d\overline{x}&=\int_{\mathbb{R}^{k}}\left|{h}(|\overline{x}|)\right|^{p(|\overline{x}|)}d\overline{x}=C\int_{0}^{\infty}|{h}(x)|^{p(x)}x^{k-1}dx\\\
&=C\int_{0}^{\infty}\left|{h}(x)x^{\frac{k-1}{p(x)}}\right|^{p(x)}dx{=C\int_{0}^{\infty}|h_{k}(x)|^{p(x)}dx,}\end{split}$

which yields
$\|\widetilde{h}\|_{L^{\overline{p}(\cdot)}(\mathbb{R}^{k},dx)}\leq
C\|h_{k}\|_{L^{p(\cdot)}(\mathbb{R}_{+},dx)}.$ We conclude that

$\left\|\mathbb{H}_{\alpha,\varepsilon}^{(2)}(h)\right\|_{L^{p(\cdot)}(\mathbb{R}_{+},dx)}\leq
C\left\|{h_{k}}\right\|_{L^{p(\cdot)}(\mathbb{R}_{+},dx)}.$

We now consider, for every $z\in\mathbb{C}$ with
$\operatorname{Re}(z)>-\frac{1}{2}$,

$\mathcal{C}_{z,\varepsilon}(f)(x)=\mathcal{H}_{z,\varepsilon}^{(2)}({f_{z}})(x),\quad
x\in\mathbb{R}_{+},$

where $f_{z}(y)=f(y)y^{-\frac{2z+1}{p(y)}}$, $y\in\mathbb{R}_{+}$.

The family
$\\{\mathcal{C}_{z,\varepsilon}\\}_{\operatorname{Re}(z)>-\frac{1}{2}}$ is an
analytic family of admissible growth in every strip
$\\{z\in\mathbb{C}:c<\operatorname{Re}(z)<d\\}$ with $-\frac{1}{2}<c<d<\infty$
([27, §3]). For every $k\in\mathbb{N}$, $k>1$, we have that

$\left\|\mathcal{C}_{\frac{k}{2}-1,\varepsilon}(f)\right\|_{L^{p(\cdot)}(\mathbb{R}_{+},dx)}\leq
C_{0}\left\|f\right\|_{L^{p(\cdot)}(\mathbb{R}_{+},dx)}$

and, for each $t\in\mathbb{R}$,

$\left\|\mathcal{C}_{\frac{k}{2}-1+it,\varepsilon}(f)\right\|_{L^{p(\cdot)}(\mathbb{R}_{+},dx)}\leq\left\|\mathcal{C}_{\frac{k}{2}-1,\varepsilon}(|f|)\right\|_{L^{p(\cdot)}(\mathbb{R}_{+},dx)}\leq
C_{0}\left\|f\right\|_{L^{p(\cdot)}(\mathbb{R}_{+},dx)}.$

According to [27, Theorem 1], for every $\alpha\geq 0$,

$\left\|\mathcal{C}_{\alpha,\varepsilon}(f)\right\|_{L^{p(\cdot)}(\mathbb{R}_{+},dx)}\leq
C_{\alpha}\left\|f\right\|_{L^{p(\cdot)}(\mathbb{R}_{+},dx)}.$

It follows that, for every $\alpha\geq 0$,

$\begin{split}\int_{0}^{\infty}\left|\mathcal{H}_{\alpha,\varepsilon}^{(2)}(f)(x)\right|^{p(x)}d\mu_{\alpha}(x)&\leq
C\int_{0}^{\infty}\left|\mathbb{H}_{\alpha,\varepsilon}^{(2)}\left(f(\cdot)e^{-\frac{|\cdot|^{2}}{p(\cdot)}}\right)(x)\right|^{p(x)}dx\\\
&=C\int_{0}^{\infty}\left|\mathcal{C}_{\alpha,\varepsilon}\left(f(\cdot)e^{-\frac{|\cdot|^{2}}{p(\cdot)}}(\cdot)^{\frac{2\alpha+1}{p(y)}}\right)(x)\right|^{p(x)}dx.\end{split}$

Then

$\begin{split}\left\|\mathcal{H}_{\alpha,\varepsilon}^{(2)}(f)\right\|_{L^{p(\cdot)}(\mathbb{R}_{+},\mu_{\alpha})}&\leq
C\left\|\mathcal{C}_{\alpha,\varepsilon}\left(f(\cdot)e^{-\frac{|\cdot|^{2}}{p(\cdot)}}(\cdot)^{\frac{2\alpha+1}{p(\cdot)}}\right)\right\|_{L^{p(\cdot)}(\mathbb{R}_{+},dx)}\\\
&\leq
C\left\|f(\cdot)e^{-\frac{|\cdot|^{2}}{p(\cdot)}}(\cdot)^{\frac{2\alpha+1}{p(\cdot)}}\right\|_{L^{p(\cdot)}(\mathbb{R}_{+},dx)}\\\
&\leq
C\left\|f\right\|_{L^{p(\cdot)}(\mathbb{R}_{+},\mu_{\alpha})}.\end{split}$

We conclude that the operator $\mathcal{H}_{\alpha,\varepsilon}$ is bounded on
$L^{p(\cdot)}(\mathbb{R}_{+},\mu_{\alpha})$.

We now prove that $\mathcal{H}_{\alpha,\varepsilon}$ is bounded on
$L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ when the dimension $n$ is
greater than one.

Let $n\in\mathbb{N}$, $n>1$. We define

$\mathbb{H}_{z,\varepsilon}^{(2)}(h)(x)=\int_{\mathbb{R}_{+}^{n}}K_{z,\varepsilon}^{(2)}(x,y)h(y)\prod_{j=1}^{n}y_{j}^{2z_{j}+1}dy\prod_{j=1}^{n}x_{j}^{\frac{2z_{j}+1}{p(x)}},$

for $x=(x_{1},\dots,x_{n})\in\mathbb{R}_{+}^{n}$ and
$z=(z_{1},\dots,z_{n})\in\mathbb{C}^{n}$ with
$\operatorname{Re}(z_{j})>-\frac{1}{2}$ for each $j=1,\dots,n$, where

$\begin{split}K_{z,\varepsilon}^{(2)}(x,y)&=\int_{(-1,1)^{n}}\chi_{F_{x}}(y,s)(1-\varphi(x,y,s))q_{+}(x,y,s)^{n+\widehat{z}}\\\
&\qquad\times\exp\left(-a_{\varepsilon}\sqrt{q_{+}(x,y,s)q_{-}(x,y,s)}\right)\prod_{j=1}^{n}(1-s_{j}^{2})^{z_{j}-1/2}ds,\end{split}$

for $x,y\in\mathbb{R}_{+}^{n}$, $z$ and $a_{\varepsilon}$ as before.

Let $k=(k_{1},\dots,k_{n})\in\mathbb{N}^{n}$, $k_{j}>1$, $j=1,\dots,n$. We
consider $\alpha_{j}=k_{j}/2-1$, $j=1,\dots,n$, and
$\alpha=(\alpha_{1},\dots,\alpha_{n})$. We have that

$\mathbb{H}_{\alpha,\varepsilon}^{(2)}(h)(x)=\int_{\mathbb{R}_{+}^{n}}K_{\alpha,\varepsilon}^{(2)}(x,y)h(y)\prod_{j=1}^{n}y_{j}^{k_{j}-1}dy\prod_{j=1}^{n}x_{j}^{\frac{k_{j}-1}{p(x)}},$

for $x=(x_{1},\dots,x_{n})\in\mathbb{R}_{+}^{n}$, and

$\begin{split}K_{\alpha,\varepsilon}^{(2)}(x,y)&=\int_{(-1,1)^{n}}\chi_{F_{x}}(y,s)(1-\varphi(x,y,s))q_{+}(x,y,s)^{\widehat{k}/2}\\\
&\quad\times\exp\left(-a_{\varepsilon}\sqrt{q_{+}(x,y,s)q_{-}(x,y,s)}\right)\prod_{j=1}^{n}(1-s_{j}^{2})^{\alpha_{j}-1/2}ds,\end{split}$

for $x,y\in\mathbb{R}_{+}^{n}$. We define
$\overline{p}(\overline{x_{1}},\dots,\overline{x_{n}})=p(x_{1},\dots,x_{n})$,
where $x_{j}=|\overline{x_{j}}|$,
$\overline{x_{j}}\in{\mathbb{R}_{+}^{k_{j}}}$, $j=1,\dots,n$. Integrating in
multi-radial polar coordinates we have that

$\left|\mathbb{H}_{\alpha,\varepsilon}^{(2)}(h)(x)\right|\leq
C\int_{|\overline{x}-\overline{y}|>\frac{C_{0}}{1+|\overline{x}|+\overline{y}|}}|\overline{x}+\overline{y}|^{{\widehat{k}}}e^{-a_{\varepsilon}|\overline{x}-\overline{y}||\overline{x}+\overline{y}|}|h(|\overline{y_{1}}|,\dots,|\overline{y_{n}}|)|d\overline{y}\prod_{j=1}^{n}x_{j}^{\frac{k_{j}-1}{p(x)}},$

for
$x=(x_{1},\dots,x_{n}){=(|\overline{x_{1}}|,\dots,|\overline{x_{n}}|)}\in\mathbb{R}_{+}^{n}$
and
$\overline{x}=(\overline{x_{1}},\dots,\overline{x_{n}})\in\prod_{j=1}^{n}\mathbb
R^{k_{j}}=\mathbb R^{{\widehat{k}}}$.

We now proceed as in the above one-dimensional case. In order to do this,
notice that if we define $\overline{p}$ by
$\overline{p}(\overline{x})=p(|\overline{x_{1}}|,\dots,|\overline{x_{n}}|)$,
for
$\overline{x}=(\overline{x_{1}},\dots,\overline{x_{n}})\in\mathbb{R}_{+}^{\widehat{k}}$,
then $\overline{p}$ belongs to $\textup{LH}(\mathbb{R}_{+}^{\widehat{k}})$,
with $1<\overline{p}^{-}\leq\overline{p}^{+}<\infty$, by virtue of Lemma A.2.
Hence, the Hardy–Littlewood maximal operator $M_{\textup{HL}}$ on
$\mathbb{R}_{+}^{\widehat{k}}$ is bounded on
$L^{\overline{p}(\cdot)}\left(\mathbb{R}_{+}^{\widehat{k}}\right)$.

We consider, for every $z=(z_{1},\dots,z_{n})\in\mathbb{C}^{n}$ such that
$\operatorname{Re}(z_{j})>-\frac{1}{2}$, for each ${j=1,\dots,n}$, the
operator

$\mathcal{C}_{z,\varepsilon}(f)(x)=\mathbb{H}_{z,\varepsilon}^{(2)}(f_{z})(x),\quad
x\in\mathbb{R}_{+}^{n},$

where $f_{z}(y)=f(y)\prod_{j=1}^{n}y_{j}^{-\frac{2z_{j}+1}{p(y)}}$ for
$y=(y_{1},\dots,y_{n})\in\mathbb{R}_{+}^{n}$. The proof can be concluded as in
the one-dimensional case by using an $n$-dimensional version of the Stein
complex interpolation with variable exponent. This result can be proved by
proceeding as in the proof of [27, Theorem 1] and by using an $n$-dimensional
version of the Three Lines Theorem (see Theorem A.1 and [4, Proposition 21]).
∎

##  4\. Proof of Theorem 1.1 for maximal operators

According to the subordination formula (1.2), since
$\frac{t}{2\sqrt{\pi}}\int_{0}^{\infty}\frac{e^{-t^{2}/(4u)}}{u^{3/2}}du=1$
for each $t>0$, we deduce that

$P_{*}^{\alpha}(f)(x)\leq W_{*}^{\alpha}(f)(x),\quad x\in\mathbb{R}_{+}^{n}.$

Hence, it suffices to see that $W_{*}^{\alpha}$ is bounded on
$L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$.

We firstly study its global part $W_{*,\text{glob}}^{\alpha}$ given, for
$x\in\mathbb{R}_{+}^{n}$, by

$\displaystyle W_{*,\text{glob}}^{\alpha}(f)(x)$
$\displaystyle=\sup_{t>0}\left|\int_{\mathbb{R}_{+}^{n}}\int_{(-1,1)^{n}}\frac{e^{-\frac{q_{-}\left(e^{-t/2}x,y,s\right)}{1-e^{-t}}+|y|^{2}}}{(1-e^{-t})^{n+\widehat{\alpha}}}(1-\varphi(x,y,s))\Pi_{\alpha}(s)dsf(y)d\mu_{\alpha}(y)\right|.$

By performing the change of variables $1-e^{-t}=u$, $t>0$, and then replacing
$u$ by $t$, we can write

$\displaystyle W_{*,\text{glob}}^{\alpha}(f)(x)$
$\displaystyle=\sup_{0<t<1}\left|\int_{\mathbb{R}_{+}^{n}}\int_{(-1,1)^{n}}\frac{e^{-\frac{q_{-}\left(\sqrt{1-t}x,y,s\right)}{t}+|y|^{2}}}{t^{n+\widehat{\alpha}}}(1-\varphi(x,y,s))\Pi_{\alpha}(s)dsf(y)d\mu_{\alpha}(y)\right|.$

Let $(x,y,s)\in G_{1}$ (recall the definition on page 2). We consider

$u(t)=\frac{(1-t)|x|^{2}+|y|^{2}-2\sum_{i=1}^{n}x_{i}y_{i}s_{i}\sqrt{1-t}}{t},\quad
t\in(0,1).$

Setting $a=|x|^{2}+|y|^{2}$ and $b=2\sum_{i=1}^{n}x_{i}y_{i}s_{i}$, we have

(4.1) $u(t)=\frac{a}{t}-\frac{\sqrt{1-t}}{t}b-|x|^{2},\quad t\in(0,1).$

We also define

$v(t)=\frac{e^{-u(t)}}{t^{n+\widehat{\alpha}}},\quad t\in(0,1).$

We are going to study the supremum of $v(t)$, for $t\in(0,1)$, by proceeding
as in the proof of [26, Proposition 2.1]. The derivative of $v$ is

$v^{\prime}(t)=-\frac{e^{-u(t)}}{t^{n+\widehat{\alpha}}}\left(u^{\prime}(t)+\frac{n+\widehat{\alpha}}{t}\right),\quad
t\in(0,1),$

where

$u^{\prime}(t)=-\frac{a}{t^{2}}+b\left(\frac{1}{2t\sqrt{1-t}}+\frac{\sqrt{1-t}}{t^{2}}\right)=\frac{-2a\sqrt{1-t}+bt+2b(1-t)}{2t^{2}\sqrt{1-t}},\quad
t\in(0,1).$

Thus,

$\displaystyle v^{\prime}(t)$
$\displaystyle=-\frac{e^{-u(t)}}{t^{n+\widehat{\alpha}}}\left(\frac{-2a\sqrt{1-t}-bt+2b}{2t^{2}\sqrt{1-t}}+\frac{n+\widehat{\alpha}}{t}\right)$
$\displaystyle=-\frac{e^{-u(t)}}{t^{n+\widehat{\alpha}}}\cdot\frac{2\sqrt{1-t}(t(n+\widehat{\alpha})-a)+b(2-t)}{2t^{2}\sqrt{1-t}},\quad
t\in(0,1).$

By choosing $C_{0}>1$ large enough, we can prove $a>n+\widehat{\alpha}$ for
any $(x,y,s)\in G_{1}$. Indeed, let us remark that

$|b|\leq 2\sum_{i=1}^{n}|x_{i}||y_{i}|\leq|x|^{2}+|y|^{2}=a.$

Besides,

$a=\frac{a-b+a+b}{2}\geq\frac{q_{-}(x,y,s)+a-|b|}{2}\geq\frac{1}{2}q_{-}(x,y,s).$

Also,

$\sqrt{a}\geq\frac{1}{\sqrt{2}}(|x|+|y|).$

Fix $(x,y,s)\in G_{1}$. If $|x|+|y|<1$ then

$a\geq\frac{1}{2}q_{-}(x,y,s)>\frac{1}{2}\frac{C_{0}^{2}}{(1+|x|+|y|)^{2}}\geq\frac{C_{0}^{2}}{8}>\frac{C_{0}}{8}$

since we shall take $C_{0}>1.$ And, if $|x|+|y|\geq 1,$ then

$a=\sqrt{a}\sqrt{a}\geq\frac{1}{\sqrt{2}}(|x|+|y|)\frac{1}{\sqrt{2}}\sqrt{q_{-}(x,y,s)}>\frac{C_{0}}{2}\frac{|x|+|y|}{1+|x|+|y|}\geq\frac{C_{0}}{2}\frac{1}{2}=\frac{C_{0}}{4}>\frac{C_{0}}{8}.$

Therefore, taking $C_{0}>8(n+\widehat{\alpha})$ we get that
$a>n+\widehat{\alpha}$ on $G_{1}$ as claimed.

Then, if $b\leq 0$, $v^{\prime}(t)>0$ for each $t\in(0,1)$, so

$\sup_{0<t<1}v(t)\leq v(1)=e^{-|y|^{2}}.$

On the other hand, if $b>0$, from the property $a>n+\widehat{\alpha}$, the
equation

$2\sqrt{1-t}(a-t(n+\widehat{\alpha}))=b(2-t)$

has a unique solution $t_{n}$. The arguments developed in [26, p. 850] allow
us to conclude that

$\sup_{0<t<1}v(t)\sim v(t_{0}),$

where
$t_{0}=2\frac{\sqrt{a^{2}-b^{2}}}{a+\sqrt{a^{2}-b^{2}}}\sim\sqrt{\frac{q_{-}(x,y,s)}{q_{+}(x,y,s)}}$.

Then,

$\sup_{0<t<1}v(t)\leq
C\left(\frac{q_{+}(x,y,s)}{q_{-}(x,y,s)}\right)^{\frac{n+\widehat{\alpha}}{2}}\exp\left(-\frac{|y|^{2}-|x|^{2}}{2}-\frac{{\sqrt{q_{+}(x,y,s)q_{-}(x,y,s)}}}{2}\right),$

provided that $C_{0}$ satisfies the above condition. From now on, $C_{0}$ will
be fixed such that the stated condition holds.

Since $q_{+}(x,y,s)q_{-}(x,y,s)\geq c$ for every $(x,y,s)\in G_{1}$ (see [21,
p. 264]) , we have that

$\sqrt{\frac{q_{+}(x,y,s)}{q_{-}(x,y,s)}}\leq Cq_{+}(x,y,s).$

Therefore, for every $(x,y,s)\in G_{1}$

$\sup_{0<t<1}v(t)\leq
Cq_{+}(x,y,s)^{n+\widehat{\alpha}}e^{-\frac{|y|^{2}-|x|^{2}}{2}-\frac{{\sqrt{q_{+}(x,y,s)q_{-}(x,y,s)}}}{2}}=CH_{\alpha,0}(x,y,s)$

where $H_{\alpha,0}$ is the function given in (3.1). Hence,
$W^{\alpha}_{*,\text{glob}}$ is pointwise smaller than a multiple of
$\mathcal{H}_{\alpha,0}$, which is a bounded operator on
$L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ by Proposition 3.1, so
$W^{\alpha}_{*,\text{glob}}$ is also bounded on
$L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$.

We now study $W^{\alpha}_{*,\text{loc}}$ defined by

$\begin{split}&W^{\alpha}_{*,\text{loc}}(f)(x)\\\
&\quad=\sup_{t>0}\left|\int_{\mathbb{R}_{+}^{n}}\int_{(-1,1)^{n}}\frac{\exp\left(\frac{-q_{-}\left(e^{-t/2}x,y,s\right)}{1-e^{-t}}+|y|^{2}\right)}{(1-e^{-t})^{n+\widehat{\alpha}}}\varphi(x,y,s)\Pi_{\alpha}(s)dsf(y)d\mu_{\alpha}(y)\right|,\end{split}$

for $x\in\mathbb{R}_{+}^{n}$. Setting $u=1-e^{-t}$ and then replacing $u$ by
$t$, we can write

$W^{\alpha}_{*,\text{loc}}(f)(x)=\sup_{0<t<1}\left|\int_{\mathbb{R}_{+}^{n}}K_{t}^{\alpha}(x,y)f(y)d\mathfrak{m}_{\alpha}(y)\right|$

for $x=(x_{1},\dots,x_{n})\in\mathbb{R}_{+}^{n}$, where

$K^{\alpha}_{t}(x,y)=\int_{(-1,1)^{n}}\frac{\exp\left(-\frac{(1-t)|x|^{2}+|y|^{2}-2\sqrt{1-t}\sum_{i=1}^{n}x_{i}y_{i}s_{i}}{t}\right)}{t^{n+\widehat{\alpha}}}\varphi(x,y,s)\Pi_{\alpha}(s)ds,$

for $x=(x_{1},\dots,x_{n})$, $y=(y_{1},\dots,y_{n})\in\mathbb{R}_{+}^{n}$ and
$t\in(0,1)$.

As it was explained in Section 2, we shall see that
$W^{\alpha}_{*,\text{loc}}$ is a bounded operator on
$L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ as a consequence of vector
valued Calderón–Zygmund theory.

According to [38, (2.6)] we have that

$q_{-}\left(\sqrt{1-t}x,y,s\right)\geq
q_{-}(x,y,s)-C\left(1-\sqrt{1-t}\right)=q_{-}(x,y,s)-C\frac{t}{1+\sqrt{1-t}},$

for $x,y\in\mathbb{R}_{+}^{n}$, $t\in(0,1)$, $s\in(-1,1)^{n}$ and $(x,y,s)\in
L_{2}$.

Then,

$|K^{\alpha}_{t}(x,y)|\leq
C\int_{(-1,1)^{n}}\frac{e^{-q_{-}(x,y,s)/t}}{t^{n+\widehat{\alpha}}}\Pi_{\alpha}(s)ds\leq
C\int_{(-1,1)^{n}}\frac{\Pi_{\alpha}(s)}{q_{-}(x,y,s)^{n+\widehat{\alpha}}}ds$

for $x,y\in\mathbb{R}_{+}^{n}$ and $t\in(0,1)$.

According to [7, Lemma 3.1] (see also [33, Lemma 2.1]), we get

(4.2)
$\sup_{t>0}|K^{\alpha}_{t}(x,y)|\leq\frac{C}{\mathfrak{m}_{\alpha}(B(x,|y-x|))},\quad
x,y\in\mathbb{R}_{+}^{n},\;x\neq y.$

Let $j=1,\dots,n$. We have that,

$\begin{split}\partial_{x_{j}}K^{\alpha}_{t}(x,y)&=\int_{(-1,1)^{n}}\left(\frac{-2x_{j}(1-t)+2y_{j}s_{j}\sqrt{1-t}}{t^{n+1+\widehat{\alpha}}}\varphi(x,y,s)+\frac{\frac{\partial\varphi}{\partial
x_{j}}(x,y,s)}{t^{n+\widehat{\alpha}}}\right)\\\
&\qquad\times\exp\left(-\frac{(1-t)|x|^{2}+|y|^{2}-2\sum_{i=1}^{n}x_{i}y_{i}s_{i}\sqrt{1-t}}{t}\right)\Pi_{\alpha}(s)ds,\end{split}$

for $x=(x_{1},\dots,x_{n})$, $y=(y_{1},\dots,y_{n})\in\mathbb{R}_{+}^{n}$, and
$t>0$.

According to the properties of $\varphi$ and using again [7, Lemma 3.1], since

(4.3)
$\begin{split}\left|x_{j}\sqrt{1-t}-y_{j}s_{j}\right|^{2}&=x_{j}^{2}(1-t)+y_{j}^{2}s_{j}^{2}-2x_{j}y_{j}s_{j}\sqrt{1-t}\\\
&\leq(1-t)|x|^{2}+|y|^{2}-2\sum_{i=1}^{n}x_{i}y_{i}s_{i}\sqrt{1-t},\end{split}$

for $x=(x_{1},\dots,x_{n})$, $y=(y_{1},\dots,y_{n})\in\mathbb{R}_{+}^{n}$,
$t\in(0,1)$ and $s\in(-1,1)^{n}$, we get

$\begin{split}|\partial_{x_{j}}K^{\alpha}_{t}(x,y)|&\leq
C\int_{(-1,1)^{n}}\frac{e^{-c\frac{q_{-}(x,y,s)}{t}}}{t^{n+\frac{1}{2}+\widehat{\alpha}}}\Pi_{\alpha}(s)ds\\\
&\leq
C\int_{(-1,1)^{n}}\frac{\Pi_{\alpha}(s)}{q_{-}(x,y,s)^{n+\frac{1}{2}+\widehat{\alpha}}}ds\\\
&\leq C\frac{1}{|x-y|\mathfrak{m}_{\alpha}(B(x,|x-y|))},\end{split}$

for $x,y\in\mathbb{R}_{+}^{n}$, $x\neq y$, and $t>0$. Hence,

(4.4)
$\sup_{t>0}|\partial_{x_{j}}K^{\alpha}_{t}(x,y)|+\sup_{t>0}|\partial_{y_{j}}K^{\alpha}_{t}(x,y)|\leq
C\frac{1}{|x-y|\mathfrak{m}_{\alpha}(B(x,|x-y|))},$

for $x,y\in\mathbb{R}_{+}^{n}$, $x\neq y$.

Let $N\in\mathbb{N}$. We consider the space $C\left(\left[1/N,N\right]\right)$
of continuous functions in $\left[1/N,N\right]$ with the usual maximum norm.
We define, for every $x,y\in\mathbb{R}_{+}^{n}$, $x\neq y$,

$\left[K^{\alpha}(x,y)\right](t)=K^{\alpha}_{t}(x,y),\quad t>0.$

By proceeding as above we can see that, for every $x\in\mathbb{R}_{+}^{n}$,
the mapping $\Phi_{x}(y)=K^{\alpha}(x,y)$, $y\in\mathbb{R}_{+}^{n}$, is
continuous from $\mathbb{R}_{+}^{n}$ into $C\left(\left[1/N,N\right]\right)$,
and then, $\Phi_{x}$ is weakly measurable. Since
$C\left(\left[1/N,N\right]\right)$ is separable, we conclude that, for every
$x\in\mathbb{R}_{+}^{n}$, $\Phi_{x}$ is strongly measurable (see [44, p.
131]). According to (4.2) and (4.4) we deduce that $K^{\alpha}$ is a
$C\left(\left[1/N,N\right]\right)$-valued Calderón–Zygmund kernel with respect
to $(\mathbb{R}_{+}^{n},|\cdot|,\mathfrak{m}_{\alpha})$.

Suppose $\lambda$ is a complex measure supported in $\left[1/N,N\right]$ and
$f\in C^{\infty}_{c}(\mathbb{R}_{+}^{n})$. By using (4.2) we obtain

$\int_{[1/N,N]}\int_{\mathbb{R}_{+}^{n}}|K^{\alpha}_{t}(x,y)||f(y)|d\mathfrak{m}_{\alpha}(y)d|\lambda|(t)<\infty,\quad
x\notin\operatorname{supp}(f),$

because $|\lambda|([1/N,N])<\infty$. Here $|\lambda|$ denotes the total
variation of $\lambda$. It follows that

(4.5)
$\begin{split}\int_{[1/N,N]}&\int_{\mathbb{R}_{+}^{n}}K^{\alpha}_{t}(x,y)f(y)d\mathfrak{m}_{\alpha}(y)d\lambda(t)\\\
&=\int_{\mathbb{R}_{+}^{n}}\int_{[1/N,N]}K^{\alpha}_{t}(x,y)f(y)d\lambda(t)d\mathfrak{m}_{\alpha}(y),\quad
x\notin\operatorname{supp}(f).\end{split}$

We define the functional $S_{\lambda}$ on $C([1/N,N])$ by

$S_{\lambda}(g)=\int_{[1/N,N]}g(t)d\lambda(t),\quad g\in C([1/N,N]).$

Equality (4.5) says that, by understanding the integral under $S_{\lambda}$ in
the $C([1/N,N])$-Bochner sense,

$S_{\lambda}\left[\int_{\mathbb{R}_{+}^{n}}[K^{\alpha}(x,y)](\cdot)f(y)d\mathfrak{m}_{\alpha}(y)\right]=\int_{[1/N,N]}W^{\alpha}_{t,\text{loc}}(f)(x)d\lambda(t),\quad
x\notin\operatorname{supp}(f).$

Since the dual of $C([1/N,N])$ is the space $\mathcal{M}([1/N,N])$ of complex
measures supported on $[1/N,N]$ we conclude that, for every
$x\notin\operatorname{supp}(f)$

$W^{\alpha}_{t,\text{loc}}(f)(x)=\left[\int_{\mathbb{R}_{+}^{n}}[K^{\alpha}(x,y)](\cdot)f(y)d\mathfrak{m}_{\alpha}(y)\right](t),\quad
t\in[1/N,N].$

According to [42, p. 73], the maximal operator $W^{\alpha}_{*}$ is bounded on
$L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$. Also, $W^{\alpha}_{*,\text{glob}}$
is bounded on $L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ (see the first part of
this proof). Then, $W^{\alpha}_{*,\text{loc}}$ is bounded on
$L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$. Hence, there exists $C>0$ such that,
for every $N\in\mathbb{N}$,

(4.6)
$\left\|\left\|W^{\alpha}_{t,\text{loc}}(f)\right\|_{C([1/N,N])}\right\|_{L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})}\leq
C\|f\|_{L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})},$

for $f\in L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$. By using (4.2), (4.4) and
(4.6) as it was explained in Section 2 we get

$\left\|\sup_{t\in[1/N,N]}|W^{\alpha}_{t,\text{loc}}(f)(x)|\right\|_{L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})}\leq
C\|f\|_{L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})}$

for $f\in L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ and $C>0$ independent
of $N\in\mathbb{N}$.

By using now the monotone convergence theorem (see [18, p. 75]), we conclude
that $W^{\alpha}_{*,\text{loc}}$ is bounded on
$L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$. Thus, the proof of Theorem
1.1 for $W^{\alpha}_{*}$ is finished.

##  5\. Proof of Theorem 1.1 for Riesz transforms

The proof of Theorem 1.1 for Riesz transforms $R^{\beta}_{\alpha}$ of order
$\beta\in\mathbb{N}^{n}\setminus\\{(0,\dots,0)\\}$, follows the same steps
done in the proof of the results in Section 4 by using some results developed
in [21] and [41]. We now sketch the proof.

Let $\beta\in\mathbb{N}^{n}\setminus\\{(0,\dots,0)\\}$ be given. For every
$f\in L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$, we have that

$R^{\beta}_{\alpha}(f)(x)=c_{\beta}f(x)+\text{p.v.}\int_{\mathbb{R}_{+}^{n}}R^{\beta}_{\alpha}(x,y)f(y)d\mathfrak{m}_{\alpha}(y),\quad\text{a.e.
}x\in\mathbb{R}_{+}^{n}$

where $c_{\beta}\in\mathbb{R}$ and

$R^{\beta}_{\alpha}(x,y)=\frac{1}{\Gamma\left(\frac{\widehat{\beta}}{2}\right)}\int_{(-1,1)^{n}}K^{\beta}_{\alpha}(x,y,s)\Pi_{\alpha}(s)\,ds,\quad
x,y\in\mathbb{R}_{+}^{n},\;x\neq y,$

with

$\displaystyle K^{\beta}_{\alpha}(x,y,s)$
$\displaystyle=\int_{0}^{1}r^{\frac{\widehat{{\beta}}-2}{2}}\left(\frac{-\log
r}{1-r}\right)^{\frac{\widehat{{\beta}}-2}{2}}\prod_{i=1}^{n}H_{{\beta}_{i}}\left(\frac{\sqrt{r}x_{i}-y_{i}s_{i}}{\sqrt{1-r}}\right)\frac{e^{-\frac{q_{-}(\sqrt{r}x,y,s)}{1-r}}}{(1-r)^{n+\widehat{\alpha}+1}}\,dr$
$\displaystyle=\int_{0}^{1}(1-t)^{\frac{\widehat{\beta}-1}{2}}\left(\frac{-\log(1-t)}{t}\right)^{\frac{\widehat{\beta}-2}{2}}\prod_{i=1}^{n}H_{{\beta}_{i}}\left(\frac{\sqrt{1-t}x_{i}-y_{i}s_{i}}{\sqrt{t}}\right)$
$\displaystyle\ \
\quad\quad\times\frac{e^{-\frac{q_{-}(\sqrt{1-t}x,y,s)}{t}}}{t^{n+\widehat{\alpha}+1}}\,\frac{dt}{\sqrt{1-t}},$

being $H_{{\beta}_{i}}$ the one-dimensional Hermite polynomial of degree
${\beta}_{i},$ $i=1,\dots,n$, and for the second equality we have made the
change of variables $t=1-r.$ In order to establish that $R_{\alpha}^{\beta}$
is bounded on $L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ we can assume
that $c_{\beta}=0$.

We define $R^{\beta}_{\alpha,\text{loc}}$ and $R^{\beta}_{\alpha,\text{glob}}$
in the usual way. Firstly, we shall prove the $L^{p(\cdot)}$-boundedness of
the global part.

Taking into account that $|\sqrt{1-t}x_{i}-y_{i}s_{i}|\leq
q_{-}^{\frac{1}{2}}(\sqrt{1-t}x,y,s)$ from (4.3), we get, for every
$\varepsilon>0$,

$\left|\prod_{i=1}^{n}H_{{\beta}_{i}}\left(\frac{\sqrt{1-t}x_{i}-y_{i}s_{i}}{\sqrt{t}}\right)\right|\leq
C\sum_{k=0}^{\widehat{{\beta}}}\left(\frac{q_{-}^{\frac{1}{2}}(\sqrt{1-t}x,y,s)}{\sqrt{t}}\right)^{k}\leq
Ce^{\varepsilon\frac{q_{-}(\sqrt{1-t}x,y,x)}{t}}.$

Also, since the function
$t\mapsto(1-t)^{\frac{\widehat{\beta}-1}{2}}\left(-\frac{\log(1-t)}{t}\right)^{\frac{\widehat{\beta}-2}{2}}$
is bounded on $[0,1]$, we have

$\displaystyle\left|R^{\beta}_{\alpha,\text{glob}}f(x)\right|$
$\displaystyle\leq
C|f(x)+C\int_{\mathbb{R}_{+}^{n}}|f(y)|\int_{(-1,1)^{n}}K_{\alpha}(x,y,s)\,\Pi_{\alpha}(s)\,ds\,d\mathfrak{m}_{\alpha}(y),$

for $x\in\mathbb{R}_{+}^{n}$, being

$K_{\alpha}(x,y,s)=\int_{0}^{1}\frac{e^{-(1-\varepsilon)\frac{q_{-}(\sqrt{1-t}x,y,s)}{t}}}{t^{n+\widehat{\alpha}+1}}\,\frac{dt}{\sqrt{1-t}}(1-\varphi(x,y,s))$

for $y\in\mathbb{R}_{+}^{n}$ and $s\in(-1,1)^{n}.$

We can see that the above kernel is, in turn, bounded by the kernel
$H_{\alpha,\varepsilon}(x,y,s)$ given in (3.1) provided that
$\varepsilon<\frac{1}{n+\widehat{\alpha}}$. When
$\displaystyle\sum_{i=1}^{n}x_{i}y_{i}s_{i}>0$ we follow closely the estimates
obtained by S. Pérez in [35], taking into account that in this case, for
$0<\varepsilon<\frac{1}{n+\widehat{\alpha}},$

$K_{\alpha}(x,y,s)\leq
C_{\varepsilon}\frac{e^{-(1-\varepsilon)u_{0}}}{t_{0}^{n+\widehat{\alpha}}}$

with $u_{0}=\frac{|y|^{2}-|x|^{2}+\sqrt{q_{+}(x,y,s)q_{-}(x,y,s)}}{2}$ and
$t_{0}=2\frac{\sqrt{a^{2}-b^{2}}}{a+\sqrt{a^{2}-b^{2}}}$, being
$a=|x|^{2}+|y|^{2}$ and $b=2\sum_{i=1}^{n}x_{i}y_{i}s_{i}$.

Indeed, by calling $u(t)=\frac{q_{-}(\sqrt{1-t}x,y,s)}{t}$, notice that $u$ is
the one given in (4.1) at the previous section. We have already proved that,
for $b>0$,

$\sup_{0<t<1}\frac{e^{-u(t)}}{t^{n+\widehat{\alpha}}}\sim\frac{e^{-u_{0}}}{t_{0}^{n+\widehat{\alpha}}}.$

Thus, for $\nu=\frac{1}{n+\widehat{\alpha}}-\varepsilon>0$ we have

$\displaystyle K_{\alpha}(x,y,s)$ $\displaystyle=\int_{0}^{1}e^{\varepsilon
u(t)}\left(\frac{e^{-u(t)}}{t^{n+\widehat{\alpha}}}\right)^{\frac{n+\widehat{\alpha}-1}{n+\widehat{\alpha}}}\left(\frac{e^{-u(t)}}{t^{n+\widehat{\alpha}}}\right)^{\frac{1}{n+\widehat{\alpha}}}\frac{dt}{t\sqrt{1-t}}$
$\displaystyle\leq
C\left(\frac{e^{-u_{0}}}{t_{0}^{n+\widehat{\alpha}}}\right)^{1-\frac{1}{n+\widehat{\alpha}}}\int_{0}^{1}e^{-\nu
u(t)}\frac{dt}{t^{2}\sqrt{1-t}}.$

By performing the change of variable $s=u(t)-u_{0}$ and following the
calculations made in [35, p. 499], the latter expression is bounded by

$\frac{e^{-\left(1-\frac{1}{n+\widehat{\alpha}}\right)u_{0}}e^{-\nu
u_{0}}}{t_{0}^{n+\widehat{\alpha}-1}}\frac{1}{t_{0}\sqrt[4]{(a-b)(a+b)}}\int_{0}^{\infty}e^{-\nu
s}\left(1+\frac{1}{\sqrt{s}}\right)\,ds.$

Moreover, recalling that $(a-b)(a+b)=q_{-}(x,y,s)q_{+}(x,y,s)\geq c$ when
$b>0$ (see [21, p. 264]) we get the estimate claimed above.

For the case $b\leq 0,$ we have that $\frac{a}{t}-|x|^{2}\leq
u(t)=\frac{q_{-}(\sqrt{1-t}x,y,s)}{t}$ like in [35, p. 500]. After making the
change of variables $a\left(\frac{1}{t}-1\right)=s$ and performing the
integration taking into account that on the global part $a\geq c,$ we get
$K_{\alpha}(x,y,s)\leq Ce^{-(1-\varepsilon)|y|^{2}}$.

Therefore, $K_{\alpha}(x,y,s)\leq CH_{\alpha,\varepsilon}(x,y,s)$ for
$0<\varepsilon<\frac{1}{n+\widehat{\alpha}}$. From Proposition 3.1 we deduce
that the operator $R^{\beta}_{\alpha,\text{glob}}$ is bounded on
$L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ by choosing
${0<\varepsilon<\frac{1}{n+\widehat{\alpha}}\wedge\frac{1}{(p^{-})^{\prime}}}$.

According to [33, p. 699] $R_{\alpha}^{\beta}$ is bounded on
$L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$. Since, as we have just proved
$R_{\alpha,\text{glob}}^{\beta}$ is bounded on
$L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$, $R_{\alpha,\text{loc}}^{\beta}$ is
also bounded on $L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$. By proceeding as in
[41, Lemma 3.3] and [7, Lemma 3.1] (see also [21, Proposition 6 and Lemma 7])
we can see that the integral kernel of $R^{\beta}_{\alpha,\text{loc}}$ is a
Calderón–Zygmund kernel with respect to $\mathfrak{m}_{\alpha}$. The procedure
developed in Section 2 leads to see that $R^{\beta}_{\alpha,\text{loc}}$ is
bounded on $L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ and with this we
finish the proof of this result.

##  6\. Proof of Theorem 1.1 for Littlewood–Paley functions

In this section we prove Theorem 1.1 for Littlewood–Paley functions
$g^{\beta,k}_{\alpha}$, with $k\in\mathbb{N}$ and $\beta\in\mathbb{N}^{n}$
such that $k+\widehat{\beta}>0$.

Let $k\in\mathbb{N}$, $k\geq 1$. We recall that
$g_{\alpha}^{k}=g_{\alpha}^{\boldsymbol{0},k},$ i.e.

$g^{k}_{\alpha}(f)(x)=\left(\int_{0}^{\infty}\left|t^{k}\partial^{k}_{t}P^{\alpha}_{t}(f)(x)\right|^{2}\frac{dt}{t}\right)^{1/2},\quad
x\in\mathbb{R}_{+}^{n},$

where

$P^{\alpha}_{t}(f)(x)=\frac{t}{2\sqrt{\pi}}\int_{0}^{\infty}\frac{e^{-\frac{t^{2}}{4u}}}{u^{\frac{3}{2}}}W^{\alpha}_{u}(f)(x)du,\quad
x\in\mathbb{R}_{+}^{n},t>0.$

We define

$P^{\alpha}_{t,\text{loc}}(f)(x)=\frac{t}{2\sqrt{\pi}}\int_{0}^{\infty}\frac{e^{-\frac{t^{2}}{4u}}}{u^{\frac{3}{2}}}W^{\alpha}_{u,\text{loc}}(f)(x)du,\quad
x\in\mathbb{R}_{+}^{n},t>0.$

and

$P^{\alpha}_{t,\text{glob}}(f)(x)=\frac{t}{2\sqrt{\pi}}\int_{0}^{\infty}\frac{e^{-\frac{t^{2}}{4u}}}{u^{\frac{3}{2}}}W^{\alpha}_{u,\text{glob}}(f)(x)du,\quad
x\in\mathbb{R}_{+}^{n},t>0.$

and consider

$g^{k}_{\alpha,\text{loc}}(f)(x)=\left(\int_{0}^{\infty}\left|t^{k}\partial^{k}_{t}P^{\alpha}_{t,\text{loc}}(f)(x)\right|^{2}\frac{dt}{t}\right)^{1/2},\quad
x\in\mathbb{R}_{+}^{n},$

and

$g^{k}_{\alpha,\text{glob}}(f)(x)=\left(\int_{0}^{\infty}\left|t^{k}\partial^{k}_{t}P^{\alpha}_{t,\text{glob}}(f)(x)\right|^{2}\frac{dt}{t}\right)^{1/2},\quad
x\in\mathbb{R}_{+}^{n}.$

We firstly prove that $g^{k}_{\alpha,\text{glob}}$ defines a bounded operator
on $L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$.

By using Minkowski inequality we get

$g^{k}_{\alpha,\text{glob}}(f)(x)\leq\int_{\mathbb{R}_{+}^{n}}|f(y)|\left(\int_{0}^{\infty}\left|t^{k}\partial^{k}_{t}P^{\alpha}_{t,\text{glob}}(x,y)\right|^{2}\frac{dt}{t}\right)^{1/2}d\mu_{\alpha}(y),$

for $x\in\mathbb{R}_{+}^{n}$, where

$P^{\alpha}_{t,\text{glob}}(x,y)=\frac{t}{2\sqrt{\pi}}\int_{0}^{\infty}\frac{e^{-\frac{t^{2}}{4u}}}{u^{\frac{3}{2}}}W^{\alpha}_{u,\text{glob}}(x,y)du\quad
x,y\in\mathbb{R}_{+}^{n},t>0.$

We have that

$\begin{split}t^{k}\partial^{k}_{t}P^{\alpha}_{t,\text{glob}}(x,y)&=t^{k}\partial^{k}_{t}\left[\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}\frac{e^{-v}}{\sqrt{v}}W^{\alpha}_{\frac{t^{2}}{4v},\text{glob}}(x,y)dv\right]\\\
&=t^{k}\partial^{k-1}_{t}\left[\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}\frac{e^{-v}}{\sqrt{v}}\partial_{t}W^{\alpha}_{\frac{t^{2}}{4v},\text{glob}}(x,y)dv\right]\\\
&=t^{k}\partial^{k-1}_{t}\left[\frac{t}{2\sqrt{\pi}}\int_{0}^{\infty}\frac{e^{-v}}{v^{3/2}}\left[\partial_{z}W^{\alpha}_{z,\text{glob}}(x,y)\right]_{z=\frac{t^{2}}{4v}}dv\right]\\\
&=t^{k}\partial^{k-1}_{t}\left[\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}\frac{e^{-\frac{t^{2}}{4z}}}{\sqrt{z}}\partial_{z}W^{\alpha}_{z,\text{glob}}(x,y)dz\right]\\\
&=\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}t^{k}\partial^{k-1}_{t}\left[e^{-\frac{t^{2}}{4z}}\right]\partial_{z}W^{\alpha}_{z,\text{glob}}(x,y)\frac{dz}{\sqrt{z}},\end{split}$

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

By using Minkowski inequality and [5, Lemma 3] we get

$\begin{split}\|t^{k}\partial^{k}_{t}P^{\alpha}_{t,\text{glob}}(x,y)\|&{}_{L^{2}\left(\mathbb{R}_{+},\frac{dt}{t}\right)}\\\
&\leq\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}|\partial_{z}W^{\alpha}_{z,\text{glob}}(x,y)|\left(\int_{0}^{\infty}\left|t^{k}\partial^{k-1}_{t}\left[e^{-\frac{t^{2}}{4z}}\right]\right|^{2}\frac{dt}{t}\right)^{\frac{1}{2}}\frac{dz}{\sqrt{z}}\\\
&\leq
C\int_{0}^{\infty}|\partial_{z}W^{\alpha}_{z,\text{glob}}(x,y)|\left(\int_{0}^{\infty}\frac{e^{-c\frac{t^{2}}{z}}}{z^{k-1}}t^{2k-1}dt\right)^{\frac{1}{2}}\frac{dz}{\sqrt{z}}\\\
&\leq C\int_{0}^{\infty}|\partial_{z}W^{\alpha}_{z,\text{glob}}(x,y)|dz\quad
x,\;y\in\mathbb{R}_{+}^{n}.\end{split}$

We recall that

$W^{\alpha}_{z,\text{glob}}(x,y)=\frac{1}{(1-e^{-z})^{\widehat{\alpha}+n}}\int_{(-1,1)^{n}}e^{-\frac{q_{-}(e^{-z/2}x,y,s)}{1-e^{-z}}+|y|^{2}}(1-\varphi(x,y,s))\Pi_{\alpha}(s)ds,$

for $x,y\in\mathbb{R}_{+}^{n}$ and $z>0$. Then,

$\partial_{z}W^{\alpha}_{z,\text{glob}}(x,y)=e^{|y|^{2}}\int_{(-1,1)^{n}}\partial_{z}\left[\frac{e^{-\frac{q_{-}(e^{-z/2}x,y,s)}{1-e^{-z}}}}{(1-e^{-z})^{\widehat{\alpha}+n}}\right](1-\varphi(x,y,s))\Pi_{\alpha}(s)ds,$

for $x,y\in\mathbb{R}_{+}^{n}$ and $z>0$.

We obtain

$\begin{split}&\left\|t^{k}\partial^{k}_{t}P^{\alpha}_{t,\text{glob}}(x,y)\right\|_{L^{2}\left(\mathbb{R}_{+},\frac{dt}{t}\right)}\\\
&\quad\leq
Ce^{|y|^{2}}\int_{(-1,1)^{n}}\int_{0}^{\infty}\left|\partial_{z}\left[\frac{e^{-\frac{q_{-}(e^{-z/2}x,y,s)}{1-e^{-z}}}}{(1-e^{-z})^{\widehat{\alpha}+n}}\right]\right|dz(1-\varphi(x,y,s))\Pi_{\alpha}(s)ds,\end{split}$

We have that

$\partial_{z}\left[\frac{e^{-\frac{q_{-}(e^{-z/2}x,y,s)}{1-e^{-z}}}}{(1-e^{-z})^{\widehat{\alpha}+n}}\right]=\frac{e^{-\frac{q_{-}(e^{-z/2}x,y,s)}{1-e^{-z}}}}{(1-e^{-z})^{\widehat{\alpha}+n}}P_{x,y,s}\left(e^{-z/2}\right),$

for $x,y\in\mathbb{R}_{+}^{n}$ and $s\in(-1,1)^{n}$, where, for every
$x,y\in\mathbb{R}_{+}^{n}$ and $s\in(-1,1)^{n}$, $P_{x,y,s}$ is a polynomial
whose degree is at most $4$. Then, for every $x,y\in\mathbb{R}_{+}^{n}$ and
$s\in(-1,1)$, the sign of $P_{x,y,s}$ changes at most four times. We obtain

$\int_{0}^{\infty}\left|\partial_{z}\left[\frac{e^{-\frac{q_{-}(e^{-z/2}x,y,s)}{1-e^{-z}}}}{(1-e^{-z})^{\widehat{\alpha}+n}}\right]\right|dz\leq
C\sup_{z\in\mathbb{R}_{+}}\frac{e^{-\frac{q_{-}(e^{-z/2}x,y,s)}{1-e^{-z}}}}{(1-e^{-z})^{\widehat{\alpha}+n}}=\sup_{0<t<1}\frac{e^{-\frac{q_{-}(\sqrt{1-t}x,y,s)}{t}}}{t^{n+\widehat{\alpha}}},$

for $x,y\in\mathbb{R}_{+}^{n}$ and $s\in(-1,1)^{n}$.

This estimate allows us to reduce the analysis of the global operator
$g^{k}_{\alpha,\text{glob}}$ to the operator considered when we studied the
operator $W^{\alpha}_{*,\text{glob}}$ in Section 4. Thus, we conclude that the
operator $g^{k}_{\alpha,\text{glob}}$ is bounded on
$L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$.

We now study the operator $g^{k}_{\alpha,\text{loc}}$. We will use vector
valued Calderón–Zygmund theory. In order to have the measurability of the
Banach valued functions that appear we are going to consider, for every
$N\in\mathbb{N}$, $N\geq 1$, the Banach space
$B_{N}=L^{2}\left((1/N,N),\frac{dt}{t}\right)$ and in the last step we pass to
the limit as $N$ goes to infinity instead of working with the Banach space
$L^{2}(\mathbb{R}_{+},\frac{dt}{t})$. Let $N\in\mathbb{N}$, $N\geq 1$. We
define the operator

$G^{k}_{\alpha,\text{loc}}(f)(x,t)=t^{k}\partial^{k}_{t}P^{\alpha}_{t,\text{loc}}(f)(x),\quad
x\in\mathbb{R}_{+},t>0.$

The integral kernel of $G^{k}_{\alpha,\text{loc}}$ with respect to
$d\mathfrak{m}_{\alpha}$ is the following

$M^{k}_{\alpha,\text{loc}}(x,y,t)=t^{k}\partial^{k}_{t}P^{\alpha}_{t,\text{loc}}(x,y)e^{-|y|^{2}},\quad
x,y\in\mathbb{R}_{+}^{n},t>0.$

Since the Poisson semigroup is a Stein symmetric diffusion semigroup, the
function $g^{k}_{\alpha}$ is bounded on
$L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$. In the first part of this proof we
establish that $g^{k}_{\alpha,\text{glob}}$ is bounded on
$L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$. Thus, there exists $C>0$ that does
not depend on $N$ such that

$\|G^{k}_{\alpha,\text{loc}}(f)\|_{L^{2}_{B_{N}}(\mathbb{R}_{+}^{n},\mu_{\alpha})}\leq
C\|f\|_{L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})}.$

By using Minkowski inequality, [5, Lemma 4] and [7, Lemma 3.1] (see also [21,
Proposition 6 and Lemma 7]) as above, we get

$\begin{split}\|M^{k}_{\alpha,\text{loc}}&(x,y,t)\|_{L^{2}\left(\mathbb{R}_{+}^{n},\frac{dt}{t}\right)}\\\
&\leq
C\int_{(-1,1)^{n}}|\varphi(x,y,s)\Pi_{\alpha}(s)\int_{0}^{\infty}\left|\partial_{z}\left[\frac{e^{-\frac{q_{-}(e^{-z/2}x,y,s)}{1-e^{-z}}}}{(1-e^{-z})^{\widehat{\alpha}+n}}\right]\right|dzds\\\
&\leq
C\int_{(-1,1)^{n}}|\varphi(x,y,s)\Pi_{\alpha}(s)\sup_{z\in\mathbb{R}_{+}}\frac{e^{-\frac{q_{-}(e^{-z/2}x,y,s)}{1-e^{-z}}}}{(1-e^{-z})^{\widehat{\alpha}+n}}dzds\\\
&\leq
C\int_{(-1,1)^{n}}\frac{\Pi_{\alpha}(s)}{q_{-}(x,y,s)^{\widehat{\alpha}+n}}ds\\\
&\leq\frac{C}{\mathfrak{m}_{\alpha}(B(x,|y-x|))},\quad
x,y\in\mathbb{R}_{+}^{n},\;x\neq y.\end{split}$

Let $j=1,\dots,n$. By proceeding in a similar way we can see that

$\displaystyle\|\partial_{x_{j}}M^{k}_{\alpha,\text{loc}}$
$\displaystyle(x,y,t)\|_{L^{2}\left(\mathbb{R}_{+}^{n},\frac{dt}{t}\right)}$
$\displaystyle\leq
C\int_{(-1,1)^{n}}|\varphi(x,y,s)\Pi_{\alpha}(s)\int_{0}^{\infty}\left|\partial_{z}\partial_{x_{j}}\left[\frac{e^{-\frac{q_{-}(e^{-z/2}x,y,s)}{1-e^{-z}}}}{(1-e^{-z})^{\widehat{\alpha}+n}}\right]\right|dzds$
$\displaystyle\quad+C\int_{(-1,1)^{n}}|\partial_{x_{j}}\varphi(x,y,s)\Pi_{\alpha}(s)\int_{0}^{\infty}\left|\partial_{z}\left[\frac{e^{-\frac{q_{-}(e^{-z/2}x,y,s)}{1-e^{-z}}}}{(1-e^{-z})^{\widehat{\alpha}+n}}\right]\right|dzds$
$\displaystyle\leq
C\int_{(-1,1)^{n}}|\varphi(x,y,s)\Pi_{\alpha}(s)\sup_{z\in\mathbb{R}_{+}}\left|\partial_{x_{j}}\left[\frac{e^{-\frac{q_{-}(e^{-z/2}x,y,s)}{1-e^{-z}}}}{(1-e^{-z})^{\widehat{\alpha}+n}}\right]\right|dzds$
$\displaystyle\quad+C\int_{(-1,1)^{n}}|\partial_{x_{j}}\varphi(x,y,s)\Pi_{\alpha}(s)\sup_{z\in\mathbb{R}_{+}}\left|\frac{e^{-\frac{q_{-}(e^{-z/2}x,y,s)}{1-e^{-z}}}}{(1-e^{-z})^{\widehat{\alpha}+n}}\right|dzds$
$\displaystyle\leq
C\int_{(-1,1)^{n}}\frac{\Pi_{\alpha}(s)}{q_{-}(x,y,s)^{\widehat{\alpha}+n+1/2}}ds$
$\displaystyle\leq\frac{C}{|x-y|\mathfrak{m}_{\alpha}(B(x,|y-x|))},\quad
x,y\in\mathbb{R}_{+}^{n},\;x\neq y.$

Hence,

(6.1)
$\|M^{k}_{\alpha,\text{loc}}(x,y)\|_{B_{N}}\leq\frac{C}{\mathfrak{m}_{\alpha}(B(x,|x-y|))}\quad
x,y\in\mathbb{R}_{+}^{n},\;x\neq y,$

and

$\begin{split}\sum_{j=1}^{n}\left(\|\partial_{x_{j}}M^{k}_{\alpha,\text{loc}}(x,y)\right.&\|_{B_{N}}+\left.\|\partial_{y_{j}}M^{k}_{\alpha,\text{loc}}(x,y)\|_{B_{N}}\right)\\\
&\leq\frac{C}{|x-y|\mathfrak{m}_{\alpha}(B(x,|x-y|))}\quad
x,y\in\mathbb{R}_{+}^{n},\;x\neq y,\end{split}$

where $C>0$ does not depend on $N$. Suppose that $h\in B_{N}$ and $g$ is a
smooth function with compact support in $\mathbb{R}_{+}^{n}$. By using (6.1)
we deduce that

$\begin{split}\int_{1/N}^{N}h(t)G^{k}_{\alpha,\text{loc}}(g)(x,t)\frac{dt}{t}&=\int_{\mathbb{R}_{+}^{n}}g(y)\int_{1/N}^{N}t^{k}\partial^{k}_{t}P^{\alpha}_{t,\text{loc}}(x,y)h(t)\frac{dt}{t}d\mathfrak{m}_{\alpha}(y)\\\
&=\int_{1/N}^{N}h(t)\left[\int_{\mathbb{R}_{+}^{n}}K(x,y)g(y)d\mathfrak{m}_{\alpha}(y)\right](t)\frac{dt}{t},\end{split}$

for $x\notin\operatorname{supp}(f)$, where, for every
$x,y\in\mathbb{R}_{+}^{n}$, $x\neq y$,

$[K(x,y)](t)=t^{k}\partial^{k}_{t}P^{\alpha}_{t,\text{loc}}(x,y),\quad\text{a.e.
}t\in(1/N,N),$

and the integral in the last line is understood in the $B_{N}$-Bochner sense.
Note that, for every $x\in\mathbb{R}_{+}^{n}$, the function $\Phi_{x}$ defined
by $\Phi_{x}(y)=K(x,y)g(y)$, $y\in\mathbb{R}_{+}^{n}$, is strongly measurable
from $\mathbb{R}_{+}^{n}$ into $B_{N}$. Indeed, let $x\in\mathbb{R}_{+}^{n}$.
Since $\Phi_{x}$ is continuous, $\Phi_{x}$ is weakly measurable. By taking
into account that $B_{N}$ is a separable Banach space, Petti’s Theorem ([44,
p. 131]) allows us to conclude that $\Phi_{x}$ is strongly measurable.

Thus, for every $x\notin\operatorname{supp}(f)$,

$G^{k}_{\alpha,\text{loc}}(f)(x,t)=\left[\int_{0}^{\infty}K(x,y)f(y)d\mathfrak{m}_{\alpha}(y)\right](t),$

in $L^{2}\left((1/N,N),\frac{dt}{t}\right)$.

The arguments explained in Section 2 allow us to conclude that there exists
$C>0$ such that, for every $N\in\mathbb{N}$, $N\geq 1$,

$\left\|\left\|G^{k}_{\alpha,\text{loc}}(f)\right\|_{L^{2}\left((1/N,N),\frac{dt}{t}\right)}\right\|_{L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})}\leq
C\left\|f\right\|_{L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})},$

for $f\in L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$,

By using the monotone convergence theorem (see [18, p. 75]) we get

$\left\|g^{k}_{\alpha,\text{loc}}(f)\right\|_{L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})}\leq
C\left\|f\right\|_{L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})},$

for $f\in L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$, and the proof of our
result is finished.

Let us consider now the Littlewood–Paley functions including also spatial
derivatives. For $\beta\in\mathbb{N}^{n}\setminus\\{(0,\dots,0)\\}$ and
$k\in\mathbb{N}$, we consider

$g^{\beta,k}_{\alpha}(f)(x)=\left(\int_{0}^{\infty}\left|t^{k+\widehat{\beta}}\partial^{k}_{t}D^{\beta}_{x}P_{t}^{\alpha}(f)(x)\right|^{2}\frac{dt}{t}\right)^{1/2},\;x\in\mathbb{R}_{+}^{n}.$

We define the local and global part of $g^{\beta,k}_{\alpha}$ as follows

$g^{\beta,k}_{\alpha,\text{loc}}(f)(x)=\left(\int_{0}^{\infty}\left|t^{k+\widehat{\beta}}\partial^{k}_{t}P^{\alpha,\beta}_{t,\text{loc}}(f)(x)\right|^{2}\frac{dt}{t}\right)^{1/2},\;x\in\mathbb{R}_{+}^{n}.$

and

$g^{\beta,k}_{\alpha,\text{glob}}(f)(x)=\left(\int_{0}^{\infty}\left|t^{k+\widehat{\beta}}\partial^{k}_{t}D^{\beta}_{x}P^{\alpha,\beta}_{t,\text{glob}}(f)(x)\right|^{2}\frac{dt}{t}\right)^{1/2},\;x\in\mathbb{R}_{+}^{n}.$

where

$P^{\alpha,\beta}_{t,\text{loc}}(f)(x)=\frac{t}{2\sqrt{\pi}}\int_{0}^{\infty}\frac{e^{-\frac{t^{2}}{4u}}}{u^{3/2}}W^{\alpha,\beta}_{u,\text{loc}}(f)(x)dx,\;x\in\mathbb{R}_{+}^{n},t>0,$

and

$P^{\alpha,\beta}_{t,\text{glob}}(f)(x)=\frac{t}{2\sqrt{\pi}}\int_{0}^{\infty}\frac{e^{-\frac{t^{2}}{4u}}}{u^{3/2}}W^{\alpha,\beta}_{u,\text{glob}}(f)(x)dx,\;x\in\mathbb{R}_{+}^{n},\
t>0.$

Here,

$W^{\alpha,\beta}_{u}(f)(x)=\int_{\mathbb{R}_{+}^{n}}D^{\beta}_{x}W^{\alpha}_{u}(x,t)f(y)d\mu_{\alpha}(y),\;x\in\mathbb{R}_{+}^{n},u>0,$

and $W^{\alpha,\beta}_{u,\text{loc}}$ and $W^{\alpha,\beta}_{u,\text{glob}}$
are defined in the usual way.

By using Minkowski inequality and [5, Lemma 4] we obtain

$\displaystyle g^{\beta,k}_{\alpha,\text{glob}}(f)(x)$ $\displaystyle\leq
C\int_{\mathbb{R}_{+}^{n}}|f(y)|(1-\varphi(x,y))$
$\displaystyle\quad\times\left(\int_{0}^{\infty}\left|t^{k+\widehat{\beta}}\partial^{k}_{t}\left[\int_{0}^{\infty}\frac{te^{-\frac{t^{2}}{4u}}}{u^{3/2}}D^{\beta}_{x}W^{\alpha}_{u}(x,y)du\right]\right|^{2}\frac{dt}{t}\right)^{1/2}d\mu_{\alpha}(y)$
$\displaystyle\leq C\int_{\mathbb{R}_{+}^{n}}|f(y)|(1-\varphi(x,y))$
$\displaystyle\quad\times\int_{0}^{\infty}\left(\int_{0}^{\infty}|t^{k+\widehat{\beta}}\partial^{k}_{t}(te^{-\frac{t^{2}}{4u}})|\frac{dt}{t}\right)^{1/2}|D^{\beta}_{x}W^{\alpha}_{u}(x,y)|\frac{du}{u^{\frac{3}{2}}}d\mu_{\alpha}(y)$
$\displaystyle\leq
C\int_{\mathbb{R}_{+}^{n}}|f(y)|(1-\varphi(x,y))\int_{0}^{\infty}u^{\widehat{\beta}/2-1}|D^{\beta}_{x}W_{u}(x,y)|dud\mu_{\alpha}(y),$

for $x\in\mathbb{R}_{+}^{n}$. From now on we follow the same steps we have
done for the higher order Riesz-Laguerre transforms restricted to the global
part in order to get the $L^{p(\cdot)}$-boundedness of this operator too,
taking into account the representation given in (1.4).

In order to study the local operator $g^{\beta,k}_{\alpha,\text{loc}}$ we use
the vector valued Calderón–Zygmund theory. We consider the operator
$G^{\beta,k}_{\alpha,\text{loc}}$ defined by

$G^{\beta,k}_{\alpha,\text{loc}}(f)(x,t)=t^{k+\widehat{\beta}}\partial^{k}_{t}P^{\alpha,\beta}_{t,\text{loc}}(f)(x),\;x\in\mathbb{R}_{+}^{n},t>0.$

The integral kernel $M^{\beta,k}_{\alpha,\text{loc}}$ of the above operator
with respect to $\mathfrak{m}_{\alpha}$ can be written as follows

$M^{\beta,k}_{\alpha,\text{loc}}(x,y,t)=\int_{(-1,1)^{n}}\varphi(x,y)M^{\beta,k}_{\alpha}(x,y,t,s)\Pi_{\alpha}(s)ds,\;x,\,y\in\mathbb{R}_{+}^{n},,t>0,$

where

$M^{\beta,k}_{\alpha}(x,y,t,s)=\frac{1}{2\sqrt{\pi}}\int_{0}^{\infty}\frac{t^{k+\widehat{\beta}}\partial^{k}_{t}\left[te^{-\frac{t^{2}}{4u}}\right]}{u^{3/2}(1-e^{-u})^{n+\widehat{\alpha}}}D^{\beta}_{x}\left[e^{-\frac{q_{-}(e^{-u/2}x,y,s)}{1-e^{-u+|y|^{2}}}}\right]du,$

for $x,y\in\mathbb{R}_{+}^{n}$, $t>0$ and $s\in(-1,1)^{n}$. By using Minkowski
inequality and [5, Lemma 4], according to [21, (2.3)], we deduce that, for
every $x,y\in\mathbb{R}_{+}^{n}$ and $s\in(-1,1)^{n}$,

$\begin{split}&\|M^{\beta,k}_{\alpha}(x,y,t,s)\|_{L^{2}\left(\mathbb{R}_{+},\frac{dt}{t}\right)}\\\
&\leq
C\int_{0}^{\infty}\frac{\left\|t^{k+\widehat{\beta}}\partial^{k}_{t}\left[te^{-\frac{t^{2}}{4u}}\right]\right\|_{L^{2}\left(\mathbb{R}_{+},\frac{dt}{t}\right)}}{u^{3/2}(1-e^{-u})^{n+\widehat{\alpha}}}\left|D^{\beta}_{x}\left[e^{-\frac{q_{-}(e^{-u/2}x,y,s)}{1-e^{-u+|y|^{2}}}}\right]\right|du\\\
&\leq
C\int_{0}^{\infty}\frac{u^{\widehat{\beta}/2-1}}{u^{3/2}(1-e^{-u})^{n+\widehat{\alpha}}}\left|D^{\beta}_{x}\left[e^{-\frac{q_{-}(e^{-u/2}x,y,s)}{1-e^{-u+|y|^{2}}}}\right]\right|du\\\
&\leq
C\int_{0}^{1}\sqrt{r}^{\widehat{\beta}-2}\left(-\frac{\log(r)}{1-r}\right)^{\frac{\widehat{\beta}-2}{2}}\prod_{i=1}^{n}\left|H_{\beta_{i}}\left(\frac{\sqrt{r}x_{i}-y_{i}s_{i}}{\sqrt{1-r}}\right)\right|\frac{e^{-\frac{q_{-}(rx,y,s)}{1-r}}}{(1-r)^{n+\widehat{\alpha}+1}}dr,\end{split}$

where we recall that, for every $j\in\mathbb{N}$, $H_{j}$ denotes the one-
dimensional Hermite polynomial of degree $j$.

As in the Riesz transform $R_{\alpha,\text{loc}}^{\beta}$ case (Section 5), we
obtain that

$\left\|M^{\beta,k}_{\alpha,\text{loc}}(x,y,\cdot)\right\|_{L^{2}\left(\mathbb{R}_{+},\frac{dt}{t}\right)}\leq\frac{C}{\mathfrak{m}_{\alpha}(B(x,|x-y|))},\;x,\,y\in\mathbb{R}_{+}^{n},\,x\neq
y.$

In a similar way we can see that

$\displaystyle\sum_{i=1}^{n}$
$\displaystyle\left(\left\|\partial_{x_{i}}M^{\beta,k}_{\alpha,\text{loc}}(x,y,\cdot)\right\|_{L^{2}\left(\mathbb{R}_{+},\frac{dt}{t}\right)}+\left\|\partial_{y_{i}}M^{\beta,k}_{\alpha,\text{loc}}(x,y,\cdot)\right\|_{L^{2}\left(\mathbb{R}_{+},\frac{dt}{t}\right)}\right)$
$\displaystyle\leq\frac{C}{|x-y|\mathfrak{m}_{\alpha}(B(x,|x-y|))},$

for $x,y\in\mathbb{R}_{+}^{n}$, $x\neq y$.

By proceeding as in the first part of the proof when $\beta=0$, we can prove
that the local operator $g^{\beta,k}_{\alpha,\text{loc}}$ is bounded on
$L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ when we show that
$g^{\beta,k}_{\alpha,\text{loc}}$ is bounded on
$L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$.

We are going to see that $g^{\beta,k}_{\alpha,\text{loc}}$ is bounded on
$L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ by proving that
$g^{\beta,k}_{\alpha}$ is bounded on $L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$.
Then, since we have proved that $g^{\beta,k}_{\alpha,\text{glob}}$ is bounded
on $L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$, we conclude that
$g^{\beta,k}_{\alpha,\text{loc}}$ is bounded on
$L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$.

According to [33, p. 699], and by performing a change of variables we obtain
that

$D^{\beta}_{x}\mathcal{L}^{\alpha}_{r}(x)=\sum_{(m,\ell)\in\mathcal{A}(\beta)}C^{\beta,\alpha}_{m,\ell}(r)\left(\prod_{i=1}^{n}x_{i}^{\beta_{i}-m_{i}}\right)\mathcal{L}^{\alpha+\beta-m}_{r-\beta+m+\ell}(x),$

for $x=(x_{1},\dots,x_{n})\in\mathbb{R}_{+}^{n}$ and $r\in\mathbb{N}^{n}$,
with

$\mathcal{A}(\beta)=\left\\{(m,\ell)\in\mathbb{N}\times\mathbb{N}^{n}:0\leq
m_{j}\leq\beta_{j},0\leq\ell_{j}\leq\frac{\beta_{j}-m_{j}}{2},j=1,\dots,n\right\\}.$

Furthermore, for every $(m,\ell)\in\mathcal{A}(\beta)$ and
$k\in\mathbb{N}^{n}$, $C^{\beta,\alpha}_{(m,\ell)}\in\mathbb{R}$ and

(6.2)
$\left|C^{\beta,\alpha}_{(m,\ell)}\right|\left\|\left(\prod_{i=1}^{n}x_{i}^{\beta_{i}-m_{i}}\right)\mathcal{L}^{\alpha+\beta-m}_{k-\beta+m+\ell}\right\|_{L^{2}(\mathbb{R}_{+},\mu_{\alpha})}\leq
C_{\beta}\lambda^{\widehat{\beta}/2}_{k}.$

Suppose that $f=\sum_{r\in\Lambda}w_{r}\mathcal{L}^{\alpha}_{r}$, where
$\Lambda$ is a finite subset of $\mathbb{N}^{n}$ and $w_{r}\in\mathbb{C}$ for
$r\in\Lambda$. Since for every $(m,\ell)\in\mathcal{A}(\beta)$, the system

$\left\\{\left(\prod_{i=1}^{n}x_{i}^{\beta_{i}-m_{i}}\right)\mathcal{L}^{\alpha+\beta-m}_{r-\beta+m+\ell}\right\\}_{k\in\Lambda_{m,\ell}}$

is orthogonal with respect to $\mu_{\alpha}$, where
$\Lambda_{m,\ell}=\\{k\in\mathbb{N}^{n}:k_{j}-\beta_{j}+m_{j}+\ell_{j}\geq
0,\,j=1,\dots,n\\}$, Bessel inequality leads, by using (6.2), to

$\begin{split}\|g^{\beta,k}_{\alpha}(f)\|^{2}_{L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})}&=\int_{0}^{\infty}t^{2(k+\widehat{\beta})-1}\int_{\mathbb{R}_{+}^{n}}\left|\sum_{r\in\Lambda}\lambda^{k/2}_{r}e^{-\sqrt{\lambda_{r}}t}c_{r}D^{\beta}_{x}\mathcal{L}^{\alpha}_{r}(x)\right|^{2}d\mu_{\alpha}(x)dt\\\
&\leq
C\int_{0}^{\infty}t^{2(k+\widehat{\beta})-1}\sum_{r\in\Lambda}|c_{r}|^{2}e^{-2\sqrt{\lambda_{r}}t}\lambda_{r}^{k+\widehat{\beta}}dt\\\
&\leq
C\sum_{r\in\Lambda}|c_{r}|^{2}=C\|f\|^{2}_{L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})}.\end{split}$

Suppose now that $f\in L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$. For every
$m\in\mathbb{N}$, we define

$f_{m}=\sum_{\gamma\in\mathbb{N}^{n},\widehat{\gamma}\leq
m}c^{\alpha}_{\gamma}(f)\mathcal{L}^{\alpha}_{\gamma}.$

We have that $f_{m}\rightarrow f$, as $m\rightarrow\infty$, in
$L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$.

It follows that

$\begin{split}|D^{\beta}_{x}W^{\alpha}_{t}(x,y)|&\leq\frac{C}{(1-e^{-t})^{n+\widehat{\alpha}}}\int_{(-1,1)^{n}}\left|\partial^{\beta}_{x}e^{-\frac{q_{-}(e^{-t/2}x,y,s)}{1-e^{-t}}+|y|^{2}}\right|\prod_{i=1}^{n}(1-s_{i}^{2})^{\alpha_{i}-1/2}ds\\\
&\leq C\frac{e^{-t/2}}{(1-e^{-t})^{r}}V(|x|,|y|),\quad
x,y\in\mathbb{R}_{+}^{n},t>0,\end{split}$

where $r\geq n+\widehat{\alpha}$ and $V$ is a polynomial with positive
coefficients. By using [5, Lemma 4] we get

$\begin{split}\left|\partial^{k}_{t}D_{x}^{\beta}P^{\alpha}_{t}(x,y)\right|&\leq
C\int_{0}^{\infty}\frac{\left|\partial^{k}_{t}\left[te^{-\frac{t^{2}}{4u}}\right]\right|}{u^{3/2}}|D^{\beta}_{x}W^{\alpha}_{u}(x,y)|du\\\
&\leq
C\int_{0}^{\infty}\frac{e^{-\frac{t^{2}}{8u}}e^{-u/2}}{u^{\frac{k+2}{2}}(1-e^{-u})^{r}}duV(|x|,|y|)\\\
&\leq
C\left(1+\int_{0}^{1}\frac{e^{-\frac{t^{2}}{8u}}}{u^{\frac{k+2}{2}+r}}du\right)V(|x|,|y|)\\\
&\leq C\left(1+t^{-k-1-2r}\right)V(|x|,|y|),\quad
x,y\in\mathbb{R}_{+}^{n},t>0.\end{split}$

Therefore,

$\begin{split}\left|\partial^{k}_{t}D_{x}^{\beta}P^{\alpha}_{t}(f_{m}-f)(x)\right|&\leq
C\left(1+t^{-k-1-2r}\right)\int_{\mathbb{R}^{n}}|f_{m}(y)-f(y)|V(|x|,|y|)d\mu_{\alpha}(y)\\\
&\leq
C\left(1+t^{-k-1-2r}\right)\left(\int_{\mathbb{R}^{n}}V^{2}(|x|,|y|)d\mu_{\alpha}(y)\right)^{1/2}\\\
&\quad\times\|f_{m}(y)-f(y)\|_{L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})},\end{split}$

for $x,y\in\mathbb{R}_{+}^{n}$, $t>0$ and $m\in\mathbb{N}$.

We deduce that

$\lim_{m\rightarrow\infty}t^{k+\widehat{\beta}}\partial^{k}_{t}D_{x}^{\beta}P^{\alpha}_{t}(f_{m})(x)=t^{k+\widehat{\beta}}\partial^{k}_{t}D_{x}^{\beta}P^{\alpha}_{t}(f)(x),$

for $x\in\mathbb{R}_{+}^{n}$ and $t>0$.

By using Fatou’s Lemma twice we get

$\begin{split}g^{\beta,k}_{\alpha}(f)(x)&=\left(\int_{0}^{\infty}\left|t^{k+\widehat{\beta}}\partial^{k}_{t}D_{x}^{\beta}P^{\alpha}_{t}(f)(x)\right|^{2}\frac{dt}{t}\right)^{1/2}\\\
&=\left(\int_{0}^{\infty}\lim_{m\rightarrow\infty}\left|t^{k+\widehat{\beta}}\partial^{k}_{t}D_{x}^{\beta}P^{\alpha}_{t}(f_{m})(x)\right|^{2}\frac{dt}{t}\right)^{1/2}\\\
&\leq\liminf_{m\rightarrow\infty}g^{\beta,k}_{\alpha}(f_{m})(x),\quad
x\in\mathbb{R}_{+}^{n},\end{split}$

and then

$\begin{split}\|g^{\beta,k}_{\alpha}(f)\|_{L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})}&\leq\left(\int_{\mathbb{R}_{+}^{n}}\liminf_{m\rightarrow\infty}|g^{\beta,k}_{\alpha}(f_{m})(x)|^{2}d\mu_{\alpha}(x)\right)^{1/2}\\\
&\leq\liminf_{m\rightarrow\infty}\|g^{\beta,k}_{\alpha}(f_{m})\|_{L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})}\\\
&\leq
C\lim_{m\rightarrow\infty}\|f_{m}\|_{L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})}\\\
&\leq C\|f\|_{L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})}.\end{split}$

Thus, we have proved that $g^{\beta,k}_{\alpha}$ is bounded on
$L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$.

##  7\. Proof of Theorem 1.1 for Laplace transform type multipliers

We recall that we have

$T^{\alpha}_{m}(f)(x)=\lim_{\epsilon\rightarrow
0^{+}}\left(f(x)\Lambda(\epsilon)+\int_{|x-y|>\epsilon}K^{\alpha}_{\phi}(x,y)f(y)d\mu_{\alpha}(y)\right),\;\;\text{a.e.
}x\in\mathbb{R}_{+}^{n},$

where $\Lambda\in L^{\infty}(\mathbb{R}_{+})$ and

$K^{\alpha}_{\phi}(x,y)=\int_{0}^{\infty}\phi(t)\left(-\frac{\partial}{\partial
t}\right)W^{\alpha}_{t}(x,y)dt,\quad x,y\in\mathbb{R}_{+}^{n},\;x\neq y,$

being $\phi\in L^{\infty}(\mathbb{R}_{+}^{n})$ and
$m(t)=t\int_{0}^{\infty}e^{-zt}\phi(z)dz$, $t\in\mathbb{R}_{+}$.

We define $T^{\alpha}_{m,\text{loc}}$, $K^{\alpha}_{\phi,\text{loc}}$,
$T^{\alpha}_{m,\text{glob}}$ and $K^{\alpha}_{\phi,\text{glob}}$ in the usual
way. We firstly observe that

$\begin{split}|K^{\alpha}_{\phi,\text{glob}}&(x,y)|\\\ &\leq
C\int_{(-1,1)^{n}}\int_{0}^{\infty}\left|\partial_{t}\left[\frac{e^{-\frac{q_{-}(e^{-t/2}x,y,s)}{1-e^{-t}}}}{(1-e^{-t})^{\widehat{\alpha}+n}}\right]\right||\phi(t)|dt|1-\varphi(x,y,s)|\Pi_{\alpha}(s)ds\\\
&\leq
C\int_{(-1,1)^{n}}\sup_{t>0}\left|\frac{e^{-\frac{q_{-}(e^{-t/2}x,y,s)}{1-e^{-t}}}}{(1-e^{-t})^{\widehat{\alpha}+n}}\right||1-\varphi(x,y,s)|\Pi_{\alpha}(s)ds.\end{split}$

By proceeding as in the proof of Section 4 we conclude that
$T^{\alpha}_{m,\text{glob}}$ is bounded on
$L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$.

We now define
$\mathbb{K}^{\alpha}_{\phi,\text{loc}}(x,y)=e^{-|y|^{2}}K^{\alpha}_{\phi,\text{loc}}(x,y)$
for $x,y\in\mathbb{R}_{+}^{n}$. By using [38, Lemma 1] and [7, Lemma 3.1] we
can see that $\mathbb{K}^{\alpha}_{\phi,\text{loc}}(x,y)$ is a scalar
Calderón–Zygmund kernel with respect to $\mathfrak{m}_{\alpha}$. According to
[42, Corollary 3, p. 121], the Laguerre multiplier $T^{\alpha}_{m}$ is bounded
on $L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$. Furthermore, as we have just
mentioned $T^{\alpha}_{m,\text{glob}}$ is bounded on
$L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$. Then, $T^{\alpha}_{m,\text{loc}}$ is
bounded on $L^{2}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ and on
$L^{2}(\mathbb{R}_{+}^{n},\mathfrak{m}_{\alpha})$.

As it was proved in Section 2, we can conclude that
$\mathbb{T}^{\alpha}_{m,\text{loc}}$ is bounded on
$L^{p(\cdot)}(\mathbb{R}_{+}^{n},\mu_{\alpha})$ and finish the proof of our
result.

##  Appendix A Auxiliary results

For the sake of completeness we include in this appendix an $n$-dimensional
version of the Three Lines Theorem in the form it was used in Section 3.
Although it can be seen as a particular case of [4, Proposition 21], we
believe that this simpler form might be enough in many circumstances.

######  Theorem A.1.

Let $n\in\mathbb{N}$, $n\geq 1$. Assume that, for every $j=1,\dots,n$,
$a_{j},\,b_{j}\in\mathbb{R}$ and $a_{j}<b_{j}$. We define
$\tau_{n}=\\{z\in\mathbb{C}^{n}:a_{j}\leq\operatorname{Re}(z_{j})\leq
b_{j},j=1,\dots,n\\}$ and
$\mathcal{F}_{n}=\\{z\in\mathbb{C}^{n}:\operatorname{Re}(z_{j})\in\\{a_{j},b_{j}\\},j=1,\dots,n\\}$.
Suppose that $U$ is an open set containing $\tau_{n}$ and $f:U\to\mathbb{C}$
is holomorphic, bounded in $\tau_{n}$, and such that $|f(z)|\leq K$ for
$z\in\mathcal{F}_{n}$. Then, $|f(z)|\leq K$ for $z\in\tau_{n}$.

###### Proof.

We will proceed by induction on the dimension $n$. The case $n=1$ corresponds
to the classical Three Lines Theorem and we refer to [34, Theorem 3.15].

Suppose the result is true for some $n\in\mathbb{N}$, $n\geq 1$. We consider
$a_{j}<b_{j}$ for $j=1,\dots,n+1$,
$\tau_{n+1}=\\{z=(z_{1},\dots,z_{n+1})\in\mathbb{C}^{n+1}:a_{j}\leq\operatorname{Re}(z_{j})\leq
b_{j},j=1,\dots,n+1\\}$,
$\mathcal{F}_{n+1}=\\{z=(z_{1},\dots,z_{n+1})\in\mathbb{C}^{n+1}:\operatorname{Re}(z_{j})\in\\{a_{j},b_{j}\\},j=1,\dots,n+1\\}$,
an open set $U$ containing $\tau_{n+1}$, and a function
$f:U\rightarrow\mathbb{C}$, holomorphic in $U$, bounded on $\tau_{n+1}$, and
such that $|f(z)|\leq K$ for $z\in\mathcal{F}_{n+1}$.

Let $t\in\mathbb{R}$. We define, $z_{n+1}(t)=a_{n+1}+it$, and
$g_{t}:U_{t}\rightarrow\mathbb{C}$ such that
$g_{t}(z_{1},\dots,z_{n})=f(z_{1},\dots,z_{n},z_{n+1}(t))$, where

$U_{t}=\\{z=(z_{1},\dots,z_{n})\in\mathbb{C}^{n}:\,(z_{1},\dots,z_{n},z_{n+1}(t))\in
U\\}.$

It is clear that $U_{t}$ is an open set in $\mathbb{C}^{n}$ that contains
$\tau_{n}$. The function $g_{t}$ is holomorphic in $U_{t}$ and bounded on
$\tau_{n}$, and if $z=(z_{1},\dots,z_{n})\in\mathcal{F}_{n}$, since
$\operatorname{Re}(z_{n+1})=a_{n+1}$,
$|g_{t}(z_{1},\dots,z_{n})|=|f(z_{1},\dots,z_{n},z_{n+1})|\leq K$. Then, using
the inductive hypothesis,

$|g_{t}(z_{1},\dots,z_{n})|=|f(z_{1},\dots,z_{n},z_{n+1}(t))|\leq K$

for $z=(z_{1},\dots,z_{n})\in\tau_{n}$. Thus, we prove that

(A.1) $|f(z_{1},\dots,z_{n+1})|\leq K\text{ if
}\operatorname{Re}(z_{j})\in[a_{j},b_{j}],\,j=1,\dots,n;\
\operatorname{Re}(z_{n+1})=a_{n+1}.$

In a similar way, we can see that

(A.2) $|f(z_{1},\dots,z_{n+1})|\leq K\text{ if
}\operatorname{Re}(z_{j})\in[a_{j},b_{j}],\,j=1,\dots,n;\
\operatorname{Re}(z_{n+1})=b_{n+1}.$

Let now $c=(c_{1},\dots,c_{n})\in\prod_{j=1}^{n}[a_{j},b_{j}]$ and
$t=(t_{1},\dots,t_{n})\in\mathbb{R}^{n}$. We consider
$\tau_{0}=\\{z\in\mathbb{C}:\operatorname{Re}(z)\in[a_{n+1},b_{n+1}]\\}$,
$\mathcal{F}_{0}=\\{z\in\mathbb{C}:\operatorname{Re}(z)\in\\{a_{n+1},b_{n+1}\\}\\}$,
and $h^{c}_{t}:U_{0}\rightarrow\mathbb{C}$ such that

$h^{c}_{t}(z)=f(c_{1}+it_{1},\dots,c_{n}+it_{n},z),\quad z\in U_{0},$

where $U_{0}=\\{z\in\mathbb{C}:\,(c_{1}+it_{1},\dots,c_{n}+it_{n},z)\in U\\}$.
The set $U_{0}$ is open in $\mathbb{C}$ and it contains $\tau_{0}$. The
function $h^{c}_{t}$ is holomorphic in $U_{0}$ and bounded on $\tau_{0}$.
Furthermore, by (A.1) and (A.2), if $z\in\mathcal{F}_{0}$,

$|h^{c}_{t}(z)|=|f(c_{1}+it_{1},\dots,c_{n}+it_{n},z)|\leq K.$

Therefore, by the one-dimensional case, we deduce that

$|h^{c}_{t}(z)|\leq K,\quad z\in\tau_{0}.$

Thus we conclude that

$|f(z)|\leq K,\quad z\in\tau_{n+1}.\qed$

######  Lemma A.2.

Let $p:\mathbb{R}_{+}^{n}\rightarrow[1,\infty)$ be a measurable function such
that $p\in\textup{LH}(\mathbb{R}_{+}^{n})$ and take
$k=(k_{1},\dots,k_{n})\in\mathbb N^{n}$ with $k_{j}\geq 1$ for each
$j=1,\dots,n$. Consider
$\overline{x}=(\overline{x_{1}},\dots,\overline{x_{n}})\in\mathbb
R^{\widehat{k}}$ with $\overline{x_{j}}\in\mathbb R^{k_{j}}$, $j=1,\dots,n$.
We define $\overline{p}:\mathbb R^{\widehat{k}}\rightarrow[1,\infty)$ by
$\overline{p}(\overline{x})=p(|\overline{x_{1}}|,\dots,|\overline{x_{n}}|)$.
Then, $\overline{p}\in\textup{LH}(\mathbb R^{\widehat{k}})$. Moreover, if
$1<p^{-}\leq p^{+}<\infty$, also
$1<\overline{p}^{-}\leq\overline{p}^{+}<\infty$.

###### Proof.

First, we shall see that $\overline{p}$ belongs to $\textup{LH}_{0}(\mathbb
R^{\widehat{k}})$, so we take
$\overline{x}=(\overline{x_{1}},\dots,\overline{x_{n}})$,
$\overline{y}=(\overline{y_{1}},\dots,\overline{y_{n}})\in\mathbb
R^{\widehat{k}}$, with $\overline{x_{j}},\overline{y_{j}}\in\mathbb
R^{k_{j}}$, $j=1,\dots,n$, and such that
$0<|\overline{x}-\overline{y}|<\frac{1}{2}$. We have that

$|(|\overline{x_{1}}|-|\overline{y_{1}}|,\dots,|\overline{x_{n}}|-|\overline{y_{n}}|)|\leq|\overline{x}-\overline{y}|.$

Indeed, if we write $\overline{x_{j}}=(x_{1}^{j},\dots,x_{{k_{j}}}^{j})$,
$\overline{y_{j}}=(y_{1}^{j},\dots,y_{{k_{j}}}^{j})$, with $j=1,\dots,n$, this
inequality is a consequence of the Cauchy–Schwarz inequality on
$\mathbb{R}^{k_{j}},$ i.e.
$|\langle\overline{x_{j}},\overline{y_{j}}\rangle|\leq|\overline{x_{j}}||\overline{y_{j}}|$,
$j=1,\dots,n$.

Since $p\in\textup{LH}_{0}(\mathbb{R}_{+}^{n})$ it follows that

$\begin{split}|\overline{p}(\overline{x})-\overline{p}(\overline{y})|&=|p(|\overline{x_{1}}|,\dots,|\overline{x_{n}}|)-p(|\overline{y_{1}}|,\dots,|\overline{y_{n}}|)|\\\
&\leq\frac{C}{-\log(|(|\overline{x_{1}}|-|\overline{y_{1}}|,\dots,|\overline{x_{n}}|-|\overline{y_{n}}|)|)}\\\
&\leq\frac{C}{-\log(|\overline{x}-\overline{y}|)}.\end{split}$

Thus, $\overline{p}\in\textup{LH}_{0}(\mathbb R^{\widehat{k}})$.

On the other hand, since $p\in\textup{LH}_{\infty}(\mathbb{R}_{+}^{n})$ and
$|(|\overline{x_{1}}|,\dots,|\overline{x_{n}}|)|=|\overline{x}|$,

$|\overline{p}(\overline{x})-p_{\infty}|=|p(|\overline{x_{1}}|,\dots,|\overline{x_{n}}|)-p_{\infty}|\leq\frac{C}{\log(e+|(|\overline{x_{1}}|,\dots,|\overline{x_{n}}|)|)}=\frac{C}{\log(e+|\overline{x}|)},$

so $\overline{p}\in\textup{LH}_{\infty}(\mathbb{R}^{\widehat{k}})$ with
$\overline{p}_{\infty}=p_{\infty}$.

Therefore, we have proved that $\overline{p}\in\textup{LH}(\mathbb
R^{\widehat{k}})$.

Finally, from the definition of $\overline{p}$, it is clear that
$\overline{p}^{-}=p^{-}$ and $\overline{p}^{+}=p^{+}$, so $1<p^{-}\leq
p^{+}<\infty$ is equivalent to
$1<\overline{p}^{-}\leq\overline{p}^{+}<\infty$. ∎

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