[2]

# BLO spaces associated with Laguerre polynomial expansions

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>and  Pablo Quijano

###### Abstract.

In this paper we introduce spaces of BLO-type related to Laguerre polynomial
expansions. We consider the probability measure on $(0,\infty)$ defined by
$d\gamma_{\alpha}(x)=\frac{2}{\Gamma(\alpha+1)}e^{-x^{2}}x^{2\alpha+1}dx$ with
$\alpha>-\frac{1}{2}$. For every $a>0$, the space
$\textup{BLO}_{a}((0,\infty),\gamma_{\alpha})$ consists of all those
measurable functions defined on $(0,\infty)$ having bounded lower oscillation
with respect to $\gamma_{\alpha}$ over an admissible family $\mathcal{B}_{a}$
of intervals in $(0,\infty)$. The space
$\textup{BLO}_{a}((0,\infty),\gamma_{\alpha})$ is a subspace of the space
$\textup{BMO}_{a}((0,\infty),\gamma_{\alpha})$ of bounded mean oscillation
functions with respect to $\gamma_{\alpha}$ and $\mathcal{B}_{a}$. The natural
$a$-local centered maximal function defined by $\gamma_{\alpha}$ is bounded
from $\textup{BMO}_{a}((0,\infty),\gamma_{\alpha})$ into
$\textup{BLO}_{a}((0,\infty),\gamma_{\alpha})$. We prove that the maximal
operator, the $\rho$-variation and the oscillation operators associated with
local truncations of the Riesz transforms in the Laguerre setting are bounded
from $L^{\infty}((0,\infty),\gamma_{\alpha})$ into
$\textup{BLO}_{a}((0,\infty),\gamma_{\alpha})$. Also, we obtain a similar
result for the maximal operator of local truncations for spectral Laplace
transform type multipliers.

###### Key words and phrases:

BLO spaces, variation operators, oscillation operators, Laguerre polynomials

###### 1991 Mathematics Subject Classification:

42B15, 42B20, 42B25, 42B35

The first author is partially supported by grant PID2019-106093GB-I00 from the
Spanish Government. The second author is partially supported by grants
PICT-2019-2019-00389 (ANPCyT), PIP-1220200101916O (CONICET) and CAI+D 2019-015
(UNL)

##  1\. Introduction

We consider, for every $\alpha>-\frac{1}{2}$, the probability measure defined
on $(0,\infty)$ by
$d\gamma_{\alpha}(x)=\frac{2}{\Gamma(\alpha+1)}e^{-x^{2}}x^{2\alpha+1}dx$.
This measure has not the doubling property with respect to the usual metric
defined by the absolute value $|\cdot|$ on $(0,\infty)$. Then, the triple
$((0,\infty),|\cdot|,\gamma_{\alpha})$ is not homogeneous in the sense of
Coifman and Weiss ([16]). Harmonic analysis in the spaces of homogeneous type
can be developed following the model of Euclidean spaces
$(\mathbb{R}^{n},\|\cdot\|,\lambda)$ where $\|\cdot\|$ denotes a norm and
$\lambda$ is the Lebesgue measure on $\mathbb{R}^{n}$. When the measure is not
doubling the situation is very different and it is necessary to introduce new
ideas (see, for instance, [13], [20], [21], [24], [25], [40], [49], [50] and
[51]).

Tolsa ([49]) defined BMO-type spaces, that he named RBMO-spaces, on
$(\mathbb{R}^{n},\mu)$ when $\mu$ is a Radon measure on $\mathbb{R}^{n}$,
which is not necessarily doubling, satisfying that $\mu(B(x,r))\leq Cr^{k}$,
$x\in\mathbb{R}^{k}$ and $r>0$, for some $k\in\\{1,\dots,n\\}$ and $C>0$. He
also proved that $\textup{RBMO}(\mathbb{R}^{n},\mu)$ has many of the
properties of the classical space $\textup{BMO}(\mathbb{R}^{n})$ of John and
Nirenberg. In particular, the integral operators defined by standard Calderón-
Zygmund kernels are bounded from $L^{\infty}(\mathbb{R}^{n},\mu)$ into
$\textup{RBMO}(\mathbb{R}^{n},\mu)$.

It is clear that, for every $0<r\leq x$, $\gamma_{\alpha}((x-r,x+r))\leq Cr$.
Then, following Tolsa’s ideas we can define the space
$\textup{RBMO}((0,\infty),\gamma_{\alpha})$ by replacing $\mathbb{R}^{n}$ by
$(0,\infty)$. However, $\textup{RBMO}((0,\infty),\gamma_{\alpha})$ is not
suitable to study harmonic analysis operators associated with Laguerre
polynomial expansions because these operators are not defined by standard
Calderón-Zygmund kernels ([19], [45] and [46]). Motivated by the results in
[37] in the Gaussian setting, the authors and R. Scotto ([5]) defined a local
BMO-type space related to the measure $\gamma_{\alpha}$ as follows.

We consider the function $m(x)=\min\\{1,1/x\\}$, $x\in(0,\infty)$. Given
$a>0$, we say that an interval $(x-r,x+r)$, with $0<r\leq x$, is
$a$-admissible, or is in the class $\mathcal{B}_{a}$, when $r\leq am(x)$. The
measure $\gamma_{\alpha}$ has the doubling property on $\mathcal{B}_{a}$, that
is, there exists $C>0$ such that, for every $0<r\leq x$ being $r\leq am(x)$,
we have that

$\gamma_{\alpha}(I(x,2r))\leq C\gamma_{\alpha}((x-r,x+r)),$

where $I(x,r):=(x-r,x+r)\cap(0,\infty)$ for $x,r>0$.

A function $f\in L^{1}((0,\infty),\gamma_{\alpha})$ is said to be in
$\textup{BMO}_{a}((0,\infty),\gamma_{\alpha})$ when

$\|f\|_{*,\alpha,a}:=\sup_{I\in\mathcal{B}_{a}}\frac{1}{\gamma_{\alpha}(I)}\int_{I}|f(y)-f_{I}|d\gamma_{\alpha}(y)<\infty,$

where $f_{I}=\fint_{I}f(y)d\gamma_{\alpha}(y)$, for every
$I\in\mathcal{B}_{a}$. For every
$f\in\textup{BMO}_{a}((0,\infty),\gamma_{\alpha})$, we define

$\|f\|_{\textup{BMO}_{a}((0,\infty),\gamma_{\alpha})}:=\|f\|_{L^{1}((0,\infty),\gamma_{\alpha})}+\|f\|_{*,\alpha,a}.$

The space $\textup{BMO}_{a}((0,\infty),\gamma_{\alpha})$ actually does not
depend on $a>0$. Then, in the sequel we will write
$\textup{BMO}((0,\infty),\gamma_{\alpha})$ and $\|\cdot\|_{*,\alpha}$ instead
of $\textup{BMO}_{a}((0,\infty),\gamma_{\alpha})$ and
$\|\cdot\|_{*,\alpha,a}$, respectively. This space can be identified with the
dual space of the Hardy space $H^{1}((0,\infty),\gamma_{\alpha})$ studied in
[5] (see [5, Theorem 1.1]).

The space $\textup{BLO}(\mathbb{R}^{n})$ of functions of bounded lower
oscillation on $\mathbb{R}^{n}$ was introduced by Coifman and Rochberg ([15]).
Later, Bennett ([1]) obtained a characterization of the functions in
$\textup{BLO}(\mathbb{R}^{n})$ by using the natural Hardy-Littlewood maximal
operators, and Leckband ([29]) proved that certain maximal operators
associated with singular integrals are bounded from $L^{p}(\mathbb{R}^{n})\cap
L^{\infty}(\mathbb{R}^{n})$ into $\textup{BLO}(\mathbb{R}^{n})$ for certain
$1\leq p<\infty$.

Based on Tolsa’s ideas, Jiang ([26]) introduced BLO-type spaces in
$(\mathbb{R}^{n},\mu)$ where $\mu$ is a positive non-doubling Radon measure
with polynomial growth. BLO-spaces in the Gaussian setting were defined by Liu
and Yang ([32]). In [25], Littlewood-Paley functions in non-doubling settings
on RBLO spaces were studied. Other results concerning RBLO spaces can be
encountered in [30] and [31]. As in happens with RBMO-spaces, RBLO-spaces for
$\gamma_{\alpha}$ do not work in a correct way in connection with harmonic
analysis operators associated to Laguerre polynomial expansions.

In this paper we introduce BLO-spaces associated with the measure
$\gamma_{\alpha}$ on $(0,\infty)$ by using admissible intervals.

Let $a>0$. We say that a function $f\in L^{1}((0,\infty),\gamma_{\alpha})$ is
in $\textup{BLO}_{a}((0,\infty),\gamma_{\alpha})$ when

$\sup_{I\in\mathcal{B}_{a}}\frac{1}{\gamma_{\alpha}(I)}\int_{I}\left(f(y)-\operatorname*{ess\
inf}_{z\in I}f(z)\right)d\gamma_{\alpha}(y)<\infty.$

For every $f\in\textup{BLO}_{a}((0,\infty),\gamma_{\alpha})$ we define

$\|f\|_{\textup{BLO}_{a}((0,\infty),\gamma_{\alpha})}:=\|f\|_{L^{1}((0,\infty),\gamma_{\alpha})}+\sup_{I\in\mathcal{B}_{a}}\frac{1}{\gamma_{\alpha}(I)}\int_{I}\left(f(y)-\operatorname*{ess\
inf}_{z\in I}f(z)\right)d\gamma_{\alpha}(y).$

It is not hard to see that

$L^{\infty}((0,\infty),\gamma_{\alpha})\subset\textup{BLO}_{a}((0,\infty),\gamma_{\alpha})\subset\textup{BMO}_{a}((0,\infty),\gamma_{\alpha}).$

The main properties of the space
$\textup{BLO}_{a}((0,\infty),\gamma_{\alpha})$ will be established in Section
2.

Our objective is to prove that maximal, variation and oscillation operators
defined by singular integrals in the Laguerre settings are bounded from
$L^{\infty}((0,\infty),\gamma_{\alpha})$ to
$\textup{BLO}_{a}((0,\infty),\gamma_{\alpha})$.

We now define the operators we are going to consider. Let
$\alpha>-\frac{1}{2}$. The Laguerre polynomial $L_{k}^{\alpha}$ of order
$\alpha$ and degree $k\in\mathbb{N}$ (see [28]) is

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

The Laguerre differential operator $\widetilde{\Delta_{\alpha}}$ is given by

$\widetilde{\Delta_{\alpha}}:=\frac{1}{2}\frac{d^{2}}{dx^{2}}+\left(\frac{2\alpha+1}{2x}-x\right)\frac{d}{dx}+\alpha+1,\quad
f\in C^{2}(0,\infty).$

We define, for every $k\in\mathbb{N}$,
$\mathcal{L}_{k}^{\alpha}(x):=L_{k}^{\alpha}(x^{2})$, $x\in(0,\infty)$. Then,
the sequence $\\{\mathcal{L}_{k}^{\alpha}\\}_{k\in\mathbb{N}}$ is an
orthonormal basis on $L^{2}((0,\infty),\gamma_{\alpha})$. For every
$k\in\mathbb{N}$, $\mathcal{L}_{k}^{\alpha}$ is an eigenfunction for
$\widetilde{\Delta_{\alpha}}$ associated with the eigenvalue
$\lambda_{k}^{\alpha}=2k+\alpha+1$.

For every $f\in L^{1}((0,\infty),\gamma_{\alpha})$, we define

$c_{k}^{\alpha}(f):=\int_{0}^{\infty}f(y)\mathcal{L}_{k}^{\alpha}(x)d\gamma_{\alpha}(x),\quad
k\in\mathbb{N}.$

We consider the operator $\Delta_{\alpha}$ given by

$\Delta_{\alpha}f=\sum_{k=0}^{\infty}\lambda_{k}^{\alpha}c_{k}^{\alpha}(f)\mathcal{L}_{k}^{\alpha},\quad
f\in D(\Delta_{\alpha}),$

being

$D(\Delta_{\alpha})=\left\\{f\in
L^{2}((0,\infty),\gamma_{\alpha}):\,\sum_{k=0}^{\infty}(\lambda_{k}^{\alpha}|c_{k}^{\alpha}(f)|)^{2}<\infty\right\\}.$

The space $C_{c}^{\infty}(0,\infty)$ of all the smooth functions with compact
support in $(0,\infty)$ is contained in $D(\Delta_{\alpha})$ and
$\Delta_{\alpha}f=\widetilde{\Delta_{\alpha}}f$, for any $f\in
C_{c}^{\infty}(0,\infty)$. The operator $\Delta_{\alpha}$ is self-adjoint and
positive in $L^{2}((0,\infty),\gamma_{\alpha})$. Furthermore, the operator
$-\Delta_{\alpha}$ generates a $C_{0}$-semigroup of operators
$\\{W_{t}^{\alpha}\\}_{t>0}$, where, for every $t>0$,

$W_{t}^{\alpha}(f)=\sum_{k=0}^{\infty}e^{-\lambda_{k}^{\alpha}t}c_{k}^{\alpha}(f)\mathcal{L}^{\alpha}_{k},\quad
f\in L^{2}((0,\infty),\gamma_{\alpha}).$

According to [28, (4.17.6)] we have that, for every $x,y,t\in(0,\infty)$,

$\displaystyle\sum_{k=0}^{\infty}e^{-kt}$
$\displaystyle\mathcal{L}_{k}^{\alpha}(x)\mathcal{L}_{k}^{\alpha}(y)$
$\displaystyle=\frac{\Gamma(\alpha+1)}{1-e^{-t}}(e^{-t/2}xy)^{-\alpha}I_{\alpha}\left(\frac{2e^{-t/2}xy}{1-e^{-t}}\right)\exp\left(-\frac{e^{-t}(x^{2}+y^{2})}{1-e^{-t}}\right),$
(1.1)

being $I_{\alpha}$ the modified Bessel function of the first kind and order
$\alpha$.

By using (1) we can write, for every $f\in L^{2}((0,\infty),\gamma_{\alpha})$
and $t>0$,

$W_{t}^{\alpha}(f)(x)=\int_{0}^{\infty}W_{t}^{\alpha}(x,y)f(y)d\gamma_{\alpha}(y),\quad
x\in(0,\infty),$ (1.2)

where

$W_{t}^{\alpha}(x,y)=\frac{\Gamma(\alpha+1)e^{-t(\alpha+1)}}{1-e^{-2t}}(e^{-t}xy)^{-\alpha}I_{\alpha}\left(\frac{2e^{-t}xy}{1-e^{-2t}}\right)\exp\left(-\frac{e^{-2t}(x^{2}+y^{2})}{1-e^{-2t}}\right),$

for $x,y,t\in(0,\infty).$

The integral in (1.2) is absolutely convergent for every $f\in
L^{p}((0,\infty),\gamma_{\alpha})$, ${1\leq p<\infty}$, and for every
$t,x\in(0,\infty)$. By defining $W_{t}^{\alpha}(f)$ by (1.2), for every $f\in
L^{p}((0,\infty),\gamma_{\alpha})$ and $t>0$, the family
$\\{W_{t}^{\alpha}\\}_{t>0}$ is a $C_{0}$-semigroup in
$L^{p}((0,\infty),\gamma_{\alpha})$, for every $1\leq p<\infty$. Thus
$\\{W_{t}^{\alpha}\\}_{t>0}$ is a symmetric diffusion semigroup in the sense
of Stein ([48]).

The study of harmonic analysis in Laguerre settings was begun by Muckenhoupt
([39]) who proved that the maximal operator $W_{*}^{\alpha}$ defined by

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

is bounded from $L^{1}((0,\infty),\gamma_{\alpha})$ into
$L^{1,\infty}((0,\infty,\gamma_{\alpha})$. This property was generalized by
Dinger ([18]) to higher dimensions.

We define the Riesz transform $R^{\alpha}$ associated with the Laguerre
operator $\Delta_{\alpha}$ by

$R^{\alpha}(f)=\sum_{k=1}^{\infty}\frac{1}{\sqrt{\lambda_{k}^{\alpha}}}c_{k}^{\alpha}(f)\frac{d}{dx}\mathcal{L}_{k}^{\alpha},\quad
f\in L^{2}((0,\infty),\gamma_{\alpha}).$

Thus $R^{\alpha}$ defines a bounded operator on
$L^{2}((0,\infty),\gamma_{\alpha})$ (see [42]). Furthermore, $R^{\alpha}$ can
be extended from $L^{2}((0,\infty),\gamma_{\alpha})\cap
L^{p}((0,\infty),\gamma_{\alpha})$ as a bounded operator on
$L^{p}((0,\infty),\gamma_{\alpha})$, for every $1<p<\infty$, and from
$L^{1}((0,\infty),\gamma_{\alpha})$ into
$L^{1,\infty}((0,\infty),\gamma_{\alpha})$ ([47]). The authors and R. Scotto
([6]) extended the above results by considering variable exponents
$L^{p(\cdot)}$-spaces. Also, in [5], endpoint estimates for Riesz transform
$R^{\alpha}$ were established proving that $R^{\alpha}$ defines a bounded
operator from $H^{1}((0,\infty),\gamma_{\alpha})$ into
$L^{1}((0,\infty),\gamma_{\alpha})$ and from
$L^{\infty}((0,\infty),\gamma_{\alpha})$ into
$\textup{BMO}((0,\infty),\gamma_{\alpha})$.

We can see that $R^{\alpha}$ is a principal value integral operator. By
proceeding as in the proof of [8, Theorem 1.1] we can see that, for every
$f\in L^{p}((0,\infty),\gamma_{\alpha})$, $1\leq p<\infty$,

$R^{\alpha}(f)(x)=\lim_{\varepsilon\to 0^{+}}\int_{|x-y|>\varepsilon,\
y\in(0,\infty)}R^{\alpha}(x,y)f(y)d\gamma_{\alpha}(y),\quad\text{a.e.
}x\in(0,\infty),$

where

$R^{\alpha}(x,y)=\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}\partial_{x}W_{t}^{\alpha}(x,y)\frac{dt}{\sqrt{t}},\quad
x,y\in(0,\infty),\ x\neq y.$

For every $\epsilon>0$, we define the $\epsilon$-truncation of the Riesz
transform $R^{\alpha}$ by

$R^{\alpha}_{\epsilon}(f)(x)=\int_{|x-y|>\varepsilon,\
y\in(0,\infty)}R^{\alpha}(x,y)f(y)d\gamma_{\alpha}(y),\quad x\in(0,\infty).$

The maximal Riesz transform $R^{\alpha}_{*}$ is defined by

$R^{\alpha}_{*}(f)=\sup_{\epsilon>0}|R^{\alpha}_{\epsilon}(f)|.$

From the results given by E. Sasso in [47] we can deduced that the maximal
operator $R^{\alpha}_{*}$ is bounded on $L^{p}((0,\infty),\gamma_{\alpha})$,
for every $1<p<\infty$, and from $L^{1}((0,\infty),\gamma_{\alpha})$ into
$L^{1,\infty}((0,\infty),\gamma_{\alpha})$.

We are going to consider the following local maximal Riesz transform
operators. For every $a>0$, we define the maximal operator $R^{\alpha}_{*,a}$
by

$R^{\alpha}_{*,a}(f)(x)=\sup_{0<\epsilon\leq
am(x)}|R^{\alpha}_{\epsilon}(f)(x)|,\quad x\in(0,\infty).$

Let $\rho>0$. If $\\{c_{t}\\}_{t>0}$ is a subset of complex numbers, we define
the $\rho$-variation $\mathcal{V}_{\rho}(\\{c_{t}\\}_{t>0})$ of
$\\{c_{t}\\}_{t>0}$ by

$\mathcal{V}_{\rho}(\\{c_{t}\\}_{t>0})=\sup_{0<t_{n}<t_{n-1}<\dots<t_{1},\
n\in\mathbb{N}}\left(\sum_{j=1}^{n-1}|c_{t_{j}}-c_{t_{j+1}}|^{\rho}\right)^{1/\rho}.$

If $\\{T_{t}\\}_{t>0}$ is a family of bounded operators in
$L^{p}((0,\infty),\gamma_{\alpha})$, with $1\leq p<\infty$, we define the
$\rho$-variation operator $\mathcal{V}_{\rho}(\\{T_{t}\\}_{t>0})$ of
$\\{T_{t}\\}_{t>0}$ by

$\mathcal{V}_{\rho}(\\{T_{t}\\}_{t>0})(f)(x)=\mathcal{V}_{\rho}(\\{T_{t}(f)(x)\\}_{t>0}).$

Since Bourgain ([10]) studied variational inequalities involving martingales
(see also [27]), $\rho$-variation operators has been extensively studied in
ergodic theory and harmonic analysis. Campbell, Jones, Reinhold and Wierdl
([11]) proved $L^{p}$-boundedness properties for $\rho$-variation operators
associated to the family of truncations for the Hilbert transform. In [12]
those results were extended by considering Riesz transforms in higher
dimensions. In order to obtain $L^{p}$-boundedness for $\rho$-variation
operators it is usual to ask for the condition $\rho>2$ (see [44]). For the
exponent $\rho=2$, oscillation operators are commonly considered.

Let $\\{t_{j}\\}_{j\in\mathbb{Z}}$ be an increasing sequence of positive real
numbers satisfying that $\lim_{j\to-\infty}t_{j}=0$ and
$\lim_{j\to+\infty}t_{j}=+\infty$. If $\\{c_{t}\\}_{t>0}$ is a set of complex
numbers, we define the oscillation with respect to
$\\{t_{j}\\}_{j\in\mathbb{Z}}$ by

$\mathcal{O}(\\{c_{t}\\}_{t>0},\\{t_{j}\\}_{j\in\mathbb{Z}})=\left(\sum_{j=-\infty}^{+\infty}\sup_{t_{j}\leq\epsilon_{j}<\epsilon_{j+1}<t_{j+1}}|c_{\epsilon_{j}}-c_{\epsilon_{j+1}}|^{2}\right)^{1/2}.$

If $\\{T_{t}\\}_{t>0}$ is a family of bounded operators in
$L^{p}((0,\infty),\gamma_{\alpha})$, with $1\leq p<\infty$, we define the
oscillation operator
$\mathcal{O}(\\{T_{t}\\}_{t>0},\\{t_{j}\\}_{j\in\mathbb{Z}})$ as follows

$\mathcal{O}(\\{T_{t}\\}_{t>0},\\{t_{j}\\}_{j\in\mathbb{Z}})(f)(x)=\mathcal{O}(\\{T_{t}(f)(x)\\}_{t>0},\\{t_{j}\\}_{j\in\mathbb{Z}}).$

$L^{p}$-boundedness properties of the oscillation operators defined by the
family of truncations of Hilbert transform and Euclidean Riesz transforms were
established in [11] and [12], respectively.

After [11] and [12], the study of $\rho$-variation and oscillation operators
defined by singular integrals has been an active working area (see, for
instance, [3], [14], [17], [22], [33], [34], [35], [36] and [38]). Variation
and oscillation operators give information about convergence properties for
the family $\\{T_{t}\\}_{t>0}$.

Being $\\{T_{t}\\}_{t>0}$ and $\\{t_{j}\\}_{j\in\mathbb{Z}}$ as above, we are
going to consider the local $\rho$-variation and oscillation operators defined
as follows. Let $a>0$. The $a$-local $\rho$-variation operator
$\mathcal{V}_{\rho,a}(\\{T_{t}\\}_{t>0})$ is given by

$\displaystyle\mathcal{V}_{\rho,a}$ $\displaystyle(\\{T_{t}\\}_{t>0})(f)(x)$
$\displaystyle=\sup_{0<t_{n}<t_{n-1}<\dots<t_{1}\leq am(x),\
n\in\mathbb{N}}\left(\sum_{j=1}^{n-1}|T_{t_{j}}(f)(x)-T_{t_{j+1}}(f)(x)|^{\rho}\right)^{1/\rho}.$

The $a$-local oscillation operator
$\mathcal{O}_{a}(\\{T_{t}\\}_{t>0},\\{t_{j}\\}_{j\in\mathbb{Z}})$ is defined
by

$\displaystyle\mathcal{O}_{a}$
$\displaystyle(\\{T_{t}\\}_{t>0},\\{t_{j}\\}_{j\in\mathbb{Z}})(f)(x)$
$\displaystyle=\left(\sum_{j\in\mathbb{Z},\,\,t_{j}\leq
am(x)}\sup_{t_{j}\leq\epsilon_{j}<\epsilon_{j+1}<t_{j+1}}|T_{\epsilon_{j}}(f)(x)-T_{\epsilon_{j+1}}(f)(x)|^{2}\right)^{1/2}.$

Our first result is the following.

######  Theorem 1.1.

Let $\alpha>-\frac{1}{2}$, $a>0$ and $\rho>2$. Suppose that
$\\{t_{j}\\}_{j\in\mathbb{Z}}$ is an increasing sequence of positive real
numbers such that $t_{j+1}\leq\theta t_{j}$, $j\in\mathbb{Z}$, for some
$\theta>1$, $\lim_{j\to-\infty}t_{j}=0$ and $\lim_{j\to+\infty}t_{j}=+\infty$.
The operators $R^{\alpha}_{*,a}$,
$\mathcal{V}_{\rho,a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0})$, and
$\mathcal{O}_{a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0},\\{t_{j}\\}_{j\in\mathbb{Z}})$
are bounded from $L^{\infty}((0,\infty),\gamma_{\alpha})$ into
$\textup{BLO}_{a}((0,\infty),\gamma_{\alpha})$.

We shall now introduce multiplier operators in the Laguerre setting. A
measurable complex function $M$ defined on $[0,\infty)$ is said to be of
Laplace transform type when

$M(x)=x\int_{0}^{\infty}\phi(t)e^{-xt}dt,\quad x>0,$

where $\phi\in L^{\infty}(0,\infty)$.

Suppose that $M$ is of Laplace transform type. We denote by $T_{M}^{\alpha}$
the spectral multiplier for the Laguerre operator $\Delta_{\alpha}$ defined by
$M-M(0)$. For every ${f\in L^{2}((0,\infty),\gamma_{\alpha})}$,
$T_{M}^{\alpha}(f)$ is given by

$T_{M}^{\alpha}(f)=\sum_{k=1}^{\infty}M(k)c_{k}^{\alpha}(f)\mathcal{L}_{k}^{\alpha}.$

Since $M$ is bounded on $(0,\infty)$, $T_{M}^{\alpha}$ is bounded on
$L^{2}((0,\infty),\gamma_{\alpha})$. Since $\\{W_{t}^{\alpha}\\}_{t>0}$ is a
symmetric diffusion semigroup, $T_{M}^{\alpha}$ is bounded on
$L^{p}((0,\infty),\gamma_{\alpha})$, for every ${1<p<\infty}$ ([48, Corollary
3, p. 121]). The authors and R. Scotto ([6, Theorem 1.1 (d)]) extended the
last result establishing variable $L^{p(\cdot)}$-boundedness properties for
$T_{M}^{\alpha}$. On the other hand, Sasso ([45]) proved that $T_{M}^{\alpha}$
defines a bounded operator from $L^{1}((0,\infty),\gamma_{\alpha})$ into
$L^{1,\infty}((0,\infty),\gamma_{\alpha})$. In [5], the authors with R. Scotto
established the endpoint estimate for $T_{M}^{\alpha}$ from
$L^{\infty}((0,\infty),\gamma_{\alpha})$ into
$\textup{BMO}((0,\infty),\gamma_{\alpha})$.

From [8, Theorem 1.1] we deduce that there exists a function $\Lambda\in
L^{\infty}(0,\infty)$ such that, for every $f\in
L^{p}((0,\infty),\gamma_{\alpha})$, $1\leq p<\infty$,

$T_{M}^{\alpha}(f)(x)=\lim_{\varepsilon\to
0^{+}}\left(\Lambda(\varepsilon)f(x)+\int_{|x-y|>\varepsilon,\
y\in(0,\infty)}K_{\phi}^{\alpha}(x,y)f(y)d\gamma_{\alpha}(y)\right),$

for a.e. $x\in(0,\infty)$, where

$K_{\phi}^{\alpha}(x,y)=-\int_{0}^{\infty}\phi(t)\partial_{t}W_{t}^{\alpha}(x,y)dt,\quad
x,y\in(0,\infty),\ x\neq y.$

A special case of $T_{M}^{\alpha}$ is the imaginary power
$\Delta_{\alpha}^{i\alpha}$ that appears when $M_{\eta}(x)=x^{i\eta}$ for
$x\in(0,\infty)$ and $\eta\in\mathbb{R}\setminus\\{0\\}$. For these values of
$\eta$,

$M_{\eta}(x)=x\int_{0}^{\infty}\phi_{\eta}(t)e^{-xt}dt,\quad x\in(0,\infty),$

where $\phi_{\eta}(t)=\frac{t^{-i\eta}}{\Gamma(1+i\eta)}$, $t>0$. Note that
$|\phi^{\prime}_{\eta}(t)|\leq C/t$, $t\in(0,\infty)$.

We define, for every $\epsilon>0$, the truncations

$Q_{\phi,\epsilon}^{\alpha}(f)(x)=\int_{|x-y|>\epsilon,\
y\in(0,\infty)}K_{\phi}^{\alpha}(x,y)f(y)d\gamma_{\alpha}(y),\quad
x\in(0,\infty),$

and consider, for every $a>0$, the $a$-local maximal operator
$Q^{\alpha}_{\phi,*,a}$, which is given by

$Q^{\alpha}_{\phi,*,a}(f)(x)=\sup_{0<\epsilon\leq
am(x)}|Q_{\phi,\epsilon}^{\alpha}(f)(x)|.$

######  Theorem 1.2.

Let $\alpha>-\frac{1}{2}$ and $a>0$. The maximal operator
$Q^{\alpha}_{\phi,*,a}$ is bounded from
$L^{\infty}((0,\infty),\gamma_{\alpha})$ into
$\textup{BLO}_{a}((0,\infty),\gamma_{\alpha})$ provided that
$|\phi^{\prime}(t)|\leq C/t$ for some $C>0$, and each $t\in(0,\infty)$.

The paper is organized as follows. In Section 2 we state the main properties
for the spaces $\textup{BLO}_{a}((0,\infty),\gamma_{\alpha})$. In the
subsequent sections we prove Theorems 1.1 and 1.2.

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

##  2\. The spaces $\textup{BLO}_{a}((0,\infty),\gamma_{\alpha})$

In this section we state the main properties of the spaces
$\textup{BLO}_{a}((0,\infty),\gamma_{\alpha})$. This properties will be useful
in the following sections and they can be proved as the corresponding
properties for the Gaussian $\textup{BLO}_{a}$ space given in [32, Theorem
3.1, Proposition 3.1 and Theorem 3.2] (see also [1] for the Euclidean case and
[23] for the non-doubling measure case).

Let $a>0$. The local natural maximal operator $\mathcal{M}^{\alpha}_{a}$
associated with the measure $\gamma_{\alpha}$ on $(0,\infty)$ is defined by

$\mathcal{M}^{\alpha}_{a}(f)(x)=\sup_{I\in\mathcal{B}_{a}(x)}\frac{1}{\gamma_{\alpha}(I)}\int_{I}f(y)d\gamma_{\alpha}(y),\;x\in(0,\infty),$

for every measurable function $f$ on $(0,\infty)$ such that
$\int_{0}^{\delta}|f(y)|d\gamma_{\alpha}(y)<\infty$, $\delta>0$.

######  Proposition 2.1.

Let $a>0$. There exists $C>0$ such that for every $I\in\mathcal{B}_{a}$ and
every measurable function $f$ on $(0,\infty)$ such that
$\|f\|_{*,\alpha}<\infty$,

$\frac{1}{\gamma_{\alpha}(I)}\int_{I}\mathcal{M}^{\alpha}_{a}(f)(y)d\gamma_{\alpha}(y)\leq
C\|f\|_{*,\alpha}+\operatorname*{ess\ inf}_{x\in
I}\mathcal{M}^{\alpha}_{a}(f)(x).$

Furthermore, the natural maximal operator $\mathcal{M}^{\alpha}_{a}$ defines a
bounded operator from $\textup{BMO}((0,\infty),\gamma_{\alpha})$ into
$\textup{BLO}_{a}((0,\infty),\gamma_{\alpha})$.

The space $\textup{BLO}_{a}((0,\infty),\gamma_{\alpha})$ can be characterized
by using the local natural maximal operator.

######  Proposition 2.2.

Let $a>0$. A measurable function $f$ belongs to
$\textup{BLO}_{a}((0,\infty),\gamma_{\alpha})$ if and only if $f\in
L^{1}((0,\infty),\gamma_{\alpha})$ and $\mathcal{M}^{\alpha}_{a}(f)-f\in
L^{\infty}((0,\infty),\gamma_{\alpha})$. In addition, we have that

$\|\mathcal{M}^{\alpha}_{a}(f)-f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}=\sup_{I\in\mathcal{B}_{a}}\left(\frac{1}{\gamma_{\alpha}(I)}\int_{I}f(y)d\gamma_{\alpha}(y)-\operatorname*{ess\
inf}_{x\in I}f(x)\right).$

By combining Proposition 2.1 and Proposition 2.2 we can establish the
following characterization of $\textup{BLO}_{a}((0,\infty),\gamma_{\alpha})$
involving the space $\textup{BMO}((0,\infty),\gamma_{\alpha})$ and the local
natural maximal operator.

######  Proposition 2.3.

Let $a>0$. A measurable function $f$ belongs to
$\textup{BLO}_{a}((0,\infty),\gamma_{\alpha})$ if and only if
$f=\mathcal{M}^{\alpha}_{a}(g)+h$, where
$g\in\textup{BMO}((0,\infty),\gamma_{\alpha})$ and $h\in
L^{\infty}((0,\infty),\gamma_{\alpha})$. Furthermore,

$\|f\|_{\textup{BLO}_{a}((0,\infty),\gamma_{\alpha})}\sim\inf\\{\|g\|_{\textup{BMO}((0,\infty),\gamma_{\alpha})}+\|h\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}\\},$

where the infimum is taken over all the pairs $(g,h)$ for which
$f=\mathcal{M}^{\alpha}_{a}(g)+h$ with
$(g,h)\in\textup{BMO}((0,\infty),\gamma_{\alpha})\times
L^{\infty}((0,\infty),\gamma_{\alpha})$.

##  3\. Proof of Theorem 1.1

###  3.1. Local variation operators

Let $f\in L^{\infty}((0,\infty),\gamma_{\alpha})$. Since the variation
operator $\mathcal{V}_{\rho}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0})$ is
bounded on $L^{2}((0,\infty),\gamma_{\alpha})$ (see [9, Theorem 1.3]) it
follows that

$\displaystyle\int_{0}^{\infty}\mathcal{V}_{\rho,a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0})(f)(x)d\gamma_{\alpha}(x)$
$\displaystyle\leq\left(\int_{0}^{\infty}\left(\mathcal{V}_{\rho}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0})(f)(x)\right)^{2}d\gamma_{\alpha}(x)\right)^{1/2}$
$\displaystyle\leq
C\left(\int_{0}^{\infty}|f(x)|^{2}d\gamma_{\alpha}(x)\right)^{1/2}$
$\displaystyle\leq C\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}.$

According to Proposition 2.2 the proof will be finished when we see that

$\|\mathcal{M}^{\alpha}_{a}(\mathcal{V}_{\rho,a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0})(f))-\mathcal{V}_{\rho,a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0})(f)\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}\leq
C\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}.$

Notice that

$\displaystyle 0$
$\displaystyle\leq\mathcal{M}^{\alpha}_{a}(\mathcal{V}_{\rho,a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0})(f))(x)-\mathcal{V}_{\rho,a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0})(f)(x)$
$\displaystyle=\sup_{I\in\mathcal{B}_{a}(x)}\frac{1}{\gamma_{\alpha}(I)}\int_{I}\mathcal{V}_{\rho,a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0})(f)(z)d\gamma_{\alpha}(z)-\mathcal{V}_{\rho,a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0})(f)(x),$

for almost every $x\in(0,\infty)$, where $I\in\mathcal{B}_{a}(x)$ indicates
that $I\in\mathcal{B}_{a}$ and $x\in I$.

Let $x$, $x_{0}$, $r_{0}\in(0,\infty)$ such that
$I=I(x_{0},r_{0})\in\mathcal{B}_{a}(x)$. We decompose $f$ as follows

$f=f\chi_{4I}+f\chi_{(0,\infty)\setminus 4I}=f_{1}+f_{2}.$

We can write

$\displaystyle\frac{1}{\gamma_{\alpha}(I)}\int_{I}$
$\displaystyle\mathcal{V}_{\rho,a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0})(f)(z)d\gamma_{\alpha}(z)-\mathcal{V}_{\rho,a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0})(f)(x)$
$\displaystyle\leq\frac{1}{\gamma_{\alpha}(I)}\int_{I}\mathcal{V}_{\rho,a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0})(f_{1})(z)d\gamma_{\alpha}(z)$
$\displaystyle\quad+\frac{1}{\gamma_{\alpha}(I)}\int_{I}\left(\mathcal{V}_{\rho,a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0})(f_{2})(z)-\mathcal{V}_{\rho,a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0})(f_{2})(x)\right)d\gamma_{\alpha}(z)$
$\displaystyle\quad+\mathcal{V}_{\rho,a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0})(f_{2})(x)-\mathcal{V}_{\rho,a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0})(f)(x)$
$\displaystyle:=J_{1}+J_{2}+J_{3}.$

By using again that the variation
$\mathcal{V}_{\rho}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0})$ is bounded on
$L^{2}((0,\infty),\gamma_{\alpha})$ we get

$\displaystyle J_{1}$
$\displaystyle\leq\left(\frac{1}{\gamma_{\alpha}(I)}\int_{I}\left(\mathcal{V}_{\rho}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0})(f_{1})(z)\right)^{2}d\gamma_{\alpha}(z)\right)^{1/2}$
$\displaystyle\leq
C\left(\frac{1}{\gamma_{\alpha}(I)}\int_{I}\left|f(z)\right|^{2}d\gamma_{\alpha}(z)\right)^{1/2}\leq
C\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}.$ (3.1)

Suppose there exists $i_{0}\in\\{1,\dots,n-1\\}$ such that
$\epsilon_{i_{0}+1}\leq am(x)<\epsilon_{i_{0}}$. Thus, for $z\in I$,

$\displaystyle\left(\sum_{j=1}^{n-1}|R^{\alpha}_{\epsilon_{j+1}}(f_{2})(z)-R^{\alpha}_{\epsilon_{j}}(f_{2})(z)|^{\rho}\right)^{1/\rho}$
$\displaystyle\leq\left(\sum_{j=1}^{i_{0}-1}|R^{\alpha}_{\epsilon_{j+1}}(f_{2})(z)-R^{\alpha}_{\epsilon_{j}}(f_{2})(z)|^{\rho}+|R^{\alpha}_{\epsilon_{i_{0}}}(f_{2})(z)-R^{\alpha}_{am(x)}(f_{2})(z)|^{\rho}\right)^{1/\rho}$
$\displaystyle\quad+\left(\sum_{j=i_{0}+1}^{n-1}|R^{\alpha}_{\epsilon_{j+1}}(f_{2})(z)-R^{\alpha}_{\epsilon_{j}}(f_{2})(z)|^{\rho}+|R^{\alpha}_{\epsilon_{i_{0}+1}}(f_{2})(z)-R^{\alpha}_{am(x)}(f_{2})(z)|^{\rho}\right)^{1/\rho}.$

Then, recalling that $m(z)\leq Cm(x)$ for every $x,z\in I$, where $C>1$, we
obtain

$\displaystyle\mathcal{V}_{\rho,a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0})(f_{2})(z)-\mathcal{V}_{\rho,a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0})(f_{2})(x)$
$\displaystyle\leq\sup_{\begin{subarray}{c}0<\epsilon_{n}<\dots<\epsilon_{1}\leq
am(x)\\\
n\in\mathbb{N}\end{subarray}}\left(\sum_{j=1}^{n-1}|R^{\alpha}_{\epsilon_{j+1}}(f_{2})(z)-R^{\alpha}_{\epsilon_{j}}(f_{2})(z)|^{\rho}\right)^{1/\rho}$
$\displaystyle\quad+\sup_{\begin{subarray}{c}am(x)\leq\epsilon_{n}<\dots<\epsilon_{1}<Cam(x)\\\
n\in\mathbb{N}\end{subarray}}\left(\sum_{j=1}^{n-1}|R^{\alpha}_{\epsilon_{j+1}}(f_{2})(z)-R^{\alpha}_{\epsilon_{j}}(f_{2})(z)|^{\rho}\right)^{1/\rho}$
$\displaystyle\quad-\sup_{\begin{subarray}{c}0<\epsilon_{n}<\dots<\epsilon_{1}\leq
am(x)\\\
n\in\mathbb{N}\end{subarray}}\left(\sum_{j=1}^{n-1}|R^{\alpha}_{\epsilon_{j+1}}(f_{2})(x)-R^{\alpha}_{\epsilon_{j}}(f_{2})(x)|^{\rho}\right)^{1/\rho}$
$\displaystyle\leq\sup_{\begin{subarray}{c}am(x)\leq\epsilon_{n}<\dots<\epsilon_{1}<Cam(x)\\\
n\in\mathbb{N}\end{subarray}}\sum_{j=1}^{n-1}|R^{\alpha}_{\epsilon_{j+1}}(f_{2})(z)-R^{\alpha}_{\epsilon_{j}}(f_{2})(z)|$
$\displaystyle\quad+\sup_{\begin{subarray}{c}0<\epsilon_{n}<\dots<\epsilon_{1}\leq
am(x)\\\
n\in\mathbb{N}\end{subarray}}\inf_{\begin{subarray}{c}0<\delta_{k}<\dots<\delta_{1}\leq
am(x)\\\
k\in\mathbb{N}\end{subarray}}\left[\left(\sum_{j=1}^{n-1}|R^{\alpha}_{\epsilon_{j+1}}(f_{2})(z)-R^{\alpha}_{\epsilon_{j}}(f_{2})(z)|^{\rho}\right)^{1/\rho}\right.$
$\displaystyle\qquad-\left.\left(\sum_{j=1}^{k-1}|R^{\alpha}_{\delta_{j+1}}(f_{2})(x)-R^{\alpha}_{\delta_{j}}(f_{2})(x)|^{\rho}\right)^{1/\rho}\right]$
$\displaystyle\leq\int_{am(x)<|z-y|<Cam(x)}|R^{\alpha}(z,y)||f_{2}(y)|d\gamma_{\alpha}(y)$
$\displaystyle\quad+\sup_{\begin{subarray}{c}0<\epsilon_{n}<\dots<\epsilon_{1}\leq
am(x)\\\
n\in\mathbb{N}\end{subarray}}\inf_{\begin{subarray}{c}0<\delta_{k}<\dots<\delta_{1}\leq
am(x)\\\
k\in\mathbb{N}\end{subarray}}\left[\left(\sum_{j=1}^{n-1}|R^{\alpha}_{\epsilon_{j+1}}(f_{2})(z)-R^{\alpha}_{\epsilon_{j}}(f_{2})(z)|^{\rho}\right)^{1/\rho}\right.$
$\displaystyle\qquad-\left.\left(\sum_{j=1}^{k-1}|R^{\alpha}_{\delta_{j+1}}(f_{2})(x)-R^{\alpha}_{\delta_{j}}(f_{2})(x)|^{\rho}\right)^{1/\rho}\right]$
$\displaystyle:=J_{2,1}(x,z)+J_{2,2}(x,z).$

If we write

$R^{\alpha}(z,y)=e^{\frac{z^{2}+y^{2}}{2}}\mathfrak{R}^{\alpha}(z,y),\quad
z,y\in(0,\infty),\,z\neq y,$

from [43, (3.3) and Proposition 3.1] we know that

$\mathfrak{R}^{\alpha}(z,y)\leq\frac{C}{\mathfrak{m}_{\alpha}(I(z,|z-y|))},\quad
z,y\in(0,\infty),\,z\neq y,$ (3.2)

where $d\mathfrak{m}_{\alpha}(x)=x^{2\alpha+1}dx$. Therefore, for $x,z\in I$,
since $m(x)\leq Cm(z)$ and using [52, (3)], we obtain

$\displaystyle J_{2,1}(x,z)$ $\displaystyle\leq C\int_{am(x)<|z-y|\leq
Cam(x)}|f_{2}(y)|e^{\frac{z^{2}-y^{2}}{2}}\frac{d\mathfrak{m}_{\alpha}(y)}{\mathfrak{m}_{\alpha}(I(z,|z-y|))}$
$\displaystyle\leq
C\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}\int_{am(x)<|z-y|\leq
Cam(x)}e^{\frac{(z+y)|z-y|}{2}}\frac{d\mathfrak{m}_{\alpha}(y)}{\mathfrak{m}_{\alpha}(I(y,|z-y|))}$
$\displaystyle\leq
C\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}\int_{am(x)<|z-y|\leq
Cam(x)}e^{Cam(x)(z+Cam(x))}\frac{dy}{|z-y|}$ $\displaystyle\leq
C\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}\int_{am(x)<|z-y|\leq
Cam(x)}\frac{dy}{|z-y|}$ $\displaystyle\leq
C\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}.$

On the other hand, for every $x,z\in I$ we have

$\displaystyle\left(\sum_{j=1}^{n-1}|R^{\alpha}_{\epsilon_{j+1}}(f_{2})(z)-R^{\alpha}_{\epsilon_{j}}(f_{2})(z)|^{\rho}\right)^{1/\rho}-\left(\sum_{j=1}^{n-1}|R^{\alpha}_{\epsilon_{j+1}}(f_{2})(x)-R^{\alpha}_{\epsilon_{j}}(f_{2})(x)|^{\rho}\right)^{1/\rho}$
$\displaystyle\leq\left(\sum_{j=1}^{n-1}\left|R^{\alpha}_{\epsilon_{j+1}}(f_{2})(z)-R^{\alpha}_{\epsilon_{j}}(f_{2})(z)-\left(R^{\alpha}_{\epsilon_{j+1}}(f_{2})(x)-R^{\alpha}_{\epsilon_{j}}(f_{2})(x)\right)\right|^{\rho}\right)^{1/\rho}$
$\displaystyle\leq\left(\sum_{j=1}^{n-1}\left|\int_{\epsilon_{j+1}<|z-y|<\epsilon_{j}}(R^{\alpha}(z,y)-R^{\alpha}(x,y))f_{2}(y)d\gamma_{\alpha}(y)\right.\right.$
$\displaystyle\quad+\left.\left.\left(\int_{\epsilon_{j+1}<|z-y|<\epsilon_{j}}R^{\alpha}(x,y)f_{2}(y)d\gamma_{\alpha}(y)\right.\right.\right.$
$\displaystyle\qquad-\left.\left.\left.\int_{\epsilon_{j+1}<|x-y|<\epsilon_{j}}R^{\alpha}(x,y)f_{2}(y)d\gamma_{\alpha}(y)\right)\right|^{\rho}\right)^{1/\rho}$
$\displaystyle\leq\sum_{j=1}^{n-1}\left|\int_{\epsilon_{j+1}<|z-y|<\epsilon_{j}}(R^{\alpha}(z,y)-R^{\alpha}(x,y))f_{2}(y)d\gamma_{\alpha}(y)\right|$
$\displaystyle\quad+\left(\sum_{j=1}^{n-1}\left|\int_{0}^{\infty}R^{\alpha}(x,y)\left(\chi_{\\{\epsilon_{j+1}<|x-y|<\epsilon_{j}\\}}(y)\right.\right.\right.$
$\displaystyle\qquad-\left.\left.\left.\chi_{\\{\epsilon_{j+1}<|z-y|<\epsilon_{j}\\}}(y)\right)f_{2}(y)d\gamma_{\alpha}(y)\right|^{\rho}\right)^{1/\rho}.$

Now, by taking supremum, we get

$\displaystyle J_{2,2}(x,z)$ $\displaystyle\leq\int_{(0,\infty)\setminus
4I}|R^{\alpha}(z,y)-R^{\alpha}(x,y)||f(y)|d\gamma_{\alpha}(y)$
$\displaystyle\quad+\sup_{\begin{subarray}{c}0<\epsilon_{n}<\dots<\epsilon_{1}\leq
am(x)\\\
n\in\mathbb{N}\end{subarray}}\left(\sum_{j=1}^{n-1}\left(\int_{0}^{\infty}\left|R^{\alpha}(x,y)\right|\left|\chi_{\\{\epsilon_{j+1}<|x-y|<\epsilon_{j}\\}}(y)\right.\right.\right.$
$\displaystyle\qquad-\left.\left.\left.\chi_{\\{\epsilon_{j+1}<|z-y|<\epsilon_{j}\\}}(y)\right||f_{2}(y)|d\gamma_{\alpha}(y)\right)^{\rho}\right)^{1/\rho}$
$\displaystyle:=J_{2,2,1}(x,z)+J_{2,2,2}(x,z).$

Since (see [4, § 4.3])

$\sup_{I\in\mathcal{B}_{a}}\sup_{x,z\in I}\int_{(0,\infty)\setminus
4I}|R^{\alpha}(z,y)-R^{\alpha}(x,y)||f(y)|d\gamma_{\alpha}(y)<\infty,$

it follows that

$J_{2,2,1}(x,z)\leq C\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})},\quad
x,z\in I.$

In order to estimate $J_{2,2,2}$ we adapt a procedure developed in [7]. From
(3.2), for $x,z\in I$, we obtain

$\displaystyle J_{2,2,2}(x,z)$ $\displaystyle\leq
C\sup_{\begin{subarray}{c}0<\epsilon_{n}<\dots<\epsilon_{1}\leq am(x)\\\
n\in\mathbb{N}\end{subarray}}\left(\sum_{j=1}^{n-1}\left(\int_{0}^{\infty}\frac{e^{\frac{x^{2}-y^{2}}{2}}}{\mathfrak{m}_{\alpha}(I(y,|x-y|))}\right.\right.$
$\displaystyle\qquad\times\left|\chi_{\\{\epsilon_{j+1}<|x-y|<\epsilon_{j}\\}}(y)-\left.\left.\chi_{\\{\epsilon_{j+1}<|z-y|<\epsilon_{j}\\}}(y)\right||f_{2}(y)|d\mathfrak{m}_{\alpha}(y)\right)^{\rho}\right)^{1/\rho}.$

Let us observe that, if $|x-y|\leq am(x)$, then

$x^{2}-y^{2}\leq|x-y||x+y|\leq am(x)(am(x)+2x)\leq C.$

Also, if $|z-y|\leq am(x)$, then

$|x-y|\leq 2r_{0}+am(x)\leq 2am(x_{0})+am(x).$

Since $x\in I\in\mathcal{B}_{a}$, $m(x_{0})\leq Cm(x)$ so $|x-y|\leq Cam(x)$,
and thus $x^{2}-y^{2}\leq C$, provided that $|z-y|\leq am(x)$.

This fact together with [52, (3)] lead to

$\displaystyle J_{2,2,2}(x,z)$
$\displaystyle\leq\sup_{\begin{subarray}{c}0<\epsilon_{n}<\dots<\epsilon_{1}\leq
am(x)\\\
n\in\mathbb{N}\end{subarray}}\left(\sum_{j=1}^{n-1}\left(\int_{0}^{\infty}\frac{|f_{2}(y)|}{|x-y|}\left|\chi_{\\{\epsilon_{j+1}<|x-y|<\epsilon_{j}\\}}(y)\right.\right.\right.$
$\displaystyle\quad-\left.\left.\left.\chi_{\\{\epsilon_{j+1}<|z-y|<\epsilon_{j}\\}}(y)\right|dy\right)^{\rho}\right)^{1/\rho},\quad
x,z\in I.$

Let us take $0<\epsilon_{n}<\dots<\epsilon_{1}\leq am(x)$ and
$j\in\\{1,\dots,n-1\\}$. Then

$\displaystyle\int_{0}^{\infty}$
$\displaystyle\frac{|f_{2}(y)|}{|x-y|}\left|\chi_{\\{\epsilon_{j+1}<|x-y|<\epsilon_{j}\\}}(y)-\chi_{\\{\epsilon_{j+1}<|z-y|<\epsilon_{j}\\}}(y)\right|dy$
$\displaystyle\leq
C\left(\int_{0}^{\infty}\frac{|f_{2}(y)|}{|x-y|}\chi_{\\{\epsilon_{j+1}<|x-y|<\epsilon_{j}\\}}(y)\chi_{\\{\epsilon_{j+1}<|x-y|<\epsilon_{j+1}+2r_{0}\\}}(y)dy\right.$
$\displaystyle\quad+\int_{0}^{\infty}\frac{|f_{2}(y)|}{|x-y|}\chi_{\\{\epsilon_{j+1}<|x-y|<\epsilon_{j}\\}}(y)\chi_{\\{\epsilon_{j}<|z-y|<\epsilon_{j}+2r_{0}\\}}(y)dy$
$\displaystyle\quad+\int_{0}^{\infty}\frac{|f_{2}(y)|}{|z-y|}\chi_{\\{\epsilon_{j+1}<|z-y|<\epsilon_{j}\\}}(y)\chi_{\\{\epsilon_{j+1}<|z-y|<\epsilon_{j+1}+2r_{0}\\}}(y)dy$
$\displaystyle\quad+\left.\int_{0}^{\infty}\frac{|f_{2}(y)|}{|z-y|}\chi_{\\{\epsilon_{j+1}<|z-y|<\epsilon_{j}\\}}(y)\chi_{\\{\epsilon_{j}<|x-y|<\epsilon_{j}+2r_{0}\\}}(y)dy\right)$
$\displaystyle=\sum_{l=1}^{4}J^{j,l}_{2,2,2}(x,z),\quad x,z\in I.$

For the above estimate, we have taken into account that, if
$\chi_{\\{\epsilon_{j+1}<|x-y|\leq\epsilon_{j}\\}}(y)-\chi_{\\{\epsilon_{j+1}<|z-y|\leq\epsilon_{j}\\}}(y)\neq
0$, then
$\chi_{\\{\epsilon_{j+1}<|x-y|\leq\epsilon_{j}\\}}(y)\chi_{\\{\epsilon_{j+1}<|z-y|\leq\epsilon_{j}\\}}(y)=0$,
with ${y\in(0,\infty)}$ and $x$, $z\in I$. Since $f_{2}(y)=0$ for $y\in 4I$,
it follows that $J^{j,l}_{2,2,2}=0$ when $l=1,3$, $z\in I$ and
$r_{0}\geq\epsilon_{j+1}$. Also, $J^{j,l}_{2,2,2}(x,z)=0$ when $l=2,4$, $z\in
I$ and $r_{0}\geq\epsilon_{j}$.

If $z\in I$ and $y\notin 4I$, then $2|x-y|\geq|z-y|\geq\frac{1}{2}|x-y|$.
Hölder inequality leads to

$J^{j,l}_{2,2,2}(x,z)\leq
C\left(\int_{0}^{\infty}\chi_{\\{\epsilon_{j+1}<|x-y|<\epsilon_{j}\\}}(y)\left(\frac{|f_{2}(y)|}{|x-y|}\right)^{2}dy\right)^{1/2}r_{0}^{1/2},\quad
z\in I,\,l=1,2;$ $J^{j,l}_{2,2,2}(x,z)\leq
C\left(\int_{0}^{\infty}\chi_{\\{\epsilon_{j+1}<|z-y|<\epsilon_{j}\\}}(y)\left(\frac{|f_{2}(y)|}{|z-y|}\right)^{2}dy\right)^{1/2}r_{0}^{1/2},\quad
z\in I,\,l=3,4.$

We obtain

$\displaystyle\left(\sum_{j=1}^{n-1}\left|\sum_{l=1}^{4}J^{j,l}_{2,2,2}(x,z)\right|^{\rho}\right)^{1/\rho}$
$\displaystyle\quad\leq
Cr_{0}^{1/2}\left[\left(\sum_{j=1}^{n-1}\left(\int_{0}^{\infty}\chi_{\\{\epsilon_{j+1}<|x-y|<\epsilon_{j}\\}}(y)\left(\frac{|f_{2}(y)|}{|x-y|}\right)^{2}dy\right)^{\rho/2}\right)^{1/\rho}\right.$
$\displaystyle\qquad+\left.\left(\sum_{j=1}^{n-1}\left(\int_{0}^{\infty}\chi_{\\{\epsilon_{j+1}<|z-y|<\epsilon_{j}\\}}(y)\left(\frac{|f_{2}(y)|}{|z-y|}\right)^{2}dy\right)^{\rho/2}\right)^{1/\rho}\right]$
$\displaystyle\quad\leq
Cr_{0}^{1/2}\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}\left[\left(\sum_{j=1}^{n-1}\int_{(0,\infty)\setminus
4I}\chi_{\\{\epsilon_{j+1}<|x-y|<\epsilon_{j}\\}}(y)\frac{dy}{|x-y|^{2}}\right)^{1/2}\right.$
$\displaystyle\qquad+\left.\left(\sum_{j=1}^{n-1}\int_{(0,\infty)\setminus
4I}\chi_{\\{\epsilon_{j+1}<|z-y|<\epsilon_{j}\\}}(y)\frac{dy}{|z-y|^{2}}\right)^{1/2}\right]$
$\displaystyle\quad\leq
Cr_{0}^{1/2}\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}\left(\int_{(0,\infty)\setminus
4I}\frac{dy}{|x-y|^{2}}+\int_{(0,\infty)\setminus
4I}\frac{dy}{|z-y|^{2}}\right)^{1/2}$ $\displaystyle\quad\leq
C\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}.$

We conclude that $J_{2,2,2}(x,z)\leq
C\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}$. By putting together the
above estimates we obtain

$\mathcal{V}_{\rho,a}\left(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0}\right)(f_{2})(z)-\mathcal{V}_{\rho,a}\left(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0}\right)(f_{2})(x)\leq
C\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})},\quad x,z\in I.$

Here, $C>0$ does not depend on $x$, $z\in I$, so it follows that

$J_{2}\leq C\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}.$

We now estimate $J_{3}$. Note first that if $|x-y|<r_{0}$, then
$|y-x_{0}|<2r_{0}$, so it is clear that

$\int_{\epsilon_{1}<|x-y|<\epsilon_{2}}R^{\alpha}(x,y)f_{2}(y)d\gamma_{\alpha}(y)=0,\quad
0<\epsilon_{1}<\epsilon_{2}\leq r_{0}.$

Suppose that $r_{0}\leq am(x)$. We have, for any $x\in I$, that

$\displaystyle\mathcal{V}_{\rho,a}$
$\displaystyle\left(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0}\right)(f_{2})(x)$
$\displaystyle\leq\sup_{0<\epsilon_{n}<\dots<\epsilon_{1}\leq r_{0},\
n\in\mathbb{N}}\left(\sum_{j=1}^{n-1}|R^{\alpha}_{\epsilon_{j+1}}(f_{2})(x)-R^{\alpha}_{\epsilon_{j}}(f_{2})(x)|^{\rho}\right)^{1/\rho}$
$\displaystyle\quad+\sup_{r_{0}<\epsilon_{n}<\dots<\epsilon_{1}\leq am(x),\
n\in\mathbb{N}}\left(\sum_{j=1}^{n-1}|R^{\alpha}_{\epsilon_{j+1}}(f_{2})(x)-R^{\alpha}_{\epsilon_{j}}(f_{2})(x)|^{\rho}\right)^{1/\rho}$
$\displaystyle=\sup_{r_{0}<\epsilon_{n}<\dots<\epsilon_{1}\leq am(x),\
n\in\mathbb{N}}\left(\sum_{j=1}^{n-1}|R^{\alpha}_{\epsilon_{j+1}}(f_{2})(x)-R^{\alpha}_{\epsilon_{j}}(f_{2})(x)|^{\rho}\right)^{1/\rho}$
$\displaystyle\leq\mathcal{V}_{\rho,a}\left(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0}\right)(f)(x)$
$\displaystyle\quad+\sup_{r_{0}<\epsilon_{n}<\dots<\epsilon_{1}\leq am(x),\
n\in\mathbb{N}}\left(\sum_{j=1}^{n-1}|R^{\alpha}_{\epsilon_{j+1}}(f_{1})(x)-R^{\alpha}_{\epsilon_{j}}(f_{1})(x)|^{\rho}\right)^{1/\rho}.$

Since, for every $y\in 4I$, $|x-y|\leq 5r_{0}\leq 5am(x)$, by using again
(3.2) and [52, (3)], we deduce that

$\displaystyle\sup_{r_{0}<\epsilon_{n}<\dots<\epsilon_{1}\leq am(x),\
n\in\mathbb{N}}$
$\displaystyle\left(\sum_{j=1}^{n-1}|R^{\alpha}_{\epsilon_{j+1}}(f_{1})(x)-R^{\alpha}_{\epsilon_{j}}(f_{1})(x)|^{\rho}\right)^{1/\rho}$
$\displaystyle\leq C\int_{|x-y|>r_{0},\ y\in
4I}\frac{e^{\frac{x^{2}+y^{2}}{2}}|f(y)|}{\mathfrak{m}_{\alpha}(I(y,|x-y|))}d\gamma_{\alpha}(y)$
$\displaystyle\leq C\int_{|x-y|>r_{0},\ y\in
4I}\frac{e^{\frac{x^{2}-y^{2}}{2}}|f(y)|}{|x-y|}dy$ $\displaystyle\leq
C\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}\int_{|x-y|>r_{0},\ y\in
4I}\frac{dy}{|x-y|}$ $\displaystyle\leq
C\frac{\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}}{r_{0}}\int_{4I}dy$
$\displaystyle\leq C\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})},$

that is, $J_{3}\leq C\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}$ for the
case $r_{0}\leq am(x)$.

When $r_{0}>am(x)$,

$\mathcal{V}_{\rho,a}\left(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0}\right)(f_{2})(x)=0,$

so $J_{3}\leq 0$ in this case.

We conclude that

$J_{3}\leq C\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}.$

By combining the above estimates, since the constant $C>0$ does not depend on
$x\in(0,\infty)$ or $I\in\mathcal{B}_{a}(x)$, we get

$\|\mathcal{M}^{\alpha}_{a}\left(\mathcal{V}_{\rho,a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0})(f)\right)-\mathcal{V}_{\rho,a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0})(f)\|_{L^{\infty}(0,\infty),\gamma_{\alpha}}\leq
C\|f\|_{L^{\infty}(0,\infty),\gamma_{\alpha}}.$

Thus the proof is finished.

###  3.2. Local oscillation operators

Theorem 1.1 for oscillation operators can be proved by using the procedure
developed in the previous section for the variation operator, so we give a
sketch of the proof.

According to [9, Theorem 1.3], the oscillation operator
$\mathcal{O}(\\{R_{\epsilon}^{\alpha}\\}_{\epsilon>0},\\{t_{j}\\}_{j\in\mathbb
Z})$ is bounded on $L^{2}((0,\infty),\gamma_{\alpha})$. This property implies
that
$\mathcal{O}_{a}(\\{R_{\epsilon}^{\alpha}\\}_{\epsilon>0},\\{t_{j}\\}_{j\in\mathbb
Z})(f)\in L^{1}((0,\infty),\gamma_{\alpha})$ for every $f\in
L^{\infty}((0,\infty),\gamma_{\alpha})$.

In order to prove our result, it is sufficient to find a positive constant $C$
such that, for every $f\in L^{\infty}((0,\infty),\gamma_{\alpha})$,

$\displaystyle\|\mathcal{M}^{\alpha}_{a}(\mathcal{O}_{a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0},\\{t_{j}\\}_{j\in\mathbb
Z})(f))-$
$\displaystyle\mathcal{O}_{a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0},\\{t_{j}\\}_{j\in\mathbb
Z})(f)\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}$ $\displaystyle\leq
C\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}.$ (3.3)

Fix $f\in L^{\infty}((0,\infty),\gamma_{\alpha})$ and let
$x,x_{0},r_{0}\in(0,\infty)$ such that
$I=I(x_{0},r_{0})\in\mathcal{B}_{a}(x)$. We write
$f=f\chi_{4I}+f\chi_{(0,\infty)\setminus 4I}:=f_{1}+f_{2}$.

$\displaystyle\frac{1}{\gamma_{\alpha}(I)}\int_{I}\mathcal{O}_{a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0},\\{t_{j}\\}_{j\in\mathbb
Z})(f)(z)d\gamma_{\alpha}(z)-\mathcal{O}_{a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0},\\{t_{j}\\}_{j\in\mathbb
Z})(f)(x)$
$\displaystyle\leq\frac{1}{\gamma_{\alpha}(I)}\int_{I}\mathcal{O}_{a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0},\\{t_{j}\\}_{j\in\mathbb
Z})(f_{1})(z)d\gamma_{\alpha}(z)$
$\displaystyle\quad+\frac{1}{\gamma_{\alpha}(I)}\int_{I}\left[\mathcal{O}_{a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0},\\{t_{j}\\}_{j\in\mathbb
Z})(f_{2})(z)-\mathcal{O}_{a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0},\\{t_{j}\\}_{j\in\mathbb
Z})(f_{2})(x)\right]d\gamma_{\alpha}(z)$
$\displaystyle\quad+\mathcal{O}_{a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0},\\{t_{j}\\}_{j\in\mathbb
Z})(f_{2})(x)-\mathcal{O}_{a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0},\\{t_{j}\\}_{j\in\mathbb
Z})(f)(x)$ $\displaystyle:=J_{1}+J_{2}+J_{3}.$

It is immediate from the $L^{2}$-boundedness of
$\mathcal{O}(\\{R_{\epsilon}^{\alpha}\\}_{\epsilon>0},\\{t_{j}\\}_{j\in\mathbb
Z})$ ([9, Theorem 1.3]) that

$J_{1}\leq C\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}.$

We now estimate the integrand of $J_{2}$. For certain $C>1$, we have

$\displaystyle\mathcal{O}_{a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0},$
$\displaystyle\\{t_{j}\\}_{j\in\mathbb
Z})(f_{2})(z)-\mathcal{O}_{a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0},\\{t_{j}\\}_{j\in\mathbb
Z})(f_{2})(x)$ $\displaystyle\leq\left(\sum_{\begin{subarray}{c}j\in\mathbb
Z\\\ t_{j}\leq
Cam(x)\end{subarray}}\sup_{t_{j-1}\leq\epsilon_{j-1}<\epsilon_{j}\leq
t_{j}}\left|R^{\alpha}_{\epsilon_{j-1}}(f_{2})(z)-R^{\alpha}_{\epsilon_{j}}(f_{2})(z)\right|^{2}\right)^{1/2}$
$\displaystyle\quad-\left(\sum_{\begin{subarray}{c}j\in\mathbb Z\\\ t_{j}\leq
am(x)\end{subarray}}\sup_{t_{j-1}\leq\epsilon_{j-1}<\epsilon_{j}\leq
t_{j}}\left|R^{\alpha}_{\epsilon_{j-1}}(f_{2})(x)-R^{\alpha}_{\epsilon_{j}}(f_{2})(x)\right|^{2}\right)^{1/2}.$

We define

$j_{0}(x)=\max\\{j\in\mathbb Z:t_{j}\leq am(x)\\}$ (3.4)

and also, provided that $t_{j_{0}(x)+1}\leq Cam(x)$, we consider

$j_{1}(x)=\max\\{j\in\mathbb Z:j>j_{0}(x),t_{j}\leq Cam(x)\\}.$ (3.5)

Thus, when $t_{j_{0}(x)+1}>Cam(x)$, we can write

$\displaystyle\mathcal{O}_{a}$
$\displaystyle(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0},\\{t_{j}\\}_{j\in\mathbb
Z})(f_{2})(z)-\mathcal{O}_{a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0},\\{t_{j}\\}_{j\in\mathbb
Z})(f_{2})(x)$ $\displaystyle\leq\left(\sum_{\begin{subarray}{c}j\in\mathbb
Z\\\ j\leq
j_{0}(x)\end{subarray}}\sup_{t_{j-1}\leq\epsilon_{j-1}<\epsilon_{j}\leq
t_{j}}\left|R^{\alpha}_{\epsilon_{j-1}}(f_{2})(z)-R^{\alpha}_{\epsilon_{j}}(f_{2})(z)\right|^{2}\right)^{1/2}$
$\displaystyle\quad-\left(\sum_{\begin{subarray}{c}j\in\mathbb Z\\\ j\leq
j_{0}(x)\end{subarray}}\sup_{t_{j-1}\leq\epsilon_{j-1}<\epsilon_{j}\leq
t_{j}}\left|R^{\alpha}_{\epsilon_{j-1}}(f_{2})(x)-R^{\alpha}_{\epsilon_{j}}(f_{2})(x)\right|^{2}\right)^{1/2}$
$\displaystyle\leq\left(\sum_{\begin{subarray}{c}j\in\mathbb Z\\\ j\leq
j_{0}(x)\end{subarray}}\sup_{t_{j-1}\leq\epsilon_{j-1}<\epsilon_{j}\leq
t_{j}}\left|D(x,z)\right|^{2}\right)^{1/2}:=\tilde{J_{2}}(x,z),$

where

$D(x,z):=R^{\alpha}_{\epsilon_{j-1}}(f_{2})(z)-R^{\alpha}_{\epsilon_{j}}(f_{2})(z)-\left(R^{\alpha}_{\epsilon_{j-1}}(f_{2})(x)-R^{\alpha}_{\epsilon_{j}}(f_{2})(x)\right).$

On the other hand, if $t_{j_{0}(x)+1}\leq Cam(x)$, we get

$\displaystyle\mathcal{O}_{a}$
$\displaystyle(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0},\\{t_{j}\\}_{j\in\mathbb
Z})(f_{2})(z)-\mathcal{O}_{a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0},\\{t_{j}\\}_{j\in\mathbb
Z})(f_{2})(x)$
$\displaystyle\leq\tilde{J_{2}}(x,z)+\left(\sum_{\begin{subarray}{c}j\in\mathbb
Z\\\ j_{0}(x)<j\leq
j_{1}(x)\end{subarray}}\sup_{t_{j-1}\leq\epsilon_{j-1}<\epsilon_{j}\leq
t_{j}}\left|R^{\alpha}_{\epsilon_{j-1}}(f_{2})(z)-R^{\alpha}_{\epsilon_{j}}(f_{2})(z)\right|^{2}\right)^{1/2}$
$\displaystyle\leq\tilde{J_{2}}(x,z)+\int_{\frac{a}{\rho}m(x)\leq|z-y|\leq
Cam(x)}|R^{\alpha}(z,y)||f_{2}(y)|d\gamma_{\alpha}(y),$

where in the last inequality we have used that $t_{j_{0}(x)}\leq am(x)\leq
t_{j_{0}(x)+1}\leq\rho t_{j_{0}(x)}$ with $\rho>1$.

Notice that we can estimate $\tilde{J_{2}}(x,z)$ in the following form

$\displaystyle\tilde{J_{2}}(x,z)$
$\displaystyle\leq\left(\sum_{\begin{subarray}{c}j\in\mathbb Z\\\ j\leq
j_{0}(x)\end{subarray}}\sup_{t_{j-1}\leq\epsilon_{j-1}<\epsilon_{j}\leq
t_{j}}\left|\int_{\epsilon_{j-1}<|z-y|<\epsilon_{j}}(R^{\alpha}(z,y)-R^{\alpha}(x,y))f_{2}(y)d\gamma_{\alpha}(y)\right.\right.$
$\displaystyle\quad+\left.\left.\int_{0}^{\infty}\left(\chi_{\\{\epsilon_{j-1}<|z-y|<\epsilon_{j}\\}}(y)-\chi_{\\{\epsilon_{j-1}<|x-y|<\epsilon_{j}\\}}(y)\right)R^{\alpha}(x,y)f_{2}(y)d\gamma_{\alpha}(y)\right|^{2}\right)^{1/2}$
$\displaystyle\leq\int_{(0,\infty)\setminus
4I}|R^{\alpha}(z,y)-R^{\alpha}(x,y)||f_{2}(y)|d\gamma_{\alpha}(y)$
$\displaystyle\quad+\left(\sum_{\begin{subarray}{c}j\in\mathbb Z\\\ j\leq
j_{0}(x)\end{subarray}}\left(\sup_{t_{j-1}\leq\epsilon_{j-1}<\epsilon_{j}\leq
t_{j}}\int_{0}^{\infty}\left|\chi_{\\{\epsilon_{j-1}<|z-y|<\epsilon_{j}\\}}(y)-\chi_{\\{\epsilon_{j-1}<|x-y|<\epsilon_{j}\\}}(y)\right|\right.\right.$
$\displaystyle\qquad\times\left.\left.|R^{\alpha}(x,y)||f_{2}(y)|d\gamma_{\alpha}(y)\right)^{2}\right)^{1/2}.$

At this point, we can proceed as in the proof of the corresponding result for
variation operators $\mathcal{V}_{\rho,a}$, by using Hölder’s inequality with
an exponent $s\in(1,2)$ instead of applying it with exponent 2. In this way,
we deduce that

$J_{2}\leq C\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}.$

In order to study $J_{3}$, we first recall that

$\int_{\epsilon_{1}<|x-y|<\epsilon_{2}}R^{\alpha}(x,y)f_{2}(y)d\gamma_{\alpha}(y),\quad
0<\epsilon_{1}<\epsilon_{2}\leq r_{0}.$

Then, if $r_{0}\geq am(x)$, we obtain

$\mathcal{O}_{a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0},\\{t_{j}\\}_{j\in\mathbb
Z})(f_{2})(x)=0.$ (3.6)

Suppose now that $r_{0}<am(x)$ and define $j_{0}(x)$ as in (3.4). If
$t_{j_{0}(x)}\leq r_{0}$, we again have (3.6). If not, we define
$j_{1}=\max\\{j\in\mathbb Z:t_{j_{1}}\leq r_{0}\\}$. Then

$\displaystyle\mathcal{O}_{a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0},$
$\displaystyle\\{t_{j}\\}_{j\in\mathbb Z})(f_{2})(x)$
$\displaystyle=\left(\sum_{j=j_{1}+1}^{j_{0}(x)}\sup_{t_{j-1}\leq\epsilon_{j-1}<\epsilon_{j}\leq
t_{j}}|R^{\alpha}_{\epsilon_{j-1}}(f_{2})(x)-R^{\alpha}_{\epsilon_{j}}(f_{2})(x)|^{2}\right)^{1/2}$
$\displaystyle\leq\mathcal{O}_{a}(\\{R^{\alpha}_{\epsilon}\\}_{\epsilon>0},\\{t_{j}\\}_{j\in\mathbb
Z})(f)(x)$
$\displaystyle\quad+\left(\sum_{j=j_{1}+1}^{j_{0}(x)}\sup_{t_{j-1}\leq\epsilon_{j-1}<\epsilon_{j}\leq
t_{j}}|R^{\alpha}_{\epsilon_{j-1}}(f_{1})(x)-R^{\alpha}_{\epsilon_{j}}(f_{1})(x)|^{2}\right)^{1/2}.$

Since $t_{j_{1}}\leq r_{0}\leq t_{j+1}\leq\rho t_{j_{1}}$, it follows that

$\displaystyle\left(\sum_{j=j_{1}+1}^{j_{0}(x)}\sup_{t_{j-1}\leq\epsilon_{j-1}<\epsilon_{j}\leq
t_{j}}|R^{\alpha}_{\epsilon_{j-1}}(f_{1})(x)-R^{\alpha}_{\epsilon_{j}}(f_{1})(x)|^{2}\right)^{1/2}$
$\displaystyle\quad\leq\sum_{j=j_{1}+1}^{j_{0}(x)}\sup_{t_{j-1}\leq\epsilon_{j-1}<\epsilon_{j}\leq
t_{j}}|R^{\alpha}_{\epsilon_{j-1}}(f_{1})(x)-R^{\alpha}_{\epsilon_{j}}(f_{1})(x)|$
$\displaystyle\quad\leq
C\int_{|x-y|>r_{0}/\rho}\frac{e^{\frac{x^{2}+y^{2}}{2}}|f(y)|}{\mathfrak{m}_{\alpha}(I(y,|x-y|))}d\gamma_{\alpha}(y)$
$\displaystyle\quad\leq C\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})},$

where we have used again the bound given in (3.2) and [52, (3)].

We conclude that

$J_{3}\leq C\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}.$

By putting together all of the above estimates, we get (3.2) and the proof of
Theorem 1.1 for local oscillation operators is completed.

###  3.3. Local maximal Riesz transform

We firstly prove that $R^{\alpha}_{*,a}$ is bounded on
$L^{p}((0,\infty),\gamma_{\alpha})$ for every $1<p<\infty$. In order to do so,
we need to decompose, for every $\epsilon>0$, the truncated integral
$R^{\alpha}_{\epsilon}$ into two parts, called local and global parts (see
[47]).

For every $\tau>0$, we consider the sets

$L_{\tau}=\left\\{(x,y,s)\in(0,\infty)\times(0,\infty)\times(-1,1):\sqrt{q_{-}(x,y,s)}\leq\frac{a(1+\alpha)\tau}{1+x+y}\right\\}$

and

$G_{\tau}=((0,\infty)\times(0,\infty)\times(-1,1))\setminus L_{\tau}.$

Here, and in the sequel, we denote $q_{\pm}(x,y,s)=x^{2}+y^{2}\pm 2xys$, for
$x,y\in(0,\infty)$ and $s\in(-1,1)$.

We choose a function $\varphi\in
C^{\infty}((0,\infty)\times(0,\infty)\times(-1,1))$ such that
$0\leq\varphi\leq 1$,

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

and

$|\partial_{x}\varphi(x,y,s)|+|\partial_{y}\varphi(x,y,s)|\leq\frac{C}{\sqrt{q_{-}(x,y,s)}},\quad
x,y\in(0,\infty),s\in(-1,1).$

We define, for each $\epsilon>0$ and $x\in(0,\infty)$

$\displaystyle R^{\alpha,\textup{loc}}_{\epsilon}(f)(x)$
$\displaystyle=\int_{|x-y|>\varepsilon,\
y\in(0,\infty)}R^{\alpha,\textup{loc}}(x,y)f(y)d\gamma_{\alpha}(y),$
$\displaystyle R^{\alpha,\textup{glob}}_{\epsilon}(f)(x)$
$\displaystyle=R^{\alpha}_{\epsilon}(f)(x)-R^{\alpha,\textup{loc}}_{\epsilon}(f)(x),\
$

where

$R^{\alpha,\textup{loc}}(x,y)=\int_{-1}^{1}R^{\alpha,\textup{loc}}(x,y,s)\Pi_{\alpha}(s)ds,\quad
x,y\in(0,\infty)$

and, for $x,y\in(0,\infty)$, $s\in(-1,1)$,

$R^{\alpha,\textup{loc}}(x,y,s)=-\frac{2}{\sqrt{\pi}}\int_{0}^{\infty}\frac{e^{-t(\alpha+2)}\left(e^{-t}x-ys\right)}{(1-e^{-2t})^{\alpha+2}}e^{-\frac{q_{-}(e^{-t}x,y,s)}{1-e^{-2t}}+y^{2}}\varphi(x,y,s)\frac{dt}{\sqrt{t}}.$

We also consider the maximal operators associated with the above,

$R^{\alpha,\textup{loc}}_{*}(f)=\sup_{\epsilon>0}\left|R^{\alpha,\textup{loc}}_{\epsilon}(f)\right|,\quad
R^{\alpha,\textup{glob}}_{*}(f)=\sup_{\epsilon>0}\left|R^{\alpha,\textup{glob}}_{\epsilon}(f)\right|,$

which clearly verify

$R^{\alpha}_{*}(f)\leq
R^{\alpha,\textup{loc}}_{*}(f)+R^{\alpha,\textup{glob}}_{*}(f).$

According to [9, § 3.1] (see also [47, Proposition 3.1]), we have that

$R^{\alpha,\textup{glob}}_{*}(f)(x)\leq
C\int_{0}^{\infty}K^{\alpha}(x,y)f(y)d\gamma_{\alpha}(y),\quad
x\in(0,\infty),$

where

$K^{\alpha}(x,y)=\int_{-1}^{1}K^{\alpha}(x,y,s)\chi_{G_{1}}(x,y,s)\Pi_{\alpha}(s)ds,\quad
x,y\in(0,\infty),$

and, for $x,y\in(0,\infty)$ and $s\in(-1,1)$,

$K^{\alpha}(x,y,s)=\begin{cases}1,&s<0,\\\
\left(\frac{q_{+}(x,y,s)}{q_{-}(x,y,s)}\right)^{\frac{\alpha+1}{2}}\exp\left(\frac{x^{2}+y^{2}-\sqrt{q_{-}(x,y,s)q_{+}(x,y,s)}}{2}\right),&s\geq
0.\end{cases}$ (3.7)

It follows that $R^{\alpha,\textup{glob}}_{*}$ is bounded on
$L^{p}((0,\infty),\gamma_{\alpha})$ for every $1<p<\infty$ (see [9, § 3.1]).

We recall that the measure $\mathfrak{m}_{\alpha}$ defined in Section 3.1 has
the doubling property on $(0,\infty)$. Therefore, by [9, (18) and (19)],
$e^{-y^{2}}R_{\alpha}^{\textup{loc}}(x,y)$, for $x,y\in(0,\infty)$, is an
$\mathfrak{m}_{\alpha}$-standard Calderón-Zygmund kernel, that is, for every
$x,y\in(0,\infty)$, $x\neq y$,

$\left|e^{-y^{2}}R_{\alpha}^{\textup{loc}}(x,y)\right|\leq\frac{C}{\mathfrak{m}_{\alpha}(I(x,|x-y|))},$

and

$\left|\partial_{x}\left[e^{-y^{2}}R_{\alpha}^{\textup{loc}}(x,y)\right]\right|+\left|\partial_{y}\left[e^{-y^{2}}R_{\alpha}^{\textup{loc}}(x,y)\right]\right|\leq\frac{C}{|x-y|\mathfrak{m}_{\alpha}(I(x,|x-y|))}.$

If we define the operators $R^{\alpha,\textup{loc}}$ and
$R^{\alpha,\textup{glob}}$ in the obvious way, we can see as above that the
later is bounded on $L^{p}((0,\infty),\gamma_{\alpha})$ for every
$1<p<\infty$. Since $R^{\alpha}$ is also bounded on
$L^{p}((0,\infty),\gamma_{\alpha})$ for every $1<p<\infty$ (see [41, Theorem
13]), we conclude that $R^{\alpha,\textup{loc}}$ is bounded on
$L^{p}((0,\infty),\gamma_{\alpha})$ for every $1<p<\infty$. By proceeding as
in [5, § 2], we deduce that $R^{\alpha,\textup{loc}}$ is bounded on
$L^{p}((0,\infty),\mathfrak{m}_{\alpha})$ for every $1<p<\infty$. Moreover,
since $R^{\alpha,\textup{loc}}$ is an $\mathfrak{m}_{\alpha}$-Calderón-Zygmund
operator, $R^{\alpha,\textup{loc}}_{*}$ is bounded on
$L^{p}((0,\infty),\mathfrak{m}_{\alpha})$ for every $1<p<\infty$. By using
again the arguments given in [5, § 2], we get that
$R^{\alpha,\textup{loc}}_{*}$ is bounded on
$L^{p}((0,\infty),\gamma_{\alpha})$ for every $1<p<\infty$.

It follows now that $R^{\alpha}_{*}$ is bounded on
$L^{p}((0,\infty),\gamma_{\alpha})$ for every $1<p<\infty$. Particularly,
using this property for $p=2$, for any $f\in
L^{\infty}((0,\infty),\gamma_{\alpha})$,

$\|R^{\alpha}_{*,a}\|_{L^{1}((0,\infty),\gamma_{\alpha})}\leq
C\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}.$

We recall that, from (3.2),

$|R^{\alpha}(x,y)|\leq
C\frac{e^{\frac{x^{2}+y^{2}}{2}}}{\mathfrak{m}_{\alpha}(I(x,|x-y|))},\quad
x,y\in(0,\infty),\ x\neq y,$

and also we can see (as in [4, § 4.3]) that

$\sup_{I\in\mathcal{B}_{a}}\sup_{x\in I}r_{0}\int_{(0,\infty)\setminus
2I}|\partial_{x}R^{\alpha}(x,y)|d\gamma_{\alpha}(y)<\infty.$

By proceeding as in the proof of [32, Theorem 4.1], it yields

$\sup_{I\in\mathcal{B}_{a}}\left\|\mathcal{M}^{\alpha}_{a}\left(R^{\alpha}_{*,a}(f)\right)-R^{\alpha}_{*,a}(f)\right\|_{L^{\infty}((0,\infty),\gamma_{\alpha})}\leq
C\|f\|_{L^{\infty}((0,\infty),\gamma_{\alpha})},$

meaning that $R^{\alpha}_{*,a}$ is bounded from
$L^{\infty}((0,\infty),\gamma_{\alpha})$ into
$\textup{BLO}_{a}((0,\infty),\gamma_{\alpha})$.

##  4\. Proof of Theorem 1.2

In this section, we will study
$L^{\infty}((0,\infty),\gamma_{\alpha})$-$\textup{BLO}_{a}((0,\infty),\gamma_{\alpha})$
estimates for the $a$-local maximal operator

$\displaystyle Q^{\alpha}_{\phi,*,a}(f)(x)$
$\displaystyle=\sup_{0<\epsilon\leq am(x)}|Q_{\phi,\epsilon}^{\alpha}(f)(x)|$
$\displaystyle=\sup_{0<\epsilon\leq am(x)}\left|\int_{|x-y|>\epsilon,\
y\in(0,\infty)}K_{\phi}^{\alpha}(x,y)f(y)d\gamma_{\alpha}(y)\right|,$

for $x\in(0,\infty)$ and $a>0$.

We recall that

$K_{\phi}^{\alpha}(x,y)=-\int_{0}^{\infty}\phi(t)\partial_{t}W_{t}^{\alpha}(x,y)dy,\quad
x,t,\in(0,\infty),\ x\neq y,$

being

$W_{t}^{\alpha}(x,y)=\left(\frac{e^{-t}}{1-e^{-2t}}\right)^{\alpha+1}\int_{-1}^{1}e^{-\frac{q_{-}(e^{-t}x,y,s)}{1-e^{-2t}}+y^{2}}\Pi_{\alpha}(s)ds,\quad
x,y,t\in(0,\infty).$

Firstly, we shall see that $Q^{\alpha}_{\phi,*,a}$ is bounded on
$L^{p}((0,\infty),\gamma_{\alpha})$ for every ${1<p<\infty}$. We define, for
$x,y,t\in(0,\infty)$,

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

and

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

In terms of these, we consider $K_{\phi}^{\alpha,\textup{loc}}$ and
$K_{\phi}^{\alpha,\textup{glob}}$ given as $K_{\phi}^{\alpha}$ but with
$W_{t}^{\alpha}$ replaced by $W_{t}^{\alpha,\textup{loc}}$ and
$W_{t}^{\alpha,\textup{glob}}$, respectively. Similarly, we define
$Q_{\phi,*,a}^{\alpha,\textup{loc}}$ and $Q_{\phi,*,a}^{\alpha,\textup{glob}}$
by putting $K_{\phi}^{\alpha,\textup{loc}}$ and
$K_{\phi}^{\alpha,\textup{glob}}$ instead of $K_{\phi}^{\alpha}$,
respectively.

We will first deal with $Q_{\phi,*,a}^{\alpha,\textup{glob}}$. Notice that,
for $x,y,t\in(0,\infty)$ and $s\in(-1,1)$

$\displaystyle\partial_{t}$
$\displaystyle\left[\left(\frac{e^{-t}}{1-e^{-2t}}\right)^{\alpha+1}\exp\left(-\frac{q_{-}(e^{-t}x,y,s)}{1-e^{-2t}}\right)\right]$
$\displaystyle=P_{x,y,s}\left(e^{-t}\right)\left(\frac{e^{-t}}{1-e^{-2t}}\right)^{\alpha+1}\exp\left(-\frac{q_{-}(e^{-t}x,y,s)}{1-e^{-2t}}\right),$

where, for every $x,y\in(0,\infty)$ and $s\in(-1,1)$, $P_{x,y,s}$ is a
polynomial whose degree is at most four. Hence,

$\displaystyle|K_{\phi}^{\alpha,\textup{glob}}(x,y)|$ $\displaystyle\leq
C\int_{-1}^{1}\sup_{t>0}\left(\frac{e^{-t}}{1-e^{-2t}}\right)^{\alpha+1}e^{-\frac{q_{-}(e^{-t}x,y,s)}{1-e^{-2t}}+y^{2}}\chi_{L_{1}^{c}}(x,y,s)\Pi_{\alpha}(s)ds$
$\displaystyle\leq
C\int_{-1}^{1}K^{\alpha}(x,y,s)\chi_{L_{1}^{c}}(x,y,s)\Pi_{\alpha}(s)ds,\quad
x,y\in(0,\infty),$

being $K^{\alpha}(x,y,s)$ as in (3.7), for $(x,y,s)\in L_{1}^{c}$.

From [9, § 3.1], it follows that the operator whose kernel is the one on the
right-hand side is bounded on $L^{p}((0,\infty),\gamma_{\alpha})$ for every
${1<p<\infty}$, and so will be $Q_{\phi,*,a}^{\alpha,\textup{glob}}$.

Furthermore, for every $f\in L^{p}((0,\infty),\gamma_{\alpha})$, $1<p<\infty$,

$\lim_{\varepsilon\to 0^{+}}\int_{|x-y|>\varepsilon,\
y\in(0,\infty)}K_{\phi}^{\alpha,\textup{glob}}(x,y)f(y)d\gamma_{\alpha}(y)=\int_{0}^{\infty}K_{\phi}^{\alpha,\textup{glob}}(x,y)f(y)d\gamma_{\alpha}(y),$

for a.e. $x\in(0,\infty)$.

We now consider the operators

$T_{M}^{\alpha,\textup{loc}}(f)(x)=\lim_{\varepsilon\to
0^{+}}\left(\Lambda(\varepsilon)f(x)+\int_{|x-y|>\varepsilon,\
y\in(0,\infty)}K_{\phi}^{\alpha,\textup{loc}}(x,y)f(y)d\gamma_{\alpha}(y)\right),$

and

$T_{M}^{\alpha,\textup{glob}}(f)(x)=\int_{0}^{\infty}K_{\phi}^{\alpha,\textup{glob}}(x,y)f(y)d\gamma_{\alpha}(y),$

for a.e. $x\in(0,\infty)$. Since $T_{M}^{\alpha}$ and
$T_{M}^{\alpha,\textup{glob}}$ are both bounded on
$L^{2}((0,\infty),\gamma_{\alpha})$ ([45, Proposition 3]), also
$T_{M}^{\alpha,\textup{loc}}$ is bounded on
$L^{2}((0,\infty),\gamma_{\alpha})$. Moreover, for every $f\in
L^{\infty}((0,\infty),\gamma_{\alpha})$

$T_{M}^{\alpha,\textup{loc}}(f)(x)=\int_{0}^{\infty}K_{\phi}^{\alpha,\textup{loc}}(x,y)f(y)d\gamma_{\alpha}(y),\quad
x\notin\operatorname{supp}(f).$

Let us now consider
$\mathbb{K}_{\phi}^{\alpha}(x,y):=e^{-y^{2}}K_{\phi}^{\alpha,\textup{loc}}(x,y)$,
for $x,y\in(0,\infty)$. We have

$\displaystyle\mathbb{K}_{\phi}^{\alpha}(x,y)$
$\displaystyle=(\alpha+1)\int_{0}^{\infty}\varphi(t)\left(\frac{e^{-t}}{1-e^{-2t}}\right)^{\alpha+1}\int_{-1}^{1}e^{-\frac{q_{-}(e^{-t}x,y,s)}{1-e^{-2t}}}\varphi(x,y,s)\Pi_{\alpha}(s)dsdt$
$\displaystyle\quad-\int_{0}^{\infty}\varphi(t)e^{-t(\alpha+1)}\int_{-1}^{1}\partial_{t}\left[\frac{e^{-\frac{q_{-}(e^{-t}x,y,s)}{1-e^{-2t}}}}{(1-e^{-2t})^{\alpha+1}}\right]\varphi(x,y,s)\Pi_{\alpha}(s)dsdt$
$\displaystyle:=\mathbb{K}_{\phi,1}^{\alpha}(x,y)+\mathbb{K}_{\phi,2}^{\alpha}(x,y),\quad
x,y\in(0,\infty).$

As in [5, § 7], we can prove that

$|\mathbb{K}_{\phi,2}^{\alpha}(x,y)|\leq\frac{C}{\mathfrak{m}_{\alpha}(I(x,|x-y|))},\quad
x,y\in(0,\infty),\ x\neq y,$

and

$|\partial_{x}\mathbb{K}_{\phi,2}^{\alpha}(x,y)|+|\partial_{y}\mathbb{K}_{\phi,2}^{\alpha}(x,y)|\leq\frac{C}{|x-y|\mathfrak{m}_{\alpha}(I(x,|x-y|))},\quad
x,y\in(0,\infty),\ x\neq y.$

On the other hand, using [45, (2.6)], i.e., $q_{-}(e^{-t}x,y,s)\geq
q_{-}(x,y,s)-2(1-e^{-2t})$, for every $(x,y,s)\in N_{1}$, and the estimates
obtained in [2, p. 12 and Lemma 3.1], we get

  1. (a)

$\displaystyle|\mathbb{K}_{\phi,1}^{\alpha}(x,y)|$ $\displaystyle\leq
C\int_{0}^{\infty}|\varphi(t)|\left(\frac{e^{-t}}{1-e^{-2t}}\right)^{\alpha+1}\int_{-1}^{1}e^{-\frac{q_{-}(x,y,s)}{1-e^{-2t}}}\Pi_{\alpha}(s)dsdt$
$\displaystyle\leq
C\int_{0}^{\infty}|\varphi(t)|e^{-t(\alpha+1)}dt\int_{-1}^{1}\frac{\Pi_{\alpha}(s)}{q_{-}(x,y,s)^{\alpha+1}}ds$
$\displaystyle\leq\frac{C}{\mathfrak{m}_{\alpha}(I(x,|x-y|))},\quad
x,y\in(0,\infty),\ x\neq y;$

  2. (b)

by [4, Lemma 3.4 (E8)],

$\displaystyle|\partial_{x}\mathbb{K}_{\phi,1}^{\alpha}(x,y)|$
$\displaystyle\leq
C\int_{0}^{\infty}|\varphi(t)|\left(\frac{e^{-t}}{1-e^{-2t}}\right)^{\alpha+1}\int_{-1}^{1}e^{-\frac{q_{-}(e^{-t}x,y,s)}{1-e^{-2t}}}$
$\displaystyle\qquad\times\left[\frac{e^{-t}|e^{-t}x-ys|}{1-e^{-2t}}\varphi(x,y,s)+|\partial_{x}\varphi(x,y,s)|\right]\Pi_{\alpha}(s)dsdt$
$\displaystyle\leq
C\int_{0}^{\infty}|\varphi(t)|\left(\frac{e^{-t}}{1-e^{-2t}}\right)^{\alpha+1}e^{-t}\int_{-1}^{1}e^{-\frac{q_{-}(e^{-t}x,y,s)}{1-e^{-2t}}}$
$\displaystyle\qquad\times\left[\frac{\sqrt{q_{-}(e^{-t}x,y,s)}}{1-e^{-2t}}+\frac{1}{\sqrt{q_{-}(x,y,s)}}\right]\Pi_{\alpha}(s)dsdt$
$\displaystyle\leq
C\int_{0}^{\infty}|\varphi(t)|e^{-t(\alpha+2)}\int_{-1}^{1}\left[\frac{e^{-\frac{q_{-}(x,y,s)}{2(1-e^{-2t})}}}{(1-e^{-2t})^{\alpha+3/2}}\right.$
$\displaystyle\qquad+\left.\frac{e^{-\frac{q_{-}(x,y,s)}{1-e^{-2t}}}}{(1-e^{-2t})^{\alpha+1}\sqrt{q_{-}(x,y,s)}}\right]\Pi_{\alpha}(s)dsdt$
$\displaystyle\leq
C\int_{0}^{\infty}|\varphi(t)|e^{-t(\alpha+2)}dt\int_{-1}^{1}\frac{\Pi_{\alpha}(s)}{q_{-}(x,y,s)^{\alpha+3/2}}ds$
$\displaystyle\leq\frac{C}{|x-y|\mathfrak{m}_{\alpha}(I(x,|x-y|))},\quad
x,y\in(0,\infty),\ x\neq y;$

  3. (c)

by [4, Lemma 3.4 (E7)] (with $x$ and $y$ interchanged), and proceeding like
before,

$\displaystyle|\partial_{y}\mathbb{K}_{\phi,1}^{\alpha}(x,y)|$
$\displaystyle\leq
C\int_{0}^{\infty}|\varphi(t)|\left(\frac{e^{-t}}{1-e^{-2t}}\right)^{\alpha+1}\int_{-1}^{1}e^{-\frac{q_{-}(e^{-t}x,y,s)}{1-e^{-2t}}}$
$\displaystyle\qquad\times\left[\frac{|y-e^{-t}xs|}{1-e^{-2t}}\varphi(x,y,s)+|\partial_{y}\varphi(x,y,s)|\right]\Pi_{\alpha}(s)dsdt$
$\displaystyle\leq
C\int_{0}^{\infty}|\varphi(t)|\left(\frac{e^{-t}}{1-e^{-2t}}\right)^{\alpha+1}\int_{-1}^{1}e^{-\frac{q_{-}(e^{-t}x,y,s)}{1-e^{-2t}}}$
$\displaystyle\qquad\times\left[\frac{\sqrt{q_{-}(e^{-t}x,y,s)}}{1-e^{-2t}}+\frac{1}{\sqrt{q_{-}(x,y,s)}}\right]\Pi_{\alpha}(s)dsdt$
$\displaystyle\leq
C\int_{0}^{\infty}|\varphi(t)|e^{-t(\alpha+1)}dt\int_{-1}^{1}\frac{\Pi_{\alpha}(s)}{q_{-}(x,y,s)^{\alpha+3/2}}ds$
$\displaystyle\leq\frac{C}{|x-y|\mathfrak{m}_{\alpha}(I(x,|x-y|))},\quad
x,y\in(0,\infty),\ x\neq y.$

All of the above proves that $T_{M}^{\alpha,\textup{loc}}$ is an
$\mathfrak{m}_{\alpha}$-Calderón-Zygmund operator. Therefore,
$T_{M}^{\alpha,\textup{loc}}$ is bounded on
$L^{p}((0,\infty),\mathfrak{m}_{\alpha})$ for every $1<p<\infty$, which yields
$Q_{\phi,*,a}^{\alpha,\textup{loc}}$ is also bounded on
$L^{p}((0,\infty),\mathfrak{m}_{\alpha})$ for every ${1<p<\infty}$. The
arguments in [5, § 2] allow us to deduce that
$Q_{\phi,*,a}^{\alpha,\textup{loc}}$ is also bounded on
$L^{p}((0,\infty),\gamma_{\alpha})$ for every ${1<p<\infty}$.

Finally, we conclude that $Q^{\alpha}_{\phi,*,a}$ is bounded on
$L^{p}((0,\infty),\gamma_{\alpha})$ for any $1<p<\infty$.

######  Remark 4.1.

We can also prove that $Q^{\alpha}_{\phi,*,a}$ is bounded from
$L^{1}((0,\infty),\gamma_{\alpha})$ to
$L^{1,\infty}((0,\infty),\gamma_{\alpha})$. Actually, it is sufficient at this
moment to know that $Q^{\alpha}_{\phi,*,a}$ is bounded on
$L^{p_{0}}((0,\infty),\gamma_{\alpha})$ for some $1<p_{0}<\infty$.

We have proved above that

$|K_{\phi}^{\alpha,\textup{loc}}(x,y)|\leq
C\frac{e^{y^{2}}}{\mathfrak{m}_{\alpha}(I(x,|x-y|))},\quad x,y\in(0,\infty),\
x\neq y.$

We also saw that

$|K_{\phi}^{\alpha,\textup{glob}}(x,y)|\leq
C\int_{-1}^{1}K^{\alpha}(x,y,s)\chi_{L_{1}^{c}}(x,y,s)\Pi_{\alpha}(s)ds,\quad
x,y\in(0,\infty),$

where $K^{\alpha}(x,y,s)$ was defined in (3.7). It is easy to see that, for
any $(x,y,s)\in L_{1}^{c}$,

$|K^{\alpha}(x,y,s)|\leq C\begin{cases}1,&s\in(-1,0),\\\
\frac{\exp\left(\frac{x^{2}+y^{2}}{2}\right)}{q_{-}(x,y,s)^{\alpha+1}},&s\in[0,1).\end{cases}$

Moreover, for any fixed constant $c>0$, if $x,y\in(0,\infty)$ with $|x-y|\leq
cam(x)$ and $s\in(-1,1)$,

$\displaystyle q_{-}(x,y,s)$
$\displaystyle=(x-y)^{2}+2xy(1-s)\leq(x-y)^{2}+4y(|x-y|+y)$ $\displaystyle\leq
5(x-y)^{2}+4y^{2}+4|x-y|x\leq C(1+y^{2}),$

which yields

$|K^{\alpha}(x,y,s)|\leq\frac{C}{q_{-}(x,y,s)^{\alpha+1}}\begin{cases}(1+y^{2})^{\alpha+1},&s\in(-1,0),\\\
\exp\left(\frac{x^{2}+y^{2}}{2}\right),&s\in[0,1),\end{cases}$

for any $(x,y,s)\in L_{1}^{c}$ with $|x-y|\leq cam(x)$.

According to [2, Lemma 3.1], we obtain

$|K_{\phi}^{\alpha,\textup{glob}}(x,y)|\leq
C\frac{e^{\frac{x^{2}+y^{2}}{2}}}{\mathfrak{m}_{\alpha}(I(x,|x-y|))},\quad
x,y\in(0,\infty),0<|x-y|\leq cam(x).$

Hence, we conclude that

$|K_{\phi}^{\alpha}(x,y)|\leq
C\frac{e^{\frac{x^{2}+y^{2}}{2}}}{\mathfrak{m}_{\alpha}(I(x,|x-y|))},\quad
x,y\in(0,\infty),0<|x-y|\leq cam(x).$

We are going to prove now that

$\sup_{I\in\mathcal{B}_{1}}\sup_{x\in I}\int_{(0,\infty)\setminus
2I}|\partial_{x}K_{\phi}^{\alpha}(x,y)|d\gamma_{\alpha}(y)<\infty.$

By partial integration, we have that

$K_{\phi}^{\alpha}(x,y)=\int_{0}^{\infty}\phi^{\prime}(t)W_{t}^{\alpha}(x,y)dt,\quad
x,y,\in(0,\infty),$

and thus,

$\displaystyle\partial_{x}K_{\phi}^{\alpha}(x,y)$
$\displaystyle=\int_{0}^{\infty}\phi^{\prime}(t)\partial_{x}W_{t}^{\alpha}(x,y)dt$
$\displaystyle=-2\int_{0}^{\infty}\phi^{\prime}(t)\left(\frac{e^{-t}}{1-e^{-2t}}\right)^{\alpha+1}$
$\displaystyle\quad\times\int_{-1}^{1}e^{-\frac{q_{-}(e^{-t}x,y,s)}{1-e^{-2t}}+y^{2}}\frac{e^{-t}(e^{-t}x-ys)}{1-e^{-2t}}\Pi_{\alpha}(s)dsdt,\quad
x,y\in(0,\infty).$

By [4, Lemma 3.4 (E8)], $|e^{-t}x-ys|\leq\sqrt{q_{-}(e^{-t}x,y,s)}$ for every
$x,y\in(0,\infty)$ and $s\in(-1,1)$. Then, using the hypothesis on
$\phi^{\prime}$,

$\displaystyle|\partial_{x}K_{\phi}^{\alpha}(x,y)|$ $\displaystyle\leq
C\int_{0}^{\infty}|\phi^{\prime}(t)|\frac{e^{-t(\alpha+2)}}{(1-e^{-2t})^{\alpha+1}}$
$\displaystyle\quad\times\int_{-1}^{1}e^{-\frac{q_{-}(e^{-t}x,y,s)}{1-e^{-2t}}+y^{2}}\sqrt{q_{-}(e^{-t}x,y,s)}\Pi_{\alpha}(s)dsdt$
$\displaystyle\leq
C\int_{0}^{\infty}\frac{1}{t}\frac{e^{-t(\alpha+2)}}{(1-e^{-2t})^{\alpha+3/2}}\int_{-1}^{1}e^{-c\frac{q_{-}(e^{-t}x,y,s)}{1-e^{-2t}}+y^{2}}\Pi_{\alpha}(s)dsdt$

for any $x,y\in(0,\infty)$.

Therefore, by [4, Lemma 3.6], there exists $C>0$ such that

$\sup_{x\in I}\int_{(0,\infty)\setminus
2I}|\partial_{x}K_{\phi}^{\alpha}(x,y)|d\gamma_{\alpha}(y)\leq C$

for every $I\in\mathcal{B}_{1}$, as claimed.

## References

  * [1] Bennett, C. Another characterization of BLO.  Proc. Amer. Math. Soc. 85, 4 (1982), 552–556.
  * [2] Betancor, J. J., Castro, A. J., and Nowak, A. Calderón–Zygmund operators in the Bessel setting.  Monatsh. Math. 167, 3-4 (2012), 375–403.
  * [3] Betancor, J. J., Crescimbeni, R., and Torrea, J. L. Oscillation and variation of the Laguerre heat and Poisson semigroups and Riesz transforms.  Acta Math. Sci. Ser. B (Engl. Ed.) 32, 3 (2012), 907–928.
  * [4] Betancor, J. J., Dalmasso, E., Quijano, P., and Scotto, R. Endpoint estimates for harmonic analysis operators associated with Laguerre polynomial expansions, 2022.  arXiv:2210.14394.
  * [5] Betancor, J. J., Dalmasso, E., Quijano, P., and Scotto, R. Harmonic analysis operators associated with Laguerre polynomial expansions on variable Lebesgue spaces, 2022.  arXiv:2202.11137.
  * [6] Betancor, J. J., Dalmasso, E., Quijano, P., and Scotto, R. Maximal function characterization of hardy spaces related to Laguerre polynomial expansions, 2022.  arXiv:2208.06498.
  * [7] Betancor, J. J., Fariña, J. C., Harboure, E., and Rodríguez-Mesa, L. Variation operators for semigroups and Riesz transforms on $BMO$ in the Schrödinger setting.  Potential Anal. 38, 3 (2013), 711–739.
  * [8] Betancor, J. J., Fariña, J. C., Rodríguez-Mesa, L., and Sanabria-García, A. Higher order Riesz transforms for Laguerre expansions.  Illinois J. Math. 55, 1 (2011), 27–68 (2012).
  * [9] Betancor, J. J., and León-Contreras, M. D. Variation inequalities for Riesz transforms and Poisson semigroups associated with Laguerre polynomial expansions, 2021.  arXiv:2110.03493.
  * [10] Bourgain, J. Pointwise ergodic theorems for arithmetic sets.  Inst. Hautes Études Sci. Publ. Math., 69 (1989), 5–45.  With an appendix by the author, Harry Furstenberg, Yitzhak Katznelson and Donald S. Ornstein.
  * [11] Campbell, J. T., Jones, R. L., Reinhold, K., and Wierdl, M. Oscillation and variation for the Hilbert transform.  Duke Math. J. 105, 1 (2000), 59–83.
  * [12] Campbell, J. T., Jones, R. L., Reinhold, K., and Wierdl, M. Oscillation and variation for singular integrals in higher dimensions.  Trans. Amer. Math. Soc. 355, 5 (2003), 2115–2137.
  * [13] Chen, W., Meng, Y., and Yang, D. Calderón-Zygmund operators on Hardy spaces without the doubling condition.  Proc. Amer. Math. Soc. 133, 9 (2005), 2671–2680.
  * [14] Chen, Y., Ding, Y., Hong, G., and Liu, H. Weighted jump and variational inequalities for rough operators.  J. Funct. Anal. 274, 8 (2018), 2446–2475.
  * [15] Coifman, R. R., and Rochberg, R. Another characterization of BMO.  Proc. Amer. Math. Soc. 79, 2 (1980), 249–254.
  * [16] Coifman, R. R., and Weiss, G. Analyse harmonique non-commutative sur certains espaces homogènes.  Lecture Notes in Mathematics, Vol. 242. Springer-Verlag, Berlin-New York, 1971.  Étude de certaines intégrales singulières.
  * [17] Crescimbeni, R., Macías, R. A., Menárguez, T., Torrea, J. L., and Viviani, B. The $\rho$-variation as an operator between maximal operators and singular integrals.  J. Evol. Equ. 9, 1 (2009), 81–102.
  * [18] Dinger, U. Weak type $(1,1)$ estimates of the maximal function for the Laguerre semigroup in finite dimensions.  Rev. Mat. Iberoamericana 8, 1 (1992), 93–120.
  * [19] Forzani, L., Sasso, E., and Scotto, R. Weak-type inequalities for higher order Riesz-Laguerre transforms.  J. Funct. Anal. 256, 1 (2009), 258–274.
  * [20] Fu, X., Lin, H., Yang, D., and Yang, D. Hardy spaces $H^{p}$ over non-homogeneous metric measure spaces and their applications.  Sci. China Math. 58, 2 (2015), 309–388.
  * [21] Fu, X., Yang, D., and Yang, D. The molecular characterization of the Hardy space $H^{1}$ on non-homogeneous metric measure spaces and its application.  J. Math. Anal. Appl. 410, 2 (2014), 1028–1042.
  * [22] Harboure, E., Macías, R. A., Menárguez, M. T., and Torrea, J. L. Oscillation and variation for the Gaussian Riesz transforms and Poisson integral.  Proc. Roy. Soc. Edinburgh Sect. A 135, 1 (2005), 85–104.
  * [23] Hu, G., Yang, D., and Yang, D. $h^{1}$, bmo, blo and Littlewood-Paley $g$-functions with non-doubling measures.  Rev. Mat. Iberoam. 25, 2 (2009), 595–667.
  * [24] Hytönen, T. A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa.  Publ. Mat. 54, 2 (2010), 485–504.
  * [25] Hytönen, T., Yang, D., and Yang, D. The Hardy space $H^{1}$ on non-homogeneous metric spaces.  Math. Proc. Cambridge Philos. Soc. 153, 1 (2012), 9–31.
  * [26] Jiang, Y. Spaces of type BLO for non-doubling measures.  Proc. Amer. Math. Soc. 133, 7 (2005), 2101–2107.
  * [27] Jones, R. L., Kaufman, R., Rosenblatt, J. M., and Wierdl, M. Oscillation in ergodic theory.  Ergodic Theory Dynam. Systems 18, 4 (1998), 889–935.
  * [28] Lebedev, N. N. Special functions and their applications.  Dover Publications, Inc., New York, 1972.  Revised edition, translated from the Russian and edited by Richard A. Silverman, Unabridged and corrected republication.
  * [29] Leckband, M. A. Structure results on the maximal Hilbert transform and two-weight norm inequalities.  Indiana Univ. Math. J. 34, 2 (1985), 259–275.
  * [30] Lin, H., Liu, Z., and Wang, C. The John-Nirenberg inequality for the regularized BLO space on non-homogeneous metric measure spaces.  Canad. Math. Bull. 63, 3 (2020), 643–654.
  * [31] Lin, H., and Yang, D. Spaces of type BLO on non-homogeneous metric measure.  Front. Math. China 6, 2 (2011), 271–292.
  * [32] Liu, L., and Yang, D. BLO spaces associated with the Ornstein-Uhlenbeck operator.  Bull. Sci. Math. 132, 8 (2008), 633–649.
  * [33] Ma, T., Torrea, J. L., and Xu, Q. Weighted variation inequalities for differential operators and singular integrals.  J. Funct. Anal. 268, 2 (2015), 376–416.
  * [34] Ma, T., Torrea, J. L., and Xu, Q. Weighted variation inequalities for differential operators and singular integrals in higher dimensions.  Sci. China Math. 60, 8 (2017), 1419–1442.
  * [35] Mas, A. Variation for singular integrals on Lipschitz graphs: $L^{p}$ and endpoint estimates.  Trans. Amer. Math. Soc. 365, 11 (2013), 5759–5781.
  * [36] Mas, A., and Tolsa, X. $L^{p}$ estimates for the variation for singular integrals on uniformly rectifiable sets.  Trans. Amer. Math. Soc. 369, 11 (2017), 8239–8275.
  * [37] Mauceri, G., and Meda, S. ${\rm BMO}$ and $H^{1}$ for the Ornstein-Uhlenbeck operator.  J. Funct. Anal. 252, 1 (2007), 278–313.
  * [38] Mirek, M., Trojan, B., and Zorin-Kranich, P. Variational estimates for averages and truncated singular integrals along the prime numbers.  Trans. Amer. Math. Soc. 369, 8 (2017), 5403–5423.
  * [39] Muckenhoupt, B. Conjugate functions for Laguerre expansions.  Trans. Amer. Math. Soc. 147 (1970), 403–418.
  * [40] Nazarov, F., Treil, S., and Volberg, A. The $Tb$-theorem on non-homogeneous spaces.  Acta Math. 190, 2 (2003), 151–239.
  * [41] Nowak, A. On Riesz transforms for Laguerre expansions.  J. Funct. Anal. 215, 1 (2004), 217–240.
  * [42] Nowak, A., and Stempak, K. $L^{2}$-theory of Riesz transforms for orthogonal expansions.  J. Fourier Anal. Appl. 12, 6 (2006), 675–711.
  * [43] Nowak, A., and Stempak, K. Riesz transforms for multi-dimensional Laguerre function expansions.  Adv. Math. 215, 2 (2007), 642–678.
  * [44] Qian, J. The $p$-variation of partial sum processes and the empirical process.  Ann. Probab. 26, 3 (1998), 1370–1383.
  * [45] Sasso, E. Spectral multipliers of Laplace transform type for the Laguerre operator.  Bull. Austral. Math. Soc. 69, 2 (2004), 255–266.
  * [46] Sasso, E. Functional calculus for the Laguerre operator.  Math. Z. 249, 3 (2005), 683–711.
  * [47] Sasso, E. Weak type estimates for Riesz-Laguerre transforms.  Bull. Austral. Math. Soc. 75, 3 (2007), 397–408.
  * [48] Stein, E. M. Topics in harmonic analysis related to the Littlewood-Paley theory.  Annals of Mathematics Studies, No. 63. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970.
  * [49] Tolsa, X. BMO, $H^{1}$, and Calderón-Zygmund operators for non doubling measures.  Math. Ann. 319, 1 (2001), 89–149.
  * [50] Tolsa, X. Littlewood-Paley theory and the $T(1)$ theorem with non-doubling measures.  Adv. Math. 164, 1 (2001), 57–116.
  * [51] Tolsa, X. The space $H^{1}$ for nondoubling measures in terms of a grand maximal operator.  Trans. Amer. Math. Soc. 355, 1 (2003), 315–348.
  * [52] Yang, D., and Yang, D. Real-variable characterizations of Hardy spaces associated with Bessel operators.  Anal. Appl. (Singap.) 9, 3 (2011), 345–368.