# The dual of the Hardy space associated with the Dunkl operators ††thanks:
Supported by the National Natural Science Foundation of China (No. 12071295).
†Corresponding author.
E-mail<EMAIL_ADDRESS>(J.-X. Jiu<EMAIL_ADDRESS>(Zh.-K. Li).

Jiaxi Jiu and Zhongkai Li†
Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

###### Abstract

The rational Dunkl operators are commuting differential-reflection operators
on the Euclidean space ${\mathbb{R}}^{d}$ associated with a root system, that
contain some non-local refection terms, and the associated Hardy space is
defined by means of the Riesz transforms with respect to the Dunkl operators.
The aim of the paper is to prove that its dual can be realized by a class of
functions on ${\mathbb{R}}^{d}$, denoted by ${\text{BMC}}_{\kappa}$ in the
text, that consists of the underlying functions of a certain type of weighted
Carleson measures. Our method is “purely analytic” and does not depend on the
atomic decomposition. As a corollary we obtain the Fefferman-Stein
decomposition of functions in ${\text{BMC}}_{\kappa}$.

2020 MS Classification: 42B30 (Primary), 42B20, 51F15 (Secondary)
Key Words and Phrases: Hardy space; Dunkl operator; Riesz transform; Carleson
measure, weighted BMO space

##  1 Introduction and main results

1.2. Let $R$ be a root system in ${\mathbb{R}}^{d}$ normalized so that
$\langle\alpha,\alpha\rangle=2$ for $\alpha\in R$ and with $R_{+}$ a fixed
positive subsystem, and $G$ the finite reflection group generated by the
reflections $\sigma_{\alpha}$ ($\alpha\in R$), where
$\sigma_{\alpha}(x)=x-\langle\alpha,x\rangle\alpha$ for $x\in{\mathbb{R}}^{d}$
and $\langle\cdot,\cdot\rangle$ stands for the usual inner product on
${\mathbb{R}}^{d}$. For a nonnegative multiplicity function $\kappa$ defined
on $R$ (invariant under $G$), the (rational) Dunkl operators $D_{j}$ ($1\leq
j\leq d)$ are defined by (cf. [11])

$\displaystyle D_{j}=\partial_{j}+\sum_{\alpha\in
R_{+}}\kappa(\alpha)\alpha_{j}\frac{1-\sigma_{\alpha}}{\langle\alpha,x\rangle},$
(1)

where $\sigma_{\alpha}f=f\circ\sigma_{\alpha}$ and $\alpha_{j}$ ($1\leq j\leq
d$) is the $j$th coordinate of $\alpha$. It is proved in [11] that the Dunkl
operators generate a commutative algebra and are anti-symmetry associated to
the measure

$\displaystyle d\omega_{\kappa}=W_{\kappa}\,dx,\qquad{\rm where}\quad
W_{\kappa}(x)=\prod_{\alpha\in
R_{+}}|\langle\alpha,x\rangle|^{2\kappa(\alpha)}.$

The weight $W_{\kappa}$ is $G$-invariant and positive homogeneous of degree
$2|\kappa|$, so that the homogeneous dimension of $d\omega_{\kappa}$ is

$\displaystyle N=2|\kappa|+d,\qquad\hbox{where}\quad|\kappa|=\sum_{\alpha\in
R_{+}}k(\alpha).$

It is also noted that ${\mathbb{R}}^{d}$, equipped with the Euclidean distance
$(x,y)\in{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}\mapsto|x-y|$ and with the
measure $\omega_{\kappa}$, is a space of homogeneous type in the sense of
Coifman and Weiss [8]. For $0<p<\infty$, the space
$L_{\kappa}^{p}({\mathbb{R}}^{d})$ consists of all measurable functions $f$ on
${\mathbb{R}}^{d}$ such that $\|f\|_{L_{\kappa}^{p}}<+\infty$, where
$\|f\|^{p}_{L_{\kappa}^{p}}=c_{\kappa}\int_{{\mathbb{R}}^{d}}|f|^{p}\,d\omega_{\kappa}$
and $c_{\kappa}^{-1}=\int_{{\mathbb{R}}^{d}}e^{-|x|^{2}/2}d\omega_{\kappa}$.
$L_{\kappa}^{\infty}({\mathbb{R}}^{d})$ is identical to the usual
$L^{\infty}({\mathbb{R}}^{d})$.

A typical example is $G=S_{d}$ (the type $A_{d-1}$ case), the symmetric group
in $d$ elements, a root system of which is given by
$R=\\{\pm(e_{i}-e_{j}):\,1\leq i<j\leq d\\}$. Since $S_{d}$ has only one orbit
in $R$, a multiplicity function $\kappa$ is constant. The Dunkl operators for
a given $\kappa\geq 0$ have the form

$\displaystyle D_{j}=\partial_{j}+\kappa\sum_{i\neq
j}\frac{1-\sigma_{ij}}{x_{j}-x_{i}},\qquad j=1,2,\cdots,d,$

where
$\sigma_{ij}(\dots,x_{i},\dots,x_{j},\dots)=(\dots,x_{j},\dots,x_{i},\dots)$.

In the general setting, the Laplacian $\Delta_{\kappa}$ associated to a root
system $R$ and a multiplicity function $\kappa$, called the
$\kappa$-Laplacian, is defined by $\Delta_{\kappa}=\sum_{j=1}^{d}D_{j}^{2}$,
which can be explicitly expressed by (cf. [10])

$\displaystyle\Delta_{\kappa}=\sum_{j=1}^{d}\partial_{j}^{2}+2\sum_{\alpha\in
R_{+}}\kappa(\alpha)\left(\frac{\langle\alpha,\nabla\rangle}{\langle\alpha,x\rangle}-\frac{1-\sigma_{\alpha}}{\langle\alpha,x\rangle^{2}}\right),$

where $\nabla$ is the usual gradient operator on ${\mathbb{R}}^{d}$. Note that
the operator $-\Delta_{\kappa}$ is well defined on
${\mathscr{S}}({\mathbb{R}}^{d})$ (the Schwartz space), symmetric and positive
on $L_{\kappa}^{2}({\mathbb{R}}^{d})$, and has a unique positive self-adjoint
extension (cf. [2]).

During the past several decades, the Dunkl operators have gained considerable
interest in various fields of mathematics and in physical applications; in
particular, they admit a far reaching generalization of Fourier analysis which
includes most special functions related to root systems, such as spherical
functions on Riemannian symmetric spaces. See, for example, [15, 47] on the
rational Dunkl theory, [43] on the trigonometric Dunkl theory, [6, 39] on
affine Hecke algebras and the $q$-Dunkl theory, [26] on the probabilistic
aspects of Dunkl theory, and [23, 34] on integrable systems related to the
Dunkl theory.

We consider the Riesz transforms associated with the Dunkl operators defined
by

${\mathcal{R}}_{j}=D_{j}(-\Delta_{\kappa})^{-1/2},\qquad j=1,\cdots,d,$

and call them the $\kappa$-Riesz transforms. They can be interpreted according
to the associated Fourier transform, i.e., the Dunkl transform; see Section 3
below. We note that these transforms were considered first in [38]. In the
rank-one case ($d=1$), the boundedness in $L_{\kappa}^{p}$ for $1<p<\infty$,
of the only one member named the $\kappa$-Hilbert transform, was proved in
[56] whenever $2|\kappa|$ is a nonnegative integer, and then in [1] without
restriction on $2|\kappa|$; furthermore, the weak (1,1) boundedness of the
maximal $\kappa$-Hilbert transform was proved in [35, 37]. In the higher-
dimensional case, the boundedness of the $\kappa$-Riesz transforms in
$L_{\kappa}^{p}$ for $1<p<\infty$ was proved in [3].

Analogously to the classical case, we define the Hardy space in the Dunkl
setting by

$\displaystyle H_{\kappa}^{1}({\mathbb{R}}^{d})=\left\\{f\in
L_{\kappa}^{1}({\mathbb{R}}^{d}):\quad{\mathcal{R}}_{j}f\in
L_{\kappa}^{1}({\mathbb{R}}^{d}),\,\,\,j=1,\cdots,d\right\\},$

and endow it with the norm
$\|f\|_{H_{\kappa}^{1}}=\|f\|_{L_{\kappa}^{1}}+\sum_{j=1}^{d}\|{\mathcal{R}}_{j}f\|_{L_{\kappa}^{1}}$.
The space $H_{\kappa}^{1}$ for the rank-one case was studied first in [35, 37]
(including $H_{\kappa}^{p}$ for some values $p<1$; also see [36] for its
analogue on the circle), and for the product case in [4, 20]. Associated to
general root systems in the higher-dimensional case, the Hardy spaces
$H_{\kappa}^{1}({\mathbb{R}}^{d})$ was recently exploited in [5, 22], and in
particular, it was proved in [22] that $H_{\kappa}^{1}({\mathbb{R}}^{d})$
admits the atomic decomposition in the sense of Coifman and Weiss [8] basing
on a square function characterization of $H_{\kappa}^{1}({\mathbb{R}}^{d})$
given in [5] and a decomposition of the tent space developed in [7, 49].
However in [5], such a square function characterization of
$H_{\kappa}^{1}({\mathbb{R}}^{d})$ was obtained with arduous effort, and
besides some subtle estimates of the associated heat kernel, through a variety
of distinct approaches from [9, 28, 49, 50], etc. For other aspects of the
Hardy spaces, see [18, 19, 24, 40, 52, 53, 54, 59] for example.

We focus on the “analytical” presentation of related theories, because in
Dunkl’s setting there are the relevant Fourier transform, the generalized
translation, Green’s formula, etc, and we expect that in this way one can
better clarify the treatment of some problems, and make more in-depth and
extensive research, or in other words, extend the theory of harmonic analysis
as far as possible to the Dunkl setting. For example, in a recent work [32] a
Lusin-type area integral is defined by

$\displaystyle\left(S_{a,h}u\right)(x)=\left(\iint_{\Gamma_{a}^{h}(0)}\left[\tau_{x}(\Delta_{\tilde{\kappa}}u^{2})\right](x_{0},t)\,x_{0}^{1-d-2|\kappa|}\,dx_{0}d\omega_{\kappa}(t)\right)^{1/2}$

for a $C^{2}$ function $u$ on the upper half-space
${\mathbb{R}}^{1+d}_{+}:=(0,\infty)\times{\mathbb{R}}^{d}$, where
$\Gamma^{h}_{a}(0)$ is the positive truncated cone of aperture $a>0$ and
height $h>0$ with vertex $(0,0)$,
$\Delta_{\tilde{\kappa}}=\partial_{x_{0}}^{2}+\Delta_{\kappa}$, and $\tau_{x}$
is the generalized translation in the Dunkl setting. We have applied the
function $S_{a,h}u$ in [32] to characterize the local existence of non-
tangential boundary values of a generalized harmonic function $u$ on
${\mathbb{R}}^{1+d}_{+}$.

In the present paper we take up the $\kappa$-Riesz transforms
${\mathcal{R}}_{j}$ ($1\leq j\leq d$) and the Hardy spaces
$H_{\kappa}^{1}({\mathbb{R}}^{d})$ from the so-called analytic point of view.
These transforms are treated in conjunction with the associated conjugate
Poisson integrals, which allows us to borrow some conclusions about the
generalized harmonic functions and yields a better understanding of them. We
give a direct proof of the boundedness of the $\kappa$-Riesz transforms from
$L^{\infty}({\mathbb{R}}^{d})$ to the space
${\text{BMO}}_{\kappa}({\mathbb{R}}^{d})$ of functions of bounded mean
oscillation with respect to the measure $d\omega_{\kappa}$. The major portion
of the the paper is to prove that the dual of the Hardy spaces
$H_{\kappa}^{1}({\mathbb{R}}^{d})$ can be realized by a class of functions on
${\mathbb{R}}^{d}$, denoted by ${\text{BMC}}_{\kappa}({\mathbb{R}}^{d})$, that
consists of the underlying functions of a certain type of Carleson measures
associated with $d\omega_{\kappa}$. The proof is based on the original ideas
of Fefferman and Stein [24] and does not depend on the atomic decomposition.
As a corollary we obtain the Fefferman-Stein decomposition of functions in
${\text{BMC}}_{\kappa}({\mathbb{R}}^{d})$. Our argument naturally implies that
${\text{BMC}}_{\kappa}({\mathbb{R}}^{d})$ is a subclass of
${\text{BMO}}_{\kappa}({\mathbb{R}}^{d})$, and the converse of this inclusion
is a consequence of the atomic decomposition in [22].

The paper is organized as follows. Some basic facts on the rational Dunkl
theory are summarized in Section 2. Section 3 is devoted to the theory of the
associated conjugate Poisson integrals, i.e., the conjugate $\kappa$-Poisson
integrals, and the $\kappa$-Riesz transforms ${\mathcal{R}}_{j}$ ($1\leq j\leq
d$); and in particular, it is concluded that the maximal $\kappa$-Riesz
transforms ${\mathcal{R}}_{j}^{*}$ are of strong type $(p,p)$ for
$1<p<\infty$, and of weak type $(1,1)$. The fundamental results of the
“analytic” Hardy space on ${\mathbb{R}}^{1+d}_{+}$ in the Dunkl setting,
denoted by ${\mathscr{H}}_{\kappa}^{1}({\mathbb{R}}^{1+d}_{+})$, and the
“real” Hardy space $H_{\kappa}^{1}({\mathbb{R}}^{d})$ are contained also in
Section 3. That the $\kappa$-Riesz transforms map
$L^{\infty}({\mathbb{R}}^{d})$ into the space
${\text{BMO}}_{\kappa}({\mathbb{R}}^{d})$ is proved in Section 4. In Section
5, we introduce the class ${\text{BMC}}_{\kappa}({\mathbb{R}}^{d})$ of
functions on ${\mathbb{R}}^{d}$ in terms of Carleson measures associated with
$d\omega_{\kappa}$ (see Definition 5.4), and prove that the $\kappa$-Riesz
transforms are bounded from $L^{\infty}({\mathbb{R}}^{d})$ to
${\text{BMC}}_{\kappa}({\mathbb{R}}^{d})$. Section 6 is devoted to the proof
of that the space ${\text{BMC}}_{\kappa}({\mathbb{R}}^{d})$ is the dual of the
Hardy spaces $H_{\kappa}^{1}({\mathbb{R}}^{d})$. Several remarks on the spaces
$H_{\kappa}^{1}({\mathbb{R}}^{d})$, ${\text{BMC}}_{\kappa}({\mathbb{R}}^{d})$
and ${\text{BMO}}_{\kappa}({\mathbb{R}}^{d})$ are given in the last section.

Throughout the paper, $X\lesssim Y$ means that $X\leq cY$ for some constant
$c>0$ independent of variables, functions, etc. We say $X\asymp Y$ if both
$X\lesssim Y$ and $Y\lesssim X$ hold, and for a measurable set
$E\subset{\mathbb{R}}^{d}$, put $|E|_{\kappa}=\int_{E}d\omega_{\kappa}$. As
usual $\SS({\mathbb{R}}^{d})$ designates the space of $C^{\infty}$ functions
on ${\mathbb{R}}^{d}$ rapidly decreasing together with their derivatives,
$C_{0}({\mathbb{R}}^{d})$ the space of continuous functions on
${\mathbb{R}}^{d}$ vanishing at infinity with norm
$\|f\|_{C_{0}}=\sup_{x\in{\mathbb{R}}^{d}}|f(x)|$, $L_{\kappa,{\rm
loc}}({\mathbb{R}}^{d})$ the set of locally integrable functions on
${\mathbb{R}}^{d}$ with respect to the measure $d\omega_{\kappa}$, and
${\mathfrak{B}}_{\kappa}({\mathbb{R}}^{d})$ the space of Borel measures $d\nu$
on ${\mathbb{R}}^{d}$ for which
$\|d\nu\|_{{\mathfrak{B}_{\kappa}}}=c_{\kappa}\int_{{\mathbb{R}}^{d}}W_{\kappa}\,|d\nu|$
is finite. We shall use the notation $B(x,r)$ (or $B((x_{0},x),r)$) to denote
the open ball in ${\mathbb{R}}^{d}$ (or ${\mathbb{R}}_{+}^{1+d}$) with radius
$r$ centered at $x\in{\mathbb{R}}^{d}$ (or $(x_{0},x)\in{\mathbb{R}}^{1+d}$).

##  2 Some facts in the Dunkl theory

In this section we give an account on results from the Dunkl theory which will
be relevant for the sequel. Concerning root systems and reflection groups, see
[29].

###  2.1 The Dunkl kernel and the Dunkl transform

As in the first section, for a root system $R$ let $G$ be the associated
reflection group and $\kappa$ a given nonnegative multiplicity function. As
shown in [12], there exists a unique degree-of-homogeneity-preserving linear
isomorphism $V_{\kappa}$ on polynomials, which intertwines the associated
commutative algebra of Dunkl operators and the algebra of usual partial
differential operators, namely, $D_{j}V_{\kappa}=V_{\kappa}\partial_{j}$ for
$1\leq i\leq d$. $V_{\kappa}$ commutes with the group action of $G$. (For a
thorough analysis on $V_{\kappa}$ with general $\kappa$, see [14].) It was
proved in [45] that $V_{\kappa}$ is positive on the space of polynomials, and
moreover, for each $x\in{\mathbb{R}}^{d}$ there exists a unique (Borel)
probability measure $\mu^{\kappa}_{x}$ of ${\mathbb{R}}^{d}$ such that

$V_{\kappa}f(x)=\int_{{\mathbb{R}}^{d}}f(\xi)\,d\mu^{\kappa}_{x}(\xi),$

where $\mu^{\kappa}_{x}$ is of compact support with

$\displaystyle\hbox{supp}\,\mu^{\kappa}_{x}\subseteq\hbox{co}\,\\{\sigma(x):\,\,\sigma\in
G\\},$ (2)

the convex hull of the orbit of $x$ under $G$. The intertwining operator
$V_{\kappa}$ has an extension to $C^{\infty}({\mathbb{R}}^{d})$ and
establishes a homeomorphism of this space (see [57, 58]). For $\delta>0$ and
$x\in{\mathbb{R}}^{d}$ we have the distribution estimate (see [31])

$\displaystyle\int_{\langle
x,\xi\rangle>|x|^{2}-\delta^{2}}\,d\mu^{\kappa}_{x}(\xi)\asymp\frac{\delta^{2|\kappa|+d}}{|{\mathbb{B}}(x,\delta)|_{\kappa}},$

which reveals the behavior of the representing measures $\mu^{\kappa}_{x}$
near the point $x$.

Associated with $G$ and $\kappa$, the Dunkl kernel is defined by

$\displaystyle
E_{\kappa}(x,z)=V_{\kappa}\left(e^{\langle\cdot,z\rangle}\right)(x)=\int_{{\mathbb{R}}^{d}}e^{\langle\xi,z\rangle}\,d\mu^{\kappa}_{x}(\xi),\qquad
x\in{\mathbb{R}}^{d},\,\,z\in{\mathbb{C}}^{d}.$

According to [42], for fixed $z\in{\mathbb{C}}^{d}$, $x\mapsto
E_{\kappa}(x,z)$ may be characterized as the unique analytic solution of the
system

$\displaystyle D^{x}_{j}E_{\kappa}(x,z)=z_{j}E_{\kappa}(x,z),\qquad
x\in{\mathbb{R}}^{d},\,\,\,j=1,\dots,d,$ (3)

with the initial value $E_{\kappa}(0,z)=1$; and $E_{\kappa}$ has a unique
holomorphic extension to ${\mathbb{C}}^{d}\times{\mathbb{C}}^{d}$ and is
symmetric in its arguments (see [30, 42]). Furthermore (cf. [30, 12, 45]), for
$z,w\in{\mathbb{C}}^{d}$, $\lambda\in{\mathbb{C}}$ and $\sigma\in G$, one has

$\displaystyle E_{\kappa}(\lambda z,w)=E_{\kappa}(z,\lambda w),\qquad
E_{\kappa}(\sigma(z),\sigma(w))=E_{\kappa}(z,w),$ (4)

and for $x\in{\mathbb{R}}^{d}$, $z\in{\mathbb{C}}^{d}$, and all multi-indices
$\nu=(\nu_{1},\dots,\nu_{d})$,

$\displaystyle\left|\partial_{z}^{\nu}E_{\kappa}(x,z)\right|\leq|x|^{|\nu|}\max_{\sigma\in
G}e^{{\rm Re}\,\langle\sigma(x),z\rangle}.$ (5)

For $f\in L_{\kappa}^{1}({\mathbb{R}}^{d})$, its Dunkl transform is defined by

$\displaystyle\left({\mathscr{F}}_{\kappa}f\right)(\xi)=c_{\kappa}\int_{{\mathbb{R}}^{d}}f(x)E_{\kappa}(-i\xi,x)\,d\omega_{\kappa}(x),\qquad\xi\in{\mathbb{R}}^{d}.$

The Dunkl transform ${\mathscr{F}}_{\kappa}$ commutes with the $G$-action, and
shares many of the important properties with the usual Fourier transform (see
[13, 30, 45, 48]), part of which are listed as follows.

###### Proposition 2.1

(i) If $f\in L_{\kappa}^{1}({\mathbb{R}}^{d})$, then
${{\mathscr{F}}}_{\kappa}f\in C_{0}({{\mathbb{R}}^{d}})$ and
$\|{{\mathscr{F}}}_{\kappa}f\|_{C_{0}}\leq\|f\|_{L_{\kappa}^{1}}$.

(ii) (Inversion) If $f\in L_{\kappa}^{1}({\mathbb{R}}^{d})$ such that
${\mathscr{F}}_{\kappa}f\in L_{\kappa}^{1}({\mathbb{R}}^{d})$, then
$f(x)=[{\mathscr{F}}_{\kappa}({\mathscr{F}}_{\kappa}f)](-x)$.

(iii) For $f\in{\mathscr{S}}({{\mathbb{R}}^{d}})$, we have
$[{\mathscr{F}}_{\kappa}(D_{j}f)](\xi)=i\xi_{j}({\mathscr{F}}_{\kappa}f)(\xi)$,
$[{\mathscr{F}}_{\kappa}(x_{j}f)](\xi)=i[D_{j}({\mathscr{F}}_{\kappa}f)](\xi)$
for $\xi\in{\mathbb{R}}^{d}$ and $1\leq j\leq d$; and ${\mathscr{F}}_{\kappa}$
is a homeomorphism of ${\mathscr{S}}({\mathbb{R}}^{d})$.

(iv) (product formula) For $f_{1},f_{2}\in L_{\kappa}^{1}({\mathbb{R}}^{d})$,
we have
$\int_{{\mathbb{R}}^{d}}f_{1}\cdot{\mathscr{F}}_{\kappa}f_{2}\,d\omega_{\kappa}=\int_{{\mathbb{R}}^{d}}f_{2}\cdot{\mathscr{F}}_{\kappa}f_{1}\,d\omega_{\kappa}$.

(v) (Plancherel) There exists a unique extension of ${\mathscr{F}}_{\kappa}$
to $L_{\kappa}^{2}({\mathbb{R}}^{d})$ with
$\|{\mathscr{F}}_{\kappa}f\|_{L_{\kappa}^{2}}=\|f\|_{L_{\kappa}^{2}}$ and
${\mathscr{F}}_{\kappa}^{-1}f=({\mathscr{F}}_{\kappa}f)(-\cdot)$.

(vi) If $f\in L_{\kappa}^{1}({\mathbb{R}}^{d})\cup
L_{\kappa}^{2}({\mathbb{R}}^{d})$ is radial, then its Dunkl transform
${{\mathscr{F}}}_{\kappa}f$ is also radial.

(vii) If $f\in L_{\kappa}^{1}({\mathbb{R}}^{d})\cup
L_{\kappa}^{2}({\mathbb{R}}^{d})$, then $[{\mathscr{F}}_{\kappa}(\sigma
f)](\xi)=({\mathscr{F}}_{\kappa}f)(\sigma(\xi))$ for $\xi\in{\mathbb{R}}^{d}$
and $\sigma\in G$; and
$[{\mathscr{F}}_{\kappa}(f(a\cdot))](\xi)=|a|^{-2|\kappa|-d}({\mathscr{F}}_{\kappa}f)(a^{-1}\xi)$
for $\xi\in{\mathbb{R}}^{d}$ and $a\in{\mathbb{R}}\setminus\\{0\\}$.

###  2.2 The generalized translation and convolution

For $x\in{\mathbb{R}}^{d}$, the generalized translation $\tau_{x}$ is defined
in $L_{\kappa}^{2}({\mathbb{R}}^{d})$ by (see [55])

$\displaystyle\left[{\mathscr{F}}_{\kappa}(\tau_{x}f)\right](\xi)=E_{\kappa}(i\xi,x)({\mathscr{F}}_{\kappa}f)(\xi),\qquad\xi\in{\mathbb{R}}^{d}.$
(6)

By Proposition 2.1(v), $\tau_{x}$ is well defined and satisfies
$\|\tau_{x}f\|_{L_{\kappa}^{2}}\leq\|f\|_{L_{\kappa}^{2}}$ in view of (5).

If $f$ is in an appropriate subclass, for example,
$A_{\kappa}({\mathbb{R}}^{d})=\left\\{f\in
L_{\kappa}^{1}({\mathbb{R}}^{d}):\,{\mathscr{F}}_{\kappa}f\in
L_{\kappa}^{1}({\mathbb{R}}^{d})\right\\}$ (cf. [55]) or
${\mathscr{S}}({\mathbb{R}}^{d})$, $\tau_{x}f$ may be expressed pointwise by
(see [44, 55])

$\displaystyle(\tau_{x}f)(t)=c_{\kappa}\int_{{\mathbb{R}}^{d}}E_{\kappa}(i\xi,x)E_{k}(i\xi,t)({\mathscr{F}}_{\kappa}f)(\xi)\,d\omega_{\kappa}(\xi),\qquad
x\in{\mathbb{R}}^{d}.$ (7)

###### Proposition 2.2

(i) If $f,g\in L_{\kappa}^{2}({\mathbb{R}}^{d})$, then for
$x\in{\mathbb{R}}^{d}$,

$\displaystyle\int_{{\mathbb{R}}^{d}}(\tau_{x}f)(t)g(t)\,d\omega_{\kappa}(t)=\int_{{\mathbb{R}}^{d}}f(t)(\tau_{-x}g)(t)\,d\omega_{\kappa}(t),$
(8)

and both sides are continuous in $x$.

(ii) For $f\in A_{\kappa}({\mathbb{R}}^{d})$, the function
$(x,t)\mapsto(\tau_{x}f)(t)$ is continuous on
${\mathbb{R}}^{d}\times{\mathbb{R}}^{d}$.

The equality (8) was proved in [55] for $f\in A_{\kappa}({\mathbb{R}}^{d})$
and $g\in L_{\kappa}^{1}({\mathbb{R}}^{d})\cap L^{\infty}({\mathbb{R}}^{d})$,
and the general case follows from a density argument; for the continuity, see
[32].

In [58] an abstract form of $\tau_{x}$ is given by
$(\tau_{x}f)(t)=V_{\kappa}^{t}V_{\kappa}^{x}\left[(V_{\kappa}^{-1}f)(x+t)\right]$
for $f\in C^{\infty}({\mathbb{R}}^{d})$ and $x,t\in{\mathbb{R}}^{d}$.

###### Proposition 2.3

([58]) (i) For fixed $x\in{\mathbb{R}}^{d}$, $\tau_{x}$ is a continuous linear
mapping from $C^{\infty}({\mathbb{R}}^{d})$ into itself, and satisfies
$D_{j}(\tau_{x}f)=\tau_{x}(D_{j}f)$ ($j=1,\dots,d$) on ${\mathbb{R}}^{d}$ for
$f\in C^{\infty}({\mathbb{R}}^{d})$;

(ii) for fixed $x,t\in{\mathbb{R}}^{d}$, the mapping $f\mapsto(\tau_{x}f)(t)$
defines a compactly supported distribution, whose support is contained in the
ball $\\{\xi\in{\mathbb{R}}^{d}:\,|\xi|\leq|x|+|t|\\}$;

(iii) for fixed $x\in{\mathbb{R}}^{d}$, if
$f\in{\mathscr{S}}({\mathbb{R}}^{d})$, then
$\tau_{x}f\in{\mathscr{S}}({\mathbb{R}}^{d})$ too, and both (6) and (7) hold.

(iv) If $f\in C^{\infty}({\mathbb{R}}^{d})$, then the function
$(x,t)\mapsto(\tau_{x}f)(t)$ is in
$C^{\infty}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d})$, and for
$x,t\in{\mathbb{R}}^{d}$, $(\tau_{t}f)(x)=(\tau_{x}f)(t)$.

###### Proposition 2.4

If $f\in C^{\infty}({\mathbb{R}}^{d})\cup L_{\kappa}^{2}({\mathbb{R}}^{d})$,
then (i) $\left[\tau_{x}(\sigma f)\right](t)=(\tau_{\sigma(x)}f)(\sigma(t))$
for $x,t\in{\mathbb{R}}^{d}$ and $\sigma\in G$; (ii)
$\left[\tau_{x}(f(a\cdot))\right](t)=(\tau_{ax}f)(at)$ for
$x,t\in{\mathbb{R}}^{d}$ and $a\in{\mathbb{R}}\setminus\\{0\\}$; (iii)
$\left[\tau_{t}(\tau_{x}f)\right](z)=\left[\tau_{x}(\tau_{t}f)\right](z)$ for
$x,t,z\in{\mathbb{R}}^{d}$.

If $f\in L_{\kappa}^{2}({\mathbb{R}}^{d})$, the required identities in the
above proposition are valid by Proposition 2.1(vii), (4) and (6). For general
$f\in C^{\infty}({\mathbb{R}}^{d})$ and for fixed $x,t\in{\mathbb{R}}^{d}$, we
take a compactly supported $\phi\in C^{\infty}({\mathbb{R}}^{d})$ with
$\phi(\xi)=1$ for $|\xi|\leq|x|+|t|$. It then follows that
$\left[\tau_{x}(\sigma(\phi f))\right](t)=\left[\tau_{\sigma(x)}(\phi
f)\right](\sigma(t))$, which is identical with $\left[\tau_{x}(\sigma
f)\right](t)=(\tau_{\sigma(x)}f)(\sigma(t))$ by Proposition 2.3(ii), so that
part (i) is done. Parts (ii) and (iii) can be verified similarly.

###### Proposition 2.5

([21, Corollary 4.21]) If $f\in{\mathscr{S}}({\mathbb{R}}^{d})$, then
$\sup_{x\in{\mathbb{R}}^{d}}\|\tau_{x}f\|_{L^{1}_{\kappa}}<\infty$.

As for radial functions in ${\mathbb{R}}^{d}$, more information about the
translation operator $\tau_{x}$ has been found. In fact, for
$x,t\in{\mathbb{R}}^{d}$, by [46, Theorem 5.1] there exists a unique compactly
supported, radial (Borel) probability measure $\rho^{\kappa}_{x,t}$ on
${\mathbb{R}}^{d}$ such that for all radial $f\in
C^{\infty}({\mathbb{R}}^{d})$,

$\displaystyle(\tau_{x}f)(t)=\int_{{\mathbb{R}}^{d}}f\,d\rho^{\kappa}_{x,t};$
(9)

and the support of $\rho^{\kappa}_{x,t}$ is contained in

$\displaystyle\left\\{\xi\in{\mathbb{R}}^{d}:\quad\min_{\sigma\in
G}|x+\sigma(t)|\leq|\xi|\leq\max_{\sigma\in G}|x+\sigma(t)|\right\\}.$ (10)

In particular, if $0\in{\rm supp}\,\rho^{\kappa}_{x,t}$, then the $G$-orbits
of $x$ and $-t$ coincide. Since ${\rm supp}\,\rho^{\kappa}_{x,t}$ is compact,
(9) leads to a natural extension of the translation operator $\tau_{x}$ to all
radial $f\in C({\mathbb{R}}^{d})$. Furthermore, if $f\in
C^{\infty}({\mathbb{R}}^{d})$ is radial, say $f(x)=f_{0}(|x|)$, from the proof
of [46, Theorem 5.1] it follows that, for $x,t\in{\mathbb{R}}^{d}$,

$\displaystyle(\tau_{x}f)(t)=\int_{{\mathbb{R}}^{d}}f_{0}(\sqrt{|x|^{2}+|t|^{2}+2\langle
t,\xi\rangle})\,d\mu^{\kappa}_{x}(\xi).$ (11)

Again, since $\mu^{\kappa}_{x}$ is of compact support, (11) allows an
extension of $\tau_{x}$ to all radial $f\in C({\mathbb{R}}^{d})$ by a density
argument.

###### Proposition 2.6

([32, Propositions 3.1 and 3.2]) (i) If $f\in C^{\infty}({\mathbb{R}}^{d})$,
and $g\in C({\mathbb{R}}^{d})$ is radial and of compact support, or if $f\in
C^{\infty}({\mathbb{R}}^{d})$ is of compact support and $g\in
C({\mathbb{R}}^{d})$ is radial, then (8) holds.

(ii) If $f\in C({\mathbb{R}}^{d})$ is radial, then
$\left[\tau_{x}(f(a\cdot))\right](t)=(\tau_{ax}f)(at)$ for
$x,t\in{\mathbb{R}}^{d}$ and $a\in{\mathbb{R}}\setminus\\{0\\}$.

###### Proposition 2.7

([55, Theorems 3.6, 3.7 and 3.8]) Let $x\in{\mathbb{R}}^{d}$ be given.

(i) The generalized translation operator $\tau_{x}$, initially defined on
$L_{\kappa}^{1}({\mathbb{R}}^{d})\cap L^{\infty}({\mathbb{R}}^{d})$, can be
extended to all radial functions $f$ in $L_{\kappa}^{p}({\mathbb{R}}^{d})$,
$1\leq p\leq 2$, and for these $f$, one has
$\|\tau_{x}f\|_{L_{\kappa}^{p}}\leq\|f\|_{L_{\kappa}^{p}}$, and moreover,
$\left[{\mathscr{F}}_{\kappa}(\tau_{x}f)\right](\xi)=E_{\kappa}(i\xi,x)({\mathscr{F}}_{\kappa}f)(\xi)$
for $\xi\in{\mathbb{R}}^{d}$.

(ii) If $f\in L_{\kappa}^{1}({\mathbb{R}}^{d})\cap
L^{\infty}({\mathbb{R}}^{d})$ is radial and nonnegative, then
$(\tau_{x}f)(t)\geq 0$ for almost every $x\in{\mathbb{R}}^{d}$.

(iii) If $f\in L_{\kappa}^{1}({\mathbb{R}}^{d})$ is radial, then
$\int_{{\mathbb{R}}^{d}}(\tau_{x}f)(t)\,d\omega_{\kappa}(t)=\int_{{\mathbb{R}}^{d}}f(t)\,d\omega_{\kappa}(t)$.

###### Corollary 2.8

If $f\in L_{\kappa}^{p}({\mathbb{R}}^{d})$ ($1\leq p\leq 2$) is radial, then
$\lim_{t\rightarrow 0}\|\tau_{t}f-f\|_{L_{\kappa}^{p}}=0$.

By Proposition 2.7(i), it suffices to show the limit for $f$ in a dense set of
$L_{\kappa}^{p}({\mathbb{R}}^{d})$. For a radial
$f\in{\mathscr{S}}({\mathbb{R}}^{d})$, the Plancherel formula (Propositions
2.1(v)) implies that

$\|\tau_{t}f-f\|_{L_{\kappa}^{2}}^{2}=c_{\kappa}\int_{{\mathbb{R}}^{d}}\left|(E_{\kappa}(i\xi,t)-1)({\mathscr{F}}_{\kappa}f)(\xi)\right|^{2}\,d\omega_{\kappa}(\xi),$

which, in view of (5), approaches to zero as $t\rightarrow 0$ by the dominated
convergence theorem. To deal with the case for $1\leq p<2$, assume that
$|f(x)|\lesssim(1+|x|^{2})^{-m}$ with a suitably large $m>0$. If $|t|<1$ and
$|x|\geq 2$, then by (2) and (11),

$\displaystyle|(\tau_{t}f)(x)|\lesssim\int_{|\xi|\leq|t|}(1+|x|^{2}+|t|^{2}+2\langle
x,\xi\rangle)^{-m}\,d\mu^{\kappa}_{t}(\xi)\lesssim(1+|x|^{2})^{-m}.$

Thus for arbitrary $M>2$, Hölder’s inequality gives

$\displaystyle\|\tau_{t}f-f\|_{L_{\kappa}^{p}}^{p}\lesssim|B_{M}(0)|_{\kappa}^{\frac{2-p}{2}}\|\tau_{t}f-f\|_{L_{\kappa}^{2}}^{p}+\int_{|x|>M}(1+|x|^{2})^{-mp}\,d\omega_{\kappa}(x),$

so that $\limsup_{t\rightarrow
0}\|\tau_{t}f-f\|_{L_{\kappa}^{p}}^{p}\lesssim\int_{|x|>M}(1+|x|^{2})^{-mp}\,d\omega_{\kappa}(x)$.
Letting $M\rightarrow\infty$ concludes the proof.

Now we consider the associated convolution $\ast_{\kappa}$. For suitable $f,g$
on ${\mathbb{R}}^{d}$, define $f\ast_{\kappa}g$ by

$\displaystyle(f\ast_{\kappa}g)(x)=c_{\kappa}\int_{{\mathbb{R}}^{d}}({\mathscr{F}}_{\kappa}f)(\xi)({\mathscr{F}}_{\kappa}g)(\xi)E_{\kappa}(i\xi,x)\,d\omega_{\kappa}(\xi),\qquad
x\in{\mathbb{R}}^{d}.$ (12)

###### Proposition 2.9

(i) If $f,g\in L_{\kappa}^{2}({\mathbb{R}}^{d})$, or if $f\in
L_{\kappa}^{1}({\mathbb{R}}^{d})$ and $g\in A_{\kappa}({\mathbb{R}}^{d})$,
then $f\ast_{\kappa}g$ is well defined and continuous on ${\mathbb{R}}^{d}$,
and moreover,

$\displaystyle(f\ast_{\kappa}g)(x)=c_{\kappa}\int_{{\mathbb{R}}^{d}}f(t)(\tau_{x}g)(-t)\,d\omega_{\kappa}(t),\qquad
x\in{\mathbb{R}}^{d}.$ (13)

(ii) If $f\in L_{\kappa}^{2}({\mathbb{R}}^{d})$ and $g\in
L_{\kappa}^{1}({\mathbb{R}}^{d})\cap L_{\kappa}^{2}({\mathbb{R}}^{d})$, then
$\|f\ast_{\kappa}g\|_{L_{\kappa}^{2}}\leq\|f\|_{L_{\kappa}^{2}}\|g\|_{L_{\kappa}^{1}}$.

(iii) If $f\in L_{\kappa}^{1}({\mathbb{R}}^{d})$, and $g\in
A_{\kappa}({\mathbb{R}}^{d})$ is radial, then
$\|f\ast_{\kappa}g\|_{L_{\kappa}^{1}}\leq\|f\|_{L_{\kappa}^{1}}\|g\|_{L_{\kappa}^{1}}$.

(iv) If $f\in L_{\kappa}^{1}({\mathbb{R}}^{d})$, and
$g\in\SS({\mathbb{R}}^{d})$, then $\|f\ast_{\kappa}g\|_{L_{\kappa}^{1}}\leq
C_{g}\|f\|_{L_{\kappa}^{1}}$, where $C_{g}>0$ is a constant depending on $g$.

(v) In all cases (i), (ii) and (iii),

$\displaystyle[{\mathscr{F}}_{\kappa}(f\ast_{\kappa}g)](\xi)=({\mathscr{F}}_{\kappa}f)(\xi)({\mathscr{F}}_{\kappa}g)(\xi),\qquad\xi\in{\mathbb{R}}^{d}.$

Part (i) follows from the product formula and the Plancherel formula, part
(ii) from (12) and the Plancherel formula, part (iii) from (13), Propositions
2.2(ii) and 2.7(i), and Fubini’s theorem, and part (iv) from (13), Proposition
2.5 and Fubini’s theorem. Part (v) is a consequence of (12) and parts
(i)-(iii).

###### Proposition 2.10

(i) If $f\in L_{\kappa}^{1}({\mathbb{R}}^{d})$ and
$\varphi\in\SS({\mathbb{R}}^{d})$, then $f\ast_{\kappa}\varphi\in
C^{\infty}({\mathbb{R}}^{d})$.

(ii) If $f\in C({\mathbb{R}}^{d})$ and radial
$\varphi\in\SS({\mathbb{R}}^{d})$ satisfy ${\rm
supp}\,f\subseteq\\{x:\,r_{2}\leq|x|\leq r_{1}\\}$ and ${\rm
supp}\,\varphi\subseteq\\{x:\,|x|\leq r_{3}\\}$ separately, where
$r_{1}>r_{2}>r_{3}>0$, then

${\rm supp}\,f\ast_{\kappa}\varphi\subseteq\\{x:\,r_{2}-r_{3}\leq|x|\leq
r_{1}+r_{3}\\}.$

For part (i), since ${{\mathscr{F}}}_{\kappa}f\in C_{0}({{\mathbb{R}}^{d}})$
and ${{\mathscr{F}}}_{\kappa}\varphi\in\SS({\mathbb{R}}^{d})$ by Proposition
2.1(i) and (iii), that $f\ast_{\kappa}\varphi\in C^{\infty}({\mathbb{R}}^{d})$
follows immediately from (5) and (12). As for part (ii), if $|x|<r_{2}-r_{3}$
or $|x|>r_{1}+r_{3}$, then for $t\in{\rm supp}\,f$,
$r_{3}<r_{2}-|x|\leq|t|-|x|\leq\min_{\sigma\in G}|x-\sigma(t)|$, or
$r_{3}<|x|-r_{1}\leq|x|-|t|\leq\min_{\sigma\in G}|x-\sigma(t)|$, so that in
view of (10), ${\rm supp}\,\varphi\cap{\rm
supp}\,\rho^{\kappa}_{x,-t}=\emptyset$, and from (9), $(\tau_{x}g)(-t)=0$.
Thus by (13), $(f\ast_{\kappa}\varphi)(x)=0$ for $|x|<r_{2}-r_{3}$ or
$|x|>r_{1}+r_{3}$, and part (ii) is done.

###### Proposition 2.11

([55, Theorems 4.1 and 4.2]) Let $\varphi\in A_{\kappa}({\mathbb{R}}^{d})$ be
radial. Then

(i) for $1\leq p\leq\infty$, the operator
$T_{\varphi}f=f\ast_{\kappa}\varphi$, initially defined on
$L_{\kappa}^{1}({\mathbb{R}}^{d})\cap L^{\infty}({\mathbb{R}}^{d})$, has a
bounded extension to $L_{\kappa}^{p}({\mathbb{R}}^{d})$, denoted by
$T_{\varphi}f=f\ast_{\kappa}\varphi$ as well, and moreover,
$\|f\ast_{\kappa}\varphi\|_{L_{\kappa}^{p}}\leq\|\varphi\|_{L_{\kappa}^{1}}\|f\|_{L_{\kappa}^{p}}$;

(ii) if, in addition,
$c_{\kappa}\int_{{\mathbb{R}}^{d}}\varphi(x)\,d\omega_{\kappa}(x)=1$, and set
$\varphi_{\epsilon}(x)=\epsilon^{-2|\kappa|-d}\varphi(\epsilon^{-1}x)$ for
$\epsilon>0$, then for $f\in X=L_{\kappa}^{p}({\mathbb{R}}^{d})$, $1\leq
p<\infty$, or $C_{0}({\mathbb{R}}^{d})$, $\lim_{\epsilon\rightarrow
0+}\|f\ast_{\kappa}\varphi_{\epsilon}-f\|_{X}=0$.

###  2.3 The generalized spherical mean operator

We note that it is still open whether the translation operator $\tau_{x}$ has
a bounded extension to non-radial functions in
$L_{\kappa}^{1}({\mathbb{R}}^{d})$. However we may consider the generalized
spherical mean operator $f\mapsto M_{f}$ associated to $G$ and $\kappa$, which
is defined in [41], for $x\in{\mathbb{R}}^{d}$ and $r\in[0,\infty)$, by

$\displaystyle
M_{f}(x,r)=d_{\kappa}\int_{S^{d-1}}(\tau_{x}f)(rt^{\prime})W_{\kappa}(t^{\prime})dt^{\prime},$

where $dt^{\prime}$ denotes the area element on the unit sphere $S^{d-1}$, and
$d_{\kappa}^{-1}=\int_{S^{d-1}}W_{\kappa}(t^{\prime})dt^{\prime}$. By
Proposition 2.3(i), $M_{f}$ is well defined for $f\in
C^{\infty}({\mathbb{R}}^{d})$ or $A_{\kappa}({\mathbb{R}}^{d})$. Moreover, by
[46, Theorems 3.1, 4.1, and Corollary 5.2] the mapping $f\mapsto M_{f}$ is
positivity-preserving on $C^{\infty}({\mathbb{R}}^{d})$, and for
$x\in{\mathbb{R}}^{d}$ and $r\in[0,\infty)$, there exists a unique compactly
supported probability (Borel) measure $\sigma^{\kappa}_{x,r}$ on
${\mathbb{R}}^{d}$ such that for all $f\in C^{\infty}({\mathbb{R}}^{d})$,

$\displaystyle M_{f}(x,r)=\int_{{\mathbb{R}}^{d}}f\,d\sigma^{\kappa}_{x,r};$
(14)

the support of $\sigma^{\kappa}_{x,r}$ is contained in

$\displaystyle\\{\xi\in{\mathbb{R}}^{d}:\,\,|\xi|\geq||x|-r|\\}\cap\left[\cup_{\sigma\in
G}\\{\xi\in{\mathbb{R}}^{d}:\,\,|\xi-\sigma(x)|\leq r\\}\right];$

the mapping $(x,r)\mapsto\sigma_{x,r}^{\kappa}$ is continuous with respect to
the weak topology on $M^{1}({\mathbb{R}}^{d})$ (the space of probability
measures); and moreover,

$\displaystyle\sigma_{\sigma(x),r}^{\kappa}(A)=\sigma_{x,r}^{\kappa}(\sigma^{-1}(A)),\qquad\sigma_{ax,ar}^{\kappa}(A)=\sigma_{x,r}^{\kappa}(a^{-1}A)$

for all $\sigma\in G$, $a>0$, and all Borel sets
$A\in{\mathcal{B}}({\mathbb{R}}^{d})$.

Since ${\rm supp}\,\rho^{\kappa}_{x,t}$ is compact, (14) gives a natural
extension of the generalized spherical mean operator $f\mapsto M_{f}$ to all
$f\in C({\mathbb{R}}^{d})$.

###### Proposition 2.12

([32, Corollary 3.6]) Assume that $\Omega$ is a $G$-invariant domain of
${\mathbb{R}}^{d}$ and $f\in C^{2}(\Omega)$. Then $f$ satisfies
$\Delta_{\kappa}f=0$ in $\Omega$ if and only if for all $x\in\Omega$,
$M_{f}(x,r)=f(x)$ provided $\overline{B(x,r)}\subset\Omega$ for $r>0$.
Moreover, in both cases, $f\in C^{\infty}(\Omega)$.

##  3 Some fundamental aspects on the $\kappa$-Riesz transforms and the Hardy
space $H_{\kappa}({\mathbb{R}}^{d})$

###  3.1 The Green formulas

In what follows, associated to the reflection group $G$ on ${\mathbb{R}}^{d}$
and the multiplicity function $\kappa$, we consider the reflection group
$\tilde{G}={\mathbb{Z}}_{2}\otimes G$ on
${\mathbb{R}}^{1+d}={\mathbb{R}}\times{\mathbb{R}}^{d}$ and the multiplicity
function $\tilde{\kappa}=(0,\kappa)$. In this case, we write
$\tilde{x}=(x_{0},x)\in{\mathbb{R}}^{1+d}$ and
$\partial_{0}=\partial/\partial_{x_{0}}$. The associated Laplacian, denoted by
$\Delta_{\tilde{\kappa}}$ and called the $\tilde{\kappa}$-Laplacian, is given
by $\Delta_{\tilde{\kappa}}=\partial_{0}^{2}+\Delta_{\kappa}$ or explicitly

$\displaystyle\Delta_{\tilde{\kappa}}=\sum_{j=0}^{d}\partial_{j}^{2}+2\sum_{\alpha\in
R_{+}}\kappa(\alpha)\left(\frac{\langle\alpha,\nabla\rangle}{\langle\alpha,x\rangle}-\frac{1-\sigma_{\alpha}}{\langle\alpha,x\rangle^{2}}\right),$

where $\nabla$ is the gradient operator on ${\mathbb{R}}^{d}$.

Since the multiplicity value associated to ${\mathbb{Z}}_{2}$ is zero, the
${\mathbb{Z}}_{2}\otimes G$-invariance of the domain $\Omega$ in [41, Theorem
4.11] and Proposition 2.12 can be relaxed to the $G$-invariance. We
reformulate the Green formulas in [41, Theorem 4.11] and [32, Proposition 3.8]
as follows.

###### Proposition 3.1

Let $\Omega$ be a bounded, regular, and $G$-invariant domain of
${\mathbb{R}}^{1+d}$. Then for $u,v\in C^{2}(\overline{\Omega})$,

$\displaystyle\iint_{\Omega}(v\Delta_{\tilde{\kappa}}u-u\Delta_{\tilde{\kappa}}v)\,d\omega_{\kappa}(x)dx_{0}=\int_{\partial\Omega}\left(v\partial_{\mathbf{n}}u-u\partial_{\mathbf{n}}v\right)W_{\kappa}(x)d\sigma(x_{0},x),$
(15)

where $\partial_{\mathbf{n}}$ denotes the directional derivative of the
outward normal, and $d\sigma(x_{0},x)$ the area element on $\partial\Omega$;
moreover the formula has the following variant

$\displaystyle\iint_{\Omega}(v\Delta_{\tilde{\kappa}}u-u\Delta_{\tilde{\kappa}}v)\,d\omega_{\kappa}(x)dx_{0}=\int_{\partial\Omega}\left(vD_{\mathbf{n}}u-uD_{\mathbf{n}}v\right)W_{\kappa}(x)d\sigma(x_{0},x),$

where $D_{\mathbf{n}}=\langle\nabla_{\tilde{\kappa}},\mathbf{n}\rangle$ is the
directional Dunkl operator and
$\nabla_{\tilde{\kappa}}=(\partial_{0},D_{1},\dots,D_{d})$ is the
$\tilde{\kappa}$-gradient.

A $C^{2}$ function $u$ defined in a $G$-invariant domain $\Omega$ of
${\mathbb{R}}^{1+d}$ is said to be $\tilde{\kappa}$-harmonic if
$\Delta_{\tilde{\kappa}}u=0$ there.

###  3.2 The $\kappa$-Poisson integral

We define the function $P$ by
$P(x)=c_{d,\kappa}(1+|x|^{2})^{-|\kappa|-(d+1)/2}$, where
$c_{d,\kappa}=2^{|\kappa|+d/2}\pi^{-1/2}\Gamma(|\kappa|+(d+1)/2)$. It follows
that $c_{\kappa}\int_{{\mathbb{R}}^{d}}P\,d\omega_{\kappa}=1$ and
$\left({\mathscr{F}}_{\kappa}P\right)(\xi)=e^{-|\xi|}$ (cf. [48, 55]). For
$f\in L_{\kappa}^{p}({\mathbb{R}}^{d})$, $1\leq p\leq\infty$, the
$\kappa$-Poisson integral of $f$ is defined by

$\displaystyle(Pf)(x_{0},x)=c_{\kappa}\int_{{\mathbb{R}}^{d}}f(t)(\tau_{x}P_{x_{0}})(-t)\,d\omega_{\kappa}(t),\qquad(x_{0},x)\in{\mathbb{R}}^{1+d}_{+},$
(16)

where $(\tau_{x}P_{x_{0}})(-t)$ is the $\kappa$-Poisson kernel with
$P_{x_{0}}(x)=x_{0}^{-2|\kappa|-d}P(x/x_{0})$ for $x_{0}>0$, i.e.,

$\displaystyle
P_{x_{0}}(x)=\frac{c_{d,\kappa}\,x_{0}}{(x_{0}^{2}+|x|^{2})^{|\kappa|+(d+1)/2}},\qquad
x\in{\mathbb{R}}^{d}.$

Note that the function $(x_{0},x)\mapsto P_{x_{0}}(x)$ is
$\tilde{\kappa}$-harmonic in ${\mathbb{R}}_{+}^{1+d}$, and so is the function
$(x_{0},x)\mapsto(\tau_{x}P_{x_{0}})(-t)$ by Propositions 2.3(i) and (iv). In
view of [5, Proposition 5.1], taking differentiation under the integral sign
on the right hand side of (16) is legitimate, and hence, the $\kappa$-Poisson
integral $(Pf)(x_{0},x)$ of $f\in L_{\kappa}^{p}({\mathbb{R}}^{d})$ is
$\tilde{\kappa}$-harmonic in ${\mathbb{R}}_{+}^{1+d}$.

Since $({\mathscr{F}}_{\kappa}P_{x_{0}})(\xi)=e^{-x_{0}|\xi|}$, from (6) and
Proposition 2.4(ii) one has

$\displaystyle{\mathscr{F}}_{\kappa}[(\tau_{x}P_{x_{0}})(-\cdot)](\xi)=e^{-x_{0}|\xi|}E_{\kappa}(-ix,\xi),$
(17)

and then for $f\in L_{\kappa}^{1}({\mathbb{R}}^{d})$, the product formula
(Propositions 2.1(iv)) gives

$\displaystyle(Pf)(x_{0},x)=c_{\kappa}\int_{{\mathbb{R}}^{d}}e^{-x_{0}|\xi|}\left({\mathscr{F}}_{\kappa}f\right)(\xi)E_{\kappa}(i\xi,x)\,d\omega_{\kappa}(\xi),\qquad(x_{0},x)\in{\mathbb{R}}^{1+d}_{+}.$
(18)

If $f\in L_{\kappa}^{2}({\mathbb{R}}^{d})$, the formula (18) holds too by the
Plancherel formula (Propositions 2.1(v)).

For the following two propositions, see [55, Theorems 4.1 and 5.4] and [5,
Theorem 7.5].

###### Proposition 3.2

For $f\in L_{\kappa}^{p}({\mathbb{R}}^{d})$ ($1\leq p\leq\infty$) and
$x_{0}>0$, $\|(Pf)(x_{0},\cdot)\|_{L_{\kappa}^{p}}\leq\|f\|_{L_{\kappa}^{p}}$;
and for $f\in X=L_{\kappa}^{p}({\mathbb{R}}^{d})$, $1\leq p<\infty$, or
$C_{0}({\mathbb{R}}^{d})$, $\lim_{x_{0}\rightarrow
0+}\|(Pf)(x_{0},\cdot)-f\|_{X}=0$.

###### Proposition 3.3

Let $u$ be $\tilde{\kappa}$-harmonic in ${\mathbb{R}}^{1+d}_{+}$. Then (i) for
$1<p<\infty$, $u$ is the $\kappa$-Poisson integral of some $f\in
L^{p}_{\kappa}({\mathbb{R}}^{d})$ if and only if $u$ satisfies

$\displaystyle A:=\sup_{x_{0}>0}\|u(x_{0},\cdot)\|_{L^{p}_{\kappa}}<+\infty,$
(19)

and moreover $\|f\|_{L^{p}_{\kappa}}=A$; and (ii) $u$ is the $\kappa$-Poisson
integral of $d\mu\in{\mathfrak{B}}_{\kappa}({\mathbb{R}}^{d})$ if and only if
$u$ satisfies (19) for $p=1$.

As usual, we denote by $\Gamma_{a}(x)$ the positive cone of aperture $a>0$
with vertex $(0,x)\in\partial{\mathbb{R}}^{1+d}_{+}={\mathbb{R}}^{d}$, and
$\Gamma^{h}_{a}(x)$ the truncated cone with height $h>0$, that is,

$\Gamma^{h}_{a}(x)=\left\\{(x_{0},t):\quad|t-x|<ax_{0}\quad\hbox{for}\,\,t\in{\mathbb{R}}^{d},\,\,0<x_{0}<h\right\\}.$

For a function $u$ on ${\mathbb{R}}^{1+d}_{+}$ and for $a>0$, the non-
tangential maximal function $u^{*}_{a}(x)$ is defined by

$u^{*}_{a}(x)=\sup_{(x_{0},t)\in\Gamma_{a}(x)}|u(x_{0},t)|;$

and for $x\in{\mathbb{R}}^{d}$, that $u$ has a non-tangential limit $A$ at
$(0,x)$ means that for every $a>0$, $\lim u(x_{0},t)=A$ as
$(x_{0},t)\in\Gamma_{a}(x)$ approaching to $(0,x)$.

If $u(x_{0},x)$ is the $\kappa$-Poisson integral of $f\in
L^{p}_{\kappa}({\mathbb{R}}^{d})$, $1\leq p\leq\infty$, we write
$(P^{*}_{a}f)(x)=u^{*}_{a}(x)$, and
$(P^{+}f)(x)=\sup_{x_{0}>0}|(Pf)(x_{0},x)|$, the associated perpendicular
maximal function.

###### Proposition 3.4

Both $P^{+}$ and $P^{*}_{a}$ for $a>0$ are bounded from
$L_{\kappa}^{p}({\mathbb{R}}^{d})$ ($1<p\leq\infty$) to itself and from
$L_{\kappa}^{1}({\mathbb{R}}^{d})$ to
$L_{\kappa}^{1,\infty}({\mathbb{R}}^{d})$, where

$L_{\kappa}^{1,\infty}({\mathbb{R}}^{d})=\\{f:\,\|f\|_{L_{\kappa}^{1,\infty}}:=\sup_{\lambda>0}\lambda|\\{x\in{\mathbb{R}}^{d}:\,|f(x)|>\lambda\\}|_{\kappa}<\infty\\}.$

The conclusions in the proposition are known. Those for $P^{+}$ are
consequences of the Hopf-Dunford-Schwartz ergodic theorem, see [51, pp. 48-9]
and [55, p. 46]. As for $P^{*}_{a}$, since
$(\tau_{x}P_{x_{0}})(-t)=(\tau_{-t}P_{x_{0}})(x)$ by (7), it follows from (11)
that, for $x,t\in{\mathbb{R}}^{d}$,

$\displaystyle(\tau_{x}P_{x_{0}})(-t)=\int_{{\mathbb{R}}^{d}}\frac{c_{d,\kappa}\,x_{0}}{(x_{0}^{2}+|x|^{2}+|t|^{2}+2\langle
x,\xi\rangle)^{|\kappa|+(d+1)/2}}\,d\mu^{\kappa}_{-t}(\xi);$

but since $|\xi|\leq|t|$ for $\xi\in\hbox{supp}\,\mu^{\kappa}_{-t}$ by (2),
one has

$x_{0}^{2}+|x|^{2}+|t|^{2}+2\langle
x,\xi\rangle\asymp\left(x_{0}+|x+\xi|+\sqrt{|t|^{2}-|\xi|^{2}}\right)^{2},$

and then, for $x^{\prime}\in{\mathbb{R}}^{d}$ satisfying
$|x^{\prime}-x|<ax_{0}$,

$\displaystyle x_{0}^{2}+|x|^{2}+|t|^{2}+2\langle
x,\xi\rangle\lesssim(a+1)^{2}\left(x_{0}^{2}+|x^{\prime}|^{2}+|t|^{2}+2\langle
x^{\prime},\xi\rangle\right).$

This implies that
$(\tau_{x^{\prime}}P_{x_{0}})(-t)\lesssim(a+1)^{2|\kappa|+d+1}(\tau_{x}P_{x_{0}})(-t)$
for $(x_{0},x^{\prime})\in\Gamma_{a}(x)$, and hence

$\displaystyle(P^{*}_{a}f)(x)\lesssim(a+1)^{2|\kappa|+d+1}\left[P^{+}(|f|)\right](x).$

The conclusions for $P^{*}_{a}$ follow immediately.

###### Corollary 3.5

If $f\in L^{p}_{\kappa}({\mathbb{R}}^{d})$, $1\leq p\leq\infty$, then its
$\kappa$-Poisson integral $Pf$ converges non-tangentially to $f$ almost every
on ${\mathbb{R}}^{d}$.

The above corollary is a direct consequence of Proposition 3.4 (cf. [5,
Corollary 5.4])), and is also a special case of the following local Fatou
theorem for $\tilde{\kappa}$-harmonic functions.

###### Theorem 3.6

([32, Theorem 2.1]) Suppose that the function $u$ is $\tilde{\kappa}$-harmonic
in ${\mathbb{R}}^{1+d}_{+}$ and $E$ is a $G$-invariant measurable subset of
$\partial{\mathbb{R}}^{d+1}_{+}={\mathbb{R}}^{d}$. Then $u$ has a finite non-
tangential limit at $(0,x)$ for almost every $x\in E$ if and only if $u$ is
non-tangentially bounded at $(0,x)$ for almost every $x\in E$.

###  3.3 The conjugate $\kappa$-Poisson integrals and the $\kappa$-Riesz
transforms

In this subsection we consider the $\kappa$-Riesz transforms by the “analytic”
approach, i.e., starting from the associated conjugate Poisson integrals. This
is different from the way in [3] and some conclusions about these transform
can be implied by the know results of the $\tilde{\kappa}$-harmonic functions.
The detailed proofs of the claims in this subsection will be given in a
separate paper (cf. [33]).

For $f\in L^{p}_{\kappa}({\mathbb{R}}^{d})$ ($1\leq p<\infty$) and for
$j=1,\cdots,d$, the $j$th conjugate $\kappa$-Poisson integral of $f$ is
defined by

$\displaystyle(Q_{j}f)(x_{0},x)=c_{\kappa}\int_{{\mathbb{R}}^{d}}f(t)(\tau_{x}Q^{(j)}_{x_{0}})(-t)\,d\omega_{\kappa}(t),\qquad(x_{0},x)\in{\mathbb{R}}^{1+d}_{+},$

where

$\displaystyle
Q^{(j)}_{x_{0}}(x)=\frac{c_{d,\kappa}x_{j}}{(x_{0}^{2}+|x|^{2})^{|\kappa|+(d+1)/2}},\qquad(x_{0},x)\in{\mathbb{R}}^{1+d}_{+}.$

The function $(\tau_{x}Q^{(j)}_{x_{0}})(-t)$ is called the $j$th conjugate
$\kappa$-Poisson kernel.

###### Proposition 3.7

(i) If we set
$u_{0}(x_{0},x)=P_{x_{0}}(x),u_{1}(x_{0},x)=Q^{(1)}_{x_{0}}(x),\cdots,u_{d}(x_{0},x)=Q^{(d)}_{x_{0}}(x)$
for $(x_{0},x)\in{\mathbb{R}}^{1+d}_{+}$, then they satisfies the generalized
Cauchy-Riemann equations (the M. Riesz system)

$\displaystyle\left\\{\begin{array}[]{l}D_{j}u_{k}=D_{k}u_{j},\qquad 0\leq
j<k\leq d,\\\ \sum_{j=0}^{d}D_{j}u_{j}=0,\end{array}\right.$ (22)

where $D_{0}=\partial_{0}$.

(ii) For $x_{0}>0$ and $j=1,\cdots,d$,
$({\mathscr{F}}_{\kappa}Q^{(j)}_{x_{0}})(\xi)=-i(\xi_{j}/|\xi|)e^{-x_{0}|\xi|}$.

(iii) For $j=1,\cdots,d$, and for $(x_{0},x)\in{\mathbb{R}}^{1+d}_{+}$, the
function $t\mapsto(\tau_{x}Q^{(j)}_{x_{0}})(-t)$ is in
$L^{q}_{\kappa}({\mathbb{R}}^{d})$ for $1<q\leq\infty$.

(iv) For $j=1,\cdots,d$, and for $(x_{0},x)\in{\mathbb{R}}^{1+d}_{+}$ and
$t\in{\mathbb{R}}^{d}$,

$\displaystyle(\tau_{x}Q^{(j)}_{x_{0}})(-t)=\frac{x_{j}-t_{j}}{x_{0}}(\tau_{x}P_{x_{0}})(-t)$
(23)

and

$\displaystyle(\tau_{x}Q^{(j)}_{x_{0}})(-t)=\frac{c_{d,\kappa}}{2|\kappa|+d-1}\int_{{\mathbb{R}}^{d}}D_{j}^{t}\left[\frac{1}{(x_{0}^{2}+A(x,t;\xi)^{2})^{|\kappa|+(d-1)/2}}\right]\,d\mu^{\kappa}_{x}(\xi),$
(24)

where

$\displaystyle A(x,t;\xi)=\sqrt{|x|^{2}+|t|^{2}-2\langle t,\xi\rangle}.$

To show part (ii), define

$\displaystyle\phi(x_{0},x)=c_{\kappa}\int_{{\mathbb{R}}^{d}}(-i\xi_{j}/|\xi|)e^{-x_{0}|\xi|}E_{\kappa}(i\xi,x)\,d\omega_{\kappa}(\xi),\qquad(x_{0},x)\in{\mathbb{R}}^{1+d}_{+}.$

The Lebesgue dominated convergence theorem asserts that
$\lim_{x_{0}\rightarrow+\infty}\phi(x_{0},x)=0$ uniformly for
$x\in{\mathbb{R}}^{d}$, and by use of (3),

$\displaystyle\partial_{x_{0}}\phi(x_{0},x)$
$\displaystyle=D_{j}^{x}\left(c_{\kappa}\int_{{\mathbb{R}}^{d}}e^{-x_{0}|\xi|}E_{\kappa}(i\xi,x)\,d\omega_{\kappa}(\xi)\right)$
$\displaystyle=D_{j}^{x}\left[P_{x_{0}}(x)\right]=-\frac{c_{d,\kappa}(2|\kappa|+d+1)\,x_{0}x_{j}}{(x_{0}^{2}+|x|^{2})^{|\kappa|+(d+3)/2}},\qquad(x_{0},x)\in{\mathbb{R}}^{1+d}_{+}.$

It follows that $\phi(x_{0},x)=Q^{(j)}_{x_{0}}(x)$, and then part (ii) is
concluded.

As for part (iv), we first prove (24). Since

$\displaystyle(\tau_{x}Q^{(j)}_{x_{0}})(t)=-\frac{c_{d,\kappa}}{2|\kappa|+d-1}\tau_{x}\left[D_{j}\left(\frac{1}{(x_{0}^{2}+|\cdot|^{2})^{|\kappa|+(d-1)/2}}\right)\right](t),$

by Proposition 2.3(i) and (11) we have

$\displaystyle(\tau_{x}Q^{(j)}_{x_{0}})(t)$
$\displaystyle=-\frac{c_{d,\kappa}}{2|\kappa|+d-1}D_{j}^{t}\left[\tau_{x}\left(\frac{1}{(x_{0}^{2}+|\cdot|^{2})^{|\kappa|+(d-1)/2}}\right)\right](t)$
$\displaystyle=-\frac{c_{d,\kappa}}{2|\kappa|+d-1}\int_{{\mathbb{R}}^{d}}D_{j}^{t}\left[\frac{1}{(x_{0}^{2}+|x|^{2}+|t|^{2}+2\langle
t,\xi\rangle)^{|\kappa|+(d-1)/2}}\right]\,d\mu^{\kappa}_{x}(\xi).$

From this the formula (24) follows.

To show (23), we use the subordination formula for the $\kappa$-Poisson kernel
(see [5, (5.1)], for example)
$(\tau_{x}P_{x_{0}})(-t)=\int_{0}^{\infty}e^{-u}h_{x_{0}^{2}/4u}(x,t)\,du/\sqrt{\pi
u}$, or its equivalent form,

$\displaystyle(\tau_{x}P_{x_{0}})(-t)=\frac{x_{0}}{2\sqrt{\pi}}\int_{0}^{\infty}e^{-\frac{x_{0}^{2}}{4v}}\,h_{v}(x,t)\,\frac{dv}{v^{3/2}},$
(25)

where $h_{v}(x,t)$ is the heat kernel associated to the Dunkl operators given
by (cf. [44])

$\displaystyle
h_{v}(x,t)=\frac{1}{(2v)^{|\kappa|+d/2}}\exp\left(-\frac{|x|^{2}+|t|^{2}}{4v}\right)E_{\kappa}\left(\frac{x}{\sqrt{2v}},\frac{t}{\sqrt{2v}}\right),\quad
x,t\in{\mathbb{R}}^{d},\,\,v>0.$ (26)

By Propositions 2.3(i), 2.4(ii) and 3.7(i),

$\partial_{x_{0}}\left[(\tau_{x}Q^{(j)}_{x_{0}})(-t)\right]=-D_{j}^{t}\left[(\tau_{x}P_{x_{0}})(-t)\right],$

and since, in view of (24),
$\lim_{x_{0}\rightarrow+\infty}(\tau_{x}Q^{(j)}_{x_{0}})(-t)=0$ uniformly for
$x,t\in{\mathbb{R}}^{d}$, it follows that

$\displaystyle(\tau_{x}Q^{(j)}_{x_{0}})(-t)=\int_{x_{0}}^{\infty}D_{j}^{t}\left[(\tau_{x}P_{\rho})(-t)\right]d\rho.$
(27)

Now applying (3), direction calculations yield

$\displaystyle D_{j}^{t}h_{v}(x,t)=\frac{x_{j}-t_{j}}{2v}h_{v}(x,t),$ (28)

and then from (17) and (27), Fubini’s theorem gives

$\displaystyle(\tau_{x}Q^{(j)}_{x_{0}})(-t)=\frac{x_{j}-t_{j}}{2\sqrt{\pi}}\int_{0}^{\infty}e^{-\frac{x_{0}^{2}}{4v}}\,h_{v}(x,t)\,\frac{dv}{v^{3/2}},$
(29)

Thus the formula (23) is proved.

###### Proposition 3.8

(i) If $f\in L^{p}_{\kappa}({\mathbb{R}}^{d})$ for $1\leq p<\infty$, then the
functions

$\displaystyle
u_{0}(x_{0},x)=(Pf)(x_{0},x),\,\,u_{1}(x_{0},x)=(Q_{1}f)(x_{0},x),\,\,\cdots,\,\,u_{d}(x_{0},x)=(Q_{d}f)(x_{0},x)$

satisfy the equations (22) in ${\mathbb{R}}^{1+d}_{+}$.

(ii) If $f\in L^{p}_{\kappa}({\mathbb{R}}^{d})$ for $1\leq p\leq 2$, then for
$j=1,\cdots,d$,

$\displaystyle(Q_{j}f)(x_{0},x)=c_{\kappa}\int_{{\mathbb{R}}^{d}}\left(-i\frac{\xi_{j}}{|\xi|}\right)e^{-x_{0}|\xi|}\left({\mathscr{F}}_{\kappa}f\right)(\xi)E_{\kappa}(i\xi,x)\,d\omega_{\kappa}(\xi),\,\,\,(x_{0},x)\in{\mathbb{R}}^{1+d}_{+}.$

The following theorem extends the theorem of M. Riesz and that of Kolmogorov
to the conjugate $\kappa$-Poisson integrals.

###### Theorem 3.9

(i) There is $C_{1}>0$ such that
$\int_{\\{x:\,|(Q_{j}f)(x_{0},x)|>s\\}}d\omega_{\kappa}\leq
C_{1}\|f\|_{L^{1}_{\kappa}}/s$ holds for all $f\in
L^{1}_{\kappa}({\mathbb{R}}^{d})$, $s>0$ and for all $x_{0}>0$, where
$j=1,\cdots,d$.

(ii) If $1<p<\infty$, then there is $C_{p}>0$ such that
$\|(Q_{j}f)(x_{0},\cdot)\|_{L^{p}_{\kappa}}\leq C_{p}\|f\|_{L^{p}_{\kappa}}$
for all $f\in L^{p}_{\kappa}({\mathbb{R}}^{d})$ and $x_{0}>0$, where
$j=1,\cdots,d$.

###### Corollary 3.10

If $1<p<\infty$, then there exists a function in
$L^{p}_{\kappa}({\mathbb{R}}^{d})$, denoted by ${\mathcal{R}}_{j}f$, such that
$(Q_{j}f)(x_{0},\cdot)$ converges to ${\mathcal{R}}_{j}f$ as $x_{0}\rightarrow
0+$, both in the $L^{p}_{\kappa}$-norm and almost everywhere non-tangentially
on ${\mathbb{R}}^{d}$, and $\|{\mathcal{R}}_{j}f\|_{L^{p}_{\kappa}}\leq
C_{p}\|f\|_{L^{p}_{\kappa}}$, where $j=1,\cdots,d$. Moreover,

$\displaystyle(Q_{j}f)(x_{0},x)=[P({\mathcal{R}}_{j}f)](x_{0},x),$ (30)

and for $1<p\leq 2$,

$\displaystyle\left[{\mathscr{F}}_{\kappa}({\mathcal{R}}_{j}f)\right](\xi)=-i(\xi_{j}/|\xi|)\left({\mathscr{F}}_{\kappa}f\right)(\xi).$
(31)

###### Corollary 3.11

If $f\in L_{\kappa}^{p}({\mathbb{R}}^{d})$ for $1<p<\infty$, and $g\in
L_{\kappa}^{p^{\prime}}({\mathbb{R}}^{d})$, then

$\displaystyle\int_{{\mathbb{R}}^{d}}({\mathcal{R}}_{j}f)(x)g(x)\,d\omega_{\kappa}=-\int_{{\mathbb{R}}^{d}}f(x)({\mathcal{R}}_{j}g)(x)\,d\omega_{\kappa}.$

From Proposition 3.7(ii) and (31) it is obvious that

${\mathcal{R}}_{j}P_{x_{0}}=Q^{(j)}_{x_{0}}.$

A stronger version of Theorem 3.9, with respect to the associated maximal
functions
$(Q^{*}_{j,a}f)(x)=\sup_{(x_{0},t)\in\Gamma_{a}(x)}|(Q_{j}f)(x_{0},t)|$ for
$a>0$ and $(Q_{j}^{+}f)(x)=\sup_{x_{0}>0}|(Q_{j}f)(x_{0},x)|$, does hold too.

###### Theorem 3.12

The operator $Q^{*}_{j,a}$ for $a>0$ is bounded from
$L_{\kappa}^{p}({\mathbb{R}}^{d})$ ($1<p<\infty$) to itself, and the operator
$Q_{j}^{+}$ is bounded from $L_{\kappa}^{1}({\mathbb{R}}^{d})$ to
$L_{\kappa}^{1,\infty}({\mathbb{R}}^{d})$.

Now we turn to a truncated version of the $\kappa$-Riesz transforms
${\mathcal{R}}_{j}$ ($1\leq j\leq d$). At first, from (9), (10) and (11) it
follows that, for $\xi\in\hbox{supp}\,\mu^{\kappa}_{x}$,

$\displaystyle\min_{\sigma\in G}|t-\sigma(x)|\leq
A(x,t;\xi)\leq\max_{\sigma\in G}|t-\sigma(x)|.$

(cf. [3, (3.9)]. Thus for $j=1,\cdots,d$, and for $x,t\in{\mathbb{R}}^{d}$
satisfying $\min_{\sigma\in G}|t-\sigma(x)|>0$, the function

$\displaystyle
K_{j}(x,t)=\frac{c_{d,\kappa}}{2|\kappa|+d-1}\int_{{\mathbb{R}}^{d}}D_{j}^{t}\left[\frac{1}{A(x,t;\xi)^{2|\kappa|+d-1}}\right]\,d\mu^{\kappa}_{x}(\xi)$

is well defined, and from (24) and (29), it can be rewritten as

$\displaystyle
K_{j}(x,t)=\frac{x_{j}-t_{j}}{2\sqrt{\pi}}\int_{0}^{\infty}h_{v}(x,t)\,\frac{dv}{v^{3/2}}.$
(32)

For $f\in L^{p}_{\kappa}({\mathbb{R}}^{d})$ ($1\leq p<\infty$) and
$\epsilon>0$, we consider

$\displaystyle({\mathcal{R}}_{j}^{\epsilon}f)(x)=c_{\kappa}\int_{d(x,t)>\epsilon}f(t)K_{j}(x,t)\,d\omega_{\kappa}(t),\qquad
x\in{\mathbb{R}}^{d},$

and call ${\mathcal{R}}_{j}^{\epsilon}$ the $j$th truncated $\kappa$-Riesz
transform ($1\leq j\leq d$). Here

$d(x,t):=\min_{\sigma\in G}|t-\sigma(x)|$

denotes the distance between the $G$-orbits of $x,t\in{\mathbb{R}}^{d}$.

###### Theorem 3.13

(i) For $1\leq p<\infty$, there is $C_{p}>0$ such that for $f\in
L^{p}_{\kappa}({\mathbb{R}}^{d})$ and $\epsilon>0$,
$\|(Q_{j}f)(\epsilon,\cdot)-{\mathcal{R}}_{j}^{\epsilon}f\|_{L^{p}_{\kappa}}\leq
C_{p}\|f\|_{L^{p}_{\kappa}}$, where $j=1,\cdots,d$; and moreover,

$\displaystyle\lim_{\epsilon\rightarrow
0+}\|(Q_{j}f)(\epsilon,\cdot)-{\mathcal{R}}_{j}^{\epsilon}f\|_{L^{p}_{\kappa}}=0.$

(ii) For $1\leq p<\infty$, there is $C_{p}^{\prime}>0$ such that for $f\in
L^{p}_{\kappa}({\mathbb{R}}^{d})$,

$\displaystyle\sup_{\epsilon>0}\left|(Q_{j}f)(\epsilon,x)-({\mathcal{R}}_{j}^{\epsilon}f)(x)\right|\leq
C_{p}^{\prime}\left[P^{+}(|f|)\right](x),\qquad x\in{\mathbb{R}}^{d},$

where $j=1,\cdots,d$; and moreover, $\lim\limits_{\epsilon\rightarrow
0+}[(Q_{j}f)(\epsilon,x)-({\mathcal{R}}_{j}^{\epsilon}f)(x)]=0$ for almost
every $x\in{\mathbb{R}}^{d}$.

###### Corollary 3.14

If $1<p<\infty$, then for $f\in L^{p}_{\kappa}({\mathbb{R}}^{d})$,
${\mathcal{R}}_{j}^{\epsilon}f$ converges to ${\mathcal{R}}_{j}f$ as
$\epsilon\rightarrow 0+$, both in the $L^{p}_{\kappa}$-norm and almost
everywhere on ${\mathbb{R}}^{d}$.

###### Corollary 3.15

The maximal $\kappa$-Riesz transform ${\mathcal{R}}_{j}^{*}$ for
$j=1,\cdots,d$, defined by
$({\mathcal{R}}_{j}^{*}f)(x)=\sup_{\epsilon>0}|({\mathcal{R}}_{j}^{\epsilon}f)(x)|$,
is bounded from $L_{\kappa}^{p}({\mathbb{R}}^{d})$ ($1<p<\infty$) to itself,
and from $L_{\kappa}^{1}({\mathbb{R}}^{d})$ to
$L_{\kappa}^{1,\infty}({\mathbb{R}}^{d})$.

Finally we formulate the $\kappa$-Riesz transforms of functions in
$L_{\kappa}^{1}({\mathbb{R}}^{d})$, in the sprite of [52, p. 221], as follows.

###### Definition 3.16

For $f\in L_{\kappa}^{1}({\mathbb{R}}^{d})$, we say that it has the $j$th
$\kappa$-Riesz transform in $L_{\kappa}^{1}({\mathbb{R}}^{d})$, if there
exists an $f_{j}\in L_{\kappa}^{1}({\mathbb{R}}^{d})$ so that

$\displaystyle\left({\mathscr{F}}_{\kappa}f_{j}\right)(\xi)=-i(\xi_{j}/|\xi|)\left({\mathscr{F}}_{\kappa}f\right)(\xi),\qquad\xi\in{\mathbb{R}}^{d}\setminus\\{0\\},$
(33)

and certainly write ${\mathcal{R}}_{j}f=f_{j}$.

###### Proposition 3.17

If $f\in L_{\kappa}^{1}({\mathbb{R}}^{d})$ and ${\mathcal{R}}_{j}f\in
L_{\kappa}^{1}({\mathbb{R}}^{d})$, then (30) holds and $(Q_{j}f)(x_{0},\cdot)$
converges to ${\mathcal{R}}_{j}f$ as $x_{0}\rightarrow 0+$, both in the
$L^{1}_{\kappa}$-norm and almost everywhere non-tangentially on
${\mathbb{R}}^{d}$.

###  3.4 The Hardy spaces ${\mathscr{H}}_{\kappa}^{1}({\mathbb{R}}^{1+d}_{+})$
and $H_{\kappa}^{1}({\mathbb{R}}^{d})$

We consider the vector-valued function $F=(u_{0},u_{1},\cdots,u_{d})$ (a
function system of Stein-Weiss type) defined on the upper space
${\mathbb{R}}^{1+d}_{+}$, that satisfies the generalized Cauchy-Riemann
equations (22). If the components of such a vector valued function $F$ are all
twice continuously differentiable in ${\mathbb{R}}^{1+d}$, i.e. $F\in
C^{2}({\mathbb{R}}^{1+d}_{+})$ says, then they are all
$\tilde{\kappa}$-harmonic because of commutativity of $D_{j}$,
$j=0,1,\cdots,d$. This further implies $F\in
C^{\infty}({\mathbb{R}}^{1+d}_{+})$ by Proposition 2.12.

The “analytic” Hardy space on ${\mathbb{R}}^{1+d}_{+}$ in the Dunkl setting,
denoted by ${\mathscr{H}}_{\kappa}^{1}({\mathbb{R}}^{1+d}_{+})$, consists of
all vector-valued functions $F\in C^{2}({\mathbb{R}}^{1+d}_{+})$ satisfying
the equations in (22) and the condition

$\displaystyle\|F\|_{{\mathscr{H}}_{\kappa}^{1}}:=\sup_{x_{0}>0}\||F(x_{0},\cdot)|\|_{L^{1}_{\kappa}}<\infty,$

where $|F(x_{0},x)|=\left(\sum_{j=0}^{d}|u_{j}(x_{0},x)|^{2}\right)^{1/2}$.

The following theorem contains the fundamental results about the Hardy space
${\mathscr{H}}_{\kappa}^{1}({\mathbb{R}}^{1+d}_{+})$.

###### Theorem 3.18

Assume that $F\in{\mathscr{H}}_{\kappa}^{1}({\mathbb{R}}^{1+d}_{+})$. Then

(i) For almost every $x\in{\mathbb{R}}^{d}$, $\lim F(x_{0},t)=\tilde{F}(x)$
exists as $(x_{0},t)$ approaches to the point $(0,x)$ nontangentially;

(ii) As $x_{0}\rightarrow 0+$, $F(x_{0},\cdot)$ converges to $\tilde{F}$ in
the $L_{\kappa}^{1}$ norm; and
$\|F\|_{{\mathscr{H}}_{\kappa}^{1}}=\|\tilde{F}\|_{L_{\kappa}^{1}}$;

(iii) $F(x_{0},x)$ is the $\kappa$-Poisson integral of $\tilde{F}$ in vector
form;

(iv) For all $a>0$,
$\|F^{*}_{a}\|_{L_{\kappa}^{1}}\lesssim\|F\|_{{\mathscr{H}}_{\kappa}^{1}}$,
where $F^{*}_{a}(x)=\sup_{(x_{0},t)\in\Gamma_{a}(x)}|F(x_{0},t)|$.

The above theorem has partly been proved in [5]. Indeed, on account of
Proposition 3.4, part (iv) is essentially the same as [5, Proposition 7.6],
which certainly implies that each component of $F$ is non-tangentially bounded
at $(0,x)$ for almost every $x\in{\mathbb{R}}^{d}$. Consequently part (i)
follows from Theorem 3.6. Furthermore, since $|F(x_{0},x)-\tilde{F}(x)|\leq
2F^{*}_{1}(x)\in L_{\kappa}^{1}({\mathbb{R}}^{d})$, the Lebesgue dominated
convergence theorem gives that
$\|F(x_{0},\cdot)-\tilde{F}\|_{L_{\kappa}^{1}}\rightarrow 0$ as
$x_{0}\rightarrow 0+$, and also
$\|\tilde{F}\|_{L_{\kappa}^{1}}\leq\|F\|_{{\mathscr{H}}_{\kappa}^{1}}$. On the
other hand, from the proof of [5, Proposition 7.6] one has, in vector form,

$F(x_{0}+\epsilon,x)=\left[P(F(\epsilon,\cdot))\right](x_{0},x),$

and letting $\epsilon\rightarrow 0+$ yields
$F(x_{0},x)=(P\tilde{F})(x_{0},x)$. Finally since

$\displaystyle|F(x_{0},x)|\leq[P(|\tilde{F}|)](x_{0},x),$ (34)

by Proposition 3.2 we get
$\|F\|_{{\mathscr{H}}_{\kappa}^{1}}\leq\|\tilde{F}\|_{L_{\kappa}^{1}}$. Thus
both parts (ii) and (iii) are proved.

###### Proposition 3.19

For
$F=(u_{0},u_{1},\cdots,u_{d})\in{\mathscr{H}}_{\kappa}^{1}({\mathbb{R}}^{1+d}_{+})$,
the functions $u_{j}$, $j=1,\cdots,d$, are uniquely determined by the first
component $u_{0}$.

  * Proof. Since $0\leq P_{x_{0}}(x)\leq x_{0}^{-2|\kappa|-d}$ for $x\in{\mathbb{R}}^{d}$ so that $0\leq(\tau_{x}P_{x_{0}})(-t)\leq x_{0}^{-2|\kappa|-d}$ for $x,t\in{\mathbb{R}}^{d}$ and $x_{0}>0$, from (16) and (34) one has $|F(x_{0},x)|\leq x_{0}^{-2|\kappa|-d}\|F\|_{{\mathscr{H}}_{\kappa}^{1}}$ for $(x_{0},x)\in{\mathbb{R}}^{1+d}_{+}$. If the vector-valued function $G=(u_{0},v_{1},\cdots,v_{d})$ is also in ${\mathscr{H}}_{\kappa}^{1}({\mathbb{R}}^{1+d}_{+})$, the generalized Cauchy-Riemann equations in (22) imply $\partial_{0}(u_{j}-v_{j})\equiv 0$ for $j=1,\cdots,d$, and so $u_{j}-v_{j}=:\varphi_{j}(x)$, independent of $x_{0}$. But by the inequality just proved, $|\varphi_{j}(x)|\leq x_{0}^{-2|\kappa|-d}(\|F\|_{{\mathscr{H}}_{\kappa}^{1}}+\|G\|_{{\mathscr{H}}_{\kappa}^{1}})$, and then letting $x_{0}\rightarrow+\infty$ proves that $u_{j}-v_{j}=\varphi_{j}(x)\equiv 0$ on ${\mathbb{R}}^{1+d}_{+}$.

We define the “real” Hardy space in the Dunkl setting by

$\displaystyle H_{\kappa}^{1}({\mathbb{R}}^{d})=\left\\{f\in
L_{\kappa}^{1}({\mathbb{R}}^{d}):\quad{\mathcal{R}}_{j}f\in
L_{\kappa}^{1}({\mathbb{R}}^{d}),\,\,\,j=1,\cdots,d\right\\},$

endowed with the norm
$\|f\|_{H_{\kappa}^{1}}=\|f\|_{L_{\kappa}^{1}}+\sum_{j=1}^{d}\|{\mathcal{R}}_{j}f\|_{L_{\kappa}^{1}}$.

###### Proposition 3.20

The space $H_{\kappa}^{1}({\mathbb{R}}^{d})$ is a Banach space.

###### Proposition 3.21

If $f\in H_{\kappa}^{1}({\mathbb{R}}^{d})$, then
$\int_{{\mathbb{R}}^{d}}f\,d\omega_{\kappa}=0$.

Since, for $f\in H_{\kappa}^{1}({\mathbb{R}}^{d})$, ${\mathscr{F}}_{\kappa}f$
and ${\mathscr{F}}_{\kappa}({\mathcal{R}}_{j}f)$ are all continuous at the
origin, the proposition follows immediately from (33) with
$f_{j}={\mathcal{R}}_{j}f$.

###### Theorem 3.22

The function $f$ defined on ${\mathbb{R}}^{d}$ is in
$H_{\kappa}^{1}({\mathbb{R}}^{d})$ if and only if there exists a vector-valued
function
$F=(u_{0},u_{1},\cdots,u_{d})\in{\mathscr{H}}_{\kappa}^{1}({\mathbb{R}}^{1+d}_{+})$
with $f=\lim u_{0}$, the boundary function of the first component of $F$ given
in Theorem 3.18. Moreover,
$\|f\|_{H_{\kappa}^{1}}\asymp\|F\|_{{\mathscr{H}}_{\kappa}^{1}}$, and for
$j=1,\cdots,d$, the boundary function $f_{j}$ of $u_{j}$ is the $j$th
$\kappa$-Riesz transform ${\mathcal{R}}_{j}f$ of $f$.

This theorem was proved in [5] for the $\kappa$-Riesz transforms defined in
the sense of distributions. Here we restate the proof based on Definition
3.16. For $f\in H_{\kappa}^{1}({\mathbb{R}}^{d})$, set $u_{0}=Pf$ and
$u_{j}=P({\mathcal{R}}_{j}f)$ for $1\leq j\leq d$. Using the expressions of
$u_{0}$ and $u_{j}$ ($1\leq j\leq d$) in the form of (18) and noting (33) with
$f_{j}={\mathcal{R}}_{j}f$, it is easy to check that the vector-valued
function $F=(u_{0},u_{1},\cdots,u_{d})$ satisfies the equations in (22); and
furthermore, by Proposition 3.2,
$\|F\|_{{\mathscr{H}}_{\kappa}^{1}}\leq\|f\|_{H_{\kappa}^{1}}$. Therefore
$F\in{\mathscr{H}}_{\kappa}^{1}({\mathbb{R}}^{1+d}_{+})$ with $f=\lim u_{0}$
by Corollary 3.5.

Conversely suppose
$F=(u_{0},u_{1},\cdots,u_{d})\in{\mathscr{H}}_{\kappa}^{1}({\mathbb{R}}^{1+d}_{+})$
with the boundary function $\tilde{F}=(f_{0},f_{1},\cdots,f_{d})$ determined
in Theorem 3.18. It suffices to show (33) for $j=1,\cdots,d$. By Theorem
3.18(iii), $u_{0}=Pf_{0}$ and $u_{j}=Pf_{j}$, and the generalized Cauchy-
Riemann equations in (22) and the expressions of $u_{0}$ and $u_{j}$ in the
form of (18) give, for $x_{0}>0$,

$\displaystyle 0$ $\displaystyle\equiv D_{j}u_{0}-D_{0}u_{j}$
$\displaystyle=c_{\kappa}\int_{{\mathbb{R}}^{d}}e^{-x_{0}|\xi|}\left[i\xi_{j}\left({\mathscr{F}}_{\kappa}f_{0}\right)(\xi)+|\xi|\left({\mathscr{F}}_{\kappa}f_{j}\right)(\xi)\right]E_{\kappa}(i\xi,x)\,d\omega_{\kappa}(\xi),\quad
x\in{\mathbb{R}}^{d}.$

It then follows from Proposition 2.1(ii) that
$i\xi_{j}\left({\mathscr{F}}_{\kappa}f_{0}\right)(\xi)+|\xi|\left({\mathscr{F}}_{\kappa}f_{j}\right)(\xi)=0$
for a.e. $x\in{\mathbb{R}}^{d}$, which is identical with (33) as desired. Thus
we have proved $f_{0}\in H_{\kappa}^{1}({\mathbb{R}}^{d})$; and finally, by
Theorem 3.18(ii),
$\|f_{0}\|_{H_{\kappa}^{1}}\lesssim\|\tilde{F}\|_{L_{\kappa}^{1}}=\|F\|_{{\mathscr{H}}_{\kappa}^{1}}$.
The proof of Theorem 3.22 is completed.

###  3.5 A dense subclass of the Hardy space
$H_{\kappa}^{1}({\mathbb{R}}^{d})$

We shall prove a density result on $H_{\kappa}^{1}({\mathbb{R}}^{d})$ that is
an analog of [52, p. 225, Lemma]). Set

$\displaystyle L_{\kappa,0}^{1}({\mathbb{R}}^{d})$ $\displaystyle=\left\\{f\in
L_{\kappa}^{1}({\mathbb{R}}^{d}):\,\,({\mathscr{F}}_{\kappa}f)(0)=0\right\\},$
$\displaystyle L_{\kappa,00}^{1}({\mathbb{R}}^{d})$
$\displaystyle=\left\\{f\in
L_{\kappa}^{1}({\mathbb{R}}^{d}):\,\,{\mathscr{F}}_{\kappa}f\,\,\hbox{has
compact support in ${\mathbb{R}}^{d}\setminus\\{0\\}$}\right\\}.$

###### Lemma 3.23

$H_{\kappa,0}^{1}({\mathbb{R}}^{d}):=\SS({\mathbb{R}}^{d})\bigcap
L_{\kappa,00}^{1}({\mathbb{R}}^{d})$ is s dense subclass of
$H_{\kappa}^{1}({\mathbb{R}}^{d})$.

  * Proof. Note that $H_{\kappa}^{1}({\mathbb{R}}^{d})\subseteq L_{\kappa,0}^{1}({\mathbb{R}}^{d})$ by Proposition 3.21. Choose a fixed radial $\eta\in\SS({\mathbb{R}}^{d})$ with the real Dunkl transform ${\mathscr{F}}_{\kappa}\eta$ satisfying $\chi_{\\{|\xi|\leq 1\\}}\leq{\mathscr{F}}_{\kappa}\eta\leq\chi_{\\{|\xi|\leq 2\\}}$, and write $\eta_{\epsilon}(x)=\epsilon^{-2|\kappa|-d}\eta(\epsilon^{-1}x)$ for $\epsilon>0$. We shall prove the following two claims:

Claim 1. $\SS({\mathbb{R}}^{d})\bigcap L_{\kappa,00}^{1}({\mathbb{R}}^{d})$ is
dense in $L_{\kappa,00}^{1}({\mathbb{R}}^{d})$ as a subspace of
$H_{\kappa}^{1}({\mathbb{R}}^{d})$.

Claim 2. $L_{\kappa,00}^{1}({\mathbb{R}}^{d})$ is dense in
$H_{\kappa}^{1}({\mathbb{R}}^{d})$.

To show Claim 1, define the operators $T_{\epsilon}$ for $\epsilon>0$ by
${\mathscr{F}}_{\kappa}(\widetilde{T_{\epsilon}}f)=\eta(0)^{-1}({\mathscr{F}}_{\kappa}f)*_{\kappa}({\mathscr{F}}_{\kappa}\eta)_{\epsilon}$.
For $f\in L_{\kappa,00}^{1}({\mathbb{R}}^{d})$, by Proposition 2.10 we have
that ${\mathscr{F}}_{\kappa}(\widetilde{T_{\epsilon}}f)\in
C^{\infty}({\mathbb{R}}^{d})$ and the supports of ${\mathscr{F}}_{\kappa}f$
and ${\mathscr{F}}_{\kappa}(\widetilde{T_{\epsilon}}f)$ for those $\epsilon$
sufficiently small are contained in a common compact set $K$ which is at a
positive distance from the origin. This asserts that
$\widetilde{T_{\epsilon}}f\in\SS({\mathbb{R}}^{d})\bigcap
L_{\kappa,00}^{1}({\mathbb{R}}^{d})$. In addition Propositions 2.1(ii) and
2.9(v) imply $(\widetilde{T_{\epsilon}}f)(x)=f(x)\eta(\epsilon x)/\eta(0)$, so
that

$\displaystyle\|\widetilde{T_{\epsilon}}f-f\|_{L^{1}_{\kappa}}=c_{\kappa}\int_{{\mathbb{R}}^{d}}|f(x)|\left|\eta(\epsilon
x)/\eta(0)-1\right|\,d\omega_{\kappa}\rightarrow
0,\qquad\hbox{as}\quad\epsilon\rightarrow 0+,$

by Lebesgue’s dominated theorem.

Now for $j=1,\cdots,d$, we select the functions
$m_{j}(\xi)\in\SS({\mathbb{R}}^{d})$ satisfying $m_{j}(\xi)=-i\xi_{j}/|\xi|$
when $\xi\in K$, and put $\psi_{j}={\mathscr{F}}_{\kappa}^{-1}m_{j}$. By
Proposition 2.9(iv), $\|f\ast_{\kappa}\psi_{j}\|_{L_{\kappa}^{1}}\leq
C_{\psi_{j}}\|f\|_{L_{\kappa}^{1}}$ and
$\|(\widetilde{T_{\epsilon}}f)\ast_{\kappa}\psi_{j}\|_{L_{\kappa}^{1}}\leq
C_{\psi_{j}}\|\widetilde{T_{\epsilon}}f\|_{L_{\kappa}^{1}}$, and then,
according to Proposition 2.9(v) and Definition 3.16,
${\mathcal{R}}_{j}f=f\ast_{\kappa}\psi_{j}$ and
${\mathcal{R}}_{j}(\widetilde{T_{\epsilon}}f)=\widetilde{T_{\epsilon}}f\ast_{\kappa}\psi_{j}$.
Thus both $f$ and $\widetilde{T_{\epsilon}}f$ belong to
$H_{\kappa}^{1}({\mathbb{R}}^{d})$, so that
$L_{\kappa,00}^{1}({\mathbb{R}}^{d})\subseteq
H_{\kappa}^{1}({\mathbb{R}}^{d})$, and again by Proposition 2.9(iv),

$\|{\mathcal{R}}_{j}(\widetilde{T_{\epsilon}}f-f)\|_{L_{\kappa}^{1}}=\|(\widetilde{T_{\epsilon}}f-f)\ast\psi_{j}\|_{L_{\kappa}^{1}}\leq
C_{\psi_{j}}\|\widetilde{T_{\epsilon}}f-f\|_{L_{\kappa}^{1}}\rightarrow
0,\qquad\hbox{as}\quad\epsilon\rightarrow 0+.$

Therefore $\lim_{\epsilon\rightarrow
0+}\|\widetilde{T_{\epsilon}}f-f\|_{H_{\kappa}^{1}}=0$, and Claim 1 is
concluded,

In order to prove Claim 2, define the operator $T_{\epsilon}$ by
$(T_{\epsilon}f)(x)=(f\ast_{\kappa}\eta_{\epsilon})(x)$, and for $f\in
L_{\kappa}^{1}({\mathbb{R}}^{d})$, consider $T_{\epsilon}f-T_{\epsilon^{-1}}f$
with $0<\epsilon<1$. By Proposition 2.9(iii),
$\|T_{\epsilon}f-T_{\epsilon^{-1}}f\|_{L_{\kappa}^{1}}\leq
2\|\eta\|_{L_{\kappa}^{1}}\|f\|_{L_{\kappa}^{1}}$, and by Proposition 2.9(v),

$[{\mathscr{F}}_{\kappa}(T_{\epsilon}f-T_{\epsilon^{-1}}f)](\xi)=({\mathscr{F}}_{\kappa}f)(\xi)[({\mathscr{F}}_{\kappa}\eta)(\epsilon\xi)-({\mathscr{F}}_{\kappa}\eta)(\epsilon^{-1}\xi)]=0$

whenever $|\xi|<\epsilon$ or $|\xi|>2\epsilon^{-1}$. Consequently
$T_{\epsilon}f-T_{\epsilon^{-1}}f\in
L_{\kappa,00}^{1}({\mathbb{R}}^{d})\subseteq
H_{\kappa}^{1}({\mathbb{R}}^{d})$.

We assert that for $f\in L_{\kappa,0}^{1}({\mathbb{R}}^{d})$,
$\lim_{\epsilon\rightarrow\infty}\|T_{\epsilon}f\|_{L^{1}_{\kappa}}=0$. Since

$(T_{\epsilon}f)(x)=c_{\kappa}\int_{{\mathbb{R}}^{d}}f(-t)[(\tau_{t}\eta_{\epsilon})(x)-\eta_{\epsilon}(x)]d\omega_{\kappa}(t),$

one has $\|T_{\epsilon}f\|_{L^{1}_{\kappa}}\leq
c_{\kappa}\int_{{\mathbb{R}}^{d}}|f(-t)|\|\tau_{t}\eta_{\epsilon}-\eta_{\epsilon}\|_{L^{1}_{\kappa}}d\omega_{\kappa}(t)$.
But by Proposition 2.4(ii),
$(\tau_{t}\eta_{\epsilon})(x)=\left(\tau_{\epsilon^{-1}t}\eta\right)_{\epsilon}(x)$,
so that for $t\in{\mathbb{R}}^{d}$,
$\|\tau_{t}\eta_{\epsilon}-\eta_{\epsilon}\|_{L^{1}_{\kappa}}=\|\tau_{\epsilon^{-1}t}\eta-\eta\|_{L^{1}_{\kappa}}\rightarrow
0$ as $\epsilon\rightarrow\infty$ by Corollary 2.8; and then, on account of
$\|\tau_{t}\eta_{\epsilon}-\eta_{\epsilon}\|_{L^{1}_{\kappa}}\leq
2\|\eta\|_{L^{1}_{\kappa}}$ by Proposition 2.7, applying Lebesgue’s dominated
theorem yields the desired assertion. Moreover $\lim_{\epsilon\rightarrow
0+}\|T_{\epsilon}f-f\|_{L^{1}_{\kappa}}=0$ by Proposition 2.11(ii), and hence,
for $f\in L_{\kappa,0}^{1}({\mathbb{R}}^{d})$,

$\|(T_{\epsilon}f-T_{\epsilon^{-1}}f)-f\|_{L^{1}_{\kappa}}\leq\|T_{\epsilon}f-f\|_{L^{1}_{\kappa}}+\|T_{\epsilon^{-1}}f\|_{L^{1}_{\kappa}}\rightarrow
0\qquad\hbox{as}\quad\epsilon\rightarrow 0+.$

Finally assume that $f\in H_{\kappa}^{1}({\mathbb{R}}^{d})$. Since
${\mathcal{R}}_{j}f\in L_{\kappa,0}^{1}({\mathbb{R}}^{d})$ for $1\leq j\leq d$
and $T_{\epsilon}f-T_{\epsilon^{-1}}f\in H_{\kappa}^{1}({\mathbb{R}}^{d})$, by
taking the Dunkl transform we get
${\mathcal{R}}_{j}(T_{\epsilon}f-T_{\epsilon^{-1}}f)=(T_{\epsilon}-T_{\epsilon^{-1}})({\mathcal{R}}_{j}f)$.
Thus

$\displaystyle\|{\mathcal{R}}_{j}(T_{\epsilon}f-T_{\epsilon^{-1}}f)-{\mathcal{R}}_{j}f\|_{L_{\kappa}^{1}}=\|(T_{\epsilon}-T_{\epsilon^{-1}})({\mathcal{R}}_{j}f)-{\mathcal{R}}_{j}f\|_{L_{\kappa}^{1}}\rightarrow
0\qquad\hbox{as}\quad\epsilon\rightarrow 0+,$

and consequently,
$\|(T_{\epsilon}f-T_{\epsilon^{-1}}f)-f\|_{H_{\kappa}^{1}}\rightarrow 0$ as
$\epsilon\rightarrow 0^{+}$. Claim 2 is proved and so is the lemma.

From the proof of the above lemma, we have the following corollary.

###### Corollary 3.24

If $f\in H_{\kappa,0}^{1}({\mathbb{R}}^{d})$, then ${\mathcal{R}}_{j}f\in
H_{\kappa,0}^{1}({\mathbb{R}}^{d})$.

Let ${\mathscr{H}}_{\kappa,0}^{1}({\mathbb{R}}^{1+d}_{+})$ be the collection
of the vector-valued functions of
${\mathscr{H}}_{\kappa}^{1}({\mathbb{R}}^{1+d}_{+})$ whose first components
have boundary values in
$H_{\kappa,0}^{1}({\mathbb{R}}^{d})=\SS({\mathbb{R}}^{d})\bigcap
L_{\kappa,00}^{1}({\mathbb{R}}^{d})$.

###### Corollary 3.25

The subclass ${\mathscr{H}}_{\kappa,0}^{1}({\mathbb{R}}^{1+d}_{+})$ is dense
in the Hardy space ${\mathscr{H}}_{\kappa}^{1}({\mathbb{R}}^{1+d}_{+})$ and
has the following properties: $F$ in
${\mathscr{H}}_{\kappa,0}^{1}({\mathbb{R}}^{1+d}_{+})$ and its every order
partial derivative $\partial^{\gamma}F$ ($\gamma$ is a $(d+1)$-tuple of non-
negative integers) in the $x_{0},x_{1},\cdots,x_{d}$ are continuous on
$\overline{{\mathbb{R}}^{1+d}_{+}}$; and they are all fast decreasing, that
means, for each polynomial $p$ in the $x_{0},x_{1},\cdots,x_{d}$, $p$ times
$F$ and each $\partial^{\gamma}F$ are bounded on
$\overline{{\mathbb{R}}^{1+d}_{+}}$.

##  4 The space ${\text{BMO}}_{\kappa}({\mathbb{R}}^{d})$ and the
$\kappa$-Riesz transforms

The BMO type space to be considered is the class of those $f\in L_{\kappa,{\rm
loc}}({\mathbb{R}}^{d})$ satisfying

$\displaystyle\|f\|_{*,\kappa}:=\sup_{B}\frac{1}{|B|_{\kappa}}\int_{B}|f-f_{B}|\,d\omega_{\kappa}<\infty,$

where the supremum ranges over all finite balls $B$ in ${\mathbb{R}}^{d}$, and
$f_{B}=|B|_{\kappa}^{-1}\int_{B}f\,d\omega_{\kappa}$. Such a class is denoted
by ${\text{BMO}}_{\kappa}({\mathbb{R}}^{d})$. As usual,
$f\in{\text{BMO}}_{\kappa}({\mathbb{R}}^{d})$ represents an element of the
quotient space of ${\text{BMO}}_{\kappa}({\mathbb{R}}^{d})$ modulo constants.

###### Proposition 4.1

There exists some $C>0$ such that for all
$f\in{\text{BMO}}_{\kappa}({\mathbb{R}}^{d})$ and
$x^{\prime}\in{\mathbb{R}}^{d}$,

$\displaystyle\int_{{\mathbb{R}}^{d}}\frac{|f(x)-f_{B_{1}(x^{\prime})}|}{(1+|x-x^{\prime}|)|B_{1+|x-x^{\prime}|}(x^{\prime})|_{\kappa}}d\omega_{\kappa}(x)\leq
C\|f\|_{*,\kappa},$ (35)

where $B_{\delta}(x^{\prime})=B(x^{\prime},\delta)$, the ball with center at
$x^{\prime}$ and radius $\delta>0$. Moreover for $\delta>0$,

$\displaystyle\int_{{\mathbb{R}}^{d}}\frac{|f(x)-f_{B_{\delta}(x^{\prime})}|}{(\delta+|x-x^{\prime}|)|B_{\delta+|x-x^{\prime}|}(x^{\prime})|_{\kappa}}\,d\omega_{\kappa}(x)\leq\frac{C}{\delta}\|f\|_{*,\kappa}.$
(36)

Observe that (36) is a consequence of (35) by dilations, while some estimates
similar to (35) are known (see [17], for example). In our style, since

$\displaystyle\left|B_{\delta}(x^{\prime})\right|_{\kappa}\asymp\delta^{d}\prod_{\alpha\in
R}(|\langle\alpha,x^{\prime}\rangle|+\delta)^{\kappa(\alpha)},$ (37)

which certainly implies the doubling property of the measure $\omega_{k}$, it
is easy to find that
$|f_{B_{2^{\ell}}(x^{\prime})}-f_{B_{1}(x^{\prime})}|\lesssim\ell\|f\|_{*,\kappa}$
for $\ell=1,2,\cdots$, and hence

$\frac{1}{|B_{2^{\ell}}(x^{\prime})|_{\kappa}}\int_{B_{2^{\ell}}(x^{\prime})}|f-f_{B_{1}(x^{\prime})}|\,d\omega_{\kappa}\lesssim(1+\ell)\|f\|_{*,\kappa}.$

Thus, with $B_{\delta}=B_{\delta}(x^{\prime})$, the left hand side of (35) is
dominated by a multiple of

$\displaystyle\frac{1}{|B_{1}|_{\kappa}}\int_{B_{1}}|f-f_{B_{1}}|d\omega_{\kappa}+\sum_{\ell=1}^{\infty}\frac{1}{2^{\ell}|B_{2^{\ell}}|_{\kappa}}\int_{B_{2^{\ell}}\setminus
B_{2^{\ell-1}}}|f-f_{B_{1}}|d\omega_{\kappa}\lesssim\|f\|_{*,\kappa},$

and then (35) is proved.

The John-Nirenberg inequality stated as follows, is a special case of that in
spaces of homogeneous type, which could be proved along the proof process of
the classical one.

###### Proposition 4.2

There exist constants $c_{0},c_{1}>0$ such that for all
$f\in{\text{BMO}}_{\kappa}({\mathbb{R}}^{d})$ and all finite balls $B$ in
${\mathbb{R}}^{d}$,

$\displaystyle\left|\left\\{x\in
B:|f(x)-f_{B}|>\lambda\right\\}\right|_{\kappa}\leq
c_{1}e^{-c_{0}\lambda/\|f\|_{*,\kappa}}|B|_{\kappa},\qquad\lambda>0.$ (38)

A direct consequence of (38) is that, for
$f\in{\text{BMO}}_{\kappa}({\mathbb{R}}^{d})$,

$\displaystyle\|f\|_{*,\kappa}\asymp\sup_{B}\left(\frac{1}{|B|_{\kappa}}\int_{B}|f-f_{B}|^{2}\,d\omega_{\kappa}\right)^{1/2}.$
(39)

For $j=1,\cdots,d$ and $\epsilon>0$, let

$K_{j}^{\epsilon}(x,t)=K_{j}(x,t)\chi_{d(x,t)>\epsilon}(x,t),$

where $\chi_{d(x,t)>\epsilon}$ denotes the indicator of the set
$\\{(x,t):\,d(x,t)>\epsilon\\}$. We consider the variant of
${\mathcal{R}}_{j}^{\epsilon}$ given by

$\displaystyle(\tilde{\mathcal{R}}_{j}^{\epsilon}f)(x)=c_{\kappa}\int_{{\mathbb{R}}^{d}}f(t)\left(K_{j}^{\epsilon}(x,t)-K_{j}^{1}(0,t)\right)\,d\omega_{\kappa}(t),\qquad
x\in{\mathbb{R}}^{d}.$ (40)

###### Proposition 4.3

For $f\in L^{\infty}({\mathbb{R}}^{d})$, the limit

$\displaystyle(\tilde{\mathcal{R}}_{j}f)(x):=\lim_{\epsilon\rightarrow
0+}(\tilde{\mathcal{R}}_{j}^{\epsilon}f)(x)$ (41)

exists for almost every $x\in{\mathbb{R}}^{d}$, and also in the
$L_{\kappa}^{2}$ norm on each finite ball.

Indeed, write $f=f_{1}+f_{2}$ with $f_{1}=f\chi_{B(0,2M)}$ for $M>0$. Then for
$x\in B(0,M)$ and $0<\epsilon<M$,
$(\tilde{\mathcal{R}}_{j}^{\epsilon}f)(x)=(\tilde{\mathcal{R}}_{j}^{\epsilon}f_{1})(x)+(\tilde{\mathcal{R}}_{j}^{M}f_{2})(x)$;
but since $f_{1}\in L_{\kappa}^{2}({\mathbb{R}}^{d})$,

$\lim_{\epsilon\rightarrow
0+}(\tilde{\mathcal{R}}_{j}^{\epsilon}f_{1})(x)=\lim_{\epsilon\rightarrow
0+}({\mathcal{R}}_{j}^{\epsilon}f_{1})(x)-c_{\kappa}\int_{{\mathbb{R}}^{d}}f_{1}(t)K_{j}^{1}(0,t)\,d\omega_{\kappa}(t)$

exists for almost every $x\in B(0,M)$, and so does $\lim_{\epsilon\rightarrow
0+}(\tilde{\mathcal{R}}_{j}^{\epsilon}f)(x)$. This certainly concludes (41)
for almost every $x\in{\mathbb{R}}^{d}$. In addition, for $x\in B(0,M)$,
$(\tilde{\mathcal{R}}_{j}f)(x)$ defined by (41) is

$({\mathcal{R}}_{j}f_{1})(x)-c_{\kappa}\int_{{\mathbb{R}}^{d}}f_{1}(t)K_{j}^{1}(0,t)\,d\omega_{\kappa}(t)+(\tilde{\mathcal{R}}_{j}^{M}f_{2})(x),$

so that, by Corollary 3.14,

$\int_{B(0,M)}|\tilde{\mathcal{R}}_{j}^{\epsilon}f-\tilde{\mathcal{R}}_{j}f|^{2}\,d\omega_{\kappa}=\int_{B(0,M)}|{\mathcal{R}}_{j}^{\epsilon}f_{1}-{\mathcal{R}}_{j}f_{1}|^{2}\,d\omega_{\kappa}\rightarrow
0\quad\hbox{as $\epsilon\rightarrow 0+$.}$

The main conclusion in this section is stated in the following theorem.

###### Theorem 4.4

The $\kappa$-Riesz transform $\tilde{\mathcal{R}}_{j}$ defined by (40) and
(41) is bounded from $L^{\infty}({\mathbb{R}}^{d})$ to
${\text{BMO}}_{\kappa}({\mathbb{R}}^{d})$, where $j=1,\cdots,d$.

We need to make some preparations for the proof of the theorem. The estimates
given in the following lemma are improvements of those in [27].

###### Lemma 4.5

(i) For $x,t\in{\mathbb{R}}^{d}$ satisfying $d(x,t)>0$,

$\displaystyle\frac{c_{1}}{\left|B(x,|x-t|)\right|_{\kappa}}\frac{|x_{j}-t_{j}|}{|x-t|}\leq|K_{j}(x,t)||\leq\frac{c_{2}|x_{j}-t_{j}|}{\left|B(x,d(x,t))\right|_{\kappa}}\frac{d(x,t)}{|x-t|^{2}},$
(42)

where the constants $c_{1},c_{2}>0$ are independent of
$x,t\in{\mathbb{R}}^{d}$.

(ii) For $x,t,t^{\prime}\in{\mathbb{R}}^{d}$ satisfying $d(x,t)\geq
2|t-t^{\prime}|$,

$\displaystyle|K_{j}(x,t)-K_{j}(x,t^{\prime})|\lesssim\frac{|t-t^{\prime}|}{|x-t|}\frac{1}{\left|B(x,d(x,t))\right|_{\kappa}}.$
(43)

  * Proof. In [5, 22], useful lower and upper bounds of the $\kappa$-Poisson kernel $(\tau_{x}P_{x_{0}})(-t)$ are obtained and they are simplified a little in [32], that is,

$\displaystyle\frac{c_{1}}{\left|B(x,x_{0}+|x-t|)\right|_{\kappa}}\frac{x_{0}}{x_{0}+|x-t|}\leq(\tau_{x}P_{x_{0}})(-t)\leq\frac{c_{2}x_{0}}{\left|B(x,x_{0}+d(x,t))\right|_{\kappa}}\frac{x_{0}+d(x,t)}{x_{0}^{2}+|x-t|^{2}},$
(44)

where the constants $c_{1},c_{2}>0$ are independent of $x_{0}>0$ and
$x,t\in{\mathbb{R}}^{d}$. From (23) and (44) it readily follows that

$\displaystyle\frac{c_{1}}{\left|B(x,x_{0}+|x-t|)\right|_{\kappa}}\frac{|x_{j}-t_{j}|}{x_{0}+|x-t|}\leq|(\tau_{x}Q^{(j)}_{x_{0}})(-t)|\leq\frac{c_{2}|x_{j}-t_{j}|}{\left|B(x,x_{0}+d(x,t))\right|_{\kappa}}\frac{x_{0}+d(x,t)}{x_{0}^{2}+|x-t|^{2}}.$

Now for $x,t\in{\mathbb{R}}^{d}$ satisfying $d(x,t)>0$, (42) is immediate by
letting $x_{0}\rightarrow 0+$.

In proving (43), we need the following estimate of the gradient of the
$\kappa$-Poisson kernel (see [5, (5.6)])

$\displaystyle\left|\nabla^{t}\left[(\tau_{x}P_{x_{0}})(-t)\right]\right|\lesssim\frac{(\tau_{x}P_{x_{0}})(-t)}{x_{0}+d(x,t)}.$
(45)

For $x,t,t^{\prime}\in{\mathbb{R}}^{d}$ and $x_{0}>0$, it follows from (23)
that $|(\tau_{x}Q^{(j)}_{x_{0}})(-t)-(\tau_{x}Q^{(j)}_{x_{0}})(-t^{\prime})|$
is dominated by

$\displaystyle\frac{|t_{j}-t_{j}^{\prime}|}{x_{0}}(\tau_{x}P_{x_{0}})(-t)+\frac{|x_{j}-t_{j}^{\prime}|}{x_{0}}\left|\left\langle\nabla\left[(\tau_{x}P_{x_{0}})(-\cdot)\right](\tilde{t}),t-t^{\prime}\right\rangle\right|,$

with some $\tilde{t}$ on the line between $t$ and $t^{\prime}$. Thus applying
(44) and (45), and noting that

$\displaystyle d(x,\tilde{t})\geq d(x,t)-|t-t^{\prime}|\geq
d(x,t)/2\quad\hbox{and}\quad|x-t^{\prime}|/2\leq|x-t|\leq 2|x-\tilde{t}|$

whenever $d(x,t)\geq 2|t-t^{\prime}|$, we get

$\displaystyle\left|(\tau_{x}Q^{(j)}_{x_{0}})(-t)-(\tau_{x}Q^{(j)}_{x_{0}})(-t^{\prime})\right|\lesssim$
$\displaystyle\frac{|t_{j}-t_{j}^{\prime}|}{\left|B(x,x_{0}+d(x,t))\right|_{\kappa}}\left[\frac{x_{0}+d(x,t)}{x_{0}^{2}+|x-t|^{2}}+\frac{|x_{j}-t_{j}^{\prime}|}{x_{0}^{2}+|x-\tilde{t}|^{2}}\right]$
$\displaystyle\lesssim$
$\displaystyle\frac{|t-t^{\prime}|}{x_{0}+|x-t|}\frac{1}{\left|B(x,x_{0}+d(x,t))\right|_{\kappa}}.$

Again by letting $x_{0}\rightarrow 0+$, the estimate in (43) is proved for
$d(x,t)\geq 2|t-t^{\prime}|$.

###### Corollary 4.6

There exists a constant $C>0$ independent of $\epsilon>0$ such that

$\displaystyle\int_{d(x,t)>2|t-t^{\prime}|}|K_{j}^{\epsilon}(x,t)-K_{j}^{\epsilon}(x,t^{\prime})|\,d\omega_{\kappa}(x)\leq
C,\qquad t,t^{\prime}\in{\mathbb{R}}^{d}.$ (46)

  * Proof. Let us first consider the $\kappa$-Riesz kernels $K_{j}(x,t)$ ($1\leq j\leq d$) themselves. Applying (43) and in view of that (see [32, (32)])

$\displaystyle\left|B(x,d(x,t))\right|_{\kappa}\asymp\left|B(t,d(x,t))\right|_{\kappa},$
(47)

we have, for $t\neq t^{\prime}$,

$\displaystyle\int_{d(x,t)>2|t-t^{\prime}|}|K_{j}(x,t)-K_{j}(x,t^{\prime})|\,d\omega_{\kappa}(x)$
$\displaystyle\lesssim$
$\displaystyle\sum_{\ell=1}^{\infty}\int_{2^{\ell}|t-t^{\prime}|<d(x,t)\leq
2^{\ell+1}|t-t^{\prime}|}\frac{|t-t^{\prime}|}{|x-t|}\frac{1}{\left|B(t,d(x,t))\right|_{\kappa}}\,d\omega_{\kappa}(x)$
$\displaystyle\lesssim$
$\displaystyle\sum_{\ell=1}^{\infty}\frac{1}{2^{\ell}}\frac{1}{\left|B(t,2^{\ell}|t-t^{\prime}|)\right|_{\kappa}}\int_{{\mathcal{O}}(B(t,2^{\ell+1}|t-t^{\prime}|))}\,d\omega_{\kappa}(x),$
(48)

where

${\mathcal{O}}(B(x^{\prime},r))=\\{x:\,d(x,x^{\prime})<r\\}$

denotes the $G$-orbit of the ball $B(x^{\prime},r)$. It follows immediately
that there exists a constant $C>0$ such that

$\displaystyle\int_{d(x,t)>2|t-t^{\prime}|}|K_{j}(x,t)-K_{j}(x,t^{\prime})|\,d\omega_{\kappa}(x)\leq
C,\qquad t,t^{\prime}\in{\mathbb{R}}^{d}.$ (49)

We now turn to the associated estimate for $K_{j}^{\epsilon}$. Assume that
$d(x,t)>2|t-t^{\prime}|$. It is obvious that
$|d(x,t)-d(x,t^{\prime})|\leq|t-t^{\prime}|$, and so $d(x,t)/2\leq
d(x,t^{\prime})\leq 2d(x,t)$.

If $d(x,t)\leq\epsilon$ and $d(x,t^{\prime})\leq\epsilon$, then
$K_{j}^{\epsilon}(x,t)=K_{j}^{\epsilon}(x,t^{\prime})=0$; if $d(x,t)>\epsilon$
and $d(x,t^{\prime})>\epsilon$, then
$K_{j}^{\epsilon}(x,t)-K_{j}^{\epsilon}(x,t^{\prime})=K_{j}(x,t)-K_{j}(x,t^{\prime})$;
if $d(x,t)>\epsilon$ and $d(x,t^{\prime})\leq\epsilon$, then
$K_{j}^{\epsilon}(x,t)-K_{j}^{\epsilon}(x,t^{\prime})=K_{j}(x,t)$ and
$\epsilon<d(x,t)\leq 2d(x,t^{\prime})\leq 2\epsilon$; and similarly, if
$d(x,t)\leq\epsilon$ and $d(x,t^{\prime})>\epsilon$, then
$K_{j}^{\epsilon}(x,t)-K_{j}^{\epsilon}(x,t^{\prime})=-K_{j}(x,t^{\prime})$
and $\epsilon<d(x,t^{\prime})\leq 2d(x,t)\leq 2\epsilon$. Thus
$\int_{d(x,t)>2|t-t^{\prime}|}|K_{j}^{\epsilon}(x,t)-K_{j}^{\epsilon}(x,t^{\prime})|\,d\omega_{\kappa}(x)$
is dominated by

$\displaystyle\int_{d(x,t)>2|t-t^{\prime}|}|K_{j}(x,t)-K_{j}(x,t^{\prime})|\,d\omega_{\kappa}(x)$
$\displaystyle\qquad\qquad\qquad+\int_{\epsilon<d(x,t)\leq
2\epsilon}|K_{j}(x,t)|\,d\omega_{\kappa}(x)+\int_{\epsilon<d(x,t^{\prime})\leq
2\epsilon}|K_{j}(x,t^{\prime})|\,d\omega_{\kappa}(x)$

From (49) the first term above is bounded by a constant; while, by use of (42)
and (47), the second term is controlled by a multiple of

$\displaystyle\int_{\epsilon<d(x,t)\leq
2\epsilon}\frac{|x_{j}-t_{j}|}{\left|B(t,d(x,t))\right|_{\kappa}}\frac{d(x,t)}{|x-t|^{2}}\,d\omega_{\kappa}(x)\lesssim\frac{1}{\left|B(t,\epsilon)\right|_{\kappa}}\int_{{\mathcal{O}}(B(t,2\epsilon))}\,d\omega_{\kappa}(x)\lesssim
1;$

and the same estimate holds for the third term. Summing these concludes (46).

We are now in a position to consider the operators
${\mathcal{R}}_{j}^{\epsilon,N}$, for $0<\epsilon<N<\infty$, defined by

$\displaystyle({\mathcal{R}}_{j}^{\epsilon,N}f)(x)=c_{\kappa}\int_{{\mathbb{R}}^{d}}f(t)K_{j}^{\epsilon,N}(x,t)\,d\omega_{\kappa}(t),\qquad
x\in{\mathbb{R}}^{d},$ (50)

where

$K_{j}^{\epsilon,N}(x,t)=K_{j}^{\epsilon}(x,t)-K_{j}^{N}(x,t).$

###### Lemma 4.7

There exists a constant $C>0$ independent of $\epsilon,N$ and $f\in
L^{\infty}({\mathbb{R}}^{d})$ such that

$\displaystyle\|{\mathcal{R}}_{j}^{\epsilon,N}f\|_{*,\kappa}\leq
C\|f\|_{L^{\infty}}.$ (51)

  * Proof. We fix a ball $B=B_{\delta}(x^{\prime})$ and decompose $f\in L^{\infty}({\mathbb{R}}^{d})$ by $f=f_{1}+f_{2}$, where $f_{1}=f\chi_{{\mathcal{O}}(3B)}$ and $f_{2}=f\chi_{[{\mathcal{O}}(3B)]^{c}}$.

By Corollary 3.15, ${\mathcal{R}}_{j}^{\epsilon,N}$ is bounded from
$L_{\kappa}^{2}({\mathbb{R}}^{d})$ to itself with a bound of operator norms
independent of $\epsilon,N$. Thus

$\displaystyle\int_{B}|{\mathcal{R}}_{j}^{\epsilon,N}f_{1}|d\omega_{\kappa}\leq|B|_{\kappa}^{1/2}\left(\int_{B}|{\mathcal{R}}_{j}^{\epsilon,N}f_{1}|^{2}d\omega_{\kappa}\right)^{1/2}$
$\displaystyle\lesssim$
$\displaystyle|B|_{\kappa}^{1/2}\left(\int_{{\mathcal{O}}(3B)}|f_{1}|^{2}d\omega_{\kappa}\right)^{1/2}\lesssim|B|_{\kappa}\|f\|_{L^{\infty}}.$
(52)

Passing to ${\mathcal{R}}_{j}^{\epsilon,N}f_{2}$, for $x\in B$ we have

$\displaystyle\left|({\mathcal{R}}_{j}^{\epsilon,N}f_{2})(x)-({\mathcal{R}}_{j}^{\epsilon,N}f_{2})(x^{\prime})\right|$
$\displaystyle=c_{\kappa}\int_{{\mathbb{R}}^{d}}|f_{2}(t)||K_{j}^{\epsilon,N}(x,t)-K_{j}^{\epsilon,N}(x^{\prime},t)|\,d\omega_{\kappa}(t)$
$\displaystyle\lesssim\|f\|_{L^{\infty}}\int_{[{\mathcal{O}}(3B)]^{c}}|K_{j}^{\epsilon,N}(x,t)-K_{j}^{\epsilon,N}(x^{\prime},t)|\,d\omega_{\kappa}(t).$

It is noted that $K_{j}(x,t)=-K_{j}(t,x)$ by (32), and then by Corollary 4.6,

$\displaystyle\left|({\mathcal{R}}_{j}^{\epsilon,N}f_{2})(x)-({\mathcal{R}}_{j}^{\epsilon,N}f_{2})(x^{\prime})\right|\lesssim\|f\|_{L^{\infty}},\qquad
x\in B.$ (53)

Combining ( ‣ 4) and (53) yields

$\displaystyle\int_{B}|({\mathcal{R}}_{j}^{\epsilon,N}f)(x)-({\mathcal{R}}_{j}^{\epsilon,N}f_{2})(x^{\prime})|d\omega_{\kappa}\lesssim|B|_{\kappa}\|f\|_{L^{\infty}}.$

Therefore ${\mathcal{R}}_{j}^{\epsilon,N}f\in{\text{BMO}}_{\kappa}$, and (51)
holds.

We now turn to the proof of Theorem 4.4.

Assume $f\in L^{\infty}({\mathbb{R}}^{d})$ and fix a ball
$B=B_{\delta}(x^{\prime})$. Note that
$|d(x,t)-d(x^{\prime},t)|\leq|x-x^{\prime}|\leq\delta$ for $x\in B$.

It follows from (40) that, for $x\in B$,

$\displaystyle\left|(\tilde{\mathcal{R}}_{j}^{N}f)(x)-(\tilde{\mathcal{R}}_{j}^{N}f)(x^{\prime})\right|\lesssim\|f\|_{L^{\infty}}\int_{{\mathbb{R}}^{d}}\left|K_{j}^{N}(x,t)-K_{j}^{N}(x^{\prime},t)\right|\,d\omega_{\kappa}(t).$
(54)

Adopting closely the argument in the proof of Corollary 4.6, the integral
above is controlled by

$\displaystyle\int_{d(x,t)>N}|K_{j}(x,t)-K_{j}(x^{\prime},t)|\,d\omega_{\kappa}(t)$
$\displaystyle\qquad\qquad+\int_{N<d(x,t)\leq
N+\delta}|K_{j}(x,t)|\,d\omega_{\kappa}(t)+\int_{N<d(x^{\prime},t)\leq
N+\delta}|K_{j}(x^{\prime},t)|\,d\omega_{\kappa}(t).$ (55)

We consider the case for $N\geq 2(|x^{\prime}|+\delta)$. The first term in
(4), after a similar treatment in ( ‣ 4), is bounded a multiple of

$\displaystyle\sum_{\ell=1}^{\infty}\int_{2^{\ell-1}N<d(x,t)\leq
2^{\ell}N}\frac{|x-x^{\prime}|}{|x-t|}\frac{1}{\left|B(x,d(x,t))\right|_{\kappa}}\,d\omega_{\kappa}(t)\lesssim\frac{\delta}{N}.$
(56)

As for the second term in (4), using (42) gives

$\displaystyle\int_{N<d(x,t)\leq
N+\delta}|K_{j}(x,t)|\,d\omega_{\kappa}(t)\lesssim$
$\displaystyle\frac{1}{\left|B(x,N)\right|_{\kappa}}\int_{N<d(x,t)\leq
N+\delta}\,d\omega_{\kappa}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\left|B(x/N,1)\right|_{\kappa}}\int_{1<d(x/N,t)\leq
1+\delta/N}\,d\omega_{\kappa}(t);$

and since $B(x/N,1)\supseteq B(x^{\prime}/N,1/2)$ and
$|d(x^{\prime}/N,t)-d(x/N,t)|\leq\delta/N$, it follows that

$\displaystyle\int_{N<d(x,t)\leq
N+\delta}|K_{j}(x,t)|\,d\omega_{\kappa}(t)\lesssim\frac{1}{\left|B(x^{\prime}/N,1/2)\right|_{\kappa}}\int_{1-\delta/N<d(x^{\prime}/N,t)\leq
1+2\delta/N}\,d\omega_{\kappa}(t).$

The same estimate holds for the third term in (4). Inserting them and (56)
into (4), and then (54), asserts that
$(\tilde{\mathcal{R}}_{j}^{N}f)(x)-(\tilde{\mathcal{R}}_{j}^{N}f)(x^{\prime})\rightarrow
0$ uniformly for $x\in B$ as $N\rightarrow\infty$.

Since, from (40) and (50),

$\tilde{\mathcal{R}}_{j}^{\epsilon}f-\tilde{\mathcal{R}}_{j}^{N}f={\mathcal{R}}_{j}^{\epsilon,N}f,$

by Lemma 4.7 and (39) we have

$\displaystyle\frac{1}{|B|_{\kappa}}\int_{B}\left|\tilde{\mathcal{R}}_{j}^{\epsilon}f-\tilde{\mathcal{R}}_{j}^{N}f-(\tilde{\mathcal{R}}_{j}^{\epsilon}f)_{B}+(\tilde{\mathcal{R}}_{j}^{N}f)_{B}\right|^{2}\,d\omega_{\kappa}\leq
C\|f\|_{L^{\infty}}^{2}.$ (57)

But, with $a=(\tilde{\mathcal{R}}_{j}^{N}f)(x^{\prime})$,

$(\tilde{\mathcal{R}}_{j}^{N}f)(x)-(\tilde{\mathcal{R}}_{j}^{N}f)_{B}=(\tilde{\mathcal{R}}_{j}^{N}f)(x)-a-[\tilde{\mathcal{R}}_{j}^{N}f-a]_{B}$

tends to zero uniformly for $x\in B$ as $N\rightarrow\infty$, as just proved.
Thus from (57) we get

$\displaystyle\frac{1}{|B|_{\kappa}}\int_{B}\left|\tilde{\mathcal{R}}_{j}^{\epsilon}f-(\tilde{\mathcal{R}}_{j}^{\epsilon}f)_{B}\right|^{2}\,d\omega_{\kappa}\leq
C\|f\|_{L^{\infty}}^{2}.$

Finally letting $\epsilon\rightarrow 0+$, Proposition 4.3 asserts
$|B|_{\kappa}^{-1}\int_{B}\left|\tilde{\mathcal{R}}_{j}f-(\tilde{\mathcal{R}}_{j}f)_{B}\right|^{2}\,d\omega_{\kappa}\leq
C\|f\|_{L^{\infty}}^{2}$. Thus, in view of (39),
$\|\tilde{\mathcal{R}}_{j}f\|_{*,\kappa}\lesssim\|f\|_{L^{\infty}}$, and the
proof of Theorem 4.4 is completed.

##  5 The associated Carleson measures

As usual, for a ball $B=B_{r}(x^{\prime})$ in ${\mathbb{R}}^{d}$ the
associated tent $T(B)$ is the closed set in ${\mathbb{R}}^{1+d}_{+}$ given by

$T(B)=\\{(x_{0},x):\,|x-x^{\prime}|\leq r-x_{0}\\}.$

###### Definition 5.1

A Borel measure $d\nu$ on ${\mathbb{R}}^{1+d}_{+}$ is called a
$\kappa$-Carleson measure if there exists some $c>0$ such that for all balls
$B$ in ${\mathbb{R}}^{d}$,

$\iint_{T(B)}|d\nu|\leq c|B|_{\kappa};$

and $\|d\nu\|_{\kappa}:=\sup_{B}|B|_{\kappa}^{-1}\iint_{T(B)}|d\nu|$ is named
the $\kappa$-Carleson norm of $d\nu$.

As in the classical case, the property underlying the definition of
$\kappa$-Carleson measures is not limited to balls. Indeed we have the
following lemma.

###### Lemma 5.2

If $d\nu$ is a $\kappa$-Carleson measure on ${\mathbb{R}}^{1+d}_{+}$ and
$E\subset{\mathbb{R}}^{d}$ is open, then

$\iint_{\widehat{E}}|d\nu|\lesssim\|d\nu\|_{\kappa}|E|_{\kappa},$

where $\widehat{E}=\\{(r,x)\in{\mathbb{R}}^{1+d}_{+}:\,\,B_{r}(x)\subset
E\\}$.

The proof of the lemma is a straightforward adaptation of that of [16, Lemma
9.4]. A $\kappa$-Carleson measure on ${\mathbb{R}}^{1+d}_{+}$ can also be
characterized as a measure $d\nu$ for which the $\kappa$-Poisson integral
defines a bounded operator from $L_{\kappa}^{p}({\mathbb{R}}^{d})$ to
$L^{p}({\mathbb{R}}^{1+d}_{+},|d\nu|)$ for $1<p<\infty$.

###### Theorem 5.3

Assume that $d\nu$ is a Borel measure on ${\mathbb{R}}^{1+d}_{+}$.

(i) If $d\nu$ is a $\kappa$-Carleson measure, then for every $p\in(1,\infty)$,

$\displaystyle\iint_{{\mathbb{R}}^{1+d}_{+}}|(Pf)(x_{0},x)|^{p}\,|d\nu|\lesssim\|d\nu\|_{\kappa}\|f\|_{L_{\kappa}^{p}}^{p},\qquad
f\in L^{p}_{\kappa}({\mathbb{R}}^{d}).$

(ii) If for some $p\in[1,\infty)$, there holds
$\iint_{{\mathbb{R}}^{1+d}_{+}}|(Pf)(x_{0},x)|^{p}\,|d\nu|\leq
c\|f\|_{L_{\kappa}^{p}}^{p}$ for $f\in L^{p}_{\kappa}({\mathbb{R}}^{d})$,
where $c>0$ is independent of $f$, then $d\nu$ is a $\kappa$-Carleson measure.

  * Proof. First suppose that $d\nu$ is a $\kappa$-Carleson measure. Let $E_{\lambda}=\\{x\in{\mathbb{R}}^{d}:(P^{*}_{1}f)(x)>\lambda\\}$. It then follows that $\\{(x_{0},x)\in{\mathbb{R}}^{1+d}_{+}:\,|(Pf)(x_{0},x)|>\lambda\\}\subset\widehat{E_{\lambda}}$, and by Lemma 5.2,

$\displaystyle\iint_{{\mathbb{R}}^{1+d}_{+}}|(Pf)(x_{0},x)|^{p}\,|d\nu|$
$\displaystyle=p\int_{0}^{\infty}\lambda^{p-1}\iint_{\\{(x_{0},x):\,|(Pf)(x_{0},x)|>\lambda\\}}|d\nu|\,d\lambda$
$\displaystyle\lesssim
p\|d\nu\|_{\kappa}\int_{0}^{\infty}\lambda^{p-1}|E_{\lambda}|_{\kappa}d\lambda,$

that is identical with
$\|d\nu\|_{\kappa}\int_{{\mathbb{R}}^{d}}(P^{*}_{1}f)(x)^{p}d\omega_{\kappa}$.
Part (i) is immediate by Proposition 3.4.

Conversely, suppose that $d\nu$ satisfies the condition in part (ii). For a
ball $B=B_{r}(x^{\prime})$ in ${\mathbb{R}}^{d}$ and for $(x_{0},x)\in T(B)$,
using the fact $B_{x_{0}}(x)\subset B$ and the first inequality in (44) we
have

$\displaystyle(P\chi_{B})(x_{0},x)\geq
c_{\kappa}\int_{B_{x_{0}}(x)}(\tau_{x}P_{x_{0}})(-t)\,d\omega_{\kappa}(t)\gtrsim\int_{B_{x_{0}}(x)}\frac{x_{0}}{x_{0}+|x-t|}\frac{d\omega_{\kappa}(t)}{\left|B(x,x_{0}+|x-t|)\right|_{\kappa}},$

which implies $(P\chi_{B})(x_{0},x)\gtrsim 1$ for $(x_{0},x)\in T(B)$.
Consequently

$\displaystyle\iint_{T(B)}|d\nu|\lesssim\iint_{T(B)}|(P\chi_{B})(x_{0},x)|^{p}|d\nu|\lesssim\|\chi_{B}\|_{L_{\kappa}^{p}}^{p}=|B|_{\kappa}.$

Hence $d\nu$ is a $\kappa$-Carleson measure.

We now introduce a class of functions on ${\mathbb{R}}^{d}$ in terms of
$\kappa$-Carleson measures.

###### Definition 5.4

A function $\varphi\in L_{\kappa,{\rm loc}}({\mathbb{R}}^{d})$ is said to be
in ${\text{BMC}}_{\kappa}({\mathbb{R}}^{d})$ if it satisfies the following two
conditions:

(i)

$\displaystyle\int_{{\mathbb{R}}^{d}}\frac{|\varphi(x)|}{(1+|x|)^{2|\kappa|+d+1}}\,d\omega_{\kappa}(x)<\infty;$
(58)

(ii)
$d\nu_{\kappa,\varphi}(x_{0},x)=x_{0}|\nabla_{\tilde{\kappa}}u_{\varphi}(x_{0},x)|^{2}\,dx_{0}d\omega_{\kappa}(x)$
is a $\kappa$-Carleson measure on ${\mathbb{R}}^{1+d}_{+}$, where
$\nabla_{\tilde{\kappa}}=(\partial_{0},D_{1},\dots,D_{d})$ is the
$\tilde{\kappa}$-gradient, and $u_{\varphi}(x_{0},x)$ is the $\kappa$-Poisson
integral of $\varphi$.

Moreover the norm of $\varphi$ in ${\text{BMC}}_{\kappa}({\mathbb{R}}^{d})$ is
defined by
$\|\varphi\|_{*,C_{\kappa}}=\|d\nu_{\kappa,\varphi}\|_{\kappa}^{1/2}$.

###### Proposition 5.5

A bounded function $\varphi$ on ${\mathbb{R}}^{d}$ belongs to
${\text{BMC}}_{\kappa}({\mathbb{R}}^{d})$ and
$\|\varphi\|_{*,C_{\kappa}}\lesssim\|\varphi\|_{L^{\infty}}$.

  * Proof. For a fixed ball $B=B_{\delta}(x^{\prime})$ in ${\mathbb{R}}^{d}$,

$\displaystyle\iint_{T(B)}d\nu_{\kappa,\varphi\chi_{{\mathcal{O}}(2B)}}(x_{0},x)\leq\iint_{{\mathbb{R}}^{1+d}_{+}}x_{0}|\nabla_{\tilde{\kappa}}u_{\varphi\chi_{{\mathcal{O}}(2B)}}(x_{0},x)|^{2}\,dx_{0}d\omega_{\kappa}(x);$
(59)

and since for $f\in L_{\kappa}^{1}({\mathbb{R}}^{d})\cup
L_{\kappa}^{2}({\mathbb{R}}^{d})$, by (3) and (18),

$\displaystyle\left[{\mathscr{F}}_{\kappa}(\partial_{x_{0}}u_{f}(x_{0},\cdot))\right](\xi)$
$\displaystyle=-|\xi|e^{-x_{0}|\xi|}\left({\mathscr{F}}_{\kappa}f\right)(\xi),$
(60)
$\displaystyle\left[{\mathscr{F}}_{\kappa}(D_{j}u_{f}(x_{0},\cdot))\right](\xi)$
$\displaystyle=i\xi_{j}e^{-x_{0}|\xi|}\left({\mathscr{F}}_{\kappa}f\right)(\xi),\quad
j=1,\cdots,d,$ (61)

the Plancherel formula asserts that the integral on the right hand side of
(59) is equal to
$2^{-1}\int_{{\mathbb{R}}^{d}}|\varphi\chi_{{\mathcal{O}}(2B)}|^{2}d\omega_{\kappa}\lesssim\|\varphi\|_{L^{\infty}}^{2}|B|_{\kappa}$.

Nevertheless, from (16),

$\displaystyle|\nabla_{\tilde{\kappa}}u_{\varphi\chi_{[{\mathcal{O}}(2B)]^{c}}}(x_{0},x)|\leq
c_{\kappa}\|\varphi\|_{L^{\infty}}\int_{[{\mathcal{O}}(2B)]^{c}}\left|\nabla_{\tilde{\kappa}}^{(x_{0},x)}[(\tau_{x}P_{x_{0}})(-t)]\right|\,d\omega_{\kappa}(t),$
(62)

and by [5, Proposition 5.1] (also [32, (32)]),

$\displaystyle\left|\nabla_{\tilde{\kappa}}^{(x_{0},x)}[(\tau_{x}P_{x_{0}})(-t)]\right|\lesssim\frac{(x_{0}+d(t,x))^{-1}}{\left|B(x,x_{0}+d(t,x))\right|_{\kappa}}.$
(63)

Since for $(x_{0},x)\in T(B)$ and $t\in[{\mathcal{O}}(2B)]^{c}$, $d(t,x)\geq
d(t,x^{\prime})/2\geq\delta$, it follows from (62) and (63) that

$\displaystyle|\nabla_{\tilde{\kappa}}u_{\varphi\chi_{[{\mathcal{O}}(2B)]^{c}}}(x_{0},x)|\lesssim\|\varphi\|_{L^{\infty}}\int_{[{\mathcal{O}}(2B)]^{c}}\frac{d\omega_{\kappa}(t)}{d(t,x)\left|B(x,d(t,x))\right|_{\kappa}}\lesssim\frac{\|\varphi\|_{L^{\infty}}}{\delta}.$
(64)

Thus

$\iint_{T(B)}d\nu_{\kappa,\varphi\chi_{[{\mathcal{O}}(2B)]^{c}}}(x_{0},x)\lesssim\left(\frac{\|\varphi\|_{L^{\infty}}}{\delta}\right)^{2}\iint_{T(B)}x_{0}\,dx_{0}d\omega_{\kappa}(x)\lesssim\|\varphi\|_{L^{\infty}}^{2}|B|_{\kappa},$

so that
$\iint_{T(B)}d\nu_{\kappa,\varphi}(x_{0},x)\lesssim\|\varphi\|_{L^{\infty}}^{2}|B|_{\kappa}$,
as desired.

Note that, for the symbol “${\text{BMC}}_{\kappa}$”, “BM” means “bounded mean”
since, at least, a function $\varphi$ with the property
$\sup_{A>0}A^{-d-2|\kappa|}\int_{|x|\leq A}|\varphi|d\omega_{\kappa}<\infty$
satisfies condition (i), and “C” certainly indicates that it is connected with
Carleson measures as stated in condition (ii).

At this stage we shall prove an analog of Theorem 4.4 for the space
${\text{BMC}}_{\kappa}({\mathbb{R}}^{d})$, that is necessary in characterizing
the dual of the Hardy space $H_{\kappa}({\mathbb{R}}^{d})$ in the next
section.

###### Theorem 5.6

The $\kappa$-Riesz transform $\tilde{\mathcal{R}}_{j}$ defined by (40) and
(41) is bounded from $L^{\infty}({\mathbb{R}}^{d})$ to
${\text{BMC}}_{\kappa}({\mathbb{R}}^{d})$, where $j=1,\cdots,d$.

If the multiplicity function $\kappa$ is identical to zero, i.e.,
$\kappa(\alpha)=0$ for all $\alpha\in R$, it is well known that
${\text{BMC}}_{0}({\mathbb{R}}^{d})={\text{BMO}}_{0}({\mathbb{R}}^{d})$ (see
[24, Theorem 3]); but for general nonnegative $\kappa$, a partial relation of
the two classes ${\text{BMC}}_{\kappa}({\mathbb{R}}^{d})$ and
${\text{BMO}}_{\kappa}({\mathbb{R}}^{d})$ is a consequence of the duality of
$H_{\kappa}({\mathbb{R}}^{d})$ and ${\text{BMC}}_{\kappa}({\mathbb{R}}^{d})$
to be proved in the next section. It is uncertain whether there was a direct
proof of the converse of this inclusion.

In proving Theorem 5.6, we shall need an estimate of the
$\tilde{\kappa}$-gradient of the conjugate $\kappa$-Poisson kernels
$(\tau_{x}Q^{(j)}_{x_{0}})(-t)$ ($j=1,2,\cdots,d$) given below.

###### Lemma 5.7

The conjugate $\kappa$-Poisson kernels $(\tau_{x}Q^{(j)}_{x_{0}})(-t)$
($j=1,2,\cdots,d$) satisfy

$\displaystyle\left|\nabla_{\tilde{\kappa}}^{(x_{0},x)}[(\tau_{x}Q^{(j)}_{x_{0}})(-t)]\right|\lesssim\frac{(x_{0}+d(t,x))^{-1}}{\left|B(x,x_{0}+d(t,x))\right|_{\kappa}}.$
(65)

  * Proof. Define the function $U_{x_{0}}(x,t)$ by

$\displaystyle
U_{x_{0}}(x,t)=\frac{1}{2}\int_{0}^{\infty}e^{-\frac{x_{0}^{2}}{4v}}\,h_{v}(x,t)\,\frac{dv}{v^{3/2}}=\frac{1}{x_{0}}\int_{0}^{\infty}e^{-u}h_{x_{0}^{2}/4u}(x,t)\,\frac{du}{\sqrt{u}},$
(66)

where $h_{v}(x,t)$ is the heat kernel associated to the Dunkl operators given
by (26).

We are going to show, for $i=1,2,\cdots,d$,

$\displaystyle\left|D_{i}^{x}\left[U_{x_{0}}(x,t)\right]\right|\lesssim\frac{(x_{0}+d(x,t))^{-1}}{\left|B(x,x_{0}+d(x,t))\right|_{\kappa}}\frac{|x_{i}-t_{i}|}{x_{0}^{2}+|x-t|^{2}}.$
(67)

From (28) and (66),

$\displaystyle
D_{i}^{x}\left[U_{x_{0}}(x,t)\right]=\frac{2(t_{i}-x_{i})}{x_{0}^{3}}\int_{0}^{\infty}e^{-u}h_{x_{0}^{2}/4u}(x,t)\sqrt{u}\,du,$
(68)

and making us of [22, Theoem 3.1]

$\displaystyle
h_{v}(x,t)\lesssim\left(1+\frac{|x-t|}{\sqrt{v}}\right)^{-2}\frac{e^{-cd(x,t)^{2}/v}}{|B(x,\sqrt{v})|_{\kappa}}$

yields

$\displaystyle\left|D_{i}^{x}\left[U_{x_{0}}(x,t)\right]\right|\lesssim\frac{|x_{i}-t_{i}|}{x_{0}^{3}}\int_{0}^{\infty}\left(1+\frac{u|x-t|^{2}}{x_{0}^{2}}\right)^{-1}\frac{e^{-u(1+4cd(x,t)^{2}/x_{0}^{2})}}{|B(x,x_{0}/\sqrt{u})|_{\kappa}}\sqrt{u}\,du.$

Since
$|B(x,r_{2})|_{\kappa}\lesssim(r_{2}/r_{1})^{d+2|\kappa|}|B(x,r_{1})|_{\kappa}$
for $r_{2}\geq r_{1}>0$, it follows that, for $u\geq u_{0}$ with
$u_{0}=\left(x_{0}/(x_{0}+d(x,t))\right)^{2}$,

$\frac{e^{-u(1+4cd(x,t)^{2}/x_{0}^{2})}}{|B(x,x_{0}/\sqrt{u})|_{\kappa}}\lesssim\frac{e^{-c^{\prime}u/u_{0}}}{|B(x,x_{0}+d(x,t))|_{\kappa}}\left(\frac{u}{u_{0}}\right)^{|\kappa|+d/2}\lesssim\frac{(u_{0}/u)^{2}}{|B(x,x_{0}+d(x,t))|_{\kappa}}.$

Consequently,

$\displaystyle\left|D_{i}^{x}\left[U_{x_{0}}(x,t)\right]\right|\lesssim\frac{x_{0}^{-3}|x_{i}-t_{i}|}{|B(x,x_{0}+d(x,t))|_{\kappa}}\left[\int_{0}^{u_{0}}\frac{x_{0}^{2}\sqrt{u}\,du}{x_{0}^{2}+u|x-t|^{2}}+u_{0}^{2}\int_{u_{0}}^{\infty}\frac{x_{0}^{2}u^{-3/2}du}{x_{0}^{2}+u|x-t|^{2}}\right].$

Thus if $|x-t|\leq x_{0}+d(x,t)$,

$\displaystyle\left|D_{i}^{x}\left[U_{x_{0}}(x,t)\right]\right|\lesssim\frac{x_{0}^{-3}|x_{i}-t_{i}|u_{0}^{3/2}}{|B(x,x_{0}+d(x,t))|_{\kappa}}\asymp\frac{(x_{0}+d(x,t))^{-1}}{\left|B(x,x_{0}+d(x,t))\right|_{\kappa}}\frac{|x_{i}-t_{i}|}{x_{0}^{2}+|x-t|^{2}},$

and if $|x-t|>x_{0}+d(x,t)$,

$\displaystyle\left|D_{i}^{x}\left[U_{x_{0}}(x,t)\right]\right|\lesssim\frac{x_{0}^{-3}|x_{i}-t_{i}|}{|B(x,x_{0}+d(x,t))|_{\kappa}}\frac{x_{0}^{2}u_{0}^{1/2}}{|x-t|^{2}}\asymp\frac{(x_{0}+d(x,t))^{-1}}{\left|B(x,x_{0}+d(x,t))\right|_{\kappa}}\frac{|x_{i}-t_{i}|}{x_{0}^{2}+|x-t|^{2}},$

so that (67) is concluded.

If $i\neq j$, it then follows from (29), (66) and (67) that

$\displaystyle\left|D_{i}^{x}[(\tau_{x}Q^{(j)}_{x_{0}})(-t)]\right|=\frac{|x_{j}-t_{j}|}{\sqrt{\pi}}\left|D_{i}^{x}\left[U_{x_{0}}(x,t)\right]\right|\lesssim\frac{(x_{0}+d(x,t))^{-1}}{\left|B(x,x_{0}+d(x,t))\right|_{\kappa}}.$
(69)

If $i=j$, from (1) and (29) direct calculations show

$\displaystyle\sqrt{\pi}D_{j}^{x}\left[(\tau_{x}Q^{(j)}_{x_{0}})(-t)\right]$
$\displaystyle=$
$\displaystyle(x_{j}-t_{j})D_{j}^{x}\left[U_{x_{0}}(x,t)\right]+U_{x_{0}}(x,t)+\sum_{\alpha\in
R_{+}}\kappa(\alpha)\alpha_{j}^{2}U_{x_{0}}(\sigma_{\alpha}(x),t).$ (70)

But from (25), (44) and (66), for each $\alpha\in R_{+}$ one has

$\displaystyle
U_{x_{0}}(\sigma_{\alpha}(x),t)=\frac{\sqrt{\pi}}{x_{0}}(\tau_{\sigma_{\alpha}(x)}P_{x_{0}})(-t)\lesssim\frac{(x_{0}+d(x,t))^{-1}}{\left|B(x,x_{0}+d(x,t))\right|_{\kappa}},$

and applying this and (67) to ( ‣ 5) gives

$\displaystyle\left|D_{j}^{x}[(\tau_{x}Q^{(j)}_{x_{0}})(-t)]\right|\lesssim\frac{(x_{0}+d(x,t))^{-1}}{\left|B(x,x_{0}+d(x,t))\right|_{\kappa}}.$
(71)

Finally, from (66) we have

$\displaystyle\partial_{x_{0}}[U_{x_{0}}(x,t)]=-\frac{x_{0}}{4}\int_{0}^{\infty}e^{-\frac{x_{0}^{2}}{4v}}\,h_{v}(x,t)\,\frac{dv}{v^{5/2}}=-\frac{2}{x_{0}^{2}}\int_{0}^{\infty}e^{-u}h_{x_{0}^{2}/4u}(x,t)\sqrt{u}\,du,$

so that from (67) and (68),

$\displaystyle\left|\partial_{x_{0}}[U_{x_{0}}(x,t)]\right|\lesssim\frac{(x_{0}+d(x,t))^{-1}}{\left|B(x,x_{0}+d(x,t))\right|_{\kappa}}\frac{x_{0}}{x_{0}^{2}+|x-t|^{2}}.$

Thus from (29) and (67),

$\displaystyle\left|\partial_{x_{0}}[(\tau_{x}Q^{(j)}_{x_{0}})(-t)]\right|=\frac{|x_{j}-t_{j}|}{\sqrt{\pi}}\left|\partial_{x_{0}}\left[U_{x_{0}}(x,t)\right]\right|\lesssim\frac{(x_{0}+d(x,t))^{-1}}{\left|B(x,x_{0}+d(x,t))\right|_{\kappa}}.$
(72)

Combining (69), (71) and (72) completes the proof of the lemma.

The following lemma is necessary too, and will also be used in the next
section.

###### Lemma 5.8

If $\psi\in L^{\infty}({\mathbb{R}}^{d})$ and $f\in
H_{\kappa}^{1}({\mathbb{R}}^{d})$ satisfying
$|f(x)|\lesssim(1+|x|)^{-2|\kappa|-d-1}$, then

$\displaystyle\int_{{\mathbb{R}}^{d}}({\mathcal{R}}_{j}f)(x)\,\psi(x)\,d\omega_{\kappa}(x)=-\int_{{\mathbb{R}}^{d}}f(x)\,(\tilde{\mathcal{R}}_{j}\psi)(x)\,d\omega_{\kappa}(x),\
\ \ j=1,2,\cdots,d.$ (73)

  * Proof. Obviously the integral on the left hand side of (73) converges absolutely; and since $\tilde{\mathcal{R}}_{j}\psi\in{\text{BMO}}_{\kappa}({\mathbb{R}}^{d})$ by Theorem 4.4, applying (35) to $\tilde{\mathcal{R}}_{j}\psi$ instead of $f$ with $x^{\prime}=0$ and in view of (37) it is asserted that the integral on the right hand side of (73) also converges absolutely.

We set $\psi_{N}=\psi\chi_{|x|\leq N}$. Then $\psi_{N}\in
L^{2}_{\kappa}({\mathbb{R}}^{d})\cap L^{\infty}({\mathbb{R}}^{d})$, and from
(40) and (41), it follows that
$(\tilde{\mathcal{R}}_{j}\psi_{N})(x)=({\mathcal{R}}_{j}\psi_{N})(x)-a_{N}$,
where
$a_{N}=c_{\kappa}\int_{{\mathbb{R}}^{d}}\psi_{N}(t)K_{j}^{1}(0,t)\,d\omega_{\kappa}(t)$.
Consequently, by Corollary 3.11 and Proposition 3.21, we have

$\displaystyle\int_{{\mathbb{R}}^{d}}({\mathcal{R}}_{j}f)(x)\,\psi_{N}(x)\,d\omega_{\kappa}(x)=-\int_{{\mathbb{R}}^{d}}f(x)\,(\tilde{\mathcal{R}}_{j}\psi_{N})(x)\,d\omega_{\kappa}(x).$

Thus (73) is concluded once it is shown that

$\displaystyle\lim_{N\rightarrow\infty}\int_{{\mathbb{R}}^{d}}f(x)\,[\tilde{\mathcal{R}}_{j}(\psi-\psi_{N})](x)\,d\omega_{\kappa}(x)=0.$
(74)

For the purpose let
$b_{N}=[\tilde{\mathcal{R}}_{j}(\psi-\psi_{N})]_{B_{\sqrt{N}}(0)}$ and insert
the number $b_{N}$ under the integral sign in (74), so that (74) is equivalent
to

$\displaystyle\lim_{N\rightarrow\infty}\int_{{\mathbb{R}}^{d}}f(x)\,\left([\tilde{\mathcal{R}}_{j}(\psi-\psi_{N})](x)-b_{N}\right)\,d\omega_{\kappa}(x)=0.$
(75)

It is easy to see that, for appropriately large $N$,

$b_{N}=\int_{|t|\geq
N}\frac{c_{\kappa}}{|B_{\sqrt{N}}(0)|_{\kappa}}\int_{B_{\sqrt{N}}(0)}[K_{j}(x^{\prime},t)-K_{j}(0,t)]\,d\omega_{\kappa}(x^{\prime})\,\psi(t)d\omega_{\kappa}(t),$

so that, for $|x|\leq\sqrt{N}$,

$\displaystyle[\tilde{\mathcal{R}}_{j}(\psi-\psi_{N})](x)-b_{N}$
$\displaystyle=$ $\displaystyle\int_{|t|\geq
N}\frac{c_{\kappa}}{|B_{\sqrt{N}}(0)|_{\kappa}}\int_{B_{\sqrt{N}}(0)}(K_{j}(x,t)-K_{j}(x^{\prime},t))d\omega_{\kappa}(x^{\prime})\,\psi(t)d\omega_{\kappa}(t).$

Now applying (43) we get, for $|x|\leq\sqrt{N}$,

$\displaystyle\left|[\tilde{\mathcal{R}}_{j}(\psi-\psi_{N})](x)-b_{N}\right|\lesssim\frac{\|\psi\|_{L^{\infty}}}{N^{|\kappa|+d/2}}\int_{|t|\geq
N}\int_{B_{\sqrt{N}}(0)}\frac{|x-x^{\prime}|}{|x-t|}\frac{d\omega_{\kappa}(x^{\prime})}{\left|B(t,d(x,t))\right|_{\kappa}}\,d\omega_{\kappa}(t).$

Note that $|x-t|\geq d(x,t)\geq|t|-|x|\geq|t|/2\geq N/2\geq 4\sqrt{N}\geq
2|x-x^{\prime}|$ for large $N$, and so

$\displaystyle\left|[\tilde{\mathcal{R}}_{j}(\psi-\psi_{N})](x)-b_{N}\right|\lesssim\int_{|t|\geq
N}\frac{\sqrt{N}\|\psi\|_{L^{\infty}}}{|t|\left|B(t,|t|)\right|_{\kappa}}d\omega_{\kappa}(t)\lesssim\frac{\|\psi\|_{L^{\infty}}}{\sqrt{N}}$

because of $\left|B(t,|t|)\right|_{\kappa}\asymp|t|^{2\kappa+d}$ according to
(37). Subsequently,

$\displaystyle\int_{|x|\leq\sqrt{N}}|f(x)|\,|[\tilde{\mathcal{R}}_{j}(\psi-\psi_{N})](x)-b_{N}|\,d\omega_{\kappa}(x)\lesssim\frac{\|f\|_{H_{\kappa}}\|\psi\|_{L^{\infty}}}{\sqrt{N}}.$
(76)

For $|x|>\sqrt{N}$, since

$|f(x)|\lesssim(\sqrt{N}+|x|)^{-2|\kappa|-d-1}\asymp(\sqrt{N}+|x|)^{-1}|B_{\sqrt{N}+|x|}(0)|_{\kappa}^{-1},$

we have

$\displaystyle\int_{|x|>\sqrt{N}}|f(x)|\,|[\tilde{\mathcal{R}}_{j}(\psi-\psi_{N})](x)-b_{N}|\,d\omega_{\kappa}(x)\lesssim\int_{{\mathbb{R}}^{d}}\frac{|[\tilde{\mathcal{R}}_{j}(\psi-\psi_{N})](x)-b_{N}|}{(\sqrt{N}+|x|)|B_{\sqrt{N}+|x|}(0)|_{\kappa}}\,d\omega_{\kappa}(x),$

which, appealing to (36), is further controlled by a multiple of
$\|\tilde{\mathcal{R}}_{j}(\psi-\psi_{N})\|_{*,\kappa}/\sqrt{N}\lesssim\|\psi\|_{L^{\infty}}/\sqrt{N}$.
Combining this with (76) proves (75), and the proof of the lemma is completed.

We now turn to the proof of Theorem 5.6.

Assume that $\psi\in L^{\infty}({\mathbb{R}}^{d})$. Obviously
$\tilde{\mathcal{R}}_{j}\psi$ satisfies condition (i) in Definition 5.4, since
$\tilde{\mathcal{R}}_{j}\psi\in{\text{BMO}}_{\kappa}({\mathbb{R}}^{d})$ by
Theorem 4.4, and Proposition 4.1 is applicable to
$\tilde{\mathcal{R}}_{j}\psi$, with $x^{\prime}=0$. It remains to show
$\|\tilde{\mathcal{R}}_{j}\psi\|_{*,C_{\kappa}}\lesssim\|\psi\|_{L^{\infty}}$.

For a fixed ball $B=B_{\delta}(x^{\prime})$ in ${\mathbb{R}}^{d}$, write
$\psi=\psi_{1}+\psi_{2}$, where $\psi_{1}=\psi\chi_{{\mathcal{O}}(2B)}$ and
$\psi_{2}=\psi\chi_{[{\mathcal{O}}(2B)]^{c}}$. Similarly to the first part of
the proof of Proposition 5.5,

$\displaystyle\iint_{T(B)}d\nu_{\kappa,\tilde{\mathcal{R}}_{j}\psi_{1}}(x_{0},x)\leq\frac{1}{2}\int_{{\mathbb{R}}^{d}}|\tilde{\mathcal{R}}_{j}\psi_{1}|^{2}d\omega_{\kappa},$

and further by (31),

$\displaystyle\iint_{T(B)}d\nu_{\kappa,\tilde{\mathcal{R}}_{j}\psi_{1}}(x_{0},x)\leq\frac{1}{2}\int_{{\mathbb{R}}^{d}}|\psi_{1}|^{2}d\omega_{\kappa}\lesssim\|\psi\|_{L^{\infty}}^{2}|B|_{\kappa}.$
(77)

In order to consider the contribution of $\psi_{2}$, we need the formula

$\displaystyle{\mathcal{R}}_{j}\left(\nabla_{\tilde{\kappa}}^{(x_{0},x)}[(\tau_{x}P_{x_{0}})(-\cdot)]\right)(t)=-\nabla_{\tilde{\kappa}}^{(x_{0},x)}[(\tau_{x}Q^{(j)}_{x_{0}})(-t)],$
(78)

which can be checked by taking the Dunkl transforms of the both sides. Indeed,
from (6), Propositions 2.4(ii) and 3.7(ii) one has

$\displaystyle{\mathscr{F}}_{\kappa}[(\tau_{x}Q^{(j)}_{x_{0}})(-\cdot)](\xi)=-{\mathscr{F}}_{\kappa}[(\tau_{-x}Q^{(j)}_{x_{0}})](\xi)=\frac{i\xi_{j}}{|\xi|}e^{-x_{0}|\xi|}E_{\kappa}(-ix,\xi),$

so that

$\displaystyle\nabla_{\tilde{\kappa}}^{(x_{0},x)}[(\tau_{x}Q^{(j)}_{x_{0}})(-t)]=c_{\kappa}\int_{{\mathbb{R}}^{d}}\frac{i\xi_{j}}{|\xi|}\nabla_{\tilde{\kappa}}^{(x_{0},x)}[e^{-x_{0}|\xi|}E_{\kappa}(-ix,\xi)]\,E_{\kappa}(it,\xi)\,d\omega_{\kappa}(\xi).$
(79)

On the other hand, it follows from (17) that
$\nabla_{\tilde{\kappa}}^{(x_{0},x)}[e^{-x_{0}|\xi|}E_{\kappa}(-ix,\xi)]$ is
the Dunkl transforms of
$\nabla_{\tilde{\kappa}}^{(x_{0},x)}[(\tau_{x}P_{x_{0}})(-\cdot)]$, and so, by
(31) the right hand side of (79) is also the inverse Dunkl transform of
$-{\mathcal{R}}_{j}\left(\nabla_{\tilde{\kappa}}^{(x_{0},x)}[(\tau_{x}P_{x_{0}})(-\cdot)]\right)(t)$.
Thus (78) is concluded.

In view of (78), the estimates (63) and (65) imply that for fixed
$(x_{0},x)\in{\mathbb{R}}^{1+d}_{+}$, each component of the vector-valued
function $t\mapsto\nabla_{\tilde{\kappa}}^{(x_{0},x)}[(\tau_{x}P_{x_{0}})(-t)$
is in the Hardy space $H_{\kappa}^{1}({\mathbb{R}}^{d})$ and satisfies the
required decreasing condition in Lemma 5.8. Thus from (16), (73) and (78),

$\displaystyle\nabla_{\tilde{\kappa}}u_{\tilde{\mathcal{R}}_{j}\psi_{2}}(x_{0},x)$
$\displaystyle=c_{\kappa}\int_{{\mathbb{R}}^{d}}(\tilde{\mathcal{R}}_{j}\psi_{2})(t)\,\nabla_{\tilde{\kappa}}^{(x_{0},x)}[(\tau_{x}P_{x_{0}})(-t)]\,d\omega_{\kappa}(t)$
$\displaystyle=c_{\kappa}\int_{{\mathbb{R}}^{d}}\psi_{2}(t)\,\nabla_{\tilde{\kappa}}^{(x_{0},x)}[(\tau_{x}Q^{(j)}_{x_{0}})(-t)]\,d\omega_{\kappa}(t),$

and by use of Lemma 5.7,

$\displaystyle\left|\nabla_{\tilde{\kappa}}u_{\tilde{\mathcal{R}}_{j}\psi_{2}}(x_{0},x)\right|\lesssim\|\psi\|_{L^{\infty}}\int_{[{\mathcal{O}}(2B)]^{c}}\frac{(x_{0}+d(t,x))^{-1}}{\left|{\mathbb{B}}(x,x_{0}+d(t,x))\right|_{\kappa}}\,d\omega_{\kappa}(t).$

Since for $(x_{0},x)\in T(B)$ and $t\in[{\mathcal{O}}(2B)]^{c}$, $d(t,x)\geq
d(t,x^{\prime})-|x^{\prime}-x|\geq d(t,x^{\prime})/2\geq\delta$, we
immediately get, for $(x_{0},x)\in T(B)$,

$\displaystyle\left|\nabla_{\tilde{\kappa}}u_{\tilde{\mathcal{R}}_{j}\psi_{2}}(x_{0},x)\right|\lesssim\|\psi\|_{L^{\infty}}/\delta,$

which takes the role of (64), and consequently,

$\displaystyle\iint_{T(B)}d\nu_{\kappa,\tilde{\mathcal{R}}_{j}\psi_{2}}(x_{0},x)\lesssim\|\psi\|_{L^{\infty}}^{2}|B|_{\kappa}.$

Combining this and (77) asserts that
$\tilde{\mathcal{R}}_{j}\psi\in{\text{BMC}}_{\kappa}({\mathbb{R}}^{d})$ and
$\|\tilde{\mathcal{R}}_{j}\psi\|_{*,C_{\kappa}}\lesssim\|\psi\|_{L^{\infty}}$,
and the proof of Theorem 5.6 is finished.

##  6 Duality of $H_{\kappa}^{1}({\mathbb{R}}^{d})$ and
${\text{BMC}}_{\kappa}({\mathbb{R}}^{d})$

For a vector valued function $F=(u_{0},u_{1},\cdots,u_{d})$ defined on
${\mathbb{R}}^{1+d}_{+}$, we construct a $(d+1)|G|$-dimensional vector valued
function $U_{F}(x_{0},x)$ following the work [5]. If
$\\{\sigma_{ij}\\}_{i,j=1}^{d}$ denotes the matrix of the group element
$\sigma\in G$ in the ordinary sense, we set
$F_{\sigma}=(u_{\sigma,0},u_{\sigma,1},\cdots,u_{\sigma,d})$, where
$u_{\sigma,0}(x_{0},x)=u_{0}(x_{0},\sigma(x))$ and

$u_{\sigma,j}(x_{0},x)=\sum_{i=1}^{d}\sigma_{ij}u_{i}(x_{0},\sigma(x)),\qquad
j=1,\cdots,d.$

The vector valued function $U_{F}(x_{0},x)$ is given by

$\displaystyle U_{F}(x_{0},x)=\left\\{F_{\sigma}(x_{0},x)\right\\}_{\sigma\in
G}.$ (80)

###### Proposition 6.1

([5, Lemma 6.1]) If $F=(u_{0},u_{1},\cdots,u_{d})$ satisfies the generalized
Cauchy-Riemann equations in (22) on ${\mathbb{R}}^{1+d}_{+}$, then for each
$\sigma\in G$, $F_{\sigma}$ also satisfies the equations in (22), and
$|F_{\sigma}(x_{0},x)|=|F(x_{0},\sigma(x))|$. Moreover
$|U_{F}(x_{0},\sigma(x))|=|U_{F}(x_{0},x)|$ for $\sigma\in G$.

###### Lemma 6.2

If $F=(u_{0},u_{1},\cdots,u_{d})$ satisfies the generalized Cauchy-Riemann
equations in (22) on ${\mathbb{R}}^{1+d}_{+}$, then there exists a constant
$C>0$ independent of $F$ and $\epsilon>0$, such that

$\displaystyle(|U_{F}|^{2}+\epsilon)^{-1/2}|\nabla_{\tilde{\kappa}}U_{F}|^{2}\leq
C\Delta_{\tilde{\kappa}}(|U_{F}|^{2}+\epsilon)^{1/2}.$ (81)

  * Proof. Direct calculations show that

$\displaystyle\Delta_{\tilde{\kappa}}(|U_{F}|^{2}+\epsilon)^{1/2}=\frac{1}{(|U_{F}|^{2}+\epsilon)^{1/2}}\left(V+\sum_{j=0}^{d}|\partial_{j}U_{F}|^{2}-\frac{1}{|U_{F}|^{2}+\epsilon}\sum_{j=0}^{d}\langle\partial_{j}U_{F},U_{F}\rangle^{2}\right),$
(82)

where

$V=\left\langle\sum_{j=0}^{d}\partial_{j}^{2}U_{F}+2\sum_{\alpha\in
R_{+}}\frac{\kappa(\alpha)}{\langle\alpha,x\rangle}\sum_{j=1}^{d}\alpha_{j}\partial_{j}U_{F},U_{F}\right\rangle.$

But by [5, (6.11) and (6.12)], $V$ can be rewritten as

$\displaystyle V=\sum_{\alpha\in
R_{+}}\frac{\kappa(\alpha)}{\langle\alpha,x\rangle^{2}}|U_{F}-\sigma_{\alpha}U_{F}|^{2},$
(83)

and there exists some $q\in(0,1)$ independent of $F$ such that

$\displaystyle\sum_{j=0}^{d}\langle\partial_{j}U_{F},U_{F}\rangle^{2}\leq\frac{|U_{F}|^{2}}{2-q}\left(V+\sum_{j=0}^{d}|\partial_{j}U_{F}|^{2}\right).$
(84)

Combining (82) and (84) yields

$\displaystyle\Delta_{\tilde{\kappa}}(|U_{F}|^{2}+\epsilon)^{1/2}\geq\frac{1-q}{2-q}\frac{1}{(|U_{F}|^{2}+\epsilon)^{1/2}}\left(V+\sum_{j=0}^{d}|\partial_{j}U_{F}|^{2}\right),$

but from (83) it is obvious that $V+\sum_{j=0}^{d}|\partial_{j}U_{F}|^{2}\geq
C_{1}|\nabla_{\tilde{\kappa}}U_{F}|^{2}$ with $C_{1}>0$ independent of $F$.
Thus (81) is proved for $C=(2-q)/C_{1}(1-q)$.

###### Lemma 6.3

If $f\in H_{\kappa,0}^{1}({\mathbb{R}}^{d})$ and
$\varphi\in{\text{BMC}}_{\kappa}({\mathbb{R}}^{d})$, then

$\displaystyle\iint_{{\mathbb{R}}^{1+d}_{+}}x_{0}\left|\big{\langle}\nabla_{\tilde{\kappa}}u_{f}(x_{0},x),\nabla_{\tilde{\kappa}}u_{\varphi}(x_{0},x)\big{\rangle}\right|d\omega_{\kappa}(x)dx_{0}\lesssim\|\varphi\|_{*,C_{\kappa}}\|f\|_{H_{\kappa}^{1}},$
(85)

where $u_{f}(x_{0},x)$ and $u_{\varphi}(x_{0},x)$ are the $\kappa$-Poisson
integrals of $f$ and $\varphi$ respectively.

  * Proof. By Corollary 3.25, $u_{f}(x_{0},x)$ is the first component of some $F\in{\mathscr{H}}_{\kappa,0}^{1}({\mathbb{R}}^{1+d}_{+})$, and by Theorem 3.22, $\|f\|_{H_{\kappa}^{1}}\asymp\|F\|_{{\mathscr{H}}_{\kappa}^{1}}$.

For $M>0$, let $B_{M}^{+}:=B^{+}((0,0),M)$ denote the upper half ball with
center $(0,0)$ and radii $M>0$ on ${\mathbb{R}}^{1+d}$. For $\epsilon>0$ we
have

$\displaystyle\iint_{B_{M}^{+}}x_{0}\left|\big{\langle}\nabla_{\tilde{\kappa}}u_{f},\nabla_{\tilde{\kappa}}u_{\varphi}\big{\rangle}\right|d\omega_{\kappa}(x)dx_{0}\leq$
$\displaystyle\iint_{B_{M}^{+}}x_{0}|\nabla_{\tilde{\kappa}}U_{F}||\nabla_{\tilde{\kappa}}u_{\varphi}|\,d\omega_{\kappa}(x)dx_{0}$
$\displaystyle\leq$ $\displaystyle J_{1}^{1/2}\times J_{2}^{1/2},$ (86)

where $U_{F}(x_{0},x)$ is given by (80), and

$\displaystyle J_{1}=$
$\displaystyle\iint_{B_{M}^{+}}x_{0}(|U_{F}|^{2}+\epsilon)^{-1/2}|\nabla_{\tilde{\kappa}}U_{F}|^{2}\,d\omega_{\kappa}(x)dx_{0},$
$\displaystyle J_{2}=$
$\displaystyle\iint_{B_{M}^{+}}x_{0}(|U_{F}|^{2}+\epsilon)^{1/2}|\nabla_{\tilde{\kappa}}u_{\varphi}|^{2}\,d\omega_{\kappa}(x)dx_{0}.$

By Lemma 6.2,

$\displaystyle
J_{1}\lesssim\iint_{B_{M}^{+}}x_{0}\Delta_{\tilde{\kappa}}(|U_{F}|^{2}+\epsilon)^{1/2}\,d\omega_{\kappa}(x)dx_{0},$

and applying the Green formula (15) we have

$\displaystyle J_{1}\lesssim$ $\displaystyle\int_{\partial
B_{M}^{+}}\left(x_{0}\partial_{\mathbf{n}}(|U_{F}|^{2}+\epsilon)^{1/2}-(|U_{F}|^{2}+\epsilon)^{1/2}\partial_{\mathbf{n}}x_{0}\right)W_{\kappa}(x)d\sigma(x_{0},x),$

where $\partial_{\mathbf{n}}$ denotes the directional derivative of the
outward normal, and $d\sigma(x_{0},x)$ the area element on $\partial\Omega$.
Since
$|\partial_{j}(|U_{F}|^{2}+\epsilon)^{1/2}|=|\langle\partial_{j}U_{F},U_{F}\rangle|/(|U_{F}|^{2}+\epsilon)^{1/2}|\leq|\partial_{j}U_{F}|$,
it follows that

$\displaystyle J_{1}\lesssim$ $\displaystyle\int_{|x|\leq
M}(|U_{F}(0,x)|^{2}+\epsilon)^{1/2}d\omega_{\kappa}(x)$
$\displaystyle\qquad+\int_{|(x_{0},x)|=M,\,x_{0}>0}\left(x_{0}|\nabla
U_{F}|+(|U_{F}|^{2}+\epsilon)^{1/2}\right)W_{\kappa}(x)d\sigma(x_{0},x).$

Furthermore, for $F\in{\mathscr{H}}_{\kappa,0}^{1}({\mathbb{R}}^{1+d}_{+})$,
one has $|U_{F}|\lesssim M^{-2|\kappa|-d-1}$ and $|\nabla U_{F}|\lesssim
M^{-2|\kappa|-d-2}$ when $|(x_{0},x)|=M$ and $x_{0}>0$, so that

$\displaystyle J_{1}\lesssim\int_{|x|\leq
M}(|U_{F}(0,x)|^{2}+\epsilon)^{1/2}d\omega_{\kappa}(x)+\frac{1}{M}+\epsilon
M^{2|\kappa|+d}.$

Inserting this into ( ‣ 6), letting $\epsilon\rightarrow 0+$ and then
$M\rightarrow\infty$, we obtain

$\displaystyle\iint_{{\mathbb{R}}_{+}^{1+d}}x_{0}\left|\big{\langle}\nabla_{\tilde{\kappa}}u_{f},\nabla_{\tilde{\kappa}}u_{\varphi}\big{\rangle}\right|d\omega_{\kappa}(x)dx_{0}\lesssim\|f\|_{H_{\kappa}^{1}}^{1/2}\,J_{3}^{1/2},$
(87)

where

$\displaystyle
J_{3}=\iint_{{\mathbb{R}}_{+}^{1+d}}x_{0}|U_{F}||\nabla_{\tilde{\kappa}}u_{\varphi}|^{2}\,d\omega_{\kappa}(x)dx_{0}.$
(88)

We now construct a $\kappa$-harmonic majorization of $|U_{F}|^{q}$. To apply
the maximum principle [44, Theorem 4.2] in our case, we give a variant of the
$\kappa$-subharmonic inequality in [5, Theorem 6.3]. Similarly to (82), for
$q>0$ and $\epsilon>0$ we have

$\displaystyle\Delta_{\tilde{\kappa}}(|U_{F}|^{2}+\epsilon)^{q/2}=\frac{q}{(|U_{F}|^{2}+\epsilon)^{1-q/2}}\left(V+\sum_{j=0}^{d}|\partial_{j}U_{F}|^{2}+\frac{q-2}{|U_{F}|^{2}+\epsilon}\sum_{j=0}^{d}\langle\partial_{j}U_{F},U_{F}\rangle^{2}\right),$

and choose $q\in(0,1)$ satisfying (84) (see [5, (6.12)]), independent of $F$
and $\epsilon$, to conclude

$\Delta_{\tilde{\kappa}}(|U_{F}|^{2}+\epsilon)^{q/2}\geq 0.$

For $\delta,\epsilon>0$ let
$f_{\delta,\epsilon}(x)=(|U_{F}(\delta,x)|^{2}+\epsilon)^{q/2}$ and set
$u_{f_{\delta,\epsilon}}(x_{0},x)=(Pf_{\delta,\epsilon})(x_{0},x)$, the
$\kappa$-Poisson integral of $f_{\delta,\epsilon}$. For
$F\in{\mathscr{H}}_{\kappa,0}^{1}({\mathbb{R}}^{1+d}_{+})$, we have
$|U_{F}(x_{0},x)|\lesssim(1+x_{0}+|x|)^{-2|\kappa|-d-2}$, and
$u_{f_{\delta,\epsilon}}(x_{0},x)$ is a continuous extension to
$\overline{{\mathbb{R}}_{+}^{1+d}}$ of $f_{\delta,\epsilon}(x)$. Consider the
function

$V_{\delta,\epsilon}(x_{0},x)=(|U_{F}(\delta+x_{0},x)|^{2}+\epsilon)^{q/2}-u_{f_{\delta,\epsilon}}(x_{0},x).$

Obviously $V_{\delta,\epsilon}(x_{0},x)$ is continuous on
$\overline{{\mathbb{R}}_{+}^{1+d}}$ and
$\Delta_{\tilde{\kappa}}V_{\delta,\epsilon}(x_{0},x)\geq 0$ in
${\mathbb{R}}_{+}^{1+d}$; moreover $V_{\delta,\epsilon}(0,x)=0$ for
$x\in{\mathbb{R}}^{d}$.

For given $\eta>0$, choose $M>0$ such that

$\displaystyle|U_{F}(x_{0},x)|<\eta\qquad\hbox{whenever}\quad|(x_{0},x)|\geq
M.$ (89)

It then follows that $|f_{\delta,\epsilon}(x)|\lesssim\eta+\epsilon$ for
$|x|\geq M$.

Now for $|(x_{0},x)|\geq 2M$, from (16) and (44) we have

$\displaystyle|u_{f_{\delta,\epsilon}}(x_{0},x)|\lesssim\eta+\epsilon+\int_{|t|\leq
M}\frac{d\omega_{\kappa}(t)}{\left|B(x,x_{0}+d(x,t))\right|_{\kappa}};$

but $x_{0}+d(x,t)\geq x_{0}+|x|-|t|\geq|(x_{0},x)|-|t|\geq|(x_{0},x)|/2$ for
$|t|\leq M$, so that

$\displaystyle|u_{f_{\delta,\epsilon}}(x_{0},x)|\lesssim\eta+\epsilon+\left(\frac{M}{|(x_{0},x)|}\right)^{2|\kappa|+d}.$

Choose $M^{\prime}>2M$ so that $(M/M^{\prime})^{2|\kappa|+d}<\eta$, and then
for $|(x_{0},x)|\geq M^{\prime}$,
$|u_{f_{\delta,\epsilon}}(x_{0},x)|\lesssim\eta+\epsilon$. From this and (89)
we get

$\displaystyle|V_{\delta,\epsilon}(x_{0},x)|\lesssim\eta+\epsilon\qquad\hbox{whenever}\quad|(x_{0},x)|\geq
M^{\prime}.$

We apply the maximum principle [44, Theorem 4.2] to
$V_{\delta,\epsilon}(x_{0},x)$ on the upper half ball
$=B^{+}((0,0),M^{\prime})$, and then obtain
$|V_{\delta,\epsilon}(x_{0},x)|\lesssim\eta+\epsilon$ for all
$(x_{0},x)\in{\mathbb{R}}_{+}^{1+d}$. Letting $\eta\rightarrow 0+$ gives

$\displaystyle(|U_{F}(\delta+x_{0},x)|^{2}+\epsilon)^{q/2}-u_{f_{\delta,\epsilon}}(x_{0},x)\lesssim\epsilon\qquad\hbox{for
all}\quad(x_{0},x)\in{\mathbb{R}}_{+}^{1+d}.$ (90)

It is easy to see that
$f_{\delta,\epsilon}(x)=(|U_{F}(\delta,x)|^{2}+\epsilon)^{q/2}$ converges to
$|U_{F}(\delta,x)|^{q}$ uniformly for $x\in{\mathbb{R}}^{d}$ as
$\epsilon\rightarrow 0+$, and so $\lim_{\epsilon\rightarrow
0+}u_{f_{\delta,\epsilon}}(x_{0},x)=u_{|U_{F}(\delta,\cdot)|^{q}}(x_{0},x)$
for $(x_{0},x)\in{\mathbb{R}}_{+}^{1+d}$. This and (90) lead to

$\displaystyle|U_{F}(\delta+x_{0},x)|^{q}\leq
u_{|U_{F}(\delta,\cdot)|^{q}}(x_{0},x)\qquad\hbox{for
all}\quad(x_{0},x)\in{\mathbb{R}}_{+}^{1+d}.$ (91)

Since $|U_{F}(\delta,\cdot)|^{q}\in L^{1/q}_{\kappa}({\mathbb{R}}^{d})$ with
$1/q>1$ and
$\sup_{\delta>0}\||U_{F}(\delta,\cdot)|^{q}\|_{L^{1/q}_{\kappa}}\lesssim\|f\|_{H_{\kappa}^{1}}^{q}$.
The *-weak compactness of $L^{1/q}_{\kappa}({\mathbb{R}}^{d})$ implies that,
there exists a sequence $\delta_{n}\rightarrow 0+$ such that
$U_{F}(\delta_{n},\cdot)$ converges *-weakly to a nonnegative function $g\in
L^{1/q}_{\kappa}({\mathbb{R}}^{d})$. Thus from (91) we get

$\displaystyle|U_{F}(x_{0},x)|^{q}\leq u_{g}(x_{0},x)\qquad\hbox{for
all}\quad(x_{0},x)\in{\mathbb{R}}_{+}^{1+d}.$

Now since
$\|u_{|U_{F}(\delta_{n},\cdot)|^{q}}(x_{0},\cdot)\|_{L^{1/q}_{\kappa}}\leq\||U_{F}(\delta_{n},\cdot)|^{q}\|_{L^{1/q}_{\kappa}}\lesssim\|f\|_{H_{\kappa}^{1}}^{q}$,
Fatou’s lemma asserts that
$\|u_{g}(x_{0},\cdot)\|_{L^{1/q}_{\kappa}}\lesssim\|f\|_{H_{\kappa}^{1}}^{q}$,
and letting $x_{0}\rightarrow 0+$ gives
$\|g\|_{L^{1/q}_{\kappa}}\lesssim\|f\|_{H_{\kappa}^{1}}^{q}$.

Finally from (88) we have

$\displaystyle
J_{3}\leq\iint_{{\mathbb{R}}_{+}^{1+d}}x_{0}u_{g}(x_{0},x)^{1/q}|\nabla_{\tilde{\kappa}}u_{\varphi}(x_{0},x)|^{2}\,d\omega_{\kappa}(x)dx_{0},$

and so, by Theorems 5.3,
$J_{3}\lesssim\|d\nu_{\kappa,\varphi}\|_{\kappa}\|g\|_{L^{1/q}_{\kappa}}^{1/q}\lesssim\|\varphi\|_{*,C_{\kappa}}^{2}\|f\|_{H_{\kappa}^{1}}$.
Substituting this into (87) proves (85), and then, the proof of the lemma is
completed.

###### Lemma 6.4

If $f\in H_{\kappa,0}^{1}({\mathbb{R}}^{d})$ and
$\varphi\in{\text{BMC}}_{\kappa}({\mathbb{R}}^{d})$, then

$\displaystyle\int_{{\mathbb{R}}^{d}}f(x)\varphi(x)d\omega_{\kappa}(x)=2\iint_{{\mathbb{R}}^{1+d}_{+}}x_{0}\big{\langle}\nabla_{\tilde{\kappa}}u_{f}(x_{0},x),\nabla_{\tilde{\kappa}}u_{\varphi}(x_{0},x)\big{\rangle}d\omega_{\kappa}(x)dx_{0},$
(92)

where $u_{f}(x_{0},x)$ and $u_{\varphi}(x_{0},x)$ are the $\kappa$-Poisson
integrals of $f$ and $\varphi$ respectively.

  * Proof. From (58) it is obvious that the integral on the left hand side of (92) converges absolutely, and so does the one on the right hand side by Lemma 6.3.

For $x_{0}>0$ we have

$\displaystyle\nabla_{\tilde{\kappa}}u_{\varphi}(x_{0},x)=c_{\kappa}\int_{{\mathbb{R}}^{d}}\varphi(t)\nabla_{\tilde{\kappa}}^{(x_{0},x)}[(\tau_{x}P_{x_{0}})(-t)]\,d\omega_{\kappa}(t)$

which is legitimate on account of (44) and (45). By use of (63),

$\displaystyle\left|\nabla_{\tilde{\kappa}}^{(x_{0},x)}[(\tau_{x}P_{x_{0}})(-t)]\right|\lesssim\left(1+\frac{1+|x|}{x_{0}}\right)^{2|\kappa|+d+1}\frac{1}{(1+|t|)^{2|\kappa|+d+1}},$

so that, in view of (58),

$\displaystyle\int_{{\mathbb{R}}^{d}}|\varphi(t)|\left|\nabla_{\tilde{\kappa}}^{(x_{0},x)}[(\tau_{x}P_{x_{0}})(-t)]\right|\,d\omega_{\kappa}(t)\lesssim\left(1+\frac{1+|x|}{x_{0}}\right)^{2|\kappa|+d+1}.$

Moreover for $f\in H_{\kappa,0}^{1}({\mathbb{R}}^{d})$,
$\int_{{\mathbb{R}}^{d}}(1+x_{0}+|x|)^{2|\kappa|+d+1}\left|\nabla_{\tilde{\kappa}}u_{f}(x_{0},x)\right|d\omega_{\kappa}(x)$
is finite. Now Fubini’s theorem yields

$\displaystyle\int_{{\mathbb{R}}^{d}}\big{\langle}\nabla_{\tilde{\kappa}}u_{f}(x_{0},x),\nabla_{\tilde{\kappa}}u_{\varphi}(x_{0},x)\big{\rangle}d\omega_{\kappa}(x)$
$\displaystyle=$ $\displaystyle
c_{\kappa}\int_{{\mathbb{R}}^{d}}\varphi(t)\int_{{\mathbb{R}}^{d}}\big{\langle}\nabla_{\tilde{\kappa}}u_{f}(x_{0},x),\nabla_{\tilde{\kappa}}^{(x_{0},x)}[(\tau_{x}P_{x_{0}})(-t)]\big{\rangle}d\omega_{\kappa}(x)d\omega_{\kappa}(t).$
(93)

By Proposition 2.1(iii) and (17), the Dunkl transform of
$x\mapsto\nabla_{\tilde{\kappa}}^{(x_{0},x)}[(\tau_{x}P_{x_{0}})(-t)]$ is

$e^{-x_{0}|\xi|}E_{\kappa}(-it,\xi)(-|\xi|,i\xi_{1},\cdots,i\xi_{d}),$

and by (60) and (61),
$\left[{\mathscr{F}}_{\kappa}(\nabla_{\tilde{\kappa}}u_{f}(x_{0},\cdot))\right](\xi)=e^{-x_{0}|\xi|}\left({\mathscr{F}}_{\kappa}f\right)(\xi)(-|\xi|,i\xi_{1},\cdots,i\xi_{d})$.
Thus the Plancherel formula (Propositions 2.1(v)) implies that the inner
integral on the right hand side of ( ‣ 6) becomes

$\displaystyle
2\int_{{\mathbb{R}}^{d}}|\xi|^{2}e^{-2x_{0}|\xi|}\left({\mathscr{F}}_{\kappa}f\right)(\xi)E_{\kappa}(it,\xi)d\omega_{\kappa}(\xi)=\frac{1}{2}c_{\kappa}^{-1}\partial_{x_{0}}^{2}[u_{f}(2x_{0},t)].$
(94)

In addition, by Lemma 6.3 the integral on the right hand side of (92)
converges absolutely, and then, Fubini’s theorem again gives

$\displaystyle\iint_{{\mathbb{R}}_{+}^{1+d}}x_{0}\big{\langle}\nabla_{\tilde{\kappa}}u_{f},\nabla_{\tilde{\kappa}}u_{\varphi}\big{\rangle}\,dx_{0}d\omega_{\kappa}(x)=\frac{1}{2}\int_{0}^{\infty}x_{0}\int_{{\mathbb{R}}^{d}}\varphi(t)\partial_{x_{0}}^{2}[u_{f}(2x_{0},t)]\,d\omega_{\kappa}(t)dx_{0}.$

Noting that
$|\Delta_{\kappa}^{t}[u_{f}(2x_{0},t)]|\lesssim(1+x_{0}+|t|)^{-2|\kappa|-d-3}$
and in view of (58), further changing order of the integration above is
permitted, and hence

$\displaystyle\iint_{{\mathbb{R}}_{+}^{1+d}}x_{0}\big{\langle}\nabla_{\tilde{\kappa}}u_{f},\nabla_{\tilde{\kappa}}u_{\varphi}\big{\rangle}d\omega_{\kappa}(x)=\frac{1}{2}\int_{{\mathbb{R}}^{d}}\varphi(t)\int_{0}^{\infty}x_{0}\partial_{x_{0}}^{2}[u_{f}(2x_{0},t)]\,dx_{0}d\omega_{\kappa}(t).$
(95)

Finally, since $f\in H_{\kappa,0}^{1}({\mathbb{R}}^{d})$ it follows from (94)
that

$\int_{0}^{\infty}x_{0}\partial_{x_{0}}^{2}[u_{f}(2x_{0},t)]\,dx_{0}=f(t),$

and inserting this into (95) finishes the proof of (92).

###### Theorem 6.5

If $\varphi\in{\text{BMC}}_{\kappa}({\mathbb{R}}^{d})$, then the linear
functional

$\displaystyle{\mathcal{L}}f=\int_{{\mathbb{R}}^{d}}f(x)\varphi(x)d\omega_{\kappa}(x),$
(96)

initially defined for $f\in H_{\kappa,0}^{1}({\mathbb{R}}^{d})$, has a unique
bounded extension to $H^{1}_{\kappa}({\mathbb{R}}^{d})$, and
$\|{\mathcal{L}}\|\lesssim\|\varphi\|_{*,C_{\kappa}}$, where the constant
$C>0$ is independent of $\varphi$.

  * Proof. It suffices to show that, for $f\in H_{\kappa,0}^{1}({\mathbb{R}}^{d})$,

$\Big{|}\int_{{\mathbb{R}}^{d}}f(x)\varphi(x)d\omega_{\kappa}(x)\Big{|}\lesssim\|\varphi\|_{*,\kappa}\|f\|_{H_{\kappa}^{1}}\lesssim\|\varphi\|_{*,C_{\kappa}}\|f\|_{H_{\kappa}},$

but that is direct consequence of Lemmas 6.3 and 6.4. The proof is completed.

###### Theorem 6.6

If $\mathcal{L}$ is a continuous linear functional on
$H^{1}_{\kappa}({\mathbb{R}}^{d})$, then there exists a unique element
$\varphi\in{\text{BMC}}_{\kappa}({\mathbb{R}}^{d})$ so that $\mathcal{L}$ can
be realized by (96), at least for $f\in H_{\kappa,0}^{1}({\mathbb{R}}^{d})$,
and $\|\varphi\|_{*,C_{\kappa}}\leq C^{\prime}\|{\mathcal{L}}\|$, where the
constant $C^{\prime}>0$ is independent of $\mathcal{L}$. Moreover, such a
function $\varphi\in{\text{BMC}}_{\kappa}({\mathbb{R}}^{d})$ can be expressed
as

$\varphi=\varphi_{0}+\sum_{j=1}^{d}\tilde{\mathcal{R}}_{j}\varphi_{j},$

where $\varphi_{0},\varphi_{1},\cdot\cdot\cdot,\varphi_{d}\in
L^{\infty}({\mathbb{R}}^{d})$.

  * Proof. As in [24, p. 145], we regard $H^{1}_{\kappa}({\mathbb{R}}^{d})$ as a closed subspace of the direct sum $X:=L^{1}_{\kappa}\oplus L^{1}_{\kappa}\oplus\cdots\oplus L^{1}_{\kappa}$ of $d+1$ copies of $L^{1}_{\kappa}({\mathbb{R}}^{d})$. If we set $X=\\{(f_{0},f_{1},\cdots,f_{d}):\,\,f_{j}\in L^{1}_{\kappa}({\mathbb{R}}^{d}),\,\,j=0,1,\cdots d\\}$ with the norm $\|(f_{0},f_{1},\cdots,f_{d})\|=\sum_{j=1}^{d}\|f_{j}\|_{L^{1}_{\kappa}}$, it is known that the dual of $X$ is equivalent to $L^{\infty}\oplus L^{\infty}\oplus\cdots\oplus L^{\infty}$. We consider the subspace $X_{0}$ of $X$ given by

$X_{0}=\\{(f,{\mathcal{R}}_{1}f,\cdots,{\mathcal{R}}_{d}f):\,\,f\in
L^{1}_{\kappa}({\mathbb{R}}^{d})\,\,\hbox{and}\,\,{\mathcal{R}}_{j}f\in
L^{1}_{\kappa}({\mathbb{R}}^{d}),\,\,j=1,\cdots d\\}.$

It is easy to see that $X_{0}$ is a closed subspace of $X$, and there is a
natural isometric mapping of $H^{1}_{\kappa}({\mathbb{R}}^{d})$ to $X_{0}$,
that is, $f\mapsto(f,{\mathcal{R}}_{1}f,\cdots,{\mathcal{R}}_{d}f)$. Thus if
$\mathcal{L}$ is a continuous linear functional on
$H^{1}_{\kappa}({\mathbb{R}}^{d})$, the one on $X_{0}$ identified by it, by
Hahn-Banach theorem, extends to a continuous linear functional on $X$, denoted
by $\mathcal{L}$ too. Therefore, there exist
$\varphi_{0},\varphi_{1},\cdot\cdot\cdot,\varphi_{d}\in
L^{\infty}({\mathbb{R}}^{d})$ such that

${\mathcal{L}}(f_{0},f_{1},\cdots,f_{d})=\sum_{j=0}^{d}\int_{{\mathbb{R}}^{d}}f_{j}\varphi_{j}d\omega_{\kappa},\qquad(f_{0},f_{1},\cdots,f_{d})\in
X,$

and $\|{\mathcal{L}}\|\asymp\sum_{j=0}^{d}\|\varphi_{j}\|_{L^{\infty}}$. In
particular, such a representation holds if
$(f,{\mathcal{R}}_{1}f,\cdots,{\mathcal{R}}_{d}f)\in X_{0}$, or equivalently,
for $f\in H^{1}_{\kappa}({\mathbb{R}}^{d})$,
${\mathcal{L}}f=\int_{{\mathbb{R}}^{d}}\varphi_{0}f\,d\omega_{\kappa}+\sum_{j=1}^{d}\int_{{\mathbb{R}}^{d}}\varphi_{j}{\mathcal{R}}_{j}f\,d\omega_{\kappa}$.
If, furthermore, we apply this to $f\in H_{\kappa,0}^{1}({\mathbb{R}}^{d})$,
and then appeal to Lemma 5.8, to get

${\mathcal{L}}f=\int_{{\mathbb{R}}^{d}}f\left(\varphi_{0}-\sum_{j=1}^{d}\tilde{\mathcal{R}}_{j}\varphi_{j}\right)\,d\omega_{\kappa}.$

According to Theorem 5.6, the function
$\varphi=\varphi_{0}-\sum_{j=1}^{d}\tilde{\mathcal{R}}_{j}\varphi_{j}$ is in
${\text{BMC}}_{\kappa}({\mathbb{R}}^{d})$ and
$\|\varphi\|_{*,C_{\kappa}}\lesssim\sum_{j=0}^{d}\|\varphi_{j}\|_{L^{\infty}}\asymp\|{\mathcal{L}}\|$.
The uniqueness is obvious, and the proof of the theorem is completed.

Finally we summarize the conclusions in the above theorem as follows.

###### Theorem 6.7

The dual of $H_{\kappa}^{1}({\mathbb{R}}^{d})$ is
${\text{BMC}}_{\kappa}({\mathbb{R}}^{d})$, and the norm of a continuous linear
functional $\mathcal{L}$ on $H^{1}_{\kappa}({\mathbb{R}}^{d})$, realized by
(96), is equivalent to the norm of the representing function
$\varphi\in{\text{BMC}}_{\kappa}({\mathbb{R}}^{d})$.

###### Theorem 6.8

A function $\varphi\in L_{\kappa,{\rm loc}}({\mathbb{R}}^{d})$ is in
${\text{BMC}}_{\kappa}({\mathbb{R}}^{d})$ if and only if it has the
representation

$\varphi=\varphi_{0}+\sum_{j=1}^{d}\tilde{\mathcal{R}}_{j}\varphi_{j},\qquad\hbox{where}\quad\varphi_{0},\varphi_{1},\cdots,\varphi_{d}\in
L^{\infty}({\mathbb{R}}^{d});$

and moreover,