# Local boundary behaviour and the area integral of generalized harmonic
functions associated with root systems ††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

(In memory of Professor Leetsch Charles Hsu (Lizhi Xu))

###### Abstract

The rational Dunkl operators are commuting differential-reflection operators
on the Euclidean space ${\mathbb{R}}^{d}$ associated with a root system. The
aim of the paper is to study local boundary behaviour of generalized harmonic
functions associated with the Dunkl operators. We introduce a Lusin-type area
integral operator $S$ by means of Dunkl’s generalized translation and the
Dunkl operators. The main results are on characterizations of local existence
of non-tangential boundary limits of a generalized harmonic function $u$ in
the upper half-space ${\mathbb{R}}^{d+1}_{+}$ associated with the Dunkl
operators, and for a subset $E$ of ${\mathbb{R}}^{d}$ invariant under the
reflection group generated by the root system, the equivalence of the
following three assertions are proved: (i) $u$ has a finite non-tangential
limit at $(x,0)$ for a.e. $x\in E$; (ii) $u$ is non-tangentially bounded for
a.e. $x\in E$; (iii) $(Su)(x)$ is finite for a.e. $x\in E$.

2020 MS Classification: 42B99(Primary), 31B25, 51F15(Secondary)
Key Words and Phrases: local non-tangential limit; area integral; root system;
Dunkl operator; generalized harmonic function

##  1 Introduction and main results

1.1. A classical result of Fatou states that a bounded harmonic function $u$
in the unit disc ${\mathbb{D}}$ has a non-tangential limit at almost every
point $e^{i\theta}$. A local version of Fatou’s theorem proved by Privalov
[43] is that if $u(z)$ is harmonic in ${\mathbb{D}}$, and at each point
$e^{i\theta}$ of a measurable subset $E$ of positive measure of the boundary
$\partial{\mathbb{D}}$, there is a cone contained in ${\mathbb{D}}$ with
vertex $e^{i\theta}$ in which $u(z)$ is bounded, then $u$ has a non-tangential
limit at almost every $e^{i\theta}\in E$. Marcinkiewicz and Zygmund [34], and
Spencer [51] obtained a completely different criterion on existence of non-
tangential limit at almost every point $e^{i\theta}$ of a measurable set
$E\subset\partial{\mathbb{D}}$ which is characterized by finiteness of Lusin’s
area integral $\int\left|\nabla u(x,y)\right|^{2}dxdy$ for almost every
$e^{i\theta}\in E$, where the integral is taken over a cone contained in
${\mathbb{D}}$ with vertex $e^{i\theta}$. One of the basic tools in these
studies is the conformal mapping, which introduces technical difficulties in
extending them to more variables and other settings. The breakthrough to this
obstacle was made by Calder${\rm\acute{o}}$n [6, 7], who generalized
Privalov’s theorem and Marcinkiewicz and Zygmund’s theorem to Euclidean half-
spaces of several variables by real method. Stein in [52] extended Spencer’s
theorem to several variables. (For a thoughtful treatment on these results,
see [53].) Since then, criteria on local existence of non-tangential boundary
limits of harmonic functions in many different contexts, in terms of non-
tangential boundedness or one-side non-tangential boundedness or finiteness of
non-tangential area integral have been intensively studied (see, among others,
[2]-[5], [8], [18], [20]-[22], [24]-[28], [30], [36]-[38], [40]-[42], [44],
[58]). Various aspects associated with the Fatou theory in one and several
complex variables are contained in the most recent paper [10].

The rational Dunkl operators are differential-reflection operators on the
Euclidean space ${\mathbb{R}}^{d}$ associated with a root system. In this
paper we study local boundary behaviour of generalized harmonic functions in
the upper half-space ${\mathbb{R}}^{d+1}_{+}={\mathbb{R}}^{d}\times(0,\infty)$
associated with the Dunkl operators. As usual we denote by
$e_{1},e_{2},\dots,e_{d}$ the standard basis vectors in ${\mathbb{R}}^{d}$.

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
$x\in{\mathbb{R}}^{d}$. For a multiplicity function $\kappa$ defined on $R$
(invariant under $G$), the Dunkl operators $D_{j}$ ($1\leq j\leq d)$ are
defined by (cf. [12])

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

where $\sigma_{\alpha}f=f\circ\sigma_{\alpha}$, $\alpha_{j}$ is the $j$th
coordinate of $\alpha$, and the associated measure is given by

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

The weight $W_{\kappa}$ is a positive homogeneous function of degree
$2|\kappa|$, where $|\kappa|=\sum_{\alpha\in R_{+}}k(\alpha)$. It is proved in
[12] that the Dunkl operators commute with each other.

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

$\displaystyle\Delta_{\kappa}=\partial_{y}^{2}+\sum_{j=1}^{d}D_{j}^{2}.$

From [11] the $\kappa$-Laplacian $\Delta_{\kappa}$ can be explicitly expressed
by

$\displaystyle\Delta_{\kappa}u=\Delta u+2\sum_{\alpha\in
R_{+}}\kappa(\alpha)\delta_{\alpha}u,$ (2)

where

$\displaystyle\delta_{\alpha}u=\frac{\langle\nabla^{(x)}u,\alpha\rangle}{\langle\alpha,x\rangle}-\frac{u-\sigma_{\alpha}u}{\langle\alpha,x\rangle^{2}},$
(3)

$\Delta=\partial_{y}^{2}+\sum_{j=1}^{d}\partial_{j}^{2}$ is the usual
Laplacian on ${\mathbb{R}}^{d+1}$, $\nabla^{(x)}$ the usual gradient operator
on ${\mathbb{R}}^{d}$, and $(\sigma u)(x,y)=u(\sigma(x),y)$ for $\sigma\in G$
and $(x,y)\in{\mathbb{R}}^{d+1}$. By [11, Proposition 1.1], if $u$ is
$\kappa$-harmonic, so is $\sigma u$ for each $\sigma\in G$.

Several examples are given as follows, see [49].

Examples. 1. The reflection group generated by the root system
$R=\\{\pm\sqrt{2}e_{j}:\,1\leq j\leq d\\}$ is denoted by $Z_{2}^{d}$. The
associated measure is
$d\omega_{\lambda}=\prod_{j=1}^{d}|x_{j}|^{2\lambda_{j}}dx$ with multiplicity
parameters $\lambda=(\lambda_{1},\dots,\lambda_{d})$, and the Dunkl operators
are given by $D_{j}=\partial_{j}+\frac{\lambda_{j}}{x_{j}}(1-\sigma_{j})$,
where $\sigma_{j}(x)=(x_{1},\dots,-x_{j},\dots,x_{d})$. In this special case,
we shall write $\Delta_{\lambda}$ instead of $\Delta_{\kappa}$, and say
“$\lambda$-harmonic” instead of “$\kappa$-harmonic”.

2. The type $A_{d-1}$ case. A root system of $S_{d}$, the symmetric group in $d$ elements, 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 is constant. The Dunkl operators for a given $\kappa\in{\mathbb{C}}$ have the form

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

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

3. The type $B_{d}$ case. The root system corresponding to the semidirect product $G=S_{d}\ltimes Z_{2}^{d}$ is given by $R=\\{\pm\sqrt{2}e_{j}:\,1\leq j\leq d;\,\,\pm(e_{i}\pm e_{j}):\,1\leq i<j\leq d\\}$. A multiplicity function has the form $\kappa=(\kappa_{0},\kappa_{1})$, with only two independent values, as there are two conjugacy classes of reflections. The associated Dunkl operators are given by

$\displaystyle
D_{j}=\partial_{j}+\frac{\kappa_{0}}{x_{j}}(1-\sigma_{j})+\kappa_{1}\sum_{i\neq
j}\left[\frac{1-\sigma_{ij}}{x_{j}-x_{i}}+\frac{1-\tau_{ij}}{x_{j}+x_{i}}\right],$

where $\tau_{ij}=\sigma_{ij}\sigma_{i}\sigma_{j}$.

During the past three decades, the Dunkl operators have gained considerable
interest in various fields of mathematics and in physical applications; they
are, in the case of the symmetric group $S_{d}$, naturally connected with the
Calogero-Sutherland models of quantum many body systems (see [29], for
example).

In what follows, we assume that the involved multiplicity functions $\kappa$
are real and nonnegative. As usual, for a subset $E$ of ${\mathbb{R}}^{d}$,
define $\sigma E=\\{\sigma(x):\,x\in E\\}$ for $\sigma\in G$, and $E$ is
called to be $G$-invariant if $\sigma E=E$ for all $\sigma\in G$; for a
function $f$ defined on a $G$-invariant subset $E$ of ${\mathbb{R}}^{d}$, set
$\sigma f=f\circ\sigma$ for $\sigma\in G$, and $f$ is called to be
$G$-invariant on $E$ if $\sigma f=f$ for all $\sigma\in G$.

1.3. Main results. We are concerned with characterizations on existence of
non-tangential boundary values, in a local sense, of $\kappa$-harmonic
functions in the upper half-space ${\mathbb{R}}^{d+1}_{+}$.

Our first result is to extend those of Privalov [43] and
Calder${\rm\acute{o}}$n [6] to the $\kappa$-harmonic functions, which
characterizes the local existence of non-tangential boundary values by non-
tangential boundedness. As usual, we denote by $\Gamma_{a}(x)$ the positive
cone of aperture $a>0$ with vertex
$(x,0)\in\partial{\mathbb{R}}^{d+1}_{+}={\mathbb{R}}^{d}$, and
$\Gamma^{h}_{a}(x)$ the truncated cone with height $h>0$, that is,

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

###### Definition 1.1

For a given function $u$ defined in ${\mathbb{R}}^{d+1}_{+}$ and for
$x\in{\mathbb{R}}^{d}$,

(i) that $u$ has a non-tangential limit at $(x,0)$ means that for every $a>0$,
$\lim u(t,y)$ exists as $(t,y)\in\Gamma_{a}(x)$ approaching to $(x,0)$; and

(ii) that $u$ is said to be non-tangentially bounded at $(x,0)$ if $u(t,y)$ is
bounded in $\Gamma^{h}_{a}(x)$ for some $a,h>0$.

###### Theorem 1.2

Suppose that $E$ is a $G$-invariant measurable subset of
$\partial{\mathbb{R}}^{d+1}_{+}={\mathbb{R}}^{d}$ and $u$ is $\kappa$-harmonic
in ${\mathbb{R}}^{d+1}_{+}$. Then the following assertions are equivalent:

(i) $u$ has a finite non-tangential limit at $(x,0)$ for almost every $x\in
E$;

(ii) $u$ is non-tangentially bounded at $(x,0)$ for almost every $x\in E$.

Our next attempt is to generalize the results of Marcinkiewicz and Zygmund
[34] and Calder${\rm\acute{o}}$n [7] to the $\kappa$-harmonic functions. For a
$C^{2}$ function $u$ in ${\mathbb{R}}^{d+1}_{+}$, we introduce, at first
formally, a Lusin-type area integral $Su=S_{a,h}u$ for some $a,h>0$ in the
Dunkl setting, that is,

$\displaystyle\left(S_{a,h}u\right)(x)=\left(\iint_{\Gamma_{a}^{h}(0)}\left[\tau_{x}(\Delta_{\kappa}u^{2})\right](t,y)\,y^{1-d-2|\kappa|}\,d\omega_{\kappa}(t)dy\right)^{1/2},$
(4)

where $\tau_{x}$ is the associated (generalized) translation in the Dunkl
setting (see Section 2), acting on the first argument. We note that $S_{a,h}u$
is first defined in Liao [33] for $G=Z_{2}$ with $d=1$, where it is used to
characterize the Hardy space $H^{1}_{\lambda}({\mathbb{R}})$ associated to the
rank-one Dunkl operator.

###### Theorem 1.3

Suppose that $E$ is a $G$-invariant measurable subset of
$\partial{\mathbb{R}}^{d+1}_{+}={\mathbb{R}}^{d}$ and $u$ is $\kappa$-harmonic
in ${\mathbb{R}}^{d+1}_{+}$. If $u$ is non-tangentially bounded at $(x,0)$ for
every $x\in E$, then for almost every $x\in E$, the area integral
$(S_{a,h}u)(x)$ is finite for some $a,h>0$.

Finally we consider the converse of Theorem 1.3, which is a generalization of
the results of Spencer [51] and Stein [52]. The answer is positive when $u$ is
$G$-invariant (i.e. $u(\sigma(x),y)=u(x,y)$ for all $\sigma\in G$).

###### Theorem 1.4

Suppose that $E$ is a $G$-invariant measurable subset of
$\partial{\mathbb{R}}^{d+1}_{+}={\mathbb{R}}^{d}$ and $u$ is $\kappa$-harmonic
in ${\mathbb{R}}^{d+1}_{+}$. If $u$ is $G$-invariant and for every $x\in E$,
its area integral $(S_{a,h}u)(x)$ is finite for some $a,h>0$, then $u$ is non-
tangentially bounded at $(x,0)$ for almost every $x\in E$.

From Theorems 1.2 to 1.4, we have

###### Corollary 1.5

Suppose that $E$ is a $G$-invariant measurable subset of
$\partial{\mathbb{R}}^{d+1}_{+}={\mathbb{R}}^{d}$ and $u$ is $\kappa$-harmonic
in ${\mathbb{R}}^{d+1}_{+}$. If $u$ is $G$-invariant, then the following
assertions are equivalent:

(i) $u$ has a finite non-tangential limit at $(x,0)$ for almost every $x\in
E$;

(ii) $u$ is non-tangentially bounded at $(x,0)$ for almost every $x\in E$;

(iii) for almost every $x\in E$, the area integral $(S_{a,h}u)(x)$ is finite
for some $a,h>0$.

For the group $G=Z_{2}^{d}$, the $G$-invariance of $u$ can be eliminated.
Indeed we have

###### Theorem 1.6

Suppose $G=Z_{2}^{d}$ with a given multiplicity parameters
$\lambda=(\lambda_{1},\dots,\lambda_{d})$, $E$ is a $G$-invariant measurable
subset of ${\mathbb{R}}^{d}$, and $u$ is $\lambda$-harmonic in
${\mathbb{R}}^{d+1}_{+}$. If for every $x\in E$, the area integral
$(S_{a,h}u)(x)$ is finite for some $a,h>0$, then $u$ is non-tangentially
bounded at $(x,0)$ for almost every $x\in E$.

Combining the conclusions in Theorems 1.2, 1.3 and 1.6, the following
corollary is immediate.

###### Corollary 1.7

Suppose $G=Z_{2}^{d}$ with a given multiplicity parameters
$\lambda=(\lambda_{1},\dots,\lambda_{d})$. If $E$ is a $G$-invariant
measurable subset of ${\mathbb{R}}^{d}$ and $u$ is $\lambda$-harmonic in
${\mathbb{R}}^{d+1}_{+}$, then the following assertions are equivalent:

(i) $u$ has a finite non-tangential limit at $(x,0)$ for almost every $x\in
E$;

(ii) $u$ is non-tangentially bounded at $(x,0)$ for almost every $x\in E$;

(iii) for almost every $x\in E$, the area integral $(S_{a,h}u)(x)$ is finite
for some $a,h>0$.

1.4. Remarks. (a) We remark that the area integral $(S_{a,h}u)(x)$ defined in
(4) is meaningful for $\kappa$-harmonic function $u$ in
${\mathbb{R}}^{d+1}_{+}$ based upon the following three points: (i) $u\in
C^{\infty}({\mathbb{R}}^{d+1}_{+})$ by Proposition 3.9; (ii)
$\left(\Delta_{\kappa}u^{2}\right)(x,y)\geq 0$ by Lemma 5.2; (iii) the
integral on the right hand side of (4) is nonnegative, since it can be written
as

$\displaystyle\lim_{\delta\rightarrow 0+}\iint_{\begin{subarray}{c}|t|\leq
ay,\\\ y\in[\delta,h]\end{subarray}}=\lim_{\delta\rightarrow
0+}\int_{\delta}^{h}\int_{0}^{ay}M_{(\Delta_{\kappa}u^{2})(\cdot,y)}(x,r)\left(\frac{r}{y}\right)^{2|\kappa|+d-1}\,drdy,$

where $M_{f}(x,r)$ is the generalized spherical mean of $f$ (see (15)) which
is positivity-preserving.

(b) In the proofs of Theorems 1.3, 1.4 and 1.6, to use the generalized
translation $\tau_{x}$ legitimately we shall work with a variant of the area
integral $(S_{a,h}u)(x)$ involving a smooth and compactly supported function
$\psi$, that is,

$\displaystyle(S^{\psi}_{a,h}u)(x)=\left(\iint_{{\mathbb{R}}^{d}\times(0,h]}\left[\tau_{x}(\Delta_{\kappa}u^{2})\right](t,y)\,\psi\left(\frac{t}{ay}\right)y^{1-d-2|\kappa|}\,d\omega_{\kappa}(t)dy\right)^{1/2},$

where $\psi\in C^{\infty}({\mathbb{R}}^{d})$ is radial and satisfies
$0\leq\psi(x)\leq 1$,

$\displaystyle\psi(x)=1\quad\hbox{for}\,\,|x|\leq
1/2,\quad\hbox{and}\quad\psi(x)=0\quad\hbox{for}\,\,|x|\geq 1.$

(c) We note that the assumption of reflection-symmetry ($G$-invariance) on the
given subset $E$ of ${\mathbb{R}}^{d}$ in the above theorems and corollaries
is reasonable, since the Dunkl operators involve values of functions at
reflection-symmetric points. Certainly, when regarding local properties, it
would be worthy of finding weaker requirements at reflection-symmetric points
then that in the position under consideration. This is still open on the non-
tangential behaviour of $\kappa$-harmonic functions.

(d) In comparison with the classical one, it seems that the more apt form of
Lusin-type area integral in the Dunkl setting would be, for $a,h>0$,

$\displaystyle\left(\iint_{\Gamma_{a}^{h}(0)}\left[\tau_{x}\left(|\nabla
u|^{2}\right)\right](t,y)\frac{d\omega_{\kappa}(t)dy}{y^{2|\kappa|+d-1}}\,\right)^{1/2}\,\,\hbox{or}\,\,\left(\iint_{\Gamma_{a}^{h}(0)}\left[\tau_{x}\left(|\nabla_{\kappa}u|^{2}\right)\right](t,y)\frac{d\omega_{\kappa}(t)dy}{y^{2|\kappa|+d-1}}\,\right)^{1/2},$

where $\nabla=\nabla^{(x,y)}$ is the $(d+1)$-dimensional gradient and
$\nabla_{\kappa}=(D_{1},\dots,D_{d},\partial_{y})$ the one associated to the
Dunkl operators (the $\kappa$-gradient). Lemma 5.2 implies that both the two
forms above are dominated by $(S_{a,h}u)(x)$, and for a $G$-invariant $u$, all
three of them are consistent (up to a constant). Therefore Theorem 1.3 is
deeper than expected, and Theorem 1.6 and Corollary 1.7 motivate that the
adoption of $S_{a,h}u$ would be also necessary for a non-$G$-invariant $u$
associated to a general reflection group $G$.

(e) In [20, 21], Hunt and Wheeden studied the local boundary behaviour of
usual harmonic functions in a bounded Lipschitz domain, and in [22], Jerison
and Kenig extended the classical results to a larger class of domains named
non-tangentially accessible domains. In the Dunkl setting, further researches
on boundary behaviour of $\kappa$-harmonic functions may be considered in a
reflection-symmetric Lipschitz domain or non-tangentially accessible domain.

The paper is organized as follows. Section 2 contains some basic knowledge on
the rational Dunkl theory, and in Section 3 we prove several propositions
which is necessary in the sequel. The proofs of Theorems 1.2 and 1.3 are given
in Sections 4 and 5 separately, and Theorems 1.4 and 1.6 are proved in Section
6.

For $1\leq p<\infty$, the space $L_{\kappa}^{p}({\mathbb{R}}^{d})$ consists of
all functions $f$ on ${\mathbb{R}}^{d}$ such that
$\|f\|_{L_{\kappa}^{p}({\mathbb{R}}^{d})}:=\left(c_{\kappa}\int_{{\mathbb{R}}^{d}}|f|^{p}\,d\omega_{\kappa}\right)^{1/p}<+\infty$,
where
$c_{\kappa}^{-1}=\int_{{\mathbb{R}}^{d}}e^{-|x|^{2}/2}d\omega_{\kappa}(x)$,
and $\|f\|_{L^{\infty}}$ is given in the usual way. For a measurable set
$E\subset{\mathbb{R}}^{d}$, set $|E|_{\kappa}=\int_{E}d\omega_{\kappa}$.
Throughout the paper, $c$, $c^{\prime}$, $c_{1}$, $c_{2}$ and so on, denote
constants which may be different in different occurrences. $X\asymp Y$ means
that $X\leq c_{1}Y$ and $Y\leq c_{2}X$ for some fixed constants
$c_{1},c_{2}>0$.

##  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
[19]. We shall use the notation ${\mathbb{B}}(x,r)$ (or
${\mathbb{B}}((x,y),r)$) to denote the open ball in ${\mathbb{R}}^{d}$ (or
${\mathbb{R}}^{d+1}$) with radius $r$ centered at $x\in{\mathbb{R}}^{d}$ (or
$(x,y)\in{\mathbb{R}}^{d+1}$).

###  2.1 The Dunkl kernel and the Dunkl transform

As in the last section, for a root system $R$ let $G$ be the associated
reflection group and $\kappa$ a given nonnegative multiplicity function. As
shown in [13], 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 [15].) It was
proved in [47] 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

$\displaystyle\hbox{supp}\,\mu^{\kappa}_{x}\subseteq\hbox{co}\,\\{\sigma(x):\,\,\sigma\in
G\\},$ (5)

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 [56, 57]).

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

$\displaystyle D^{x}_{j}E_{\kappa}(x,z)=z_{j}E_{\kappa}(x,z),\qquad
x\in{\mathbb{R}}^{d},\,\,\,j=1,\dots,d,$ (6)

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 [9, 39]). Furthermore (cf. [9, 13, 47]), for
$z,w\in{\mathbb{C}}^{d}$, $\lambda\in{\mathbb{C}}$ and $\sigma\in G$, one has

$\displaystyle E_{\kappa}(\lambda z,w)=E_{\kappa}(z,\lambda w),\qquad
E_{\kappa}(\sigma(z),\sigma(w))=E_{\kappa}(z,w),$ (7)

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

$\displaystyle\left|\partial_{z}^{\nu}E_{\kappa}(x,z)\right|\leq|x|^{|\nu|}\max_{\sigma\in
G}e^{{\rm Re}\,\langle\sigma(x),z\rangle}.$ (8)

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
[14, 9, 47, 50]), 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
$\|{{\mathscr{F}}}_{\kappa}f\|_{\infty}\leq\|f\|_{L_{\kappa}^{1}}$.

(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 $f\in{\mathscr{S}}({{\mathbb{R}}^{d}})$ (the Schwartz space), 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
$\|{\mathscr{F}}_{\kappa}f\|_{L_{\kappa}^{2}}=\|f\|_{L_{\kappa}^{2}}$.

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

###  2.2 The generalized translation

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

$\displaystyle\left[{\mathscr{F}}_{\kappa}(\tau_{x}f)\right](\xi)=E_{\kappa}(i\xi,x)({\mathscr{F}}_{\kappa}f)(\xi).$
(9)

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 (8).

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
${\mathscr{S}}({\mathbb{R}}^{d})$, $(\tau_{x}f)(t)$ may be expressed pointwise
by (see [46, 55])

$\displaystyle(\tau_{x}f)(t)=c_{\kappa}\int_{{\mathbb{R}}^{d}}E_{\kappa}(i\xi,x)E_{k}(i\xi,t)({\mathscr{F}}_{\kappa}f)(\xi)\,d\omega_{\kappa}(\xi).$
(10)

###### Proposition 2.2

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

$\displaystyle\int_{{\mathbb{R}}^{d}}(\tau_{x}f)(t)g(t)\,d\omega_{\kappa}(t)=\int_{{\mathbb{R}}^{d}}f(t)(\tau_{-x}g)(t)\,d\omega_{\kappa}(t),$
(11)

and both sides are continuous in $x$.

The equality (11) was proved in [55] for $f\in A_{\kappa}({\mathbb{R}}^{d})$
and $g\in L_{\kappa}^{2}({\mathbb{R}}^{d})\cap L^{\infty}({\mathbb{R}}^{d})$.
The general case, for $f,g\in L_{\kappa}^{2}({\mathbb{R}}^{d})$, follows from
a density argument. On the continuity of
$F(x):=\int_{{\mathbb{R}}^{d}}\tau_{x}f\cdot g\,d\omega_{\kappa}$, for
$x,x^{0}\in{\mathbb{R}}^{r}$ one has

$\displaystyle\left|F(x)-F(x^{0})\right|\leq\|\tau_{x}f-\tau_{x^{0}}f\|_{L_{\kappa}^{2}}\|g\|_{L_{\kappa}^{2}}=\left\|\left(E_{\kappa}(ix,\cdot)-E_{\kappa}(ix^{0},\cdot)\right){\mathscr{F}}_{\kappa}f\right\|_{L_{\kappa}^{2}}\|g\|_{L_{\kappa}^{2}}$

by Proposition 2.1(v) and (9); and then, in view of (8), Lebesgue’s dominated
convergence theorem yields the desired continuity.

In [57] 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

([57]) (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 (9) and (10) 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 $f\in C^{\infty}({\mathbb{R}}^{d})$, 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
$x,t,z\in{\mathbb{R}}^{d}$.

In consideration of radial functions in ${\mathbb{R}}^{d}$, one may learn more
about the translation operator $\tau_{x}$. In fact, for
$x,t\in{\mathbb{R}}^{d}$, by [48, 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 $f\in
C^{\infty}({\mathbb{R}}^{d})$,

$\displaystyle(\tau_{x}f)(t)=\int_{{\mathbb{R}}^{d}}f\,d\rho^{\kappa}_{x,t};$
(12)

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

$\displaystyle\left\\{\xi\in{\mathbb{R}}^{d}:\quad\min_{\sigma\in
G}|x+\sigma(t)|\leq|\xi|\leq\max_{\sigma\in G}|x+\sigma(t)|\right\\}.$ (13)

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,
(12) 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 [48, Theorem 5.1] it follows that, for $x,t\in{\mathbb{R}}^{d}$,

$\displaystyle(\tau_{x}f)(t)=\int_{{\mathbb{R}}^{d}}f_{0}(\sqrt{|x|^{2}+|t|^{2}+2\langle
t,\xi\rangle})\,d\mu^{\kappa}_{x}(\xi).$ (14)

Again, since $\mu^{\kappa}_{x}$ is of compact support, (14) allows an
extension of $\tau_{x}$ to all radial $f\in C({\mathbb{R}}^{d})$ by a density
argument.

###### Proposition 2.5

([55, Theorems 3.7 and 3.8]) Assume $x\in{\mathbb{R}}^{d}$ and $1\leq p\leq
2$.

(i) The generalized translation operator $\tau_{x}$, initially defined on
$L_{\kappa}^{1}({\mathbb{R}}^{d})\cap L_{\kappa}^{2}({\mathbb{R}}^{d})$, 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
$\|\tau_{x}f\|_{L_{\kappa}^{p}}\leq\|f\|_{L_{\kappa}^{p}}$;

(ii) if $f\in L_{\kappa}^{p}({\mathbb{R}}^{d})$ is radial and nonnegative,
then $\tau_{x}f$ is nonnegative;

(iii) if $f\in L_{\kappa}^{1}({\mathbb{R}}^{d})$ is radial, then

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

###  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 [35], for $x\in{\mathbb{R}}^{d}$ and $r\in[0,\infty)$, by

$\displaystyle
M_{f}(x,r)=d_{\kappa}\int_{{\mathbb{S}}^{d-1}}(\tau_{x}f)(rt^{\prime})W_{\kappa}(t^{\prime})dt^{\prime},$
(15)

where $dt^{\prime}$ denotes the area element on ${\mathbb{S}}^{d-1}$, and
$d_{\kappa}^{-1}=\int_{{\mathbb{S}}^{d-1}}W_{\kappa}(t^{\prime})dt^{\prime}$.
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
[48, 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 $f\in C^{\infty}({\mathbb{R}}^{d})$,

$\displaystyle M_{f}(x,r)=\int_{{\mathbb{R}}^{d}}f\,d\sigma^{\kappa}_{x,r};$
(16)

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

$\displaystyle\\{\xi\in{\mathbb{R}}^{d}:\,\,|\xi|\geq||x|-r|\\}\cap\left[\cup_{\sigma\in
G}\\{\xi\in{\mathbb{R}}^{d}:\,\,|\xi-\sigma(x)|\leq r\\}\right];$ (17)

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, (16) gives a natural
extension of the generalized spherical mean operator $f\mapsto M_{f}$ to all
$f\in C({\mathbb{R}}^{d})$.

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

Associated to the reflection group $G$ on ${\mathbb{R}}^{d}$ and the
multiplicity function $\kappa$, we define
$P(x)=c_{d,\kappa}(1+|x|^{2})^{-|\kappa|-(d+1)/2}$, 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. [50, 55]). The
associated Poisson kernel $P_{y}$ for $y>0$ is given by
$P_{y}(x)=y^{-2|\kappa|-d}P(x/y)$, i.e.

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

For $f\in L_{\kappa}^{p}({\mathbb{R}}^{d})$, $1\leq p\leq\infty$, the Poisson
integral of $f$ is defined as

$\displaystyle(Pf)(x,y)=c_{\kappa}\int_{{\mathbb{R}}^{d}}f(t)(\tau_{x}P_{y})(-t)\,d\omega_{\kappa}(t),\qquad(x,y)\in{\mathbb{R}}^{d+1}_{+}.$
(18)

For convenience, we call $P_{y}(x)$ the $\kappa$-Poisson kernel, and
$(Pf)(x,y)$ the $\kappa$-Poisson integral of $f$. Note that the function
$(x,y)\mapsto P_{y}(x)$ is $\kappa$-harmonic in ${\mathbb{R}}_{+}^{d+1}$, and
so is the function $(x,y)\mapsto(\tau_{x}P_{y})(-t)$ by Propositions 2.3(i)
and (iv). Appealing to [1, Proposition 5.1], the following proposition is
immediate.

###### Proposition 2.6

For $f\in L_{\kappa}^{p}({\mathbb{R}}^{d})$, $1\leq p\leq\infty$, its
$\kappa$-Poisson integral $(Pf)(x,y)$ is $\kappa$-harmonic in
${\mathbb{R}}_{+}^{d+1}$.

The next proposition is a consequence of Proposition 2.4 and the
$G$-invariance of $P_{y}$.

###### Proposition 2.7

If $f\in L_{\kappa}^{p}({\mathbb{R}}^{d})$ ($1\leq p\leq\infty$) is
$G$-invariant, then its $\kappa$-Poisson integral $(Pf)(x,y)$ is also
$G$-invariant in $x$.

The conclusions in the following two propositions can be found in [1, 55], and
for $d=1$, $G={\mathbb{Z}}_{2}$, see [31, Proposition 3.2 and Corollary 3.9].

###### Proposition 2.8

([55, Theorems 4.1 and 5.4]) For $f\in L_{\kappa}^{p}({\mathbb{R}}^{d})$
($1\leq p\leq\infty$) and $y>0$,
$\|(Pf)(\cdot,y)\|_{L_{\kappa}^{p}({\mathbb{R}}^{d})}\leq\|f\|_{L_{\kappa}^{p}({\mathbb{R}}^{d})}$;
and for $f\in X=L_{\kappa}^{p}({\mathbb{R}}^{d})$, $1\leq p<\infty$, or
$C_{0}({\mathbb{R}}^{d})$, $\lim_{y\rightarrow 0+}\|(Pf)(\cdot,y)-f\|_{X}=0$.

###### Proposition 2.9

([1, Corollary 5.4]) Assume that $a>0$. If $f\in
L^{p}_{\kappa}({\mathbb{R}}^{d})$, $1\leq p\leq\infty$, then for almost every
$x\in{\mathbb{R}}^{d}$, its $\kappa$-Poisson integral $(Pf)(t,y)$ converges to
$f(x)$ as $(t,y)\in\Gamma_{a}(x)$ approaching to $(x,0)$.

The next proposition is the (global) Fatou-type theorem for $\kappa$-harmonic
functions.

###### Proposition 2.10

([1, Corollary 7.4]) If $u$ is $\kappa$-harmonic and bounded on
${\mathbb{R}}^{d+1}_{+}$, then $u$ has a non-tangential limit at almost every
point of the boundary.

##  3 Some elementary propositions

In this section we present several propositions which will be useful
subsequently. Some conclusions may be found elsewhere, but here we treat them
as elementarily as possible. In particular, it will be proved that, on a
general $G$-invariant domain $\Omega$ of ${\mathbb{R}}^{d}$, a twice-
differentiable function $f$, harmonic associated to the Dunkl operators, is
infinitely differentiable and satisfies the mean value formula
$M_{f}(x,r)=f(x)$ for all $x\in\Omega$ provided
$\overline{{\mathbb{B}}(x,r)}\subset\Omega$ with $r>0$.

###### Proposition 3.1

If (i) $f\in C^{\infty}({\mathbb{R}}^{d})$, and $g\in C({\mathbb{R}}^{d})$ is
radial and of compact support, or (ii) $f\in C^{\infty}({\mathbb{R}}^{d})$
with compact support, and $g\in C({\mathbb{R}}^{d})$ is radial, then (11)
holds.

  * Proof. Assume that ${\rm supp}\,g\subset\\{\xi\in{\mathbb{R}}^{d}:\,|\xi|\leq r_{0}\\}$. For fixed $x\in{\mathbb{R}}^{d}$, we take a compactly supported $\phi\in C^{\infty}({\mathbb{R}}^{d})$ with $\phi(\xi)=1$ for $|\xi|\leq|x|+r_{0}$. It then follows from Proposition 2.3(ii) that $[\tau_{x}(\phi f)](t)=(\tau_{x}f)(t)$ for $|t|\leq r_{0}$, and from (12) and (13), $(\tau_{-x}g)(t)=0$ for $|t|>|x|+r_{0}$. Thus the assertion of part (i) is validated by applying Proposition 2.2 to $\phi f$ and $g$. A similar treatment verifies the assertion of part (ii). $\square$

###### Proposition 3.2

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 $a\in{\mathbb{R}}\setminus\\{0\\}$.

Indeed, by Proposition 2.4 the assertion is true for
$f\in{\mathscr{S}}({\mathbb{R}}^{d})$. For general radial $f\in
C({\mathbb{R}}^{d})$, we take a radial and compactly supported $\phi\in
C^{\infty}({\mathbb{R}}^{d})$ with $\phi(\xi)=1$ for $|\xi|\leq|a|(|x|+|t|)$,
and then, choose a sequence of radial functions
$\\{\phi_{j}\\}\subset{\mathscr{S}}({\mathbb{R}}^{d})$ converging uniformly to
$\phi f$ on ${\mathbb{R}}^{d}$. Letting $j\rightarrow\infty$ on the both sides
of $\left[\tau_{x}(\phi_{j}(a\cdot))\right](t)=(\tau_{ax}\phi_{j})(at)$ yields
$\left[\tau_{x}((\phi f)(a\cdot))\right](t)=(\tau_{ax}(\phi f))(at)$, which is
identical with $\left[\tau_{x}(f(a\cdot))\right](t)=(\tau_{ax}f)(at)$ on
account of (13).

In what follows, we set $\widetilde{\Delta}_{\kappa}=\sum_{j=1}^{d}D_{j}^{2}$.

###### Proposition 3.3

(i) If $f\in A_{\kappa}({\mathbb{R}}^{d})$, then

$\displaystyle
M_{f}(x,r)=c_{\kappa}\int_{{\mathbb{R}}^{d}}j_{|\kappa|+(d-2)/2}(r|\xi|)E_{\kappa}(i\xi,x)({\mathscr{F}}_{\kappa}f)(\xi)\,d\omega_{\kappa}(\xi).$
(19)

(ii) If $f\in C^{\infty}({\mathbb{R}}^{d})$, then for $x\in{\mathbb{R}}^{d}$
and $r>0$,

$\displaystyle
M_{f}(x,r)-f(x)=\int_{0}^{r}\frac{1}{s^{2|\kappa|+d-1}}\int_{0}^{s}\tilde{s}^{2|\kappa|+d-1}M_{\widetilde{\Delta}_{\kappa}f}(x,\tilde{s})\,d\tilde{s}.$
(20)

  * Proof. Part (i) follows from (10), [48, Corollary 2.5] and Fubini’s theorem.

As for part (ii), if $f\in{\mathscr{S}}({\mathbb{R}}^{d})$, from (19) it is
obvious that $\lim_{r\rightarrow 0+}M_{f}(x,r)=f(x)$, and by (6) and
Proposition 2.1(iii),
$\widetilde{\Delta}^{x}_{\kappa}\left[M_{f}(x,r)\right]=M_{\widetilde{\Delta}_{\kappa}f}(x,r)$.
From [48, (4.5)] we have

$\displaystyle\frac{\partial}{\partial
r}\left[r^{2|\kappa|+d-1}\frac{\partial}{\partial
r}M_{f}(x,r)\right]=r^{2|\kappa|+d-1}\widetilde{\Delta}^{x}_{\kappa}\left[M_{f}(x,r)\right],$

so that (20) follows immediately. For general $f\in
C^{\infty}({\mathbb{R}}^{d})$ and for $x\in{\mathbb{R}}^{d}$, $r>0$, we choose
a compactly supported $\phi\in C^{\infty}({\mathbb{R}}^{d})$ with
$\phi(\xi)=1$ for $|\xi|\leq|x|+r$. It then follows that $\phi
f\in{\mathscr{S}}({\mathbb{R}}^{d})$ and (20) holds with $\phi f$ instead of
$f$. But by (16) and (17) $M_{\phi f}(x,r)=M_{f}(x,r)$ and
$M_{\widetilde{\Delta}_{\kappa}(\phi
f)}(x,s)=M_{\widetilde{\Delta}_{\kappa}f}(x,s)$ for $s\in[0,r)$, so that (20)
holds for $f$ itself. $\square$

###### Proposition 3.4

(i) If $f\in C({\mathbb{R}}^{d})$, then

$\displaystyle\lim_{r\rightarrow 0+}M_{f}(x,r)=f(x)$ (21)

for $x\in{\mathbb{R}}^{d}$, and the limit holds uniformly for $x$ in a compact
subset of ${\mathbb{R}}^{d}$;

(ii) if $f\in C({\mathbb{R}}^{d})$, and $g\in C({\mathbb{R}}^{d})$ is radial
and of compact support, then for $x\in{\mathbb{R}}^{d}$,

$\displaystyle
d_{\kappa}^{-1}\int_{0}^{\infty}M_{f}(x,r)g_{0}(r)\,r^{2|\kappa|+d-1}dr=\int_{{\mathbb{R}}^{d}}f(t)(\tau_{-x}g)(t)\,d\omega_{\kappa}(t),$
(22)

where $g(t)=g_{0}(|t|)$ for $t\in{\mathbb{R}}^{d}$.

  * Proof. Assume that ${\rm supp}\,g\subset\\{\xi\in{\mathbb{R}}^{d}:\,|\xi|\leq r_{0}\\}$ and $x\in\overline{{\mathbb{B}}(0,r_{1})}$, where $r_{0},r_{1}>0$. If $f\in C^{\infty}({\mathbb{R}}^{d})$, (21) follows from (20), where the limit holds uniformly for all $x\in\overline{{\mathbb{B}}(0,r_{1})}$; and (22) follows from Proposition 3.1 by using the spherical coordinates to the left hand side of (11). For general $f\in C({\mathbb{R}}^{d})$, we take a compactly supported $\phi\in C^{\infty}({\mathbb{R}}^{d})$ with $\phi(\xi)=1$ for $|\xi|\leq r_{0}+r_{1}$, and then, choose a sequence of functions $\\{\phi_{j}\\}\subset{\mathscr{S}}({\mathbb{R}}^{d})$ converging uniformly to $\phi f$ on ${\mathbb{R}}^{d}$. Note that, by (16) and (17) $M_{\phi f}(x,r)=M_{f}(x,r)$ for $r\in[0,r_{0}]$ and $x\in\overline{{\mathbb{B}}(0,r_{1})}$, and $M_{\phi_{j}}(x,r)$ converges to $M_{\phi f}(x,r)$ uniformly for $r\in[0,r_{0}]$ as $j$ tends to infinity; and further, by (12) and (13) $(\tau_{-x}g)(t)=0$ for $|t|>|x|+r_{0}$. Now applying (22) to $\phi_{j}$ and letting $j\rightarrow\infty$, (22) is proved for $f\in C({\mathbb{R}}^{d})$. Finally for $x\in\overline{{\mathbb{B}}(0,r_{1})}$ and $r\in[0,r_{0}]$,

$\displaystyle|M_{f}(x,r)-f(x)|\leq 2\|\phi
f-\phi_{j}\|_{C_{0}({\mathbb{R}}^{d})}+|M_{\phi_{j}}(x,r)-\phi_{j}(x)|,$

and then, letting $r\rightarrow 0+$ and $j\rightarrow\infty$ successively
proves (21) for $f\in C({\mathbb{R}}^{d})$ and the limit in (21) holds
uniformly for all $x\in\overline{{\mathbb{B}}(0,r_{1})}$. $\square$

We are now in a position to prove a mean-value property of functions
satisfying $\widetilde{\Delta}_{\kappa}f=0$ in a general case. Such a problem
was earlier considered in [17, 35].

###### Proposition 3.5

Let $\Omega$ be a $G$-invariant domain of ${\mathbb{R}}^{d}$. If $f\in
C^{2}(\Omega)$ satisfies $\widetilde{\Delta}_{\kappa}f=0$ in $\Omega$, then
$f\in C^{\infty}(\Omega)$, and for $x\in\Omega$,

$\displaystyle M_{f}(x,r)=f(x)$ (23)

provided $\overline{{\mathbb{B}}(x,r)}\subset\Omega$ for $r>0$.

  * Proof. Let $\phi\in C^{\infty}({\mathbb{R}}^{d})$ be a radial function satisfying $\int_{{\mathbb{R}}^{d}}\phi\,d\omega_{\kappa}=1$ and ${\rm supp}\,\phi\subseteq{\mathbb{B}}(0,1)$, and write $\phi(t)=\tilde{\phi}(|t|)$ for $t\in{\mathbb{R}}^{d}$. For given $\overline{{\mathbb{B}}(x,r)}\subset\Omega$ with $r>0$, choose $r^{\prime\prime}>r^{\prime}>r$ such that $\overline{{\mathbb{B}}(x,r^{\prime\prime})}\subset\Omega$. By Urysohn’s lemma, there exists a function $\psi\in C^{\infty}({\mathbb{R}}^{d})$ such that $\psi(t)=1$ for $t\in\cup_{\sigma\in G}\overline{{\mathbb{B}}(\sigma(x),r^{\prime})}$ and $\psi(t)=0$ for $t\notin\cup_{\sigma\in G}{\mathbb{B}}(\sigma(x),r^{\prime\prime})$. Thus $\psi f$ has a natural $C^{2}$ extension to ${\mathbb{R}}^{d}$ with compact support.

For $\epsilon>0$, define

$\displaystyle
f_{\epsilon}(\tilde{x})=\int_{{\mathbb{R}}^{d}}\psi(t)f(t)(\tau_{-\tilde{x}}\phi_{\epsilon})(t)\,d\omega_{\kappa}(t),$
(24)

where $\phi_{\epsilon}(t)=\epsilon^{-2|\kappa|-d}\phi(\epsilon^{-1}t)$.
Obviously $f_{\epsilon}\in C^{\infty}({\mathbb{R}}^{d})$.

Since by (10),
$\widetilde{\Delta}^{x}_{\kappa}\left[(\tau_{-x}\phi_{\epsilon})(t)\right]=\widetilde{\Delta}^{t}_{\kappa}\left[(\tau_{-x}\phi_{\epsilon})(t)\right]$,
the Green formula in [35, Theorem 4.11] gives

$\displaystyle\left(\widetilde{\Delta}_{\kappa}f_{\epsilon}\right)(\tilde{x})$
$\displaystyle=\int_{{\mathbb{R}}^{d}}\psi(t)f(t)\,\widetilde{\Delta}^{t}_{\kappa}\left[(\tau_{-\tilde{x}}\phi_{\epsilon})(t)\right]\,d\omega_{\kappa}(t)$
$\displaystyle=\int_{{\mathbb{R}}^{d}}(\tau_{-\tilde{x}}\phi_{\epsilon})(t)\left[\widetilde{\Delta}_{\kappa}(\psi
f)\right](t)\,d\omega_{\kappa}(t).$ (25)

We assume that $\tilde{x}\in\cup_{\sigma\in
G}\overline{{\mathbb{B}}(\sigma(x),r)}$, says
$\tilde{x}\in\overline{{\mathbb{B}}(\sigma_{1}(x),r)}$ for some $\sigma_{1}\in
G$. For $0<\epsilon<r^{\prime}-r$, ${\rm
supp}\,\phi_{\epsilon}\subseteq{\mathbb{B}}(0,r^{\prime}-r)$, and hence, (13)
implies

${\rm supp}\,\phi_{\epsilon}\cap{\rm
supp}\,\rho^{\kappa}_{-\tilde{x},t}=\emptyset\qquad\hbox{for}\quad
t\notin\cup_{\sigma\in G}{\mathbb{B}}(\sigma(\tilde{x}),r^{\prime}-r),$

so that $(\tau_{-\tilde{x}}\phi_{\epsilon})(t)=0$ by (12). But for
$t\in\cup_{\sigma\in G}{\mathbb{B}}(\sigma(\tilde{x}),r^{\prime}-r)$, letting
$|t-\sigma_{2}(\tilde{x})|<r^{\prime}-r$ for some $\sigma_{2}\in G$, it
follows that
$|t-\sigma_{2}(\sigma_{1}(x))|\leq|t-\sigma_{2}(\tilde{x})|+|\sigma_{2}(\tilde{x}-\sigma_{1}(x))|<r^{\prime}$,
i.e., $t\in\cup_{\sigma\in G}{\mathbb{B}}(\sigma(x),r^{\prime})$, and hence
$[\widetilde{\Delta}_{\kappa}(\psi f)](t)=(\widetilde{\Delta}_{\kappa}f)(t)=0$
by the assumption. Thus ( ‣ 3) implies
$(\widetilde{\Delta}_{\kappa}f_{\epsilon})(\tilde{x})=0$ for
$\tilde{x}\in\cup_{\sigma\in G}\overline{{\mathbb{B}}(\sigma(x),r)}$, and
consequently, $M_{\widetilde{\Delta}_{\kappa}f_{\epsilon}}(x,s)=0$ for
$s\in(0,r)$ by (16) and (17). Now applying (20) to $f_{\epsilon}$ we get

$\displaystyle M_{f_{\epsilon}}(x,r)=f_{\epsilon}(x).$ (26)

Also, for $\tilde{x}\in\cup_{\sigma\in
G}\overline{{\mathbb{B}}(\sigma(x),r)}$, by Proposition 3.4 and (24) we have

$\displaystyle f_{\epsilon}(\tilde{x})=d_{\kappa}^{-1}\int_{0}^{\infty}M_{\psi
f}(\tilde{x},\epsilon s)\tilde{\phi}(s)\,s^{2|\kappa|+d-1}ds,$ (27)

and since ${\rm supp}\,\tilde{\phi}\subseteq[0,1]$, Proposition 3.4(i) implies
that $f_{\epsilon}(\tilde{x})$ tends to
$\psi(\tilde{x})f(\tilde{x})=f(\tilde{x})$ as $\epsilon\rightarrow 0+$
uniformly for $\tilde{x}\in\cup_{\sigma\in
G}\overline{{\mathbb{B}}(\sigma(x),r)}$. Then by (16) and (17), (23) is
obtained by taking $\epsilon\rightarrow 0+$ in (26).

Finally we fix a number $\epsilon\in(0,r^{\prime}-r)$. It has been shown that
$f_{\epsilon}\in C^{\infty}({\mathbb{B}}(x,r))$, and for
$\tilde{x}\in{\mathbb{B}}(x,r)$ and $s\in[0,1]$, $\cup_{\sigma\in
G}{\mathbb{B}}(\sigma(\tilde{x}),\epsilon s)\subset\cup_{\sigma\in
G}{\mathbb{B}}(\sigma(x),r^{\prime})$, so that $M_{\psi f}(\tilde{x},\epsilon
s)=M_{f}(\tilde{x},\epsilon s)$. Thus by (23) and (27), we have
$f(\tilde{x})=f_{\epsilon}(\tilde{x})$ for $\tilde{x}\in{\mathbb{B}}(x,r)$,
and hence $f\in C^{\infty}({\mathbb{B}}(x,r))$. The proof of the proposition
is completed. $\square$

Notice that, the final part of the proof of the above proposition implies that
a function $f\in C(\Omega)$ satisfying the mean value property (23) must be
infinitely differentiable. Further, if a function $f\in C^{2}(\Omega)$
satisfying (23), Proposition 3.3(ii) gives that
$M_{\widetilde{\Delta}_{\kappa}f}(x,r)\equiv 0$ for $r>0$ with
$\overline{{\mathbb{B}}(x,r)}\subset\Omega$, and so
$\widetilde{\Delta}_{\kappa}f(x)=0$ by Proposition 3.4(ii).

###### Corollary 3.6

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

Such a characterization for $\Omega={\mathbb{R}}^{d}$ and $f\in
C^{\infty}({\mathbb{R}}^{d})$ was proved in [35]. For a general $G$-invariant
domain $\Omega$ and $f\in C^{2}(\Omega)$, a different characterization, in
terms of a generalized volume mean value property, was given in [17];
moreover, by [17, Remark 3.2], the method there also yields a characterization
by means of the generalized spherical mean value property $M_{f}(x,\rho)=f(x)$
for $0<\rho\leq r/3$ provided $\overline{{\mathbb{B}}(x,r)}\subset\Omega$ with
$r>0$.

In what follows we consider the reflection group
$\tilde{G}=G\otimes{\mathbb{Z}}_{2}$ on ${\mathbb{R}}^{d+1}$ and the
multiplicity function $\tilde{\kappa}=(\kappa,0)$. In this case, since the
multiplicity value associated to ${\mathbb{Z}}_{2}$ is zero, the
$G\otimes{\mathbb{Z}}_{2}$-invariance of the domain $\Omega$ in [35, Theorem
4.11] and Proposition 3.5 can be relaxed to the $G$-invariance. We reformulate
the Green formula in [35, Theorem 4.11] as follows.

###### Proposition 3.7

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

$\displaystyle\iint_{\Omega}(v\Delta_{\kappa}u-u\Delta_{\kappa}v)\,d\omega_{\kappa}(x)dy=\int_{\partial\Omega}\left(v\partial_{\mathbf{n}}u-u\partial_{\mathbf{n}}v\right)W_{\kappa}(x)d\sigma(x,y),$
(28)

where $\partial_{\mathbf{n}}$ denotes the directional derivative of the
outward normal, and $d\sigma(x,y)$ the area element on $\partial\Omega$.

We have a variant of (28) with the directional Dunkl operator $D_{\mathbf{n}}$
instead of the usual directional derivative $\partial_{\mathbf{n}}$. For
$\alpha\in R$, write $\tilde{\alpha}=(\alpha,0)$ as a vector in
${\mathbb{R}}^{d+1}$; and for a unit vector $\mathbf{n}$ in
${\mathbb{R}}^{d+1}$, the directional Dunkl operator $D_{\mathbf{n}}$ is given
by $D_{\mathbf{n}}=\langle\nabla_{\kappa},\mathbf{n}\rangle$, where
$\nabla_{\kappa}=(D_{1},\dots,D_{d},\partial_{y})$ is the $\kappa$-gradient.
From (1),

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

and for $\alpha\in R_{+}$, since $\partial\Omega$ is $G$-invariant, the change
of variables $x\rightarrow\sigma_{\alpha}(x)$ shows that

$\displaystyle\int_{\partial\Omega}\langle\tilde{\alpha},{\mathbf{n}}\rangle\left(v\frac{u-\sigma_{\alpha}u}{\langle\alpha,x\rangle}-u\frac{v-\sigma_{\alpha}v}{\langle\alpha,x\rangle}\right)W_{\kappa}(x)d\sigma(x,y)=0.$

Summing the above expression with the multiple $\kappa(\alpha)$ over
$\alpha\in R_{+}$ and inserting the result into the right hand side of (28),
we obtain the following form of the Green formula.

###### Proposition 3.8

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

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

where ${\mathbf{n}}$ denotes the outward normal of $\partial\Omega$.

The next proposition is a consequence of Proposition 3.5 with
$\Omega={\mathbb{R}}^{d+1}_{+}={\mathbb{R}}^{d}\times(0,\infty)$.

###### Proposition 3.9

If the $C^{2}$ function $u$ on the upper-space ${\mathbb{R}}^{d+1}_{+}$ is
$\kappa$-harmonic, then $u\in C^{\infty}({\mathbb{R}}^{d+1}_{+})$.

##  4 The proof of Theorem 1.2

That part (i) implies part (ii) in Theorem 1.2 is obvious; what is left is to
show that part (ii) implies part (i) in Theorem 1.2, which is restated as
follows.

###### Theorem 4.1

Suppose that $E$ is a $G$-invariant measurable subset of
$\partial{\mathbb{R}}^{d+1}_{+}={\mathbb{R}}^{d}$ and $u$ is $\kappa$-harmonic
in ${\mathbb{R}}^{d+1}_{+}$. If $u$ is non-tangentially bounded at $(x,0)$ for
every $x\in E$, then there exists a subset $E_{0}$ of full measure of $E$ such
that for every $a>0$ and each $x\in E_{0}$, $u(t,y)$ has a finite limit as
$(t,y)\in\Gamma_{a}(x)$ approaches to $(x,0)$.

For a subset $E$ of $\partial{\mathbb{R}}^{d+1}_{+}={\mathbb{R}}^{d}$ and for
fixed $a,h>0$, we always use the notation

$\displaystyle\Omega^{E}(a,h):=\bigcup_{x^{0}\in E}\Gamma^{h}_{a}(x^{0}).$

We shall need a series of lemmas.

###### Lemma 4.2

([53, p. 201, Lemma]) Let $u$ be a continuous function in
${\mathbb{R}}^{d+1}_{+}$, and $E$ a measurable set of
$\partial{\mathbb{R}}^{d+1}_{+}={\mathbb{R}}^{d}$ with $0<|E|_{0}<\infty$,
where $|\cdot|_{0}$ denotes the Lebesgue measure on ${\mathbb{R}}^{d}$. If $u$
is non-tangentially bounded at $(x,0)$ for every $x\in E$, then for any
$\epsilon>0$, there exists a compact set $E_{\epsilon}$ satisfying

(i) $E_{\epsilon}\subset E$, $|E-E_{\epsilon}|_{0}<\epsilon$;

(ii) for any $a>0$ and $h>0$, there is a contant $c_{a,h,\epsilon}>0$, so that
$|u(x,y)|\leq c_{a,h,\epsilon}$, $(x,y)\in\Omega^{E_{\epsilon}}(a,h)$.

In [1, 16], useful lower and upper bounds of the translated $\kappa$-Poisson
kernel $(\tau_{x}P_{y})(-t)$ are obtained, namely, there are two constants
$c_{1},c_{2}>0$, independent of $y>0$ and $x,t\in{\mathbb{R}}^{d}$, such that

$\displaystyle\frac{c_{1}}{V(x,t,y+|x-t|)}\frac{y}{y+|x-t|}\leq(\tau_{x}P_{y})(-t)\leq\frac{c_{2}y}{V(x,t,y+d(x,t))}\frac{y+d(x,t)}{y^{2}+|x-t|^{2}},$
(29)

where
$V(x,t,r)=\max\\{\left|{\mathbb{B}}(x,r)\right|_{\kappa},\left|{\mathbb{B}}(t,r)\right|_{\kappa}\\}$,
and $d(x,t)=\min_{\sigma\in G}|x-\sigma(t)|$. We note that the bounds given in
(29) have simpler but equivalent forms, which are stated as follows.

###### Lemma 4.3

There exist two constants $c_{1},c_{2}>0$, independent of $y>0$ and
$x,t\in{\mathbb{R}}^{d}$, such that

$\displaystyle\frac{c_{1}}{\left|{\mathbb{B}}(x,y+|x-t|)\right|_{\kappa}}\frac{y}{y+|x-t|}\leq(\tau_{x}P_{y})(-t)\leq\frac{c_{2}y}{\left|{\mathbb{B}}(x,y+d(x,t))\right|_{\kappa}}\frac{y+d(x,t)}{y^{2}+|x-t|^{2}}.$
(30)

To verify (30), it suffices to show

$\displaystyle\left|{\mathbb{B}}(x,y+|x-t|)\right|_{\kappa}$
$\displaystyle\asymp\left|{\mathbb{B}}(t,y+|x-t|)\right|_{\kappa},$ (31)
$\displaystyle\left|{\mathbb{B}}(x,y+d(x,t))\right|_{\kappa}$
$\displaystyle\asymp\left|{\mathbb{B}}(t,y+d(x,t))\right|_{\kappa}.$ (32)

The equation (31) is obvious since

$\displaystyle\left|{\mathbb{B}}(x,r)\right|_{\kappa}\asymp
r^{d}\prod_{\alpha\in R}(|\langle\alpha,x\rangle|+r)^{\kappa(\alpha)}.$ (33)

As for (32), suppose $\tilde{\sigma}\in G$ such that
$|x-\tilde{\sigma}(t)|=d(x,t)$. It follows that

$\displaystyle\prod_{\alpha\in
R}(|\langle\alpha,t\rangle|+y+d(x,t))^{k(\alpha)}\leq\prod_{\alpha\in
R}(|\langle\tilde{\sigma}(\alpha),x\rangle|+y+(1+\sqrt{2})|x-\tilde{\sigma}(t)|)^{k(\alpha)},$
(34)

and since $k(\alpha)=k(\tilde{\sigma}(\alpha))$, $\tilde{\sigma}(R)=R$, (34)
implies that $\left|{\mathbb{B}}(t,y+d(x,t))\right|_{\kappa}\leq
c\left|{\mathbb{B}}(x,y+d(x,t))\right|_{\kappa}$ with some fixed $c>0$. By
symmetry of $x$ and $t$, (32) is verified.

The next lemma is a maximum principle which is a little stronger than that in
[46, Theorem 4.2].

###### Lemma 4.4

(maximum principle) Let $\Omega$ be a $G$-invariant bounded domain in
${\mathbb{R}}^{d+1}$, and let $u(x,y)$ be continuous on $\overline{\Omega}$.
Assume that $u$ is of class $C^{2}$ in the region where $u>0$ and satisfies
$\Delta_{\kappa}u\geq 0$ there. If $u|_{\partial\Omega}\leq 0$, then $u\leq 0$
on the whole $\Omega$.

To show this, set
$u_{\epsilon}=u+\epsilon(|x|^{2}+y^{2})-\epsilon\max_{\overline{\Omega}}(|x|^{2}+y^{2})$
for small $\epsilon>0$. It is clear that $u_{\epsilon}|_{\partial\Omega}\leq
0$, and in the region where $u_{\epsilon}>0$,
$\Delta_{\kappa}u_{\epsilon}=\Delta_{\kappa}u+2\epsilon(2|\kappa|+d+1)>0$. We
claim that $u_{\epsilon}\leq 0$ on whole $\Omega$, so that $u\leq 0$.
Otherwise, $u_{\epsilon}$ would attain its absolute (positive) maximum at some
point $(x_{0},y_{0})$ in the interior of $\Omega$; but by [46, Lemma 4.1],
$\Delta_{\kappa}u_{\epsilon}(x_{0},y_{0})\leq 0$. This leads to a
contradiction.

For a positive integer $m$, consider an $m$-tuple
$F=(u_{1},u_{2},\dots,u_{m})$ of real-valued functions defined on
${\mathbb{R}}^{d+1}_{+}$. We shall use the the following notations:
$|F|=(\sum_{j=1}^{m}u_{j}^{2})^{1/2}$, $\sigma F=(\sigma u_{1},\sigma
u_{2},\dots,\sigma u_{m})$, $\langle F,\sigma
F\rangle=\sum_{j=1}^{m}u_{j}\sigma u_{j}$, and $\langle
F,\partial_{x_{i}}F\rangle=\sum_{j=1}^{m}u_{j}\partial_{x_{i}}u_{j}$ for
$0\leq i\leq d$ with $x_{0}=y$. The following proposition is a crucial element
in the proof of Theorem 4.1.

###### Proposition 4.5

If $u_{1},u_{2},\dots,u_{m}$ are all $\kappa$-harmonic in
${\mathbb{R}}^{d+1}_{+}$, then, with the notations as above,
$\Delta_{\kappa}|F|\geq 0$ in the region where $|F|>0$.

  * Proof. In the region where $|F|>0$, direct calculations show that $\partial_{x_{i}}|F|=|F|^{-1}\langle F,\partial_{x_{i}}F\rangle$ and

$\partial_{x_{i}}^{2}|F|=|F|^{-1}|\partial_{x_{i}}F|^{2}-|F|^{-3}\langle
F,\partial_{x_{i}}F\rangle+|F|^{-1}\langle F,\partial_{x_{i}}^{2}F\rangle.$

Since every $u_{j}$ is $\kappa$-harmonic, from (2) and (3) it follows that

$\displaystyle\Delta|F|=\frac{1}{|F|^{3}}\sum_{i=0}^{d}\left(|F|^{2}|\partial_{x_{i}}F|^{2}-\langle
F,\partial_{x_{i}}F\rangle^{2}\right)-\frac{2}{|F|}\sum_{\alpha\in
R_{+}}\kappa(\alpha)\langle F,\delta_{\alpha}F\rangle$

and

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

Summing them gives

$\displaystyle\Delta_{\kappa}|F|=\frac{1}{|F|^{3}}\sum_{i=0}^{d}\left(|F|^{2}|\partial_{x_{i}}F|^{2}-\langle
F,\partial_{x_{i}}F\rangle^{2}\right)+\frac{2}{|F|}\sum_{\alpha\in
R_{+}}\kappa(\alpha)\frac{|F||\sigma_{\alpha}F|-\langle
F,\sigma_{\alpha}F\rangle}{\langle\alpha,x\rangle^{2}}.$

Obviously both terms on the right hand above are nonnegative. $\square$

For $m=(d+1)|G|$ (where $|G|$ is the order of $G$) and
$u_{1},u_{2},\dots,u_{d+1}$ satisfying the generalized Cauchy-Riemann
equations in terms of the Dunkl operators, a sharper conclusion was obtained
in [1], that is, there exists some $q\in(0,1)$ such that
$\Delta_{\kappa}\left(\sum_{\sigma\in G}\sum_{j=1}^{d+1}|\sigma
u_{j}|^{2}\right)^{q/2}\geq 0$ in the region where
$\sum_{j=1}^{d+1}u_{j}^{2}\neq 0$. The difference between this and Proposition
4.5 is that, we need not restrict the number of functions, while the
assumption that the Cauchy-Riemann equations are satisfied is, in fact,
replaced by the fact that the functions are $\kappa$-harmonic.

We will adopt the strategy of the proof of [53, p. 201, Theorem 3] in proving
Theorem 4.1. For convenience we extract the following two lemmas. In the proof
of the later one, we need a $G$-invariant auxiliary function, by which an
associated domain defined is $G$-invariant. This makes Lemma 4.4 and
Proposition 4.5 applicable.

###### Lemma 4.6

Let $E$ be a $G$-invariant measurable subset of
$\partial{\mathbb{R}}^{d+1}_{+}={\mathbb{R}}^{d}$, and for $a>0$,
$\Omega=\Omega^{E}(a,1)$. Then there exists a nonnegative $\kappa$-harmonic
function $H$ on ${\mathbb{R}}^{d+1}_{+}$ satisfying

(i) $H(x,y)$ is $G$-invariant in $x$ and $H(x,y)\geq 2$ for
$(x,y)\in{\mathbb{R}}^{d+1}_{+}\bigcap\partial\Omega$;

(ii) $H$ has non-tangential limit $0$ at $(x,0)$ for almost every $x\in E$.

  * Proof. We define

$\displaystyle H_{0}(x,y)=\left(P\chi_{E^{c}}\right)(x,y)+y,$

where $\chi_{E^{c}}$ is the characteristic function of the complement $E^{c}$
of $E$. It is obvious that $H_{0}$ is nonnegative and $\kappa$-harmonic in
${\mathbb{R}}^{d+1}_{+}$, and also $G$-invariant in $x$ by Proposition 2.7.
From Proposition 2.9, $H$ has non-tangential limit $0$ at $(x,0)$ for almost
every $x\in E$.

Now we prove that $H_{0}$ has a positive lower bound on
${\mathbb{R}}^{d+1}_{+}\bigcap\partial\Omega$. For $(x,y)\in\partial\Omega$
with $y=1$, $H(x,y)\geq 1$. If $(x,y)\in\partial\Omega$ with $0<y<1$, then
$\\{t:\,|t-x|<ay\\}\subset E^{c}$; this is because, $t^{\prime}\in E$ with
$|t^{\prime}-x|<ay$ implies that
$(x,y)\in\Gamma^{1}_{a}(t^{\prime})\subset\Omega$. Thus, by (18) we have

$\displaystyle H_{0}(x,y)\geq
c_{\kappa}\int_{|t-x|<ay}(\tau_{x}P_{y})(-t)\,d\omega_{\kappa}(t),$

and using the first inequality in (30),

$\displaystyle H_{0}(x,y)\geq
c\int_{|t-x|<ay}\frac{d\omega_{\kappa}(t)}{\left|{\mathbb{B}}(x,y+|x-t|)\right|_{\kappa}}\geq\frac{c\left|{\mathbb{B}}(x,ay)\right|_{\kappa}}{\left|{\mathbb{B}}(x,(a+1)y)\right|_{\kappa}}.$

This shows that $H_{0}$ has a lower bound $c_{0}>0$ on
${\mathbb{R}}^{d+1}_{+}\bigcap\partial\Omega$. Finally the function
$H=2H_{0}/c_{0}$ is desired. $\square$

###### Lemma 4.7

Suppose that $E$ is a $G$-invariant compact subset of
$\partial{\mathbb{R}}^{d+1}_{+}={\mathbb{R}}^{d}$ and the function $u$ is
$\kappa$-harmonic in ${\mathbb{R}}^{d+1}_{+}$. If

$\displaystyle|u(x,y)|\leq 1,\qquad(x,y)\in\Omega,$ (35)

where $\Omega=\Omega^{E}(a,2)$ for some $a>0$, then for almost every $x\in E$,
$u(t,y)$ has a finite limit as $(t,y)\in\Gamma_{a}(x)$ approaching to $(x,0)$.

  * Proof. We first note that $\Omega$ is an open, $G$-invariant, bounded domain in ${\mathbb{R}}^{d+1}_{+}$. Choose a sequence $\\{y_{k}\\}_{k=1}^{\infty}\subset(0,1)$ such that $y_{k}\rightarrow 0$, and for $x\in{\mathbb{R}}^{d}$, define

$\varphi_{k}(x)=\left\\{\begin{aligned} &u(x,y_{k}),\quad\text{if}\
(x,y_{k})\in\Omega,\\\ &0,\qquad\qquad\text{otherwise}.\\\
\end{aligned}\right.$

Obviously $|\varphi_{k}|\leq 1$ on ${\mathbb{R}}^{d}$ for all $k\geq 1$, and
thus, one can find a function $\varphi$ with $|\varphi|\leq 1$ on
${\mathbb{R}}^{d}$ and a subsequence $\\{\varphi_{k_{j}}\\}$ so that
$\\{\varphi_{k_{j}}\\}$ converges weakly∗ to $\varphi$. In particular, for
their $\kappa$-Poisson integrals, we have
$\lim_{j\rightarrow\infty}(P\varphi_{k_{j}})(x,y)=(P\varphi)(x,y)$,
$(x,y)\in{\mathbb{R}}^{d+1}_{+}$.

If we write $\psi_{k}(x,y)=u(x,y+y_{k})-(P\varphi_{k})(x,y)$, then

$\displaystyle\psi(x,y):=\lim_{j\rightarrow\infty}\psi_{k_{j}}(x,y)=u(x,y)-(P\varphi)(x,y),\qquad(x,y)\in{\mathbb{R}}^{d+1}_{+}.$
(36)

Further we claim that, for each $k$,

$\displaystyle|\psi_{k}(x,y)|\leq\sqrt{|G|}H(x,y),\qquad(x,y)\in\Omega_{1}:=\Omega^{E}(a,1),$
(37)

where $H$ is the function given in Lemma 4.6. In fact, we shall prove the
following stronger version of (37)

$\displaystyle\Psi_{k}(x,y)\leq\sqrt{|G|}H(x,y),\qquad(x,y)\in\Omega_{1},$
(38)

where $\Psi_{k}(x,y)=\left(\sum_{\sigma\in
G}|\psi_{k}(\sigma(x),y)|^{2}\right)^{1/2}$.

If there were some $(x^{0},y^{0})\in\Omega_{1}$ so that
$\Psi_{k}(x^{0},y^{0})>\sqrt{|G|}H(x^{0},y^{0})$, we consider the set

$\Omega_{0}=\\{(x,y)\in\Omega_{1}:\,\Psi_{k}(x,y)-\sqrt{|G|}H(x,y)>\epsilon_{0}\\},$

where
$\epsilon_{0}=\frac{1}{2}(\Psi_{k}(x^{0},y^{0})-\sqrt{|G|}H(x^{0},y^{0}))$. It
follows that $\Omega_{0}$ is non-empty, open, and also $G$-invariant, since
both $\Psi_{k}$ and $H$ are $G$-invariant. We assert that

$\displaystyle\partial\Omega_{0}\bigcap\partial\Omega_{1}\neq\emptyset.$

Otherwise, $\overline{\Omega_{0}}\subset\Omega_{1}$ and $\tilde{\Psi}_{k}=0$
on $\partial\Omega_{0}$, where
$\tilde{\Psi}_{k}=\Psi_{k}-\sqrt{|G|}H-\epsilon_{0}$. It follows that
$\tilde{\Psi}_{k}\in C^{2}(\overline{\Omega_{0}})$ and
$\Delta_{\kappa}\tilde{\Psi}_{k}\geq 0$ in $\Omega_{0}$ by Proposition 4.5 and
Lemma 4.6. Lemma 4.4 (the maximum principle) yields that $\tilde{\Psi}_{k}\leq
0$ on the whole $\Omega_{0}$, which contradicts the definition of
$\Omega_{0}$.

For $(x^{*},y^{*})\in\partial\Omega_{0}\bigcap\partial\Omega_{1}$, there
exists a sequence of points
$\left\\{(\hat{x}_{\ell},\hat{y}_{\ell})\right\\}\subset\Omega_{0}$ converging
to $(x^{*},y^{*})$, so that

$\displaystyle\sqrt{|G|}H(\hat{x}_{\ell},\hat{y}_{\ell})+\epsilon_{0}<\Psi_{k}(\hat{x}_{\ell},\hat{y}_{\ell}),\qquad\ell=1,2,\dots.$
(39)

If $y^{*}>0$, letting $\ell\rightarrow\infty$ gives
$\sqrt{|G|}H(x^{*},y^{*})+\epsilon_{0}\leq\Psi_{k}(x^{*},y^{*})$; but by
Proposition 2.8 and (35),
$|\psi_{k}(\sigma(x^{*}),y^{*})|\leq|u(\sigma(x^{*}),y^{*}+y_{k})|+|(P\varphi_{k})(\sigma(x^{*}),y^{*})|\leq
2$, which implies $\Psi_{k}(x^{*},y^{*})\leq 2\sqrt{|G|}$ and leads to a
contradiction with the fact $H(x^{*},y^{*})\geq 2$ by Lemma 4.6. Hence
$y^{*}=0$, i.e.
$\lim_{\ell\rightarrow\infty}(\hat{x}_{\ell},\hat{y}_{\ell})=(x^{*},0)$, and
further $\sigma(x^{*})\in E$ for all $\sigma\in G$.

Since $\varphi_{k}(x)=u(x,y_{k})$ for $|x-x^{*}|<ay_{k}$, $\varphi_{k}$ is
continuous at $x^{*}$. For given $\epsilon>0$, take $\delta_{1}>0$ so that
$|\varphi_{k}(t)-\varphi_{k}(x^{*})|<\epsilon$ whenever
$|t-x^{*}|\leq\delta_{1}$, and then, choose $N_{1}\in{\mathbb{N}}$ such that
$|\hat{x}_{\ell}-x^{*}|\leq\delta_{1}/2$ for $\ell\geq N_{1}$. Now from (18),
for $\ell\geq N_{1}$ we have

$\displaystyle\left|(P\varphi_{k})(\hat{x}_{\ell},\hat{y}_{\ell})-\varphi_{k}(x^{*})\right|$
$\displaystyle\leq
c_{\kappa}\int_{{\mathbb{R}}^{d}}\left|\varphi_{k}(t)-\varphi_{k}(x^{*})\right|\left(\tau_{\hat{x}_{\ell}}P_{\hat{y}_{\ell}}\right)(-t)\,d\omega_{\kappa}(t)$
$\displaystyle\leq\epsilon+2c_{\kappa}\int_{|t-\hat{x}_{\ell}|>\delta_{1}/2}\left(\tau_{\hat{x}_{\ell}}P_{\hat{y}_{\ell}}\right)(-t)\,d\omega_{\kappa}(t).$

By [1, Proposition 5.2], there exists $\delta_{2}>0$ such that for
$0<\hat{y}_{\ell}<\delta_{2}$, the last integral above is controlled by
$\epsilon/2c_{\kappa}$. Now choose $N\geq N_{1}$ so that
$0<\hat{y}_{\ell}<\delta_{2}$ for $\ell>N$. It follows that for $\ell>N$,
$\left|(P\varphi_{k})(\hat{x}_{\ell},\hat{y}_{\ell})-\varphi_{k}(x^{*})\right|\leq
2\epsilon$. This shows that $(P\varphi_{k})(\hat{x}_{\ell},\hat{y}_{\ell})$
tends to $\varphi_{k}(x^{*})$ as $\ell\rightarrow\infty$. (Note that we could
not refer to [1, Corollary 5.3] directly because the function $\varphi_{k}$ is
not continuous on ${\mathbb{R}}^{d}$.) Therefore
$\lim_{\ell\rightarrow\infty}\psi_{k}(\hat{x}_{\ell},\hat{y}_{\ell})=u(x^{*},y_{k})-\varphi_{k}(x^{*})=0$,
and similarly
$\lim_{\ell\rightarrow\infty}\psi_{k}(\sigma(\hat{x}_{\ell}),\hat{y}_{\ell})=0$
for all $\sigma\in G$. Thus
$\lim_{\ell\rightarrow\infty}\Psi_{k}(\hat{x}_{\ell},\hat{y}_{\ell})=0$, and
from (39) we have

$\limsup_{\ell\rightarrow\infty}\sqrt{|G|}H(\hat{x}_{\ell},\hat{y}_{\ell})\leq-\epsilon_{0},$

which contradicts with nonnegativity of $H$ by Lemma 4.6. The claim (38), and
so (37), are proved.

Finally, taking $k=k_{j}$ in (37) and letting $j\rightarrow\infty$ yields
$|\psi(x,y)|\leq\sqrt{|G|}H(x,y)$ for $(x,y)\in\Omega_{1}$, and by Lemma 4.6,
for almost every $x\in E$, $\psi(t,y)$ tends to zero as
$(t,y)\in\Gamma_{a}(x)$ approaching to $(x,0)$. Further by Proposition 2.9,
$(P\varphi)(x,y)$ has a finite non-tangential limit for almost every $x\in E$,
and hence, from (36), $u(x,y)=\psi(x,y)+(P\varphi)(x,y)$ has the desired
assertion in the lemma. $\square$

Now we turn to the proof of Theorem 4.1.

We assume that the set $E$ is bounded, without loss of generality. By Lemma
4.2, for each $k\in{\mathbb{N}}$, there exists a compact set $E_{k}\subset E$,
such that $|E\setminus E_{k}|_{0}<1/k$, and for any $a>0$, there is a constant
$c_{a,k}>0$, so that $|u(x,y)|\leq c_{a,k}$, $(x,y)\in\cup_{x^{0}\in
E_{k}}\Gamma^{2}_{a}(x^{0})$. If we put $E_{0}=\cup_{k=1}^{\infty}E_{k}$, then
$|E\setminus E_{0}|_{0}=0$. Since $E$ is $G$-invariant, we may choose each
$E_{k}$ preserving this property; and otherwise, $\cap_{\sigma\in G}(\sigma
E_{k})$ could be used instead of $E_{k}$ and
$\left|E\setminus\left[\cap_{\sigma\in G}(\sigma
E_{k})\right]\right|_{0}<|G|/k$. Thus applying Lemma 4.7 to $u/c_{a,k}$, it
follows that, for almost every $x\in E_{k}$, $u(t,y)/c_{a,k}$ has a finite
limit as $(t,y)\in\Gamma_{a}(x)$ approaching to $(x,0)$, and hence, $u$ has a
finite non-tangential limit at $(x,0)$ for almost every $x\in E$. The proof is
completed. $\square$

##  5 The proof of Theorem 1.3

In order to prove Theorem 1.3 (and also Theorems 1.4 and 1.6 in the next
section), we introduce a variant of the area integral $S_{a,h}u$, that is,

$\displaystyle(S^{\psi}_{a,h}u)(x)=\left(\iint_{{\mathbb{R}}^{d}\times(0,h]}\left[\tau_{x}(\Delta_{\kappa}u^{2})\right](t,y)\,\psi\left(\frac{t}{ay}\right)y^{1-d-2|\kappa|}\,d\omega_{\kappa}(t)dy\right)^{1/2},$
(40)

where $\psi\in C^{\infty}({\mathbb{R}}^{d})$ is radial and satisfies
$0\leq\psi(x)\leq 1$,

$\displaystyle\psi(x)=1\quad\hbox{for}\,\,|x|\leq
1/2,\quad\hbox{and}\quad\psi(x)=0\quad\hbox{for}\,\,|x|\geq 1.$

The use of $\psi$ is to ensure that various operations on the generalized
translation $\tau_{x}$ are more legitimate. Otherwise, we will often face the
operation of this operator on some discontinuous functions such as the
characteristic functions of balls, which will bring confusion or complexity in
some cases.

We shall prove the following theorem, which implies Theorem 1.3 by Proposition
5.3 below.

###### Theorem 5.1

Suppose that $E$ is a $G$-invariant measurable subset of
$\partial{\mathbb{R}}^{d+1}_{+}={\mathbb{R}}^{d}$ and $u$ is $\kappa$-harmonic
in ${\mathbb{R}}^{d+1}_{+}$. If $u$ is non-tangentially bounded at $(x,0)$ for
every $x\in E$, then for $a,h>0$, the area integral $(S^{\psi}_{a,h}u)(x)$ is
finite for almost every $x\in E$

Since the claim $(S_{a,h}u)(x)<\infty$ in Theorem 1.3 allows the truncated
cones to be different at different points, the above theorem is a stronger
version then Theorem 1.3.

###### Lemma 5.2

If $u$ is $\kappa$-harmonic in ${\mathbb{R}}^{d+1}_{+}$, then

$\displaystyle\left(\Delta_{\kappa}u^{2}\right)(x,y)=2\left|\nabla
u(x,y)\right|^{2}+2\sum_{\alpha\in
R_{+}}\kappa(\alpha)\left(\frac{u(x,y)-u(\sigma_{\alpha}(x),y)}{\langle\alpha,x\rangle}\right)^{2},$
(41)

where $\nabla=\nabla^{(x,y)}$ is the $(d+1)$-dimensional gradient.

The lemma is known in some literature. Indeed, since $u$ is $\kappa$-harmonic,
from (2) and (3) direct calculations show that

$\displaystyle\Delta u^{2}=2\left|\nabla^{(x,y)}u\right|^{2}-4u\sum_{\alpha\in
R_{+}}\kappa(\alpha)\delta_{\alpha}u,$

and

$\displaystyle 2\sum_{\alpha\in
R_{+}}\kappa(\alpha)\delta_{\alpha}u^{2}=2\sum_{\alpha\in
R_{+}}\kappa(\alpha)\left(2u\frac{\langle\nabla^{(x)}u,\alpha\rangle}{\langle\alpha,x\rangle}-\frac{u^{2}-\sigma_{\alpha}u^{2}}{\langle\alpha,x\rangle^{2}}\right).$

The equality (41) follows by summing the above two equalities.

###### Proposition 5.3

If $u$ is $\kappa$-harmonic in ${\mathbb{R}}^{d+1}_{+}$, then the area
integrals $S_{a,h}u$ and $S^{\psi}_{a,h}u$ are well defined, and for $a,h>0$
and $x\in{\mathbb{R}}^{d}$, we have

$\displaystyle\left(S^{\psi}_{a,h}u\right)(x)\leq\left(S_{a,h}u\right)(x)\leq\left(S^{\psi}_{2a,h}u\right)(x)$
(42)

and

$\displaystyle(S^{\psi}_{a,h}u)(x)=\left(\iint_{{\mathbb{R}}^{d}\times(0,h]}\left(\Delta_{\kappa}u^{2}\right)(t,y)\,\left(\tau_{-\frac{x}{ay}}\psi\right)\left(\frac{t}{ay}\right)\frac{d\omega_{\kappa}(t)dy}{y^{2|\kappa|+d-1}}\right)^{1/2}.$
(43)

  * Proof. Since $u$ is $\kappa$-harmonic in ${\mathbb{R}}^{d+1}_{+}$, by Proposition 3.9 it is infinitely differentiable in ${\mathbb{R}}^{d+1}_{+}$, so that for fixed $x\in{\mathbb{R}}^{d}$, $\tau_{x}(\Delta_{\kappa}u^{2})\in C^{\infty}({\mathbb{R}}^{d+1}_{+})$ by Proposition 2.3(i). We write the integral in (40) as

$\displaystyle\lim_{\delta\rightarrow
0+}\iint_{{\mathbb{R}}^{d}\times(\delta,h]}\left[\tau_{x}(\Delta_{\kappa}u^{2})\right](t,y)\,\psi\left(\frac{t}{ay}\right)\frac{d\omega_{\kappa}(t)dy}{y^{2|\kappa|+d-1}},$
(44)

and obviously, it can be further written as

$\displaystyle\lim_{\delta\rightarrow
0+}\int_{\delta}^{h}\int_{0}^{\infty}M_{(\Delta_{\kappa}u^{2})(\cdot,y)}(x,r)\psi_{0}\left(\frac{|t|}{ay}\right)\left(\frac{r}{y}\right)^{2|\kappa|+d-1}\,drdy,$

where $\psi(x)=\psi_{0}(|x|)$ for some $\psi_{0}\in C^{\infty}({\mathbb{R}})$.
Similarly the integral in (4) could be written into

$\displaystyle\lim_{\delta\rightarrow
0+}\int_{\delta}^{h}\int_{0}^{ay}M_{(\Delta_{\kappa}u^{2})(\cdot,y)}(x,r)\left(\frac{r}{y}\right)^{2|\kappa|+d-1}\,drdy.$

The last two expressions imply that the integrals in (4) and (40) both are
nonnegative since the generalized spherical mean $M_{f}$ is positivity-
preserving, and hence both $S_{a,h}u$ and $S^{\psi}_{a,h}u$ are well defined.
The inequalities in (42) are immediate since
$\psi_{0}(\cdot/ay)\leq\chi_{(0,ay)}\leq\psi_{0}(\cdot/2ay)$, and (43) follows
from (44) by Propositions 2.4(ii) and 3.1. $\square$

###### Lemma 5.4

Let $E$ be a $G$-invariant compact subset of
$\partial{\mathbb{R}}^{d+1}_{+}={\mathbb{R}}^{d}$ and $u$ a $\kappa$-harmonic
function in ${\mathbb{R}}^{d+1}_{+}$. Then for $a,h>0$,

$\displaystyle\int_{E}(S^{\psi}_{a,h}u)^{2}(x)\,d\omega_{\kappa}(x)\leq
c\iint_{\Omega}y(\Delta_{\kappa}u^{2})(t,y)\,d\omega_{\kappa}(t)dy$ (45)

whenever the right hand side above is finite, where $\Omega=\Omega^{E}(a,h)$
and $c=a^{2|\kappa|+d}\|\psi\|_{L_{\kappa}^{1}({\mathbb{R}}^{d})}$.

  * Proof. If for $x\in E$ and $y\in(0,h]$, $(t,y)\notin\cup_{\sigma\in G}\Gamma_{a}^{h}(\sigma(x))$, then

$|t-\sigma(x)|\geq ay\qquad\hbox{for all}\,\,\,\sigma\in G,$

and so, for $|\xi|<1$,

$|\xi|<\min_{\sigma\in
G}\left|\frac{t}{ay}-\sigma\left(\frac{x}{ay}\right)\right|.$

Thus by (12) and (13) it follows that

$\displaystyle\left(\tau_{-\frac{x}{ay}}\psi\right)\left(\frac{t}{ay}\right)=0\qquad\hbox{for}\,\,\,(t,y)\notin\cup_{\sigma\in
G}\Gamma_{a}^{h}(\sigma(x)),$ (46)

and then, from (43) we have

$\displaystyle\int_{E}(S^{\psi}_{a,h}u)^{2}(x)d\omega_{\kappa}(x)=\iint_{\Omega}(\Delta_{\kappa}u^{2})(t,y)\,k_{1}(t,y)\,\frac{d\omega_{\kappa}(t)dy}{y^{2|\kappa|+d-1}},$
(47)

where

$\displaystyle
k_{1}(t,y)=\int_{E}\left(\tau_{-\frac{x}{ay}}\psi\right)\left(\frac{t}{ay}\right)d\omega_{\kappa}(x).$

But by Propositions 2.3(iv) and 2.5(iii),

$k_{1}(t,y)\leq\int_{{\mathbb{R}}^{d}}\left(\tau_{\frac{t}{ay}}\psi\right)\left(-\frac{x}{ay}\right)d\omega_{\kappa}(x)=(ay)^{2|\kappa|+d}\|\psi\|_{L_{\kappa}^{1}({\mathbb{R}}^{d})}.$

Inserting this into (47) yields (45). $\square$

###### Lemma 5.5

Let $E$ be a $G$-invariant compact subset of
$\partial{\mathbb{R}}^{d+1}_{+}={\mathbb{R}}^{d}$, and for $a,h>0$, put
$\Omega=\Omega^{E}(a,h)$. Then there exists a family of $G$-invariant regions
$\\{\Omega_{\epsilon}\\}_{\epsilon\in(0,h/3)}$, with the following properties:

(i) $\overline{\Omega_{\epsilon}}\subset\Omega$, and
$\Omega_{\epsilon_{1}}\subset\Omega_{\epsilon_{2}}$ if
$\epsilon_{2}<\epsilon_{1}$;

(ii) $\Omega_{\epsilon}\rightarrow\Omega$ as $\epsilon\rightarrow 0+$ (i.e.
$\cup\Omega_{\epsilon}=\Omega)$;

(iii) the boundary $\partial\Omega_{\epsilon}$ is the union of two parts,
$\partial\Omega_{\epsilon}={\mathcal{C}}_{\epsilon}^{1}\cup{\mathcal{C}}_{\epsilon}^{2}$,
so that ${\mathcal{C}}_{\epsilon}^{2}$ is a portion of the hyperplane
$y=h-\epsilon$; and

(iv) ${\mathcal{C}}_{\epsilon}^{1}$ is a portion of the hypersurface
$y=a^{-1}\delta_{\epsilon}(x)$ where $\delta_{\epsilon}\in
C^{\infty}({\mathbb{R}}^{d})$, and $|\partial_{j}\delta_{\epsilon}(x)|\leq 1$,
$j=1,\dots,d$.

The lemma is an analog of [53, p. 206, Lemma 2.2.1], and we only need to check
the $G$-invariance of $\Omega_{\epsilon}$. As in [53, pp. 206-7], set
$\delta(x)={\mathrm{d}ist}(x,E)$. Then $\delta$ is a Lipschitz function, and
since $E$ is $G$-invariant, so is $\delta$. We choose a nonnegative radial
$\varphi\in C^{\infty}({\mathbb{R}}^{d})$, satisfying ${\rm
supp}\,\varphi\subset\overline{{\mathbb{B}}(0,1)}$ and
$\int_{{\mathbb{R}}^{d}}\phi(x)dx=1$, and for $\epsilon\in(0,h/3)$, define
$\delta_{\epsilon}(x)=\left(\delta\ast\varphi_{a\epsilon}\right)(x)+2a\epsilon$
and

$\Omega_{\epsilon}=\left\\{(x,y):\,\delta_{\epsilon}(x)<ay,\,0<y<h-\epsilon\right\\}.$

Obviously $\delta_{\epsilon}$ is $G$-invariant and so $\Omega_{\epsilon}$ is
$G$-invariant. Since $\delta(x)<\delta_{\epsilon}(x)$,
$\delta_{\epsilon_{2}}(x)<\delta_{\epsilon_{1}}(x)$ for
$\epsilon_{2}<\epsilon_{1}$, and $\delta_{\epsilon}(x)$ tends to $\delta(x)$
as $\epsilon\rightarrow 0+$ uniformly, these $\Omega_{\epsilon}$’s satisfy the
requirements (i)-(iv).

###### Lemma 5.6

Let $b,\eta>0$ be given and $u$ a $\kappa$-harmonic function in
$\cup_{\sigma\in G}\Gamma_{b}^{\eta}(\sigma(x^{0}))$. If $|u|\leq 1$ in
$\cup_{\sigma\in G}\Gamma_{b}^{\eta}(\sigma(x^{0}))$, then for fixed
$a\in(0,b)$ and $h\in(0,\eta)$, there exists a constant $c=c(a,b,h,\eta)>0$,
so that $y|\nabla u|\leq c$ in $\Gamma_{a}^{h}(x^{0})$.

  * Proof. We apply Proposition 3.5 to the case when $\Omega=\cup_{\sigma\in G}\Gamma_{b}^{\eta}(\sigma(x^{0}))\subset{\mathbb{R}}^{d+1}_{+}$, $\tilde{G}=G\otimes{\mathbb{Z}}_{2}$ with the multiplicity function $\tilde{\kappa}=(\kappa,0)$. In this case, for $(x,y)\in\Omega$, the associated generalized translation $\tilde{\tau}_{(x,y)}$ is given by $(\tilde{\tau}_{(x,y)}u)(t,s)=(\tau_{x}u)(t,y+s)$, where $\tau_{x}$ acts on the first argument as before. For $u$ is $\kappa$-harmonic in $\Omega$, one has $u\in C^{\infty}(\Omega)$ and

$\displaystyle
u(x,y)=\tilde{M}_{u}((x,y),r)\quad\hbox{with}\,\,\,\tilde{M}_{u}((x,y),r)=\tilde{d}_{\kappa}\int_{{\mathbb{S}}^{d}}(\tilde{\tau}_{(x,y)}u)(r(t^{\prime},s^{\prime}))W_{\kappa}(t^{\prime})d(t^{\prime},s^{\prime})$
(48)

provided $\overline{{\mathbb{B}}((x,y),r)}\subset\Omega$ for $r>0$, where
$d(t^{\prime},s^{\prime})$ denotes the area element on ${\mathbb{S}}^{d}$, and
$\tilde{d}_{\kappa}^{-1}=\int_{{\mathbb{S}}^{d}}W_{\kappa}(t^{\prime})d(t^{\prime},s^{\prime})$.

We take a nonnegative even $\phi_{0}\in C^{\infty}({\mathbb{R}})$ with ${\rm
supp}\,\phi_{0}\subseteq[-1,1]$ and decreasing on $[0,1]$, and define
$\phi(x,y)=\phi_{0}(\sqrt{|x|^{2}+y^{2}})$ normalized so that
$\int_{{\mathbb{R}}^{d+1}}\phi(x,y)d\omega_{\kappa}(x)dy=1$, and
$\phi_{\epsilon}(x,y)=\epsilon^{-2|\kappa|-d-1}\phi(\epsilon^{-1}x,\epsilon^{-1}y)$
for $\epsilon>0$. Thus for $(x,y)\in\Omega$, from (48) it follows that

$\displaystyle
u(x,y)=\int_{{\mathbb{R}}^{d+1}}\left(\tilde{\tau}_{(x,y)}u\right)(t,s)\phi_{\epsilon}(t,s)\,d\omega_{\kappa}(t)ds$
(49)

for $\epsilon>0$ so that
$\overline{{\mathbb{B}}((x,y),\epsilon)}\subset\Omega$, and moreover, by
Proposition 3.1(i),

$\displaystyle
u(x,y)=\int_{{\mathbb{R}}^{d+1}}u(t,s)\left(\tilde{\tau}_{(-x,-y)}\phi_{\epsilon}\right)(t,s)\,d\omega_{\kappa}(t)ds.$
(50)

Note that by (16), (17), and in view of (48), the two integrals in (49) and
(50) only use those values of $u$ in the set

$\displaystyle\bigcup_{\sigma\in
G}\\{(t,s)\in{\mathbb{R}}^{d+1}:\,\,|(t,s)-(\sigma(x),y)|\leq\epsilon\\}.$
(51)

Now for fixed $a\in(0,b)$ and $h\in(0,\eta)$ and for
$(x,y)\in\Gamma_{a}^{h}(x^{0})$, choose $\epsilon=c_{1}y$ for suitable
$c_{1}>0$ depending only on $a,b,h$ and $\eta$, such that
$\overline{{\mathbb{B}}((x,y),\epsilon)}\subset\Gamma_{b}^{\eta}(x^{0})$. Thus
the set given in (51) is contained in $\Omega$, and for $j=1,2,\dots,d$, by
the assumption we have

$\displaystyle|(\partial_{x_{j}}u)(x,y)|\leq\int_{{\mathbb{R}}^{d+1}}\left|\partial_{x_{j}}\left[\left(\tilde{\tau}_{(-x,-y)}\phi_{\epsilon}\right)(t,s)\right]\right|d\omega_{\kappa}(t)ds.$
(52)

From Proposition 2.3 and (14),

$\displaystyle\left(\tilde{\tau}_{(-x,-y)}\phi_{\epsilon}\right)(t,s)=\int_{{\mathbb{R}}^{d}}\phi_{0,\epsilon}\left(A(x,y,t,s,\xi)\right)\,d\mu^{\kappa}_{t}(\xi),$
(53)

where $\phi_{0,\epsilon}(r)=\epsilon^{-2|\kappa|-d-1}\phi_{0}(\epsilon^{-1}r)$
and

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

and hence

$\displaystyle\partial_{x_{j}}\left[\left(\tilde{\tau}_{(-x,-y)}\phi_{\epsilon}\right)(t,s)\right]=\frac{1}{\epsilon^{2|\kappa|+d+2}}\int_{{\mathbb{R}}^{d}}\frac{\phi^{\prime}_{0}\left(\epsilon^{-1}A(x,y,t,s,\xi)\right)}{A(x,y,t,s,\xi)}(x_{j}-\xi_{j})\,d\mu^{\kappa}_{t}(\xi),$

But since $A(x,y,t,s,\xi)\geq\sqrt{|t|^{2}-|\xi|^{2}+|x-\xi|^{2}}\geq|x-\xi|$
by (5), one has

$\displaystyle\left|\partial_{x_{j}}\left[\left(\tilde{\tau}_{(-x,-y)}\phi_{\epsilon}\right)(t,s)\right]\right|\leq\epsilon^{-1}\left(\tilde{\tau}_{(-x,-y)}\tilde{\phi}_{\epsilon}\right)(t,s),$

where $\tilde{\phi}(x,y)=|\phi^{\prime}_{0}(\sqrt{|x|^{2}+y^{2}})|$. Inserting
this into (52) and appealing to Proposition 2.5(iii), we obtain

$\displaystyle|(\partial_{x_{j}}u)(x,y)|\leq\epsilon^{-1}\|\tilde{\phi}\|_{L^{1}({\mathbb{R}}^{d+1},d\omega_{\kappa}(t)ds)},$

and so, $y|(\partial_{x_{j}}u)(x,y)|\leq c$ for
$(x,y)\in\Gamma_{a}^{h}(x^{0})$. Similarly, $y|(\partial_{y}u)(x,y)|\leq c$
for $(x,y)\in\Gamma_{a}^{h}(x^{0})$ and the lemma is proved. $\square$

After preparing these lemmas, one may complete the proof of Theorem 5.1 along
the way of [53, pp. 208-9].

Proof of Theorem 5.1.

For given $a,h>0$, we fix $b>a$ and $\eta>h$. We may assume that $E$ is
bounded, without loss of generality. By Lemma 4.2, for each
$j\in{\mathbb{N}}$, there exists a compact set $E_{j}\subset E$, such that
$|E\setminus E_{j}|_{0}<1/j$, and there is a constant $c_{b,\eta,j}>0$, so
that $|u(x,y)|\leq c_{b,\eta,j}$, $(x,y)\in\Omega^{E_{j}}(b,\eta)$. If we put
$E_{0}=\bigcup_{j=1}^{\infty}E_{j}$, then $|E\setminus E_{0}|_{0}=0$. Since
$E$ is $G$-invariant, we may choose each $E_{j}$ preserving this property as
in the proof of Theorem 4.1. Thus, the proof of the theorem would be completed
once we prove that for a $G$-invariant compact set
$E\subset\partial{\mathbb{R}}_{+}^{d+1}={\mathbb{R}}^{d}$,

$\int_{E}(S^{\psi}_{a,h}u)^{2}(x)\,d\omega_{\kappa}(x)<\infty$

under the condition

$\displaystyle|u(x,y)|\leq 1\qquad\text{for}\,\,\,(x,y)\in\Omega^{E}(b,\eta);$
(54)

and by Lemma 5.4, it suffices to show

$\displaystyle\iint_{\Omega^{E}(a,h)}y(\Delta_{\kappa}u^{2})(t,y)\,d\omega_{\kappa}(t)dy<\infty.$

Moreover, for $\Omega=\Omega^{E}(a,h)$, there exists a family of $G$-invariant
regions $\\{\Omega_{\epsilon}\\}_{\epsilon\in(0,h/3)}$ satisfying (i)-(iv) in
Lemma 5.5, and hence, we only need to prove that, there exists some constant
$c>0$ independent of $\epsilon\in(0,h/3)$, so that

$\displaystyle\iint_{\Omega_{\epsilon}}y(\Delta_{\kappa}u^{2})(t,y)\,d\omega_{\kappa}(t)dy\leq
c.$ (55)

Taking integration over $\Omega_{\epsilon}$ instead of $\Omega$ is for
legitimate use of Green’s formula. In fact, for $U,V\in
C^{2}(\overline{\Omega_{\epsilon}})$, the Green formula (28) reads as

$\displaystyle\iint_{\Omega_{\epsilon}}(V\Delta_{\kappa}U-U\Delta_{\kappa}V)\,d\omega_{\kappa}(x)dy=\int_{\partial\Omega_{\epsilon}}\left(V\partial_{\mathbf{n}}U-U\partial_{\mathbf{n}}V\right)W_{\kappa}(x)d\sigma(x,y).$

If $U=u^{2}$ and $V=y$, then

$\displaystyle\iint_{\Omega_{\epsilon}}y(\Delta_{\kappa}u^{2})(x,y)\,d\omega_{\kappa}(x)dy=\int_{\partial\Omega_{\epsilon}}\left(y\frac{\partial
u^{2}}{\partial\mathbf{n}}-u^{2}\frac{\partial
y}{\partial\mathbf{n}}\right)W_{\kappa}(x)d\sigma(x,y).$ (56)

Since $\partial\Omega_{\epsilon}\subset\Omega^{E}(b,\eta)$, it follows from
Lemma 5.6(i) and (54) that

$\displaystyle\left|y\frac{\partial u^{2}}{\partial\mathbf{n}}\right|\leq
2y|u||\nabla u|\leq c\qquad\mathrm{on}\,\,\,\partial\Omega_{\epsilon};$

and since $\left|\partial y/\partial\mathbf{n}\right|\leq 1$, we have
$\left|u^{2}\partial y/\partial\mathbf{n}\right|\leq 1$ on
$\partial\Omega_{\epsilon}$. Applying these estimates to the right hand side
of (56) and on account of boundedness of $\Omega=\Omega^{E}(a,h)$, we get

$\displaystyle\iint_{\Omega_{\epsilon}}y(\Delta_{\kappa}u^{2})(x,y)\,d\omega_{\kappa}(x)dy\leq
c\int_{\partial\Omega_{\epsilon}}W_{\kappa}(x)d\sigma(x,y)\leq
c^{\prime}\int_{\partial\Omega_{\epsilon}}d\sigma(x,y).$ (57)

Since $E$ is bounded, it follows from [53, p. 209] that the area of
$\partial\Omega_{\epsilon}$ is bounded by a constant independent of
$\epsilon$. This proves (55) and end the proof of Theorem 5.1. $\square$

##  6 The proofs of Theorems 1.4 and 1.6

We shall prove the following analogs of Theorems 1.4 and 1.6 with
$S^{\psi}_{a,h}u$ instead of $S_{a,h}u$. Theorems 1.4 and 1.6 follow
immediately by Proposition 5.3.

###### Theorem 6.1

Suppose that $E$ is a $G$-invariant measurable subset of
$\partial{\mathbb{R}}^{d+1}_{+}={\mathbb{R}}^{d}$ and $u$ is $\kappa$-harmonic
in ${\mathbb{R}}^{d+1}_{+}$. If $u$ is $G$-invariant and for every $x\in E$,
its area integral $(S^{\psi}_{a,h}u)(x)$ is finite for some $a,h>0$, then $u$
is non-tangentially bounded at $(x,0)$ for almost every $x\in E$.

###### Theorem 6.2

Suppose $G=Z_{2}^{d}$ with a given multiplicity parameters
$\lambda=(\lambda_{1},\dots,\lambda_{d})$, $E$ is a $G$-invariant measurable
subset of ${\mathbb{R}}^{d}$, and $u$ is $\lambda$-harmonic in
${\mathbb{R}}^{d+1}_{+}$. If for every $x\in E$, the area integral
$(S^{\psi}_{a,h}u)(x)$ is finite for some $a,h>0$, then $u$ is non-
tangentially bounded at $(x,0)$ for almost every $x\in E$.

We need several lemmas.

###### Lemma 6.3

Let $E$ be a $G$-invariant, bounded and measurable set of ${\mathbb{R}}^{d}$.
Then for any $\epsilon>0$, there exists a $G$-invariant compact set
$E_{\epsilon}$ satisfying

(i) $E_{\epsilon}\subset E$, $|E-E_{\epsilon}|_{\kappa}<\epsilon$;

(ii) for $\theta\in(0,1)$, there exists some $\delta>0$, such that for $x\in
E_{\epsilon}$ and $0<r<\delta$,

$\displaystyle\left|{\mathbb{B}}(x,r)\cap
E\right|_{\kappa}>\theta|{\mathbb{B}}(x,r)|_{\kappa}.$

  * Proof. By (33), $d\omega_{\kappa}$ is a doubling measure on ${\mathbb{R}}^{d}$, and if $f$ is locally integrable with respect to $d\omega_{\kappa}$, then by [54, p. 13, Corollary], $\lim_{r\rightarrow 0}|{\mathbb{B}}(x,r)|_{\kappa}^{-1}\int_{{\mathbb{B}}(x,r)}fd\omega_{\kappa}=f(x)$ for almost every $x\in{\mathbb{R}}^{d}$. Taking $f=\chi_{E}$, it follows that

$\lim_{r\rightarrow 0+}\frac{\left|{\mathbb{B}}(x,r)\cap
E\right|_{\kappa}}{|{\mathbb{B}}(x,r)|_{\kappa}}=1\qquad\textrm{for
a.e.}\,\,x\in E.$

By Egorov’s theorem, for given $\epsilon>0$ there exists $E_{\epsilon}\subset
E$, $|E-E_{\epsilon}|_{\kappa}<\epsilon$, so that $\left|{\mathbb{B}}(x,r)\cap
E\right|_{\kappa}/|{\mathbb{B}}(x,r)|_{\kappa}$ tends to $1$ uniformly for
$x\in E_{\epsilon}$ as $r\rightarrow 0+$. Clearly one may take $E_{\epsilon}$
to be a closed subset of $E$. Since $E$ is $G$-invariant, the uniform
convergence above is true also on $\sigma E_{\epsilon}$ for each $\sigma\in
G$, and so is on $\cup_{\sigma\in G}(\sigma E_{\epsilon})$. Thus we may
further take $E_{\epsilon}$ to be $G$-invariant, as desired. $\square$

The next lemma gives an equivalent estimate of $(\tau_{-x}\phi)(t)$ for radial
$\phi\in C({\mathbb{R}}^{d})$ with $\phi(0)>0$ and for $t$ close to $x$.

###### Lemma 6.4

Assume that $\phi\in C({\mathbb{R}}^{d})$ is radial. If $\phi(0)>0$, then
there exist some $c_{1},c_{2},\delta_{0}>0$, all independent of $x$, such that

$\displaystyle\frac{c_{1}}{|{\mathbb{B}}(x,1)|_{\kappa}}\leq(\tau_{-x}\phi)(t)\leq\frac{c_{2}}{|{\mathbb{B}}(x,1)|_{\kappa}}\qquad\hbox{for}\,\,\,|t-x|<\delta_{0}.$
(58)

We note that the first inequality in (58) is a sharper form of [1, (4.15)].
Its proof depends on a distribution inequality for the representing measures
of Dunkl’s intertwining operator proved in [23].

###### Lemma 6.5

([23]) There exists a constant $c>0$ independent of $\delta>0$ and
$x\in{\mathbb{R}}^{d}$ such that

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

Proof of Lemma 6.4.

By continuity there exists some $r_{0}>0$ so that $\phi(\xi)\geq\phi(0)/2$
whenever $|\xi|<r_{0}$. It follows from Proposition 3.2 and (14) that

$\displaystyle(\tau_{-x}\phi)(t)=(\tau_{x}\phi)(-t)=\int_{{\mathbb{R}}^{d}}\phi_{0}(\sqrt{|x|^{2}+|t|^{2}-2\langle
t,\xi\rangle})\,d\mu^{\kappa}_{x}(\xi),$ (59)

where $\phi(x)=\phi_{0}(|x|)$ for some $\phi_{0}\in C({\mathbb{R}})$.

If $\langle x,\xi\rangle>|x|^{2}-r_{0}^{2}/8$ for
$\xi\in\hbox{supp}\,\mu^{\kappa}_{x}$, from (5) we have

$\displaystyle|x-\xi|^{2}<|\xi|^{2}-|x|^{2}+r_{0}^{2}/4\leq r_{0}^{2}/4,$

and so

$\displaystyle|x|^{2}+|t|^{2}-2\langle t,\xi\rangle$
$\displaystyle=|x-t|^{2}-2\langle x-\xi,x-t\rangle+2|x|^{2}-2\langle
x,\xi\rangle$ $\displaystyle\leq|x-t|^{2}+r_{0}|x-t|+r_{0}^{2}/4.$

Now we set $\delta_{0}=r_{0}/2$. It follows that
$\sqrt{|x|^{2}+|t|^{2}-2\langle t,\xi\rangle}\leq r_{0}$ whenever
$|t-x|\leq\delta_{0}$, and then, from (59),

$\displaystyle(\tau_{-x}\phi)(t)\geq\frac{\phi(0)}{2}\int_{\langle
x,\xi\rangle>|x|^{2}-r_{0}^{2}/8}\,d\mu^{\kappa}_{x}(\xi).$

By Lemma 6.5, this proves the first inequality in (58) for
$|t-x|\leq\delta_{0}$.

The second inequality in (58) follows from [16, Corollary 3.5] since
$|{\mathbb{B}}(t,1)|_{\kappa}\asymp|{\mathbb{B}}(x,1)|_{\kappa}$ when
$|t-x|<\delta_{0}$. $\square$

###### Lemma 6.6

Let $b,\eta>0$ be given and $u$ a $\kappa$-harmonic function on
$\cup_{\sigma\in G}\Gamma_{b}^{\eta}(\sigma(x^{0}))$. If
$(S^{\psi}_{b,\eta}u)(x^{0})\leq 1$, then for fixed $a\in(0,b/2)$ and
$h\in(0,\eta)$, there exists a constant $c=c(a,b,h,\eta)>0$, so that
$y|\nabla_{\kappa}u|\leq c$ in $\Gamma_{a}^{h}(x^{0})$.

  * Proof. We use the notation in the proof of Lemma 5.6. Applying (50) to $D_{j}u$ for $1\leq j\leq d$ instead of $u$ and noting that $\int_{{\mathbb{R}}^{d+1}}\left(\tilde{\tau}_{(-x,-y)}\phi_{\epsilon}\right)(t,s)d\omega_{\kappa}(t)ds=1$ by Proposition 2.5(iii), we have

$\displaystyle\left|(D_{j}u)(x,y)\right|^{2}\leq\int_{{\mathbb{R}}^{d+1}}\left|(D_{j}u)(t,s)\right|^{2}\left(\tilde{\tau}_{(-x,-y)}\phi_{\epsilon}\right)(t,s)\,d\omega_{\kappa}(t)ds$
(60)

for $\epsilon>0$ so that
$\overline{{\mathbb{B}}((x,y),\epsilon)}\subset\Omega:=\cup_{\sigma\in
G}\Gamma_{b}^{\eta}(\sigma(x^{0}))$.

For fixed $a\in(0,b/2)$ and $h\in(0,\eta)$ and for
$(x,y)\in\Gamma_{a}^{h}(x^{0})$, take $\epsilon=c_{2}y$ for suitable
$0<c_{2}<(b-2a)/(b+2)$, independent of $(x,y)$, such that
$\overline{{\mathbb{B}}((x,y),\epsilon)}\subset\Gamma_{b}^{\eta}(x^{0})\subset\Omega$.
We claim that, with the choice of $\epsilon$,

$\displaystyle\left(\tilde{\tau}_{(-x,-y)}\phi_{\epsilon}\right)(t,s)\leq\frac{cy^{-2}}{s^{2|\kappa|+d-1}}\left(\tau_{-\frac{x^{0}}{bs}}\psi\right)\left(\frac{t}{bs}\right).$
(61)

Indeed, for $\xi\in\hbox{supp}\,\mu^{\kappa}_{t}$ satisfying
$A(x,y,t,s,\xi)\leq\epsilon$, one has $(1-c_{2})y\leq
s\leq(1+c_{2})y\leq\eta$, and

$\displaystyle\sqrt{|t|^{2}+|x^{0}|^{2}-2\langle x^{0},\xi\rangle}\leq
A(x,y,t,s,\xi)+|x-x^{0}|\leq(c_{2}+a)y\leq\frac{c_{2}+a}{1-c_{2}}s\leq\frac{1}{2}bs.$

Thus for $\xi\in\hbox{supp}\,\mu^{\kappa}_{t}$,

$\displaystyle\phi_{0}\left(\frac{1}{\epsilon}A(x,y,t,s,\xi)\right)\leq
c^{\prime}\psi_{0}\left(\frac{1}{bs}\sqrt{|t|^{2}+|x^{0}|^{2}-2\langle
x^{0},\xi\rangle}\right)$

with $\psi_{0}$ such that $\psi(x)=\psi_{0}(|x|)$, and so, from (14), (53) and
Proposition 2.4(ii),

$\displaystyle\left(\tilde{\tau}_{(-x,-y)}\phi_{\epsilon}\right)(t,s)\leq\frac{c^{\prime}}{\epsilon^{2|\kappa|+d+1}}\left(\tau_{-x^{0}}\psi\left(\frac{\cdot}{bs}\right)\right)(t)\leq\frac{c^{\prime\prime}y^{-2}}{s^{2|\kappa|+d-1}}\left(\tau_{-\frac{x^{0}}{bs}}\psi\right)\left(\frac{t}{bs}\right),$

which verifies the claim (61).

Applying (61) to (60) we obtain

$\displaystyle y^{2}\left|(D_{j}u)(x,y)\right|^{2}\leq
c\int_{{\mathbb{R}}^{d}\times(0,\eta]}\left|(D_{j}u)(t,s)\right|^{2}\left(\tau_{-\frac{x^{0}}{bs}}\psi\right)\left(\frac{t}{bs}\right)\frac{d\omega_{\kappa}(t)ds}{s^{2|\kappa|+d-1}}.$
(62)

From (46),
$\left(\tau_{-\frac{x^{0}}{bs}}\psi\right)\left(\frac{t}{bs}\right)=0$ for
$(t,s)\notin\Omega$, and since $u$ is $\kappa$-harmonic in $\Omega$, (1) and
(41) imply $\left|(D_{j}u)(t,s)\right|^{2}\leq
c\left(\Delta_{\kappa}u^{2}\right)(t,s)$ for $(t,s)\in\Omega$. Thus from (62),
$y\left|(D_{j}u)(x,y)\right|\leq c(S^{\psi}_{b,\eta}u)(x^{0})\leq c$ for
$(x,y)\in\Gamma_{a}^{h}(x^{0})$, and so does
$y\left|(\partial_{y}u)(x,y)\right|$ similarly. The lemma is proved. $\square$

###### Lemma 6.7

Let $E$ be a $G$-invariant, bounded and measurable set of
$\partial{\mathbb{R}}^{d+1}_{+}={\mathbb{R}}^{d}$, $u$ a $\kappa$-harmonic
function in ${\mathbb{R}}^{d+1}_{+}$, and let $b,\eta>0$ be given. Then for
any $\epsilon>0$, there exists a $G$-invariant compact set $E_{\epsilon}$
satisfying the following conditions:

(i) $E_{\epsilon}\subset E$, $|E-E_{\epsilon}|_{\kappa}<\epsilon$;

(ii) There are some $a\in(0,b)$, $h\in(0,\eta)$, and
$c=c(\epsilon,a,b,h,\eta)>0$, such that

$\displaystyle\iint_{\Omega^{E_{\epsilon}}(a,h)}y(\Delta_{\kappa}u^{2})(x,y)\,d\omega_{\kappa}(x)dy\leq
c\int_{E}(S^{\psi}_{b,\eta}u)^{2}(x)\,d\omega_{\kappa}(x),$ (63)

whenever the right hand side above is finite.

  * Proof. First applying Lemma 6.4 to $\psi$, there exist some $c_{0}>0$, $\delta_{0}\in(0,1)$, both independent of $x$, such that

$\displaystyle(\tau_{-x}\psi)(t)\geq\frac{c_{0}}{|{\mathbb{B}}(x,1)|_{\kappa}}\qquad\hbox{for}\,\,|t-x|<\delta_{0}.$
(64)

For given $\epsilon>0$, by Lemma 6.3 there there exist a $G$-invariant compact
subset $E_{\epsilon}$ of $E$ and a number $\delta_{1}>0$, satisfying
$|E-E_{\epsilon}|_{\kappa}<\epsilon$ and

$\displaystyle\left|{\mathbb{B}}(x,r)\cap
E\right|_{\kappa}>\frac{1}{2}|{\mathbb{B}}(x,r)|_{\kappa}\qquad\hbox{for}\,\,x\in
E_{\epsilon}\,\,\hbox{and}\,\,r\in(0,\delta_{1}).$ (65)

Now we fix $a<\delta_{0}b$ and $h<\eta$. Since $(t,y)\in\cup_{\sigma\in
G}\Gamma_{b}^{\eta}(\sigma(x))$ is equivalent to $(x,y)\in\cup_{\sigma\in
G}\Gamma_{b}^{\eta}(\sigma(t))$, from (43) and (46) and by Propositions
2.4(ii), we have

$\displaystyle\int_{E}(S^{\psi}_{b,\eta}u)^{2}(x)\,d\omega_{\kappa}(x)$
$\displaystyle=\int_{E}\iint_{{\mathbb{R}}^{d}\times(0,\eta]}\left(\Delta_{\kappa}u^{2}\right)(t,y)\,\left(\tau_{-\frac{x}{by}}\psi\right)\left(\frac{t}{by}\right)\frac{d\omega_{\kappa}(t)dy}{y^{2|\kappa|+d-1}}\,d\omega_{\kappa}(x)$
$\displaystyle\geq\iint_{\Omega^{E_{\epsilon}}(a,h)}(\Delta_{\kappa}u^{2})(t,y)\,k_{2}(t,y)\,\frac{d\omega_{\kappa}(t)dy}{y^{2|\kappa|+d-1}},$
(66)

where

$\displaystyle
k_{2}(t,y)=\int_{E}\left(\tau_{-\frac{x}{by}}\psi\right)\left(\frac{t}{by}\right)\chi_{\cup_{\sigma\in
G}\Gamma_{b}^{\eta}(\sigma(t))}(x,y)d\omega_{\kappa}(x).$ (67)

For given $(t,y)\in\Omega^{E_{\epsilon}}(a,h)$, there exists some
${\bar{x}}\in E_{\epsilon}$ such that $(t,y)\in\Gamma_{a}^{h}({\bar{x}})$.
Thus, when

$\displaystyle|x-\bar{x}|<\gamma
y,\qquad\gamma=\min\\{(\delta_{0}b-a)/2,\delta_{1}/h\\},$

we have $|x-t|<a^{\prime}y$ with $a^{\prime}=(a+\delta_{0}b)/2$, which
certainly implies that $(x,y)\in\Gamma_{b}^{\eta}(t)$.

Since for $(x,y)\in\Gamma_{a^{\prime}}^{h}(t)$,
$|\frac{t}{by}-\frac{x}{by}|<a^{\prime}y/by<\delta_{0}$, by (33) and (64) we
have

$\displaystyle\left(\tau_{-\frac{x}{by}}\psi\right)\left(\frac{t}{by}\right)\geq\frac{c\,y^{2|\kappa|+d}}{\left|{\mathbb{B}}(x,by)\right|_{\kappa}}\geq\frac{c\,y^{2|\kappa|+d}}{\left|{\mathbb{B}}(\bar{x},\gamma
y)\right|_{\kappa}}$ (68)

whenever $|x-\bar{x}|<\gamma y$. Thus from (67) and (68) it follows that

$\displaystyle k_{2}(t,y)$ $\displaystyle\geq\int_{{\mathbb{B}}(\bar{x},\gamma
y)\cap
E}\left(\tau_{-\frac{x}{by}}\psi\right)\left(\frac{t}{by}\right)d\omega_{\kappa}(x)$
$\displaystyle\geq\frac{c\,y^{2|\kappa|+d}}{\left|{\mathbb{B}}(\bar{x},\gamma
y)\right|_{\kappa}}\left|{\mathbb{B}}(\bar{x},\gamma y)\cap
E\right|_{\kappa}.$

Since for $(t,y)\in\Omega^{E_{\epsilon}}(a,h)$, $\gamma y<\gamma
h\leq\delta_{1}$, we use (65) to get $k_{2}(t,y)\geq(c/2)y^{2|\kappa|+d}$.
Inserting this into ( ‣ 6) proves (63) immediately. The proof of the lemma is
completed. $\square$

Proof of Theorem 6.1.

Again, we assume that $E$ is bounded without loss of generality. By the
assumption, $E=\bigcup E_{j}$, where $E_{j}=\cap_{\sigma\in G}\left\\{x\in
E:\,(S^{\psi}_{j^{-1},j^{-1}}u)(\sigma(x))\leq j\right\\}$. Note that for each
$j$, $E_{j}$ is $G$-invariant, and by Lemma 6.7, there exists a sequence of
$G$-invariant compact subsets $E_{j,k}$ ($k=1,2,\cdots$) of $E_{j}$ such that
$|E_{j}\setminus E_{j,k}|_{\lambda}<1/k$, and there are some $a\in(0,1/j)$,
$h\in(0,1/j)$, and $c=c(a,h,j,k)>0$, such that

$\displaystyle\iint_{\Omega^{E_{j,k}}(a,h)}y(\Delta_{\kappa}u^{2})(x,y)\,d\omega_{\kappa}(x)dy\leq
c\int_{E_{j}}(S^{\psi}_{j^{-1},j^{-1}}u)^{2}(x)\,d\omega_{\kappa}(x).$

For $(S^{\psi}_{j^{-1},j^{-1}}u)(x)\leq j$ ($x\in E_{j}$) and $E_{j}$ is a
bounded set, the left hand side above is finite; and further, by Lemma 6.6
(with $0<a<1/2j$) there exists some $c_{1}=c_{1}(a,h,j,k)>0$ such that
$y|\nabla_{\kappa}u(x,y)|\leq c_{1}$ for $(x,y)\in\Omega^{E_{j}}(a,h)$. Since
$u$ is $G$-invariant, this is identical to $y|\nabla u(x,y)|\leq c_{1}$ for
$(x,y)\in\Omega^{E_{j}}(a,h)$.

Since $|E_{j}\setminus\bigcup_{k=1}^{\infty}E_{j,k}|_{\kappa}=0$, the proof of
the theorem would be completed once we prove that, for a $G$-invariant compact
set $E\subset\partial{\mathbb{R}}^{d+1}_{+}={\mathbb{R}}^{d}$, $u$ is non-
tangentially bounded at $(x,0)$ for almost every $x\in E$ under the conditions

$\displaystyle\iint_{\Omega^{E}(a,h)}y(\Delta_{\kappa}u^{2})(x,y)\,d\omega_{\kappa}(x)dy\leq
1$ (69)

for some $a,h>0$, and

$\displaystyle y|\nabla u(x,y)|\leq
1\qquad{\mathrm{f}or}\,\,\,(x,y)\in\Omega^{E}(a,h).$ (70)

Note that the conditions (69) and (70) come from the normalization of the
conclusions deduced in the previous paragraph.

We fix $a_{1}\in(0,a)$ and $h_{1}\in(0,h)$, and work with $\Omega_{\epsilon}$
($\epsilon\in(0,h_{1}/3)$) given by Lemma 5.5 associated with
$\Omega=\Omega^{E}(a_{1},h_{1})$. By Green’s formula (56), (69) implies

$\displaystyle\int_{\partial\Omega_{\epsilon}}\left(y\frac{\partial
u^{2}}{\partial\mathbf{n}}-u^{2}\frac{\partial
y}{\partial\mathbf{n}}\right)W_{\kappa}(x)d\sigma(x,y)\leq 1.$ (71)

From Lemma 5.5, the boundary $\partial\Omega_{\epsilon}$ of
$\Omega_{\epsilon}$ consists of two parts,
$\partial\Omega_{\epsilon}={\mathcal{C}}_{\epsilon}^{1}\cup{\mathcal{C}}_{\epsilon}^{2}$,
where ${\mathcal{C}}_{\epsilon}^{1}$ is a portion of the smooth hypersurface
$y=a_{1}^{-1}\delta_{\epsilon}(x)$ with
$|\partial_{j}\delta_{\epsilon}(x)|\leq 1$, $j=1,\dots,d$, and
${\mathcal{C}}_{\epsilon}^{2}$ is a portion of the hyperplane
$y=h_{1}-\epsilon$.

We extract the term $\int_{{\mathcal{C}}_{\epsilon}^{1}}u^{2}\frac{\partial
y}{\partial\mathbf{n}}W_{\kappa}(x)d\sigma(x,y)$ from (71), so that

$\displaystyle-\int_{{\mathcal{C}}_{\epsilon}^{1}}u^{2}\frac{\partial
y}{\partial\mathbf{n}}W_{\kappa}(x)d\sigma(x,y)\leq\int_{{\mathcal{C}}_{\epsilon}^{2}}u^{2}\frac{\partial
y}{\partial\mathbf{n}}W_{\kappa}(x)d\sigma(x,y)-\int_{\partial\Omega_{\epsilon}}y\frac{\partial
u^{2}}{\partial\mathbf{n}}W_{\kappa}(x)d\sigma(x,y)+1.$

Since $E$ is bounded and $2h_{1}/3\leq y\leq h_{1}$ for
$(x,y)\in{\mathcal{C}}_{\epsilon}^{2}$, the first term on the right hand side
above is bounded by a constant independent of $\epsilon$, and so is the
contribution coming from $(x,y)\in{\mathcal{C}}_{\epsilon}^{2}$ in the second
term. Thus we get

$\displaystyle-\int_{{\mathcal{C}}_{\epsilon}^{1}}u^{2}\frac{\partial
y}{\partial\mathbf{n}}W_{\kappa}(x)d\sigma(x,y)\leq-\int_{{\mathcal{C}}_{\epsilon}^{1}}y\frac{\partial
u^{2}}{\partial\mathbf{n}}W_{\kappa}(x)d\sigma(x,y)+c.$ (72)

Since the hypersurface ${\mathcal{C}}_{\epsilon}^{1}$ is determined by the
equation $\delta_{\epsilon}(x)-a_{1}y=0$ and so the direction ${\mathbf{n}}$
is given by
$(\nabla^{(x)}\delta_{\epsilon},-a_{1})/\sqrt{|\nabla^{(x)}\delta_{\epsilon}|^{2}+a_{1}^{2}}$,
we have $\partial
y/\partial{\mathbf{n}}=-a_{1}/\sqrt{|\nabla^{(x)}\delta_{\epsilon}|^{2}+a_{1}^{2}}\leq-
a_{1}/\sqrt{d+a_{1}^{2}}$; and since
$\Omega_{\epsilon}\subset\Omega^{E}(a_{1},h_{1})$ is at a positive distance
from the $x$-space ${\mathbb{R}}^{d}$, (70) implies that $y\left|\partial
u^{2}/\partial\mathbf{n}\right|=2y|u|\left|\partial
u/\partial\mathbf{n}\right|\leq 2y|u||\nabla u|\leq 2|u|$ on
${\mathcal{C}}_{\epsilon}^{1}$. Applying these estimates to (72) gives

$\displaystyle\frac{a_{1}}{\sqrt{d+a_{1}^{2}}}\int_{{\mathcal{C}}_{\epsilon}^{1}}u^{2}\,W_{\kappa}(x)d\sigma(x,y)$
$\displaystyle\leq
2\int_{{\mathcal{C}}_{\epsilon}^{1}}|u|\,W_{\kappa}(x)d\sigma(x,y)+c$
$\displaystyle\leq
c^{\prime}\left(\int_{{\mathcal{C}}_{\epsilon}^{1}}u^{2}\,W_{\kappa}(x)d\sigma(x,y)\right)^{1/2}+c,$

where the last inequality is due to the fact that
$\int_{{\mathcal{C}}_{\epsilon}^{1}}W_{\kappa}(x)d\sigma(x,y)$ is bounded by a
constant independent of $\epsilon$ as in (57). This certainly implies that

$\displaystyle\int_{{\mathcal{C}}_{\epsilon}^{1}}u^{2}(x,y)W_{\kappa}(x)d\sigma(x,y)\leq
c,$ (73)

where $c>0$ is independent of $\epsilon$.

If we define $f_{\epsilon}(x)=|u(x,a_{1}^{-1}\delta_{\epsilon}(x))|$ for $x$
satisfying $(x,y)\in{\mathcal{C}}_{\epsilon}^{1}$ with some $y>0$, and
$f_{\epsilon}(x)=0$ otherwise, then from (73), one has, for
$\epsilon\in(0,h_{1}/3)$,

$\displaystyle\int_{{\mathbb{R}}^{d}}|f_{\epsilon}(x)|^{2}\,d\omega_{\kappa}(x)\leq\int_{{\mathcal{C}}_{\epsilon}^{1}}u^{2}(x,y)W_{\kappa}(x)d\sigma(x,y)\leq
c.$ (74)

Now for $(x,y)\in{\mathcal{C}}_{\epsilon}^{1}\subset\Omega^{E}(a_{1},h_{1})$
with $0<y<h_{1}/2$, one can choose a constant $c\in(0,1)$ independent of
$(x,y)$, so that the ball ${\mathbb{B}}((x,y),cy)\subset\Omega^{E}(a,h)$. It
follows that, for $(t,s)\in{\mathbb{B}}((x,y),cy)$, $|u(x,y)-u(t,s)|\leq
cy\sup|\nabla u|$, where the supremum is taken over the line segment joining
$(x,y)$ and $(t,s)$, and by (70),

$\displaystyle|u(x,y)|\leq|u(t,s)|+c,\qquad(t,s)\in{\mathbb{B}}((x,y),cy).$
(75)

Since $|\partial_{j}\delta_{\epsilon}(x)|\leq 1$ ($j=1,\dots,d$), for
$(x,y),(t,s)\in{\mathcal{C}}_{\epsilon}^{1}$ one has

$|(t,s)-(x,y)|=\sqrt{|t-x|^{2}+a_{1}^{-2}|\delta_{\epsilon}(t)-\delta_{\epsilon}(x)|^{2}}\leq\tilde{c}|t-x|$

with $\tilde{c}=\sqrt{1+a_{1}^{-2}d}$, so that

$\displaystyle\int_{{\mathcal{C}}_{\epsilon}^{1}\cap{\mathbb{B}}((x,y),cy)}W_{\kappa}(t)d\sigma(t,s)\geq\int_{{\mathbb{B}}(x,c_{1}y)}W_{\kappa}(t)dt=|{\mathbb{B}}(x,c_{1}y)|_{\kappa},$
(76)

where $c_{1}=c/\tilde{c}<c$. In addition, it follows from (30) that

$\displaystyle(\tau_{x}P_{y})(-t)\geq
c^{\prime}/\left|{\mathbb{B}}(x,(c+1)y)\right|_{\kappa},\qquad(t,s)\in{\mathbb{B}}((x,y),cy).$
(77)

We then take the surface integration to (75) over
${\mathcal{C}}_{\epsilon}^{1}\cap{\mathbb{B}}((x,y),cy)$ and apply (76) and
(77), to get

$\displaystyle|u(x,y)|$ $\displaystyle\leq
c_{2}\int_{{\mathcal{C}}_{\epsilon}^{1}\cap{\mathbb{B}}((x,y),cy)}|u(t,s)|(\tau_{x}P_{y})(-t)W_{\kappa}(t)d\sigma(t,s)+c$
$\displaystyle\leq c_{3}v_{\epsilon}(x,y)+c$ (78)

for $(x,y)\in{\mathcal{C}}_{\epsilon}^{1}$ with $0<y<h_{1}/2$, where
$v_{\epsilon}(x,y)$ is the $\kappa$-Poisson integral of $f_{\epsilon}(x)$.
Further, since $u$ has a bound on
$\\{(x,y)\in\partial\Omega_{\epsilon}:\,h_{1}/2\leq y\leq h_{1}-\epsilon\\}$
independent of $\epsilon$, we could choose the constant $c$ suitably large, so
that (6) is true for all $(x,y)\in\partial\Omega_{\epsilon}$.

Considering the function $U(x,y)=|u(x,y)|-c_{3}v_{\epsilon}(x,y)-c$, by Lemma
4.5 we have $\Delta_{\kappa}U(x,y)=\Delta_{\kappa}|u(x,y)|\geq 0$ in the
region where $U(x,y)>0$ which implies $|u(x,y)|>0$. Since
$U|_{\partial\Omega_{\epsilon}}\leq 0$ from (6), by the maximum principle
(Lemma 4.4) we assert that (6) holds on the whole $\Omega_{\epsilon}$.

Finally, since $\\{f_{\epsilon}:\,\epsilon\in(0,h_{1}/3)\\}$ is a bounded set
in $L_{\kappa}^{2}({\mathbb{R}}^{d})$ by (74), there exists a sequence
$\\{f_{\epsilon_{k}}\\}_{k=1}^{\infty}$, so that $f_{\varepsilon_{k}}$
converges weakly to a function $f\in L_{\kappa}^{2}({\mathbb{R}}^{d})$ as
$k\rightarrow\infty$; and in particular, if $v(x,y)$ denotes the
$\kappa$-Poisson integral of $f$, then $v_{\epsilon_{k}}(x,y)$ converges
pointwise to $v(x,y)$ for $(x,y)\in{\mathbb{R}}^{d+1}_{+}$. Thus, since by
Lemma 5.5, $\Omega_{\epsilon_{k}}$ approaches increasingly to
$\Omega^{E}(a_{1},h_{1})$, we conclude from (6) that

$\displaystyle|u(x,y)|\leq
c^{\prime\prime}v(x,y)+c,\qquad(x,y)\in\Omega^{E}(a_{1},h_{1}).$

By Proposition 2.9, $v(x,y)$ is non-tangentially bounded at $(x,0)$ for almost
every $x\in{\mathbb{R}}^{d}$, and hence, the same is true for $u(x,y)$ and for
almost every $x\in E$. The proof of Theorem 6.1 is completed. $\square$

In order to prove Theorem 6.2, we need the following lemma for $G=Z_{2}^{d}$,
that is an analog of Lemma 6.6.

###### Lemma 6.8

Suppose $G=Z_{2}^{d}$ with a given multiplicity parameters
$\lambda=(\lambda_{1},\dots,\lambda_{d})$. Let $b,\eta>0$ be given and $u$ a
$\lambda$-harmonic function on $\Omega=\cup_{\sigma\in
G}\Gamma_{b}^{\eta}(\sigma(x^{0}))$. If
$(S^{\psi}_{b,\eta}u)(\sigma(x^{0}))\leq 1$ for all $\sigma\in G$, then for
fixed $a\in(0,b/2)$ and $h\in(0,\eta)$, there exists a constant
$c=c(a,b,h,\eta)>0$, so that $y|\nabla u|\leq c$ in $\Gamma_{a}^{h}(x^{0})$.

  * Proof. By Lemma 6.6, it suffices to show that, for $j=1,2,\dots,d$,

$\displaystyle y\left|\frac{u(x,y)-u(\sigma_{j}(x),y)}{x_{j}}\right|\leq
c,\qquad(x,y)\in\Gamma_{a}^{h}(x^{0}),$ (79)

where $\sigma_{j}(x)=(x_{1},\dots,-x_{j},\dots,x_{d})$.

We continue to use the notation in the proof of Lemma 5.6. From (50) we have

$\displaystyle u(x,y)-u(\sigma_{j}(x),y)$ $\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}^{d+1}}u(t,s)\left[\left(\tilde{\tau}_{(-x,-y)}\phi_{\epsilon}\right)(t,s)-\left(\tilde{\tau}_{(-x,-y)}\phi_{\epsilon}\right)(\sigma_{j}(t),s)\right]\,d\omega_{\lambda}(t)ds$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int_{{\mathbb{R}}^{d+1}}(u(t,s)-u(\sigma_{j}(t),s)\left[\left(\tilde{\tau}_{(-x,-y)}\phi_{\epsilon}\right)(t,s)-\left(\tilde{\tau}_{(-x,-y)}\phi_{\epsilon}\right)(\sigma_{j}(t),s)\right]\,d\omega_{\lambda}(t)ds.$
(80)

Recall that the intertwining operator $V_{\lambda}$ for $G=Z_{2}^{d}$ has the
form (cf. [13, 59])

$\displaystyle(V_{\lambda}f)(x)=\int_{[-1,1]^{d}}f(x_{1}\theta_{1},\dots,x_{d}\theta_{d})\,dm_{\lambda}(\theta),$
(81)

where $dm_{\lambda}(\theta)=dm_{\lambda_{1}}(\theta_{1})\dots
dm_{\lambda_{d}}(\theta_{d})$, and for $1\leq i\leq d$,
$dm_{\lambda_{i}}(\theta_{i})=c_{\lambda_{i}}(1+\theta_{i})(1-\theta_{i}^{2})^{\lambda_{i}-1}d\theta_{i}$,
$c_{\lambda_{i}}=\Gamma(\lambda_{i}+1/2)/\Gamma(\lambda_{i})\Gamma(1/2)$. In
view of (14) we have

$\displaystyle\left(\tilde{\tau}_{(-x,-y)}\phi_{\epsilon}\right)(t,s)-\left(\tilde{\tau}_{(-x,-y)}\phi_{\epsilon}\right)(\sigma_{j}(t),s)$
$\displaystyle=\frac{1}{\epsilon^{2|\lambda|+d+1}}\int_{[-1,1]^{d}}\left[\phi_{0}\left(\frac{1}{\epsilon}B(x,y,t,s,\theta)\right)-\phi_{0}\left(\frac{1}{\epsilon}B(x,y,\sigma_{j}(t),s,\theta)\right)\right]\,dm_{\lambda}(\theta),$

where

$B(x,y,t,s,\theta)=\sqrt{|x|^{2}+|t|^{2}-2(x_{1}t_{1}\theta_{1}+\dots+x_{d}t_{d}\theta_{d})+(s-y)^{2}}.$

The change of variables $\theta_{j}\mapsto-\theta_{j}$ to the second integrand
above yields

$\displaystyle\left(\tilde{\tau}_{(-x,-y)}\phi_{\epsilon}\right)(t,s)-\left(\tilde{\tau}_{(-x,-y)}\phi_{\epsilon}\right)(\sigma_{j}(t),s)$
$\displaystyle=\frac{2c_{\lambda_{j}}}{\epsilon^{2|\lambda|+d+1}}\int_{[-1,1]^{d-1}}\int_{-1}^{1}\phi_{0}\left(\frac{1}{\epsilon}B(x,y,t,s,\theta)\right)\theta_{j}(1-\theta_{j}^{2})^{\lambda_{j}-1}d\theta_{j}\,d\widetilde{m_{\lambda}}(\theta),$

where $d\widetilde{m_{\lambda}}(\theta)=\prod_{k\neq
j}dm_{\lambda_{k}}(\theta_{k})$. Now we take integration by parts for
$\theta_{j}$ to get

$\displaystyle\left|\left(\tilde{\tau}_{(-x,-y)}\phi_{\epsilon}\right)(t,s)-\left(\tilde{\tau}_{(-x,-y)}\phi_{\epsilon}\right)(\sigma_{j}(t),s)\right|$
$\displaystyle\leq\frac{c_{\lambda_{j}}\lambda_{j}^{-1}|x_{j}t_{j}|}{\epsilon^{2|\lambda|+d+2}}\int_{[-1,1]^{d-1}}\int_{-1}^{1}\frac{\left|\phi^{\prime}_{0}\left(\epsilon^{-1}B(x,y,t,s,\theta)\right)\right|}{B(x,y,t,s,\theta)}(1-\theta_{j}^{2})^{\lambda_{j}}d\theta_{j}\,d\widetilde{m_{\lambda}}(\theta),$
(82)

We consider the generalized translation $\tau^{+}_{x}$ in ${\mathbb{R}}^{d}$
associated the parameters

$\lambda^{+}=(\lambda_{1},\dots,\lambda_{j-1},\lambda_{j}+1,\lambda_{j+1},\dots,\lambda_{d}).$

As before, the associated translation $\tilde{\tau}^{+}_{(x,y)}$ for
$(x,y)\in{\mathbb{R}}^{d+1}$ is given by
$(\tilde{\tau}^{+}_{(x,y)}u)(t,s)=(\tau^{+}_{x}u)(t,y+s)$, where
$\tau^{+}_{x}$ acts on the first argument. Note that
$\tilde{\phi}_{0}(r):=|\phi^{\prime}_{0}(r)/r|$ is in
$C^{\infty}({\mathbb{R}})$ and even. If we set
$\tilde{\phi}(x,y)=\tilde{\phi}_{0}(\sqrt{|x|^{2}+y^{2}})$ and
$\tilde{\phi}_{\epsilon}(x,y)=\epsilon^{-2|\lambda^{+}|-d-1}\tilde{\phi}(\epsilon^{-1}x,\epsilon^{-1}y)$,
then from (14), (81) and ( ‣ 6),

$\displaystyle\left|\left(\tilde{\tau}_{(-x,-y)}\phi_{\epsilon}\right)(t,s)-\left(\tilde{\tau}_{(-x,-y)}\phi_{\epsilon}\right)(\sigma_{j}(t),s)\right|$
$\displaystyle\leq\frac{|x_{j}t_{j}|}{2\lambda_{j}+1}\left[\left(\tilde{\tau}^{+}_{(-x,-y)}\tilde{\phi}_{\epsilon}\right)(t,s)+\left(\tilde{\tau}^{+}_{(-\sigma_{j}(x),-y)}\tilde{\phi}_{\epsilon}\right)(t,s)\right].$
(83)

In what follows we set

$\Psi_{j}(x,y)=\left|\frac{u(x,y)-u(\sigma_{j}(x),y)}{x_{j}}\right|.$

Inserting ( ‣ 6) into ( ‣ 6) gives

$\displaystyle(4\lambda_{j}+2)\Psi_{j}(x,y)\leq\int_{{\mathbb{R}}^{d+1}}\Psi_{j}(t,s)\left[\left(\tilde{\tau}^{+}_{(-x,-y)}\tilde{\phi}_{\epsilon}\right)(t,s)+\left(\tilde{\tau}^{+}_{(-\sigma_{j}(x),-y)}\tilde{\phi}_{\epsilon}\right)(t,s)\right]\,d\omega_{\lambda^{+}}(t)ds,$

and by Hölder’s inequality and Proposition 2.5(iii),

$\displaystyle\Psi_{j}(x,y)^{2}\leq
c\int_{{\mathbb{R}}^{d+1}}\Psi_{j}(t,s)^{2}\left[\left(\tilde{\tau}^{+}_{(-x,-y)}\tilde{\phi}_{\epsilon}\right)(t,s)+\left(\tilde{\tau}^{+}_{(-\sigma_{j}(x),-y)}\tilde{\phi}_{\epsilon}\right)(t,s)\right]\,d\omega_{\lambda^{+}}(t)ds,$

where
$c=\|\tilde{\phi}\|_{L^{1}({\mathbb{R}}^{d+1},d\omega_{\lambda^{+}}(t)ds)}/2(2\lambda_{j}+1)^{2}$.
Furthermore, by Proposition 3.1(i),

$\displaystyle\Psi_{j}(x,y)^{2}\leq
c\int_{{\mathbb{R}}^{d+1}}\left[\left(\tilde{\tau}^{+}_{(x,y)}+\tilde{\tau}^{+}_{(\sigma_{j}(x),y)}\right)\Psi_{j}^{2}\right](t,s)\tilde{\phi}_{\epsilon}(t,s)\,d\omega_{\lambda^{+}}(t)ds,$

and since

$t_{j}^{2}\tilde{\phi}_{\epsilon}(t,s)\leq\epsilon^{-2|\lambda|-d-1}\Phi(\epsilon^{-1}t,\epsilon^{-1}s),\qquad\Phi(t,s)=\sqrt{|t|^{2}+s^{2}}|\phi^{\prime}_{0}(\sqrt{|t|^{2}+s^{2}})|,$

it follows that

$\displaystyle\epsilon^{2|\lambda|+d+1}\Psi_{j}(x,y)^{2}\leq
c\int_{{\mathbb{R}}^{d+1}}\left[\left(\tilde{\tau}^{+}_{(x,y)}+\tilde{\tau}^{+}_{(\sigma_{j}(x),y)}\right)\Psi_{j}^{2}\right](t,s)\Phi\left(\frac{t}{\epsilon},\frac{s}{\epsilon}\right)\,d\omega_{\lambda}(t)ds.$
(84)

We note that the translation operation $\tilde{\tau}_{(x,y)}$ can be
iteratively written into
$\tilde{\tau}_{(x,y)}=\tau_{x_{1}}\dots\tau_{x_{d}}\tau_{y}$, where $\tau_{y}$
is the usual translation, and $\tau_{x_{i}}$ ($1\leq i\leq d$) denotes the
one-dimensional generalized translation in the $x_{i}$ variable associated to
the parameter $\lambda_{i}$, which is given by (cf. [32, 45])

$\displaystyle(\tau_{x_{i}}f)(t_{i})=\int_{-1}^{1}\left(f_{e}(\tilde{B}_{x_{i},t_{i}}(\theta_{i}))+f_{o}(\tilde{B}_{x_{i},t_{i}}(\theta_{i}))\frac{x_{i}+t_{i}}{\tilde{B}_{x_{i},t_{i}}(\theta_{i})}\right)dm_{\lambda_{i}}(\theta_{i}),$
(85)

where $f$ is a function defined on ${\mathbb{R}}^{1}$, $f_{e}$ and $f_{o}$ are
its even part and odd part respectively, and
$\tilde{B}_{x_{i},t_{i}}(\theta_{i})=\sqrt{x_{i}^{2}+t_{i}^{2}+2x_{i}t_{i}\theta_{i}}$.
If we use $\tau^{+}_{x_{j}}$ to denote the generalized translation in the
$x_{j}$ variable associated to the parameter $\lambda_{j}+1$, then
$\tilde{\tau}^{+}_{(x,y)}=\tau_{x_{1}}\dots\tau_{x_{j-1}}\tau^{+}_{x_{j}}\tau_{x_{j+1}}\dots\tau_{x_{d}}\tau_{y}$.
But from (85), one has

$\displaystyle\left[(\tau^{+}_{x_{j}}+\tau^{+}_{-x_{j}})f\right](t_{j})=c_{\lambda_{j}+1}\int_{-1}^{1}\left[f(\tilde{B}_{x_{j},t_{j}}(\theta_{j}))\left(1+\frac{t_{j}+x_{j}\theta_{j}}{\tilde{B}_{x_{j},t_{j}}(\theta_{j})}\right)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.+f(-\tilde{B}_{x_{j},t_{j}}(\theta_{j}))\left(1-\frac{t_{j}+x_{j}\theta_{j}}{\tilde{B}_{x_{j},t_{j}}(\theta_{j})}\right)\right](1-\theta_{j}^{2})^{\lambda_{j}}d\theta_{j},$

which implies that $\tau^{+}_{x_{j}}+\tau^{+}_{-x_{j}}$ is a positive
operator, and if $f$ is nonnegative, then

$\displaystyle\left[(\tau^{+}_{x_{j}}+\tau^{+}_{-x_{j}})f\right](t_{j})\leq\frac{2\lambda_{j}+1}{2\lambda_{j}}\left[(\tau_{x_{j}}+\tau_{-x_{j}})f\right](t_{j}).$
(86)

Now we rewrite (84) as

$\displaystyle\epsilon^{2|\lambda|+d+1}\Psi_{j}(x,y)^{2}\leq
c\int_{{\mathbb{R}}^{d+1}}\left[\left(\tau_{x_{j}}^{+}+\tau_{-x_{j}}^{+}\right)\Psi_{j}^{2}\right](t,s)K_{\epsilon}(x,y,t,s)\,d\omega_{\lambda}(t)ds,$

where
$K_{\epsilon}(x,y,t,s)=\left[\tau_{-x_{1}}\dots\tau_{-x_{j-1}}\tau_{-x_{j+1}}\dots\tau_{-x_{d}}\tau_{-y}\left(\Phi\left(\epsilon^{-1}\cdot\right)\right)\right](t,s)$,
and applying (86), we get

$\displaystyle\epsilon^{2|\lambda|+d+1}\Psi_{j}(x,y)^{2}\leq\frac{c(2\lambda_{j}+1)}{2\lambda_{j}}\int_{{\mathbb{R}}^{d+1}}\Psi_{j}(t,s)^{2}$
$\displaystyle\qquad\qquad\times\left[\left(\tilde{\tau}_{(-x,-y)}\left(\Phi\left(\epsilon^{-1}\cdot\right)\right)\right)(t,s)+\left(\tilde{\tau}_{(-\sigma_{j}(x),-y)}\left(\Phi\left(\epsilon^{-1}\cdot\right)\right)\right)(t,s)\right]\,d\omega_{\lambda}(t)ds,$
(87)

As in the proof of Lemma 6.6, for fixed $a\in(0,b/2)$ and $h\in(0,\eta)$ and
for $(x,y)\in\Gamma_{a}^{h}(x^{0})$, take $\epsilon=c_{2}y$ for suitable
$0<c_{2}<(b-2a)/(b+2)$, independent of $(x,y)$, such that
$\overline{{\mathbb{B}}((x,y),\epsilon)}\subset\Gamma_{b}^{\eta}(x^{0})\subset\Omega$.
Similarly to (61) we have

$\displaystyle\left(\tilde{\tau}_{(-x,-y)}\left(\Phi\left(\epsilon^{-1}\cdot\right)\right)\right)(t,s)$
$\displaystyle\leq\frac{cy^{2|\lambda|+d-1}}{s^{2|\lambda|+d-1}}\left(\tau_{-\frac{x^{0}}{bs}}\psi\right)\left(\frac{t}{bs}\right),$
(88)
$\displaystyle\left(\tilde{\tau}_{(-\sigma_{j}(x),-y)}\left(\Phi\left(\epsilon^{-1}\cdot\right)\right)\right)(t,s)$
$\displaystyle\leq\frac{cy^{2|\lambda|+d-1}}{s^{2|\lambda|+d-1}}\left(\tau_{-\frac{\sigma_{j}(x^{0})}{bs}}\psi\right)\left(\frac{t}{bs}\right),$
(89)

and for $s>\eta$, both the quantities on the left hand side above vanish.

Applying (88) and (89) to (6) we obtain

$\displaystyle y^{2}\Psi_{j}(x,y)^{2}\leq
c\int_{{\mathbb{R}}^{d}\times(0,\eta]}\Psi_{j}(t,s)^{2}\left[\left(\tau_{-\frac{x^{0}}{bs}}\psi\right)\left(\frac{t}{bs}\right)+\left(\tau_{-\frac{\sigma_{j}(x^{0})}{bs}}\psi\right)\left(\frac{t}{bs}\right)\right]\frac{d\omega_{\lambda}(t)ds}{s^{2|\lambda|+d-1}}.$
(90)

From (46),
$\left(\tau_{-\frac{x^{0}}{bs}}\psi\right)\left(\frac{t}{bs}\right)=\left(\tau_{-\frac{\sigma_{j}(x^{0})}{bs}}\psi\right)\left(\frac{t}{bs}\right)=0$
for $(t,s)\notin\Omega$, and since $u$ is $\lambda$-harmonic in $\Omega$, from
(41) it follows that
$\Psi_{j}(t,s)^{2}\leq(2\lambda_{j})^{-1}\left(\Delta_{\kappa}u^{2}\right)(t,s)$
for $(t,s)\in\Omega$. Thus from (43) and (90),

$y^{2}\Psi_{j}(x,y)^{2}\leq
c\left[(S^{\psi}_{b,\eta}u)^{2}(x^{0})+(S^{\psi}_{b,\eta}u)^{2}(\sigma_{j}(x^{0}))\right]\leq
2c$

for $(x,y)\in\Gamma_{a}^{h}(x^{0})$. Thus (79) is proved and the proof of the
lemma is finished. $\square$

We remark that for general groups $G$, it is uncertain whether the analog to
Lemma 6.8 holds, because further properties of the intertwining operator are
needed in its proof, which are not known yet.

Now one may complete the proof of Theorem 6.2 by following the line of that of
Theorem 6.1. It should be noted that, in the current situation
($G=Z_{2}^{d}$), by Lemma 6.8 the condition (70) is true without
$G$-invariance of $u$ under the assumptions of Theorem 6.2.

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## Declarations

### Funding

The research is supported by the National Natural Science Foundation of China
(No. 12071295).

### Conflicts of interest/Competing interests

None of the authors have any conflicts of interest/competing interests in the
manuscript.

### Availability of data and materials

Not applicable.

### Code availability

Not applicable.

### Author’s contributions

All authors contributed equally and significantly in conducting this research
work and writing this paper. All authors read and approved the final
manuscript.