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HARMONIC ANALYSIS IN DUNKL SETTINGS
THE ANH BUI
ABSTRACT. Let 𝐿be the Dunkl Laplacian on the Euclidean space ℝ𝑁associated with a normalized root 𝑅and
a multiplicity function 𝑘(𝜈)≥0, 𝜈 ∈𝑅. In this paper, we first prove that the Besov and Triebel-Lizorkin spaces
associated with the Dunkl Laplacian 𝐿are identical to the Besov and Triebel-Lizorkin spaces defined in the space of
homogeneous type (ℝ𝑁,⋅, 𝑑𝑤), where 𝑑𝑤(x) = 𝜈∈𝑅𝜈, x𝑘(𝜈)𝑑x. Next, consider the Dunkl transform denoted
by . We introduce the multiplier operator 𝑇𝑚, defined as 𝑇𝑚𝑓=−1(𝑚𝑓), where 𝑚is a bounded function defined
on ℝ𝑁. Our second aim is to prove multiplier theorems, including the Hörmander multiplier theorem, for 𝑇𝑚on
the Besov and Tribel-Lizorkin spaces in the space of homogeneous type (ℝ𝑁,⋅, 𝑑𝑤). Importantly, our findings
present novel results, even in the specific case of the Hardy spaces.
CONTENTS
1. Introduction 1
2. Preliminaries 7
2.1. Dyadic cube systems in the Dunkl setting 7
2.2. Some kernel estimates 9
2.3. Spaces of test functions and Calderón reproducing formulae 12
3. Besov and Triebel-Lizorkin spaces in the Dunkl setting 29
3.1. Atomic decomposition of Besov and Triebel-Lizorkin spaces on (ℝ𝑁,⋅, 𝑑𝑤)29
3.2. Besov and Triebel-Lizorkin spaces associated to the Dunkl operator 34
4. Multiplier theorem for the Dunkl transform 48
References 55
1. INTRODU CTION
The Dunkl theory in an interesting subject in analysis and can be view as a generalization of Fourier analysis
in the Euclidean setting. This foundation can be tracked back by the influential paper of Dunkl [14]. Since then,
significant progress has been made in its development. See for example [13, 15, 16, 32, 37, 38, 39, 49, 50, 33]
and the references therein. For further information, we refer the reader to [40, 41] for more information and
references. Motivated by the extensive research conducted in the Dunkl settings, in this paper we will develop
the harmonic analysis on the Dunkl setting. Firstly we would like to introduce the framework of the Dunkl
setting in [40, 41] (see also [3, 2, 19, 20]).
On the Euclidean space ℝ𝑁we consider a normalized root system 𝑅and a multiplicity function 𝑘(𝜈)≥0, 𝜈 ∈
𝑅. Define
𝑑𝑤(x) =
𝜈∈𝑅x, 𝜈𝑘(𝜈)𝑑x.
In this paper, for 𝑝∈ (0,∞) we denote by 𝐿𝑝(𝑑𝑤)the set of all measurable functions 𝑓such that
𝑓𝐿𝑝(𝑑𝑤)∶=
ℝ𝑁𝑓(x)𝑝𝑑𝑤(x)1∕𝑝<∞.
We consider the Euclidean space ℝ𝑁with the usual scalar product and the induced norm. For a nonzero
vector 𝜈∈ℝ𝑁the reflection 𝜎𝜈with respect to the hyperplane 𝜈⟂orthogonal to 𝜈is given by
(1) 𝜎𝜈x = x − 2x, 𝜈
𝜈2𝜈.
2010 Mathematics Subject Classification. 42B35, 42B15, 33C52, 35K08.
Key words and phrases. Dunkl Laplacian, Dunkl transform, Besov and Triebel space, space of homogeneous type, heat kernel.
1
2 THE ANH BUI
We denote by 𝐺the Weyl group generated by the reflections 𝜎𝜈, 𝜈 ∈𝑅.
A finite set 𝑅 ⊂ ℝ𝑁∖{0} is called a root system if 𝜎𝜈(𝑅) = 𝑅for every 𝜈∈𝑅. We shall consider normalized
reduced root systems, that is, 𝜈2= 2 for every 𝜈∈𝑅. The finite group 𝐺generated by the reflections
𝜎𝜈is called the Weyl group (reflection group) of the root system. A multiplicity function is a 𝐺-invariant
function 𝑘∶𝑅→[0,∞) which will be fixed and be nonnegative throughout this paper. We will denote by
(x) = {𝜎(x) ∶ 𝜎∈𝑅}the 𝐺-orbit of a point x ∈ ℝ𝑁, and (𝐸)={(x) ∶ x ∈ 𝐸}the 𝐺-orbit of 𝐸 ⊂ ℝ𝑁.
The number 𝔑∶= 𝑁+𝜈∈𝑅𝑘(𝜈)is called the homogeneous dimension of the system, since
(2) 𝑤(𝐵(𝑡x, 𝑡𝑟)) = 𝑡𝔑𝑤(𝐵(x, 𝑟)) for x ∈ ℝ𝑁, 𝑡, 𝑟 > 0,
where 𝐵(x, 𝑟) = {y ∈ ℝ𝑁∶x − y≤𝑟}denotes the (closed) Euclidean ball centered at xwith radius 𝑟 > 0.
Observe that
(3) 𝑤(𝐵(x, 𝑟)) ≃ 𝑟𝑁
𝜈∈𝑅
(x, 𝜈+𝑟)𝑘(𝜈)≳ 𝑟𝔑,
so 𝑑𝑤 is a doubling measure, that is, there exists a constant 𝐶 > 0such that
(4) 𝑤(𝐵(x,2𝑟)) ≤𝐶𝑤(𝐵(x, 𝑟)) for x ∈ ℝ𝑁, 𝑟 > 0.
Moreover, by (3),
(5) 𝑅
𝑟𝑁≲𝑤(𝐵(x, 𝑅))
𝑤(𝐵(x, 𝑟)) ≲𝑅
𝑟𝔑
for x ∈ ℝ𝑁,0< 𝑟 < 𝑅.
Finally, define
(6) 𝑑(x,y) = min
𝜎∈𝐺x − 𝜎(y)
as the distance between two 𝐺-orbits (x) and (y). Obviously, (𝐵(x, 𝑟)) = {y ∈ ℝ𝑁∶𝑑(x,y) < 𝑟};
moreover,
𝑤(𝐵(x, 𝑟)) ≤𝑤((𝐵(x, 𝑟))) ≤𝐺𝑤(𝐵(x, 𝑟))
for all x ∈ ℝ𝑁and 𝑟 > 0. We also denote 𝐵𝑑(x, 𝑟) ∶= {y ∈ ℝ𝑁∶𝑑(x,y) < 𝑟}, i.e., 𝐵𝑑(x, 𝑟) = (𝐵(x, 𝑟)).
Let 𝐸(x,y) be the associated Dunkl kernel, which was introduced in [15]. It is known that 𝐸(x,y) has a
unique extension to a holomorphic function in ℂ𝑁×ℂ𝑁. The Dunkl transform is defined by
(7) 𝑓(𝜉) = 𝑐−1
𝑘
ℝ𝑁
𝐸(−𝑖𝜉, x)𝑓(x)𝑑𝑤(x),
where
𝑐𝑘=
ℝ𝑁
𝑒−x2
2𝑑𝑤(x) >0.
It is know that the Dunkl transform, which is initially defined for 𝑓∈𝐿1(𝑑𝑤), is an isometry on 𝐿2(𝑑𝑤)and
preserves the Schwartz class of functions (ℝ𝑁). The inverse of the Dunkl transform is defined by
−1𝑔(x) = 𝑐−1
𝑘
ℝ𝑁
𝐸(𝑖𝜉, x)𝑔(𝜉)𝑑𝑤(𝜉).
For x ∈ ℝ𝑁, the Dunkl translation 𝜏xis defined by
(8) 𝜏x𝑓(y) = 𝑐−1
𝑘
ℝ𝑁
𝐸(𝑖𝜉, x)𝐸(𝑖𝜉 , y)𝑓(𝜉)𝑑𝑤(𝜉)
is bounded on 𝐿2(𝑑𝑤).
In what follows, we use the notation
𝑔(x,y) = 𝜏x𝑔(−y) = 𝜏−y 𝑔(𝑥).
The Dunkl convolution of two appropriate functions is defined by
(9) 𝑓∗𝑔(x) = 𝑐𝑘−1[𝑓𝑔](x) =
ℝ𝑁
𝑓(𝜉)𝑔(𝜉)𝐸(x, 𝑖𝜉)𝑑𝑤(𝜉),
or equivalently,
𝑓∗𝑔(x) =
ℝ𝑁
𝑓(y)𝜏x𝑔(−y)𝑑𝑤(y) =
ℝ𝑁
𝑓(y)𝑔(x,y)𝑑𝑤(y).
HARMONIC ANALYSIS IN DUNKL SETTINGS 3
We now consider the Dunkl operators 𝑇𝜉defined by
𝑇𝜉𝑓(x) = 𝜕𝜉𝑓(x) +
𝜈∈𝑅
𝑘(𝜈)
2𝜈, 𝜉 𝑓(x) − 𝑓(𝜎𝜈(x))
𝜈, x,
where 𝜕𝜉𝑓denotes the directional derivative of 𝑓in the direction 𝜉.
Set 𝑇𝑗=𝑇𝑒𝑗, where {𝑒1,…, 𝑒𝑁}is the canonical basis for ℝ𝑁. The Dunkl Laplacian is defined by
𝐿∶=−
𝑁
𝑗=1
𝑇2
𝑗
= −Δℝ𝑁−
𝜈∈𝑅
𝑘(𝜈)𝛿𝜈𝑓(x),
where Δℝ𝑁is the Laplacian on ℝ𝑁and 𝛿𝜈is defined by
𝛿𝜈𝑓(x) = 𝜕𝜈𝑓(x)
𝛼, x−𝑓(x) − 𝑓(𝜎𝜈(x))
𝛼, x2.
It is well-known that the operator 𝐿is a non-negative and self-adjoint operator on 𝐿2(𝑑𝑤)and generates a
semigroup 𝑒−𝑡𝐿 whose kernel ℎ𝑡(x,y) is defined by
ℎ𝑡(x,y) = 𝑐−1
𝑘(2𝑡)−𝔑∕2 exp −x2+y2
4𝑡𝐸x
2𝑡
,y
2𝑡,x,y ∈ ℝ𝑁, 𝑡 > 0.
See for example [1, 37].
The paper focuses on two primary objectives. Firstly, we will investigate some characterizations of the Besov
spaces and Triebel-Lizorkin spaces in (ℝ𝑁,⋅, 𝑑𝑤)via the heat semigroup 𝑒−𝑡𝐿 . Secondly, we will prove the
boundedness of the multiplier of the Dunkl transform on these function spaces.
For the first aim, we would like to go back the theory of Hardy spaces in (ℝ𝑁,⋅, 𝑑 𝑤). Since (ℝ𝑁,⋅, 𝑑 𝑤)
is a homogneous type in the sense of Coifman and Weiss, one can define the Hardy spaces 𝐻𝑝
CW(𝑑𝑤)with
𝔑
𝔑+1 < 𝑝 ≤1via the atomic decomposition. For convenience, we provide the definition of the Hardy spaces
𝐻𝑝
CW(𝑑𝑤)with 𝔑
𝔑+1 < 𝑝 ≤1in [10], which can be viewed as extensions to the Hardy spaces on ℝ𝑁in the
influential paper of Fefferman and Stein [21] (see also [46, 45]). For 0< 𝑝 ≤1, we say that a function 𝑎is a
(𝑝, 2) atom if there exists a ball 𝐵such that
(i) supp 𝑎⊂𝐵;
(ii) 𝑎𝐿2(𝑑𝑤)≤𝑤(𝐵)1∕2−1∕𝑝;
(iii)
𝑎(x)𝑑𝑤(x) = 0.
For 𝑝= 1 the atomic Hardy space 𝐻1
CW(𝑑𝑤)is defined as follows. We say that a function 𝑓∈𝐻1
CW(𝑑𝑤),
if 𝑓∈𝐿1(𝑑𝑤)and there exist a sequence (𝜆𝑗)𝑗∈ℕ∈𝓁1and a sequence of (𝑝, 2)-atoms (𝑎𝑗)𝑗∈ℕsuch that
𝑓=𝑗𝜆𝑗𝑎𝑗. We set
𝑓𝐻1
CW(𝑑 𝑤)= inf
𝑗𝜆𝑗∶𝑓=
𝑗
𝜆𝑗𝑎𝑗,
where the infimum is taken over all possible atomic decomposition of 𝑓.
For 0< 𝑝 < 1, as in [10], we need to introduce the Lipschitz space 𝔏𝛼. We say that the function 𝑓∈𝔏𝛼if
there exists a constant 𝑐 > 0, such that
𝑓(x) − 𝑓(y)≤𝑐𝑤(𝐵)𝛼
for all ball 𝐵and x,y ∈ 𝐵. The best constant 𝑐above can be taken to be the norm of 𝑓and is denoted by 𝑓𝔏𝛼.
Now let 0<𝑝<1and 𝛼= 1∕𝑝− 1. We say that a function 𝑓∈𝐻𝑝
CW(𝑑𝑤), if 𝑓∈ (𝔏𝛼)∗and there is a
sequence (𝜆𝑗)𝑗∈ℕ∈𝓁𝑝and a sequence of (𝑝, 2)-atoms (𝑎𝑗)𝑗∈ℕsuch that 𝑓=𝑗𝜆𝑗𝑎𝑗. Furthermore, we set
𝑓𝐻𝑝
CW(𝑑 𝑤)= inf
𝑗𝜆𝑗𝑝1∕𝑝
∶𝑓=
𝑗
𝜆𝑗𝑎𝑗,
where the infimum is taken over all possible atomic decomposition of 𝑓.
4 THE ANH BUI
Apart from the Hardy spaces 𝐻𝑝
CW(𝑑𝑤), we can consider the Hardy spaces 𝐻𝑝
𝐿(𝑑𝑤)associated to the Dunkl
Laplacian operator 𝐿. For 0< 𝑝 ≤1, the Hardy space 𝐻𝑝
𝐿(𝑑𝑤)is defined as the completion of the set
𝑓∈𝐿2(𝑑𝑤) ∶ 𝐿𝑓∈𝐿𝑝(𝑑𝑤)
under the norm 𝑓𝐻𝑝
𝐿(𝑑𝑤)=𝐿𝑓𝐿𝑝(𝑑𝑤)where
𝐿𝑓(𝑥) =
∞
0
𝑑(x,y)<𝑡 𝑡2𝐿𝑒−𝑡𝐿2𝑓(y)2𝑑𝑤(y)𝑑𝑡
𝑡𝑤(𝐵(x, 𝑡))1∕2.
It is important to highlight that in the definition of the square function 𝐿we employ the distance 𝑑defined by
(6) instead of the Euclidean distance ⋅. A good reason for this choice is that while the heat kernel ℎ𝑡(x,y)
does not satisfy the Gaussian upper bound with respect to the Euclidean distance ⋅, it satisfies the Gaussian
upper bound with respect to the distance 𝑑. See Lemma 2.5. As a result, the theory of the Hardy space 𝐻𝑝
𝐿(𝑑𝑤)
follows directly from existing results in [30, 43, 4, 18].
So it is natural to question if the Hardy spaces 𝐻𝑝
CW and 𝐻𝑝
𝐿(𝑑𝑤)are the same when 𝔑
𝔑+1 < 𝑝 ≤1. This
question is not straightforward since the Hardy space 𝐻𝑝
CW corresponds to the Euclidean distance ⋅, while
𝐿does not satisfy the Gaussian upper bound with respect to the Euclidean distance; moreover, the Hardy space
𝐻𝑝
𝐿(𝑑𝑤)corresponds to the Euclidean distance 𝑑. For the case 𝑝= 1, a positive answer was carried out in [20].
It remains unclear if the approach in [20] can be extended to the case 𝔑
𝔑+1 <𝑝<1.
Motivated by the aforementioned research, our first primary objective is to establish a more comprehensive
result that encompasses not only the Hardy spaces but also the Besov and Triebel-Lizorkin spaces in (ℝ𝑁,⋅
, 𝑑𝑤). It is worth noting that the theory of Besov and Triebel-Lizorkin spaces holds significant interest in
the field of harmonic analysis and has garnered considerable attention. See for example [27, 26, 28, 29] and
the references therein. In the light of the fact that (ℝ𝑁,⋅, 𝑑𝑤)is a space of homogeneous type satisfying
the reverse doubling property (i.e., the first inequality in (5)), we can define the Besov space
𝐵𝑠
𝑝,𝑞(𝑑𝑤)with
𝑠∈ (−1,1), 𝑝(𝑠, 1) <𝑝<∞,0< 𝑞 < ∞and the Triebel-Lizorkin space
𝐹𝑠
𝑝,𝑞(𝑑𝑤)with 𝑠∈ (−1,1), 𝑝(𝑠, 1) <
𝑝, 𝑞 < ∞(see Definition 3.2), following the framework established in [26]. Here, 𝑝(𝑠, 1) = max{ 𝔑
𝔑+1 ,𝔑
𝔑+𝑠+1 }.
As far as we know, these ranges of the indices 𝑠, 𝑝, 𝑞 are the best in the setting of the space of homogeneous
type except from the case 𝑝= ∞ or 𝑞= ∞.
Moreover, based on the general findings in [6], we can also define the Besov space
𝐵𝑠,𝐿
𝑝,𝑞 (𝑑𝑤)and the Triebel-
Lizorkin space
𝐹𝑠,𝐿
𝑝,𝑞 (𝑑𝑤)associated to the Dunkl Laplacian operator 𝐿. See Definition 3.9. It is intriguing
to observe that these function spaces exhibit similar properties to their counterparts in the Euclidean setting.
For example,
𝐹0,𝐿
𝑝,2(𝑑𝑤)coincide with the 𝐿𝑝(𝑑𝑤)space if 1<𝑝<∞and
𝐹0,𝐿
𝑝,2(𝑑𝑤)coincide with the Hardy
space 𝐻𝑝
𝐿(𝑑𝑤)for 𝑝∈ (0,1]. For further properties such as atomic decomposition and interpolation we refer to
Section 3.2 (see also [6]). Drawing inspiration from these two research streams, our primary result is as follows.
Theorem 1.1. (a) For 𝑠∈ (−1,1),𝑝(𝑠, 1) <𝑝<∞and 0< 𝑞 < ∞, the Besov spaces
𝐵𝑠
𝑝,𝑞(𝑑𝑤)and
𝐵𝑠,𝐿
𝑝,𝑞 (𝑑𝑤)
are identical with the equivalent norms.
(b) For 𝑠∈ (−1,1) and 𝑝(𝑠, 1) < 𝑝, 𝑞 < ∞, the Triebel-Lizorkin spaces
𝐹𝑠
𝑝,𝑞(𝑑𝑤)and
𝐹𝑠,𝐿
𝑝,𝑞 (𝑑𝑤)are identical
with the equivalent norms.
To evaluate the impact of Theorem 1.1, we only consider the particular case when 𝑠= 0, 𝑞 = 2 and
𝔑
𝔑+1 < 𝑝 ≤1. In this case, Theorem 1.1 (b) reads
𝐹0
𝑝,2(𝑑𝑤)≡
𝐹0,𝐿
𝑝,2(𝑑𝑤). This, along with the fact that
𝐹0
𝑝,2(𝑑𝑤)≡𝐻𝑝
CW(𝑑𝑤)(see [25, Remark 5.5]) and
𝐹0,𝐿
𝑝,2(𝑑𝑤)≡𝐻𝑝
𝐿(𝑑𝑤)(see Theorem 3.21), implies that
𝐻𝑝
CW(𝑑𝑤)≡𝐻𝑝
𝐿(𝑑𝑤)for 𝔑
𝔑+1 < 𝑝 ≤1. This, along with the maximal function characterizations of the Hardy
space 𝐻𝑝
𝐿(ℝ𝑁)via heat kernels of 𝐿in [43], extends the result in [20, 52] to the best range of 𝑝in the theory
of Hardy space on the space of homogeneous type. In addition, Theorem 1.1, together with known results in
[6], deduces the heat kernel characterizations for the Besov spaces
𝐹𝑠,𝐿
𝑝,𝑞 (𝑑𝑤)and Triebel-Lizorkin spaces. See
Corollary 3.29.
HARMONIC ANALYSIS IN DUNKL SETTINGS 5
One of the main difficulties of the theory of the Besov and Triebel-Lizorkin spaces is the constructions of
distributions. See for example [35, 6]. In the Dunkl setting, with the ranges of indices in Theorem 1.1 we
can use the distributions in [26, 27] to define our function spaces. See Theorem 3.22. This leads to some
advantages to prove the coincidence between the function spaces associated to 𝐿and the function spaces in the
space (ℝ𝑁,⋅, 𝑑𝑤). Moreover, our approach relies on the atomic decomposition of the Besov and Triebel-
Lizorkin spaces. See Section 3. We note that even the atomic decomposition results for the spaces
𝐵𝑠
𝑝,𝑞(𝑑𝑤)
and
𝐹𝑠
𝑝,𝑞(𝑑𝑤)are new in the literature. See Theorems 3.6 and 3.7.
The second main result focuses on multiplier theorems for the Dunkl transform. Let us provide a brief
overview of the problem. Let 𝑚be a bounded function on ℝ𝑁. We define
𝑇𝑚𝑓=−1(𝑚𝑓).
In the particular case when the multiplicity function 𝑘≡0, the Dunkl transform turns out to be the Fourier
transform, and the Dunkl multiplier 𝑇𝑚boils down to the Fourier multiplier. Regarding the boundedness of
the Dunkl multiplier 𝑇𝑚we recall the main result in [19]. In what follows, we define 𝑓𝑊𝑞
𝑠(ℝ𝑁)=(𝐼−
Δ)𝑠∕2𝑓𝐿𝑞(ℝ𝑁,𝑑 x) for 𝑠 > 0and 𝑞∈ (1,∞], and 𝑎∨𝑏= max{𝑎, 𝑏}and 𝑎∧𝑏= min{𝑎, 𝑏}.
Theorem 1.2 ([19]).Let 𝜓be a smooth radial function on ℝ𝑁such that supp 𝜓 ⊂ {𝜉∶ 1∕4 ≤𝜉≤4} and
𝜓≡1on {𝜉∶ 1∕2 ≤𝜉≤2}. If 𝑚is a function on ℝ𝑁satisfying the following condition
(10) sup
𝑡>0𝜓(⋅)𝑚(𝑡⋅)𝑊2
𝛼(ℝ𝑁)<∞
for some 𝛼 > 𝔑. Then we have:
(a) the multiplier 𝑇𝑚is weak type of (1,1);
(b) the multiplier 𝑇𝑚is bounded on 𝐿𝑝(𝑑𝑤)for 1<𝑝<∞;
(c) the multiplier 𝑇𝑚is bounded on the Hardy space 𝐻1
CW(𝑑𝑤).
Moreover, if we assume the additional condition
(11) sup
y∈ℝ𝑁𝜏y𝑓𝐿1(𝑑𝑤)≤𝐶𝑓𝐿1(𝑑 𝑤),
then the conclusions (a), (b) and (c) above hold true provided that (12)is valid for some 𝛼 > 𝔑∕2.
The condition (10) with 𝛼 > 𝔑is unexpected but is reasonable since we need an extra order of 𝔑∕2 in the
proof of the boundedness of the Dunkl translation on 𝐿𝑝(𝑑𝑤), 𝑝 ≠2. The extra number of derivatives of 𝔑∕2
can be removed under the condition (11). This condition holds true in the rank-one case and the product case
as in [37].
The second main aim in this paper is as follows. Firstly, we aim to reduce the number of the derivative imposed
in the function 𝑚to 𝔑∕2 without the extra condition (11); however, we have to use the norm ⋅𝑊∞
𝛼(ℝ𝑁)rather
than ⋅𝑊2
𝛼(ℝ𝑁). Secondly, we extend the boundedness to the Besov space
𝐵𝑠
𝑝,𝑞(𝑑𝑤)and Triebel-Lizorkin space
𝐹𝑠
𝑝,𝑞(𝑑𝑤). More precisely, we have the following result.
Theorem 1.3. Let 𝛼 > 𝔑
2. Let 𝜓be a smooth radial function defined on ℝ𝑁such that supp 𝜓 ⊂ {𝜉∶ 1∕4 ≤
𝜉≤4} and 𝜓≡1on {𝜉∶ 1∕2 ≤𝜉≤2}. If 𝑚is a function on ℝ𝑁satisfying the following condition
(12) sup
𝑡>0𝜓(⋅)𝑚(𝑡⋅)𝑊∞
𝛼(ℝ𝑁)<∞.
Then the following hold true.
(a) The multiplier 𝑇𝑚is bounded on
𝐹𝑠
𝑝,𝑞(𝑑𝑤)provided that 𝑠∈ (−1,1),𝑝(𝑠, 1) < 𝑝, 𝑞 < ∞and 𝛼 > 𝔑(1
1∧𝑝∧𝑞−
1
2), i.e.,
𝑇𝑚
𝐹𝑠
𝑝,𝑞(𝑑 𝑤)→
𝐹𝑠
𝑝,𝑞(𝑑 𝑤)≲𝑚(0)+ sup
𝑡>0𝜓(⋅)𝑚(𝑡⋅)𝑊∞
𝛼(ℝ𝑁).
(b) The multiplier 𝑇𝑚is bounded on
𝐵𝑠
𝑝,𝑞(𝑑𝑤)provided that 𝑠∈ (−1,1),𝑝(𝑠, 1) < 𝑝, 𝑞 < ∞and 𝛼 > 𝔑(1
1∧𝑝∧𝑞−
1
2), i.e.,
𝑇𝑚
𝐵𝑠
𝑝,𝑞(𝑑 𝑤)→
𝐵𝑠
𝑝,𝑞(𝑑 𝑤)≲𝑚(0)+ sup
𝑡>0𝜓(⋅)𝑚(𝑡⋅)𝑊∞
𝛼(ℝ𝑁).
6 THE ANH BUI
Moreover, if (11)is assumed, the estimates (a) and (b) hold true with 𝑊2
𝛼(ℝ𝑁)taking place of 𝑊∞
𝛼(ℝ𝑁)in
(12).
Note that our approach can prove that (a) and (b) hold true under the condition (10) (without the additional
condition (11)) with 𝛼 > 𝔑(1
1∧𝑝∧𝑞−1
2) + 𝔑
2.
In the particular case when 𝑠= 0, 𝑞 = 2 and 𝑝∈ (0,1], the estimate (a) in Theorem 1.3 boils down to
𝑇𝑚
𝐹0
𝑝,2(𝑑𝑤)→
𝐹0
𝑝,2(𝑑𝑤)≲𝑚(0)+ sup
𝑡>0𝜓(⋅)𝑚(𝑡⋅)𝑊∞
𝛼(ℝ𝑁),
as long as 𝛼 > 𝔑(1
𝑝−1
2). This, along with the fact that
𝐹0
𝑝,2(𝑑𝑤)≡𝐻𝑝
CW(𝑑𝑤)for 𝔑
𝔑+1 < 𝑝 ≤1(see [25, Remark
5.5]), implies that
𝑇𝑚𝐻𝑝
CW(𝑑 𝑤)→𝐻𝑝
CW(𝑑 𝑤)≲𝑚(0)+ sup
𝑡>0𝜓(⋅)𝑚(𝑡⋅)𝑊∞
𝛼(ℝ𝑁),
as long as 𝛼 > 𝔑(1
𝑝−1
2)and 𝔑
𝔑+1 < 𝑝 ≤1. It is worth noting that this estimate improves (c) in Theorem 1.2 for
𝔑
𝔑+1 < 𝑝 < 1. It is important to emphasize that it seems that the approach in [19] can be applied to prove the
estimate for 𝑝 < 1.
In addition, since
𝐹0
𝑝,2(𝑑𝑤)≡𝐿𝑝(𝑑𝑤)(see [26]), the estimate (a) reads
𝑇𝑚𝐻𝑝
CW(𝑑 𝑤)→𝐻𝑝
CW(𝑑 𝑤)≲𝑚(0)+ sup
𝑡>0𝜓(⋅)𝑚(𝑡⋅)𝑊∞
𝛼(ℝ𝑁),
as long as 𝛼 > 𝔑∕2. This improves the main result in [44, Theorem 3] even when 𝑁= 1.
Our techniques regarding the proof of Theorem 1.3 are built upon the framework of function spaces associated
to the operator 𝐿in [6, 7]. We begin by proving the sharp estimates for the multiplier 𝑇𝑚on the Hardy spaces
𝐻𝑝
𝐿(𝑑𝑤), while obtaining the (un-sharp) estimates for the multiplier 𝑇𝑚on the Besov/Triebel-Lizorkin spaces.
The sharpness will follow by utilizing the duality and interpolation arguments as in [7].
Once we have established the boundedness of 𝑇𝑚on the Besov and Triebel-Lizorkin spaces associated with
𝐿, as stated in Theorem 4.1, the conclusion of Theorem 1.3 follows directly from this result, in conjunction with
Theorem 1.1. It is worth noting that the boundedness of the multiplier operator 𝑇𝑚on the Besov and Triebel-
Lizorkin spaces associated with 𝐿, with the full range of indices as outlined in Theorem 4.1, is of considerable
significance in its own right.
We now consider the Riesz transform define by 𝑗∶= 𝑇𝑗𝐿−1∕2, 𝑗 = 1,…, 𝑁 . It is clear that for each
𝑗= 1,…, 𝑁 the Riesz transform can be viewed as the Dunkl multiplier 𝑇𝑚𝑗with 𝑚𝑗(𝜉)=−𝑖𝜉𝑗
𝜉. See for
example [49, 3]. As a direct consequence of Theorem 1.1, we have:
Corollary 1.4. For each 𝑗= 1,…, 𝑁, the Riesz transform 𝑗∶= 𝑇𝑗𝐿−1∕2 is bounded on the Triebel-Lizorkin
space
𝐹𝑠
𝑝,𝑞(𝑑𝑤)with 𝑠∈ (−1,1),𝑝(𝑠, 1) < 𝑝, 𝑞 < ∞, and is bounded on the Besov space with 𝑠∈ (−1,1),
𝑝(𝑠, 1) <𝑝<∞,0< 𝑞 < ∞.
By using Theorem 4.1 we also obtain the boundedness of the Riesz transforms 𝑗on the Besov and Triebel-
Lizorkin associated to the operator 𝐿with full ranges of indices 𝑠∈ℝand 0< 𝑝, 𝑞 < ∞.
The organization of the paper is as follows. In Section 2, we first recall results in [31] on the dyadic systems in
the Dunkl setting. Then some kernel estimates of the functional calculus of 𝐿will be establish. In addition, we
also prove some Calderón reproducing formulae related to the functional calculus of 𝐿under the distributions
in [26, 27]. Section 3 will provide the proof of Theorem 1.1. The proof of Theorem 1.3 will be given in Section 4.
Throughout this paper, we use 𝐶to denote positive constants, which are independent of the main parameters
involved and whose values may vary at every occurrence. By writing 𝑓 ≲ 𝑔, we mean that 𝑓≤𝐶𝑔. We also use
𝑓≃𝑔to denote that 𝐶−1𝑔≤𝑓≤𝐶 𝑔. We also use the following notations through the paper, 𝑎∨𝑏= max{𝑎, 𝑏}
and 𝑎∧𝑏= min{𝑎, 𝑏}.
HARMONIC ANALYSIS IN DUNKL SETTINGS 7
2. PRELIMINARIES
2.1. Dyadic cube systems in the Dunkl setting. Let 𝑋be a metric space, with metric 𝑑and 𝜇is a nonnegative
Borel measure on 𝑋, which satisfies the doubling property below. For x ∈ 𝑋and 𝑟 > 0we set 𝐵(x, 𝑟) = {y ∈
𝑋∶𝑑(x,y) < 𝑟}to be the open ball with radius 𝑟 > 0and center x ∈ 𝑋. The doubling property of 𝜇provides
a constant 𝐶 > 0such that
(13) 𝜇(𝐵(x,2𝑟)) ≤𝐶𝜇(𝐵(x, 𝑟))
for all x ∈ 𝑋and 𝑟 > 0.
Let 0<𝑟<∞. The Hardy-Littlewood maximal function 𝑟is defined by
(14) 𝑟𝑓(x) = sup
x∈𝐵1
𝜇(𝐵)
𝐵𝑓(y)𝑟𝑑𝜇(y)1∕𝑟
where the sup is taken over all balls 𝐵containing x. We will drop the subscripts 𝑟when 𝑟= 1.
Let 0<𝑟<∞. It is well-known that
(15) 𝑟𝑓𝑝≲𝑓𝑝
for all 𝑝>𝑟.
We recall the Fefferman-Stein vector-valued maximal inequality in [24]. For 0< 𝑝 < ∞,0< 𝑞 ≤∞and
0<𝑟<min{𝑝, 𝑞}, we then have for any sequence of measurable functions {𝑓𝜈},
(16)
𝜈𝑟𝑓𝜈𝑞1∕𝑞
𝑝≲
𝜈𝑓𝜈𝑞1∕𝑞
𝑝.
The Young’s inequality and (16) imply the following inequality: If {𝑎𝜈} ∈ 𝓁𝑞∩𝓁1, then
(17)
𝑗
𝜈𝑎𝑗−𝜈𝑟𝑓𝜈𝑞1∕𝑞
𝑝≲
𝜈𝑓𝜈𝑞1∕𝑞
𝑝.
The following result on the existence of a system of dyadic cubes in a space of homogeneous type is just a
combination of [31, Theorem 2.2] and [11, Proposition 2.5] .
Lemma 2.1. Fix 𝛿∈ (0,1) such that 𝛿≤1∕24. Then there exist a family of set {𝑄𝑘
𝛼∶𝑘∈ℤ, 𝛼 ∈𝐼𝑘}and a
set of points {x𝑄𝑘
𝛼∶𝑘∈ℤ, 𝛼 ∈𝐼𝑘}and satisfying
(i) 𝑑(x𝑄𝑘
𝛼,x𝑄𝑘
𝛽)≥𝛿𝑘for all 𝑘∈ℤand 𝛼, 𝛽 ∈𝐼𝑘with 𝛼≠𝛽;
(ii) min𝛼∈𝐼𝑘𝑑(x,x𝑄𝑘
𝛼)≤𝛿𝑘for all x ∈ 𝑋;
(iii) for any 𝑘∈ℤ,𝛼∈𝐼𝑘𝑄𝑘
𝛼=ℝ𝑁and {𝑄𝑘
𝛼∶𝛼∈𝐼𝑘}is disjoint;
(iv) if 𝑘, 𝓁∈ℤand 𝑘≥𝓁, then either 𝑄𝑘
𝛼⊂ 𝑄𝓁
𝛽or 𝑄𝑘
𝛼∩𝑄𝓁
𝛽= ∅ for every 𝛼∈𝐼𝑘and 𝛽∈𝐼𝓁;
(v) for any 𝑘∈ℤand 𝛼∈𝐼𝑘,𝐵(x𝑄𝑘
𝛼, 𝛿𝑘∕6) ⊂ 𝑄𝑘
𝛼⊂ 𝐵(x𝑄𝑘
𝛼,2𝛿𝑘).
We now fix 𝛿∈ (0,1∕24) through the paper. Return to the Dunkl setting, we have two metrics defined in
ℝ𝑁: the Euclidean metric induced by the Euclidean norm ⋅and the orbit distance 𝑑defined by (6).
Since both (ℝ𝑁,⋅, 𝑑𝑤)is a space of homogeneous type, we can apply Lemma 2.1 to extract a system
of dyadic cubes corresponding to each metric. For the homogeneous space (ℝ𝑁,⋅, 𝑑𝑤), denote by 𝒟∶=
{𝑄𝑘
𝛼∶𝑘∈ℤ, 𝛼 ∈𝐼𝑘}the set constructed by Lemma 2.1. The set 𝒟is called the system of dyadic cubes in
(ℝ𝑁,⋅, 𝑑𝑤). For each 𝑘∈ℤ, we denote 𝒟𝑘= {𝑄𝑘
𝛼∶𝛼∈𝐼𝑘}. For each 𝑘∈ℤand 𝛼∈𝐼𝑘, we denote
𝓁(𝑄𝑘
𝛼) = 𝛿𝑘and 𝜆𝑄𝑘
𝛼=𝐵(x𝑄𝑘
𝛼, 𝜆𝛿𝑘), 𝜆 > 0.
Although (ℝ𝑁, 𝑑, 𝑑𝑤)is not a space of homogeneous type, (ℝ𝑁∕𝐺, 𝑑 , 𝑑𝜇)is a space of homogeneous type,
where 𝜇(𝐸) = 𝑤(∪(x)∈𝐸(x)). Therefore, there exists a family 𝒟𝑑∶= {𝑄𝑘,𝑑
𝛼∶𝑘∈ℤ, 𝛼 ∈𝐼𝑑
𝑘}satisfying (i)-
(v) in Lemma 2.1. The set 𝒟𝑑is called the system of dyadic cubes in (ℝ𝑁, 𝑑, 𝑑𝑤). For each 𝑘∈ℤ, we denote
𝒟𝑑
𝑘= {𝑄𝑘,𝑑
𝛼∶𝛼∈𝐼𝑑
𝑘}. For each 𝑘∈ℤand 𝛼∈𝐼𝑑
𝑘, we denote 𝓁(𝑄𝑘,𝑑
𝛼) = 𝛿𝑘and 𝜆𝑄𝑘
𝛼=𝐵𝑑(x𝑄𝑘,𝑑
𝛼, 𝜆𝛿𝑘) ∶=
(𝐵(x𝑄𝑘,𝑑
𝛼, 𝜆𝛿𝑘)), 𝜆 > 0. Note that we use the subscript 𝑑in dyadic cubes to indicate the metric 𝑑.
8 THE ANH BUI
In what follows, for 𝑟 > 0we still denote by 𝑟the maximal function defined by (14) in (ℝ𝑁,⋅, 𝑑𝑤)and
by 𝑑
𝑟the maximal function defined by (14) in (ℝ𝑁, 𝑑, 𝑑 𝑤). Obviously,
𝑑
𝑟𝑓(x) ≲
𝑦∈(x)
𝑟𝑓(y)
for all x ∈ ℝ𝑁. Hence, 𝑑
𝑟also enjoys the inequalities (15), (16) and (17).
We have the following technical lemmas.
Lemma 2.2 ([23]).Let 𝒟𝑑= {𝒟𝑑
𝑘}𝑘∈ℤbe the system of dyadic cubes in (ℝ𝑁, 𝑑, 𝑑𝑤). Let 𝑀 > 𝔑,𝜅∈ [0,1],
and 𝜂, 𝑘 ∈ℤand 𝑘≥𝜂. Assume that {𝑓𝑄𝑑}𝑄𝑑∈𝒟𝑑
𝑘is a sequence of functions satisfying
𝑓𝑄𝑑(x)≲𝑤(𝑄𝑑)
𝑤(𝐵(x𝑄𝑑, 𝛿𝜂)) 𝜅1 + 𝑑(x,x𝑄𝑑)
𝛿𝜂−𝑀.
Then for 𝔑
𝑀< 𝑟 ≤1and a sequence of numbers {𝑠𝑄𝑑}𝑄𝑑∈𝒟𝑑
𝑘, we have
𝑄𝑑∈𝒟𝑑
𝑘𝑠𝑄𝑑𝑓𝑄𝑑(x)≲ 𝛿−𝔑(𝑘−𝜂)(1∕𝑟−𝜅)𝑑
𝑟
𝑄𝑑∈𝒟𝑑
𝑘𝑠𝑄𝑑𝜒𝑄𝑑(x).
The statement still holds true if we replace all dyadic cubes 𝒟𝑑= {𝒟𝑑
𝑘}𝑘∈ℤ, the distance 𝑑and the maximal
function 𝑑
𝑟by the dyadic cubes 𝒟= {𝒟𝑘}𝑘∈ℤ, the distance ⋅and the maximal function 𝑟, respectively.
Lemma 2.3. Let 𝒟= {𝒟𝑘}𝑘∈ℤbe the system of dyadic cubes in (ℝ𝑁,⋅, 𝑑𝑤). Let 𝑀 > 𝔑,𝜅∈ [0,1], and
𝜂, 𝑘 ∈ℤ,𝑘≥𝜂. Assume that {𝑓𝑄}𝑄∈𝒟𝑘is a sequence of functions satisfying
(18) 𝑓𝑄(x)≲𝑤(𝑄)
𝑤(𝐵(x𝑄, 𝛿𝜂)) 𝜅1 + 𝑑(x,x𝑄)
𝛿𝜂−𝑀.
where 𝑑is the distance defined by (6). Then for 𝔑
𝑀< 𝑟 ≤1and a sequence of numbers {𝑠𝑄}𝑄∈𝒟𝑘, we have
𝑄∈𝒟𝑘𝑠𝑄𝑓𝑄(x)≲ 𝛿−𝔑(𝑘−𝜂)(1∕𝑟−𝜅)𝑑
𝑟
𝑄∈𝒟𝑘𝑠𝑄𝜒𝑄(x).
Proof. Before coming to the proof we would like to point out the main issue of the lemma lies on the fact that
the dyadic system corresponds to the metric ⋅, while the estimate (18) corresponds to the metric 𝑑.
Fix x ∈ ℝ𝑁. We set
0= {𝑄∈𝒟𝑘∶𝑑(x,x𝑄)≤𝛿𝜂}, 𝑄0=
𝑄∈0
𝑄
and
𝑘= {𝑄∈𝒟𝑘∶𝛿−𝑘+𝜂+1 < 𝑑(x,x𝑄)≤𝛿−𝑘+𝜂}, 𝑄𝑘=
𝑗∶𝑗≤𝑘
𝑄∈𝑗
𝑄, 𝑘 ∈ℕ+.
Then we write
𝑄∈𝒟𝑘𝑠𝑄𝑓𝑄(x)=
𝑘∈ℕ
𝑄∈𝑘𝑠𝑄𝑓𝑄(x)≤
𝑘∈ℕ
𝑄∈𝑘𝑠𝑄𝑤(𝑄)
𝑤(𝐵(x𝑄, 𝛿𝜂)) 𝜅1 + 𝑑(x, 𝑥𝑄)
𝛿𝜂−𝑀
=∶
𝑘∈ℕ
𝐸𝑘.
For each 𝑘∈ℕ, we have
(19)
𝐸𝑘≲
𝑄∈𝑘
𝛿𝑘𝑀 𝑤(𝑄)
𝑤(𝐵(x𝑄, 𝛿𝜂)) 𝜅
𝑠𝑄≤𝛿𝑘𝑀
𝑄∈𝑘𝑤(𝑄)
𝑤(𝐵(x𝑄, 𝛿𝜂)) 𝜅𝑟
𝑠𝑄𝑟1∕𝑟
≲ 𝛿𝑘𝑀
𝑄𝑘
𝑄∈𝑘𝑤(𝑄)
𝑤(𝐵(𝑥𝑄, 𝛿𝜂)) 𝜅𝑤(𝑄)−1∕𝑟𝑠𝑄𝜒𝑄(𝑦)𝑟
𝑑𝑤(𝑦)1∕𝑟
≲ 𝛿𝑘𝑀 1
𝑤(𝑄𝑘)
𝑄𝑘
𝑄∈𝑘𝑤(𝑄𝑘)
𝑤(𝑄)1∕𝑟𝑤(𝑄)
𝑤(𝐵(𝑥𝑄, 𝛿𝜂)) 𝜅
𝑠𝑄𝜒𝑄(𝑦)𝑟
𝑑𝑤(𝑦)1∕𝑟
.
HARMONIC ANALYSIS IN DUNKL SETTINGS 9
It is easy to see that 𝑤(𝑄𝑘) ≃ 𝑤(𝐵(x, 𝛿𝜂−𝑘)) ≃ 𝑤(𝐵(x𝑄, 𝛿𝜂−𝑘)), for each 𝑄∈𝑘. Therefore,
𝑤(𝑄𝑘)
𝑤(𝑄)1∕𝑟𝑤(𝑄)
𝑤(𝐵(𝑥𝑄, 𝛿𝜂)) 𝜅≲ 𝛿−𝑘𝔑∕𝑟𝛿−𝔑(𝑘−𝜂)(1∕𝑟−𝜅).
Inserting this into (19) gives
𝐸𝑘≲ 𝛿𝑘𝑁 𝛿−𝑘𝔑∕𝑟𝛿−𝔑(𝑘−𝜂)(1∕𝑟−𝜅)1
𝑤(𝑄𝑘)
𝑄𝑘
𝑄∈𝑘𝑠𝑄𝜒𝑄(𝑦)𝑟𝑑𝑤(y)1∕𝑟
≲ 𝛿𝑘(𝑀−𝔑∕𝑟)𝛿−𝔑(𝑘−𝜂)(1∕𝑟−𝜅)𝑑
𝑄∈𝑘𝑠𝑄𝜒𝑄(x).
Since 𝑟 > 𝔑
𝑀, we find that
𝑘∈ℕ
𝐸𝑘≲ 𝛿−𝔑(𝑘−𝜂)(1∕𝑟−𝜅)𝑑
𝑄∈𝒟𝑘𝑠𝑄𝜒𝑄(x).
This completes our proof. □
2.2. Some kernel estimates. In this section, we will prove some kernel estimates for the functional calculus
of 𝐿. We now recall the following result in [20].
Lemma 2.4 ([20]).(a) For x,y ∈ ℝ𝑁and 𝑡 > 0,ℎ(x,y) ≥0. In addition,
ℝ𝑁
ℎ𝑡(x,y)𝑑𝑤(y) =
ℝ𝑁
ℎ𝑡(y,x)𝑑𝑤(y) = 1
for all x ∈ ℝ𝑁and 𝑡 > 0.
(b) For every nonnegative integer 𝑚and every multi-indices 𝛼, 𝛽 there are constants 𝐶, 𝑐 > 0such that
𝜕𝑚
𝑡𝜕𝛼
x𝜕𝛽
yℎ𝑡(x,y)≤𝐶𝑡−(𝑚+𝛼∕2+𝛽∕2)1 + x−y
𝑡−2 1
𝑤(𝐵(x,𝑡))
exp 𝑑(x,y)2
𝑐𝑡 .
Moreover, if y − y′≤𝑡, then
𝜕𝑚
𝑡ℎ𝑡(x,y) − 𝜕𝑚
𝑡ℎ𝑡(x,y′)≤𝐶𝑡−𝑚y − y′
𝑡1 + x−y
𝑡−2 1
𝑤(𝐵(x,𝑡))
exp 𝑑(x,y)2
𝑐𝑡 .
Note that the extra decay 1 + x−y
𝑡−2
in (b) plays an essential role in the proofs of our main results.
Since 𝐿is a nonnegative self-adjoin operator on 𝐿2(𝑑𝑤), for each bounded Borel function 𝐹∶ [0,∞) →ℂ,
we can defined
𝐹(𝐿) =
∞
0
𝐹(𝜆)𝑑𝐸(𝜆)
is bounded on 𝐿2(𝑑𝑤), where 𝐸(𝜆)is the spectral resolution of 𝐿.
The following lemma addresses some kernel estimates for the functional calculus of 𝐿.
Lemma 2.5. Let 𝜑∈𝒮(ℝ)be an even function. Then the following estimates hold true.
(a) For every 𝑀 > 0and every multi-indices 𝛼, 𝛽 , the kernel 𝜑(𝑡𝐿)(x,y) of 𝜑(𝑡𝐿)satisfies the following
estimate
𝜕𝛼
x𝜕𝛽
y𝜑(𝑡𝐿)(x,y)≲ 𝑡−(𝛼+𝛽)1 + x−y
𝑡−2 1
𝑤(𝐵(x, 𝑡 +𝑑(x,y)))𝑡
𝑡+𝑑(x,y) 𝑀
for all x,y ∈ ℝ𝑁and 𝑡 > 0.
(b) For every 𝑀 > 0,
𝜑(𝑡𝐿)(x,y) − 𝜑(𝑡𝐿)(x,y′)≲y − y′
𝑡1 + x−y
𝑡−2 1
𝑤(𝐵(x, 𝑡 +𝑑(x,y)))𝑡
𝑡+𝑑(x,y) 𝑀
for all 𝑡 > 0and x,y,y′∈ℝ𝑁with y − y′≤𝑡.
10 THE ANH BUI
(c) If 𝜑(0) = 0, then
ℝ𝑁
𝜑(𝑡𝐿)(x,y)𝑑𝑤(y) =
ℝ𝑁
𝜑(𝑡𝐿)(y,x)𝑑𝑤(y) = 0
for all x ∈ ℝ𝑁and 𝑡 > 0.
We note that the estimates of Lemma 2.5 without the extra decay 1 + x−y
𝑡−2
is standard. See for
example [35, Theorem 3.1]. The presence of the extra decay needs special properties of the operator 𝐿.
Proof. Item (c) follows from Lemma 2.4 (a) by applying [35, Theorem 3.1], while item (b) follows directly from
(a). Hence, it suffices to prove (a).
It was proved in [19] that
𝐿ℎ𝑡(x,y) = x−y2
4𝑡2ℎ𝑡(x,y) − 𝑁
2𝑡ℎ𝑡(x,y) − 1
2𝑡
𝜈∈𝑅
𝑘(𝜈)ℎ𝑡(𝜎𝜈(x),y),
or equivalently,
x−y2
4𝑡ℎ𝑡(x,y) = −𝑡𝐿ℎ𝑡(x,y) + 𝑁
2ℎ𝑡(x,y) + 1
2
𝜈∈𝑅
𝑘(𝜈)ℎ𝑡(𝜎𝜈(x),y).
Since ℎ𝑡(x,y) is analytic on ℂ+= {𝑧∶ℜ𝑧≥0}, we have for 𝑧∈ℂ+,
x−y2
4𝑧ℎ𝑧(x,y) = −𝑧𝐿ℎ𝑧(x,y) + 𝑁
2ℎ𝑧(x,y) + 1
2
𝜈∈𝑅
𝑘(𝜈)ℎ𝑧(𝜎𝜈(x),y).
It follows that
x−y2
4𝑧ℎ𝑧(x,y)≲𝑧𝐿ℎ𝑧(x,y)+ℎ𝑧(x,y)+
𝜈∈𝑅
𝑘(𝜈)ℎ𝑧(𝜎𝜈(x),y).
Arguing similarly to the proof of [9, Lemma 4.1], we have
ℎ𝑧(𝜎𝜈(x), 𝑦)≲1
𝑤(𝐵(x,𝑧
cos 𝜃))𝑤(𝐵(y,𝑧
cos 𝜃))1∕2 exp −𝑐𝑑(x,y)2
𝑧cos 𝜃1
(cos 𝜃)𝔑, 𝜈 ∈𝑅,
and
𝑧𝐿ℎ𝑧(x,y)≲1
𝑤(𝐵(x,𝑧
cos 𝜃))𝑤(𝐵(y,𝑧
cos 𝜃))1∕2 exp −𝑐𝑑(x,y)2
𝑧cos 𝜃1
(cos 𝜃)𝔑+1
for all x,y ∈ ℝ𝑁and 𝑧∈ℂ+= {𝑧∈ℂ∶ℜ𝑧 > 0} where 𝜃= arg 𝑧.
Consequently,
x−y2
4𝑧ℎ𝑧(x,y)≲1
𝑤(𝐵(x,𝑧
cos 𝜃))𝑤(𝐵(y,𝑧
cos 𝜃))1∕2 exp −𝑐𝑑(x,y)2
𝑧cos 𝜃1
(cos 𝜃)𝔑+1 ,
which implies that
ℎ𝑧(x,y)≲1 + x−y2
𝑧−1 1
𝑤(𝐵(x,𝑧
cos 𝜃))𝑤(𝐵(y,𝑧
cos 𝜃))1∕2 exp −𝑐𝑑(x,y)2
𝑧cos 𝜃1
(cos 𝜃)𝔑+1
for all x,y ∈ ℝ𝑁and 𝑧∈ℂ+= {𝑧∈ℂ∶ℜ𝑧 > 0} where 𝜃= arg 𝑧.
At this stage, by the argument use in the proof of [8, Lemma 2.1], we obtain that for an even function 𝜑∈
𝒮(ℝ)and every 𝑀 > 0,
(20) 𝜑(𝑡𝐿)(x,y)≲1 + x−y
𝑡−2 1
𝑤(𝐵(x, 𝑡 +𝑑(x,y)))𝑡
𝑡+𝑑(x,y) 𝑀
for all x,y ∈ ℝ𝑁and 𝑡 > 0.
HARMONIC ANALYSIS IN DUNKL SETTINGS 11
Recall that for any 𝑚∈ℕ,
(𝐼+𝑡𝐿)−𝑚=1
𝑚!
∞
0
𝑠𝑚−1𝑒−𝑠𝑒−𝑠𝑡𝐿 𝑑𝑠.
From this formula and Lemma 2.4, by a simple calculation it can be verified that for any 𝑀 > 0and multi-indices
𝛼, 𝛽 , there exists 𝑚such that
(21) 𝜕𝛼
x𝜕𝛽
y𝐾(𝐼+𝑡𝐿)−𝑚(x,y)≲ 𝑡−(𝛼+𝛽)1 + x−y
𝑡−2 1
𝑤(𝐵(x, 𝑡 +𝑑(x,y)))𝑡
𝑡+𝑑(x,y) 𝑀,
where 𝐾(𝐼+𝑡𝐿)−𝑚(x,y) denotes the kernel of (𝐼+𝑡𝐿)−𝑚.
For any even function 𝜑∈𝒮(ℝ), we write
𝜑(𝑡𝐿)=(𝐼+𝑡2𝐿)−𝑚𝜑(𝑡𝐿),
where 𝜑(𝜉) = (1 + 𝜉2)𝑚𝜑(𝜉)and 𝑚is a sufficiently large number which will be fixed later.
Then we have
𝜑(𝑡𝐿)(x,y) =
ℝ𝑁
𝐾(𝐼+𝑡2𝐿)−𝑚(x,z)𝜑(𝑡𝐿)(z,y)𝑑𝑤(z).
From this, (20) and (21), for any 𝑀 > 0and multi-indices 𝛼by taking 𝑚sufficiently large, we obtain
(22) 𝜕𝛼
x𝜑(𝑡𝐿)(x,y)≲ 𝑡−𝛼1 + x−y
𝑡−2 1
𝑤(𝐵(x, 𝑡 +𝑑(x,y)))𝑡
𝑡+𝑑(x,y) 𝑀.
We now write
𝜑(𝑡𝐿) = 𝜑(𝑡𝐿)(𝐼+𝑡2𝐿)−𝑚
so that
𝜑(𝑡𝐿)(x,y) =
ℝ𝑁
𝜑(𝑡𝐿)(x,z)𝐾(𝐼+𝑡2𝐿)−𝑚(z,y)𝑑𝑤(z).
Then we apply (21) and (22) to obtain
𝜕𝛼
x𝜕𝛽
y𝜑(𝑡𝐿)(x,y)≲ 𝑡−(𝛼+𝛽)1 + x−y
𝑡−2 1
𝑤(𝐵(x, 𝑡 +𝑑(x,y)))𝑡
𝑡+𝑑(x,y) 𝑀.
This completes our proof. □
Since 𝐿is non-negative self-adjoint and satisfies the Gaussian upper bound with respect to the distance 𝑑,
arguing similarly to the proof of [42, Theorem 2] we obtain that the kernel 𝐾cos(𝑡𝐿)of cos(𝑡𝐿)satisfies
(23) supp 𝐾cos(𝑡𝐿)⊂{(x,y) ∈ ℝ𝑁×ℝ𝑁∶𝑑(x,y) ≤𝑡}.
We have the following useful lemma.
Lemma 2.6. Let 𝜑∈𝐶∞
0(ℝ)be an even function with supp 𝜑 ⊂ (−1,1) and
𝜑= 2𝜋. Denote by Φthe
Fourier transform of 𝜑. Then the kernel 𝑡2𝐿Φ(𝑡𝐿)(x,y) of 𝑡2𝐿Φ(𝑡𝐿)satisfies
(24)
ℝ𝑁
𝑡2𝐿Φ(𝑡𝐿)(x,y)𝑑𝑤(y) =
ℝ𝑁
𝑡2𝐿Φ(𝑡𝐿)(y,x)𝑑𝑤(y) = 0
(25) supp 𝑡2𝐿Φ(𝑡𝐿)(⋅,⋅)⊂{(x,y) ∈ ℝ𝑁×ℝ𝑁∶𝑑(x,y) ≤𝑡},
(26) 𝑡2𝐿Φ(𝑡𝐿)(x,y)≲1 + x−y
𝑡−2 1
𝑤(𝐵(y, 𝑡))
for all x,y ∈ ℝ𝑁and 𝑡 > 0; moreover,
(27) 𝑡2𝐿Φ(𝑡𝐿)(x,y) − 𝑡2𝐿Φ(𝑡𝐿)(x′,y)≲x−x′
𝑡1 + x−y
𝑡−2 1
𝑤(𝐵(y, 𝑡)) ,
for all x,x′,y′∈ℝ𝑁and 𝑡 > 0.
Proof. The proof of (25) is similar to that of [42, Lemma 3]. The remaining properties (24), (26) and (27)
follows directly from Lemma 2.5 and (25).
This completes our proof. □
12 THE ANH BUI
2.3. Spaces of test functions and Calderón reproducing formulae. We first begin with concepts on test
functions and distributions on the space of homogeneous type in [26].
For x,y ∈ ℝ𝑁and 𝑟, 𝛾 > 0, we set
𝑃𝛾(x,y; 𝑟) = 1
𝑤(𝐵(x, 𝑟 +x−y)) 𝑟