Abstract
On \( {\mathbb {R}}^N\) equipped with a root system R and a multiplicity function \(k>0\), we study the generalized (Dunkl) translations \(\tau _{{\textbf{x}}}g(-{\textbf{y}})\) of not necessarily radial kernels g. Under certain regularity assumptions on g, we derive bounds for \(\tau _{{\textbf{x}}}g(-{\textbf{y}})\) by means the Euclidean distance \(\Vert {\textbf{x}}-{\textbf{y}}\Vert \) and the distance \(d({\textbf{x}},{\textbf{y}})=\min _{\sigma \in G} \Vert {\textbf{x}}-\sigma ({\textbf{y}})\Vert \), where G is the reflection group associated with R. Moreover, we prove that \(\tau \) does not preserve positivity, that is, there is a non-negative Schwartz class function \(\varphi \), such that \(\tau _{{\textbf{x}}}\varphi (-{\textbf{y}})<0\) for some points \({\textbf{x}},{\textbf{y}}\in {\mathbb {R}}^N\).
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
We consider \({\mathbb {R}}^N\) equipped with a root system R and a multiplicity function \(k>0\). Behavior of the generalized Dunkl translations \(\tau _{\textbf{x}}g(-\textbf{y})\) and, consequently, boundedness of the generalized convolution operators
on various function spaces are ones of the main problems in the harmonic analysis in the Dunkl setting. Here and subsequently, dw is the measure associated with the system (R, k) (see (2.2)). If \(f\in L^p(dw)\), \(g\in L^1(dw)\) and one of them is radial then, thanks to the Rösler formula (see (2.22)) on translations of radial functions, one has
with \(C=1\). Further, since the generalized translations of any radial non-negative function g are non-negative, some pointwise estimates for \(\tau _{\textbf{x}}g(-\textbf{y})\) can be derived from the bounds of the heat kernel \(h_t(\textbf{x},\textbf{y})\) (see Proposition 4.3). In particular, if g is a radial function such that \(|g(\textbf{x})|\le C_M(1+\Vert \textbf{x}\Vert )^{-M}\) for all \(M>0\), then for any \(M'>0\),
where \(d(\textbf{x},\textbf{y})=\min _{\sigma \in G} \Vert {\textbf{x}}-\sigma ({\textbf{y}})\Vert \), G is the reflection group associated with R (see (2.29)).
On the other hand, the \(L^p(dw)\)-bounds for the generalized translations \(\tau _{\textbf{x}}g\) of non-radial \(L^p\)-functions for \(p\ne 2\) is an open problem as well as the inequality (1.1). However, if we assume some regularity of a (non-radial) function g in its smoothness and decay, then
(see [11, Proposition 5.1]) and, consequently,
The estimates of the form (1.3), which make use of the distance \(d(\textbf{x},\textbf{y})\) of the orbits and the measures of the balls, seem to be useful, because they allow one to reduce some problems to the setting of spaces of homogeneous type and apply tools from the theory of these spaces for obtaining some analytic-spirit results. For instance, in [4] this approach was used in order to define and characterize the real (Dunkl) Hardy space \(H^1_{\Delta _k}\) by means of boundary values of the Dunkl conjugate harmonic functions, maximal functions associated with radial kernels, the relevant Riesz transforms, square functions and atoms (which were defined in the spirit of [16]). From the point of view of non-radial kernels g, in some cases, the estimates (1.3) can be used as a substitute for \(L^p\)-boundedness of the Dunkl translations (see [8]).
On the other hand, it was noticed that in some cases the estimates of the form (1.3) are not strong enough to obtain some harmonic analysis results involving Dunkl translations and convolutions. For example in order to prove that the Hardy space \(H^1_{\Delta _k}\) admits atomic decomposition into Coifman–Weiss atoms, the authors of [9] needed the following estimates for the generalized translations of radial continuous functions supported in the unit ball:
The estimate (1.5) is a slightly weaker version of (1.2) because the factor \((1+\Vert {\textbf{x}}-{\textbf{y}}\Vert )\) is raised to the power negative one, however its presence is crucial for the proof the atomic decomposition. Further, a presence of the factor \((1+\Vert {\textbf{x}}-{\textbf{y}}\Vert )^{-\delta }\) (or its scaled version) in estimates of some integral kernels helps to handle harmonic analysis problems in the Dunkl setting (see e.g. [10, Section 5] and [25] for a study of singular integrals).
Another question can be asked for the exponent(s) associated with the Euclidean distance(s) in estimates of generalized translations of g. It was proved in [12] that for the Dunkl heat kernel \(h_t({\textbf{x}},{\textbf{y}})\) the exponents depend on sequences of reflections needed to move \({\textbf{y}}\) to a Weyl chamber of \({\textbf{x}}\). To be more precise, the following upper and lower bounds for \(h_t({\textbf{x}},{\textbf{y}})\) hold: for all \(c_l>1/4\) and \(0<c_u<1/4\) there are constants \(C_l,C_u>0\) such that
where \(\Lambda ({\textbf{x}},{\textbf{y}},t)\) is expressed as a sum of products of specially selected factors of the form \((1+\Vert {\textbf{x}}-\sigma ({\textbf{y}})\Vert /\sqrt{t})^{-2}\) (see Sect. 2.7 for details). The estimate (1.6) improves the known bound
(see [9, Theorem 3.1] for a proof of (1.7)), which can be used, as we remarked out, for proving estimates for translations of radial kernels. An alternative proof of (1.7) which uses a Poincaré inequality was announced by W. Hebisch. Let us also point out the presence of the same function \(\Lambda ({\textbf{x}},{\textbf{y}}, t)\) in the upper and lower bounds (1.6). Thus if \(d({\textbf{x}},{\textbf{y}})^2\le t\), the estimates (1.6) are sharp.
The goal of this paper is to present some properties of the generalized translations \(\tau _{\textbf{x}} g(-\textbf{y})\) of non-radial kernels g, and, in particular, propose some methods which allow to one to derive estimates for \(\tau _{\textbf{x}} g(-\textbf{y})\) and express them in terms of measures w(B) of appropriate balls and the distances \(\Vert {\textbf{x}}-{\textbf{y}}\Vert \) and \(d({\textbf{x}},{\textbf{y}})\). We expect that information about generalized translations of non-radial functions can be useful in further development of real harmonic analysis in the Dunkl setting; for example in characterizations of some function spaces by means of non-radial kernels. We prove that if a (non-radial) function g is sufficiently regular, then
(see Theorem 4.1).
Further we aim to obtain estimates for \(\tau _{\textbf{x}} g(-\textbf{y})\) for non-radial g and interpret them in the context of (1.6). From one point of view, one can expect the upper estimates making use of the same function \(\Lambda (\textbf{x},\textbf{y},1)\). Since in the case of non-radial kernels the Rösler’s formula is not available, we need a different approach, which is presented in Sect. 3, for obtaining estimates for the generalized translations of any non-radial Schwartz-class function \(\varphi \), which involve the function \(\Lambda ^{1/2}\), that is,
(see Theorem 4.5). Let us note that (1.9) improve the bounds (1.8).
Then we use the methods described in Sect. 3 in order to unify two approaches to the theory of singular integrals from [10] and [25]. We prove that for a large class of singular integral operators of convolution type (including the Dunkl transform multiplier operators), their corresponding integral kernels \({\mathcal {K}}({\textbf{x}},{\textbf{y}})\) satisfy Calderón-Zygmund type conditions, which are expressed by means of the distances \(\Vert {\textbf{x}}-{\textbf{y}}\Vert \) and \(d({\textbf{x}},{\textbf{y}})\) and \(w(B({\textbf{x}},d({\textbf{x}},{\textbf{y}}))\) (see Sects. 4.2 and 4.3).
Further, it turns out that our approach developed in Sect. 3 can be used in order to prove non-positivity of the Dunkl translations. We show that for any root system \(R \ne \emptyset \) and \(k>0\) there is a non-negative Schwartz class function \(\varphi \) such that \(\tau _{{\textbf{x}}}\varphi (-{\textbf{y}})<0\) for some \({\textbf{x}},{\textbf{y}}\) (see Sect. 4.4 for details).
2 Preliminaries and Notation
2.1 Dunkl Theory
In this section we present basic facts concerning the theory of the Dunkl operators. For more details we refer the reader to [6, 21, 23, 24].
We consider the Euclidean space \({\mathbb {R}}^N\) with the scalar product \(\langle \textbf{x},{\textbf{y}}\rangle =\sum _{j=1}^N x_jy_j \), where \({\textbf{x}}=(x_1,\ldots ,x_N)\), \({\textbf{y}}=(y_1,\ldots ,y_N)\), and the norm \(\Vert {\textbf{x}}\Vert ^2=\langle {\textbf{x}},{\textbf{x}}\rangle \).
A normalized root system in \({\mathbb {R}}^N\) is a finite set \(R\subset {\mathbb {R}}^N\setminus \{0\}\) such that \(R \cap \alpha \mathbb {R} = \{\pm \alpha \}\), \(\sigma _\alpha (R)=R\), and \(\Vert \alpha \Vert =\sqrt{2}\) for all \(\alpha \in R\), where \(\sigma _\alpha \) is defined by
The finite group G generated by the reflections \(\sigma _{\alpha }\), \(\alpha \in R\), is called the Coxeter group (reflection group) of the root system.
A multiplicity function is a G-invariant function \(k:R\rightarrow {\mathbb {C}}\) which will be fixed and positive throughout this paper.
The associated measure dw is defined by \(dw({\textbf{x}})=w({\textbf{x}})\, d{\textbf{x}}\), where
Let \(\textbf{N}=N+\sum _{\alpha \in R}k(\alpha )\). Then,
where, here and subsequently, \(B({\textbf{x}},r)=\{{\textbf{y}}\in {\mathbb {R}}^N: \Vert {\textbf{x}}-{\textbf{y}}\Vert \le r\}\). Observe that there is a constant \(C>0\) such that for all \(\textbf{x} \in \mathbb {R}^N\) and \(r>0\) we have
so \(dw({\textbf{x}})\) is doubling, that is, there is a constant \(C>0\) such that
Moreover, there exists a constant \(C\ge 1\) such that, for every \(\textbf{x}\in \mathbb {R}^N\) and for all \(r_2\ge r_1>0\),
For \(\xi \in \mathbb {R}^N\), the Dunkl operators \(T_\xi \) are the following k-deformations of the directional derivatives \(\partial _\xi \) by difference operators:
where, here and subsequently,
The Dunkl operators \(T_{\xi }\), which were introduced in [6], commute and are skew-symmetric with respect to the G-invariant measure dw, i.e. for reasonable functions f, g (for instance, \(f,g \in \mathcal {S}(\mathbb {R}^N)\)) we have
Let us denote \(T_j=T_{e_j}\), where \(\{e_j\}_{1 \le j \le N}\) is a canonical orthonormal basis of \(\mathbb {R}^N\).
For \(f,g \in C^1(\mathbb {R}^N)\), we have the following Leibniz-type rule
For multi-index \({\beta }=(\beta _1,\beta _2,\ldots ,\beta _N) \in {\mathbb {N}}_0^N=(\mathbb {N} \cup \{0\})^{N} \), we denote
For fixed \({\textbf{y}}\in {\mathbb {R}}^N\), the Dunkl kernel \(\textbf{x} \mapsto E({\textbf{x}},{\textbf{y}})\) is a unique analytic solution to the system
The function \(E({\textbf{x}},{\textbf{y}})\), which generalizes the exponential function \(e^{\langle {\textbf{x}},{\textbf{y}}\rangle }\), has a unique extension to a holomorphic function on \({\mathbb {C}}^N\times {\mathbb {C}}^N\). It was proved in [21, Corollary 5.3] that for all \(\textbf{x},\textbf{y} \in \mathbb {R}^N\) and \(\nu \in \mathbb {N}_0^{N}\) we have
2.2 Dunkl Transform
Let \(f \in L^1(dw)\). We define the Dunkl transform \(\mathcal {F}f\) of f by
where
is so called Mehta-Macdonald integral. The Dunkl transform is a generalization of the Fourier transform on \(\mathbb {R}^N\). It was introduced in [7] for \(k \ge 0\) and further studied in [5] in a more general context. It was proved in [7, Corollary 2.7] (see also [5, Theorem 4.26]) that it extends uniquely to an isometry on \(L^2(dw)\), i.e.,
We have also the following inversion theorem ( [5, Theorem 4.20]): for all \(f \in L^1(dw)\) such that \(\mathcal {F}f \in L^1(dw)\) we have
So, the inverse \({\mathcal {F}}^{-1}\) of \(\mathcal {F}\) has the form
It can be easily checked using (2.12) that for compactly supported \(f \in L^1(dw)\) we have
2.3 Dunkl translations
Suppose that \(f \in \mathcal {S}(\mathbb {R}^N)\) (the Schwartz class of functions on \({\mathbb {R}}^N)\) and \(\textbf{x} \in \mathbb {R}^N\). We define the Dunkl translation \(\tau _{\textbf{x}}f\) of f to be
The Dunkl translation was introduced in [20]. The definition can be extended to the functions which are not necessary in \(\mathcal {S}(\mathbb {R}^N)\). For instance, thanks to the Plancherel’s theorem (see (2.15)), one can define the Dunkl translation of any \(L^2(dw)\) function f by
(see [20] and [26, Definition 3.1]). In particular, it follows from (2.20), (2.13), and (2.15) that for all \(\textbf{x} \in \mathbb {R}^N\) the operators \(f \mapsto \tau _{\textbf{x}}f\) are contractions on \(L^2(dw)\). Here and subsequently, we write \(g({\textbf{x}},{\textbf{y}}):=\tau _{{\textbf{x}}}g(-{\textbf{y}})\).
We will need the following result concerning the support of the Dunkl translated of a compactly supported function.
Theorem 2.1
([8] Theorem 1.7) Let \(f \in L^2(dw)\), \(\textrm{supp}\, f \subseteq B(0,r)\), and \(\textbf{x} \in \mathbb {R}^N\). Then
Here and subsequently, for a measurable set \(A \subseteq \mathbb {R}^N\) we denote
2.4 Dunkl Translations of Radial Functions
The following specific formula was obtained by Rösler [22] for the Dunkl translations of (reasonable) radial functions \(f({\textbf{x}})=\tilde{f}({\Vert \textbf{x}\Vert })\):
Here
and \(\mu _{\textbf{x}}\) is a probability measure, which is supported in the set \({\text {conv}}\mathcal {O}(\textbf{x})\) (the convex hull of the orbit of \({\textbf{x}}\) under the action of G).
2.5 Dunkl Convolution
Assume that \(f,g \in L^2(dw)\). The generalized convolution (or Dunkl convolution) \(f*g\) is defined by the formula
equivalently, by
Generalized convolution of \(f,g \in \mathcal {S}(\mathbb {R}^N)\) was considered in [20, 28], the definition was extended to \(f,g \in L^2(dw)\) in [26].
2.6 Generalized Heat Semigroup and Heat Kernel
The Dunkl Laplacian associated with R and k is the differential-difference operator \(\Delta _k=\sum _{j=1}^N T_{j}^2\), which acts on \(C^2(\mathbb {R}^N)\)-functions by
The operator \(\Delta _k\) is essentially self-adjoint on \(L^2(dw)\) (see for instance [1, Theorem 3.1]) and generates a semigroup \(H_t\) of linear self-adjoint contractions on \(L^2(dw)\). The semigroup has the form
where the heat kernel
is a \(C^\infty \)-function of all the variables \({\textbf{x}},{\textbf{y}} \in \mathbb {R}^N\), \(t>0\), and satisfies
In terms of the generalized translations we have
and, in terms of the Dunkl transform,
2.7 Upper and Lower Heat Kernel Bounds
The closures of connected components of
are called (closed) Weyl chambers. We define the distance of the orbit of \({\textbf{x}}\) to the orbit of \({\textbf{y}}\) by
For a finite sequence \(\varvec{\alpha }=(\alpha _1,\alpha _2,\ldots ,\alpha _m)\) of elements of R, \({\textbf{x}},{\textbf{y}}\in {\mathbb {R}}^N\) and \(t>0\), let \(\ell ({\varvec{\alpha }}):=m\) be the length of \(\varvec{\alpha }\),
and
For \({\textbf{x}},{\textbf{y}}\in {\mathbb {R}}^N\), let \( n ({\textbf{x}},{\textbf{y}})=0\) if \( d({\textbf{x}},{\textbf{y}})=\Vert {\textbf{x}}-{\textbf{y}}\Vert \) and
otherwise. In other words, \(n({\textbf{x}},{\textbf{y}})\) is the smallest number of reflections \(\sigma _\alpha \) which are needed to move \({\textbf{y}}\) to a (closed) Weyl chamber of \({\textbf{x}}\). We also allow \(\varvec{\alpha }\) to be the empty sequence, denoted by \(\varvec{\alpha } =\emptyset \). Then for \(\varvec{\alpha }=\emptyset \), we set: \(\sigma _{\varvec{\alpha }}=\textrm{id}\) (the identity operator), \(\ell (\varvec{\alpha })=0\), and \(\rho _{\varvec{\alpha }}(\textbf{x},\textbf{y},t)=1\) for all \(\textbf{x},\textbf{y} \in \mathbb {R}^N\) and \(t>0\).
We say that a finite sequence \(\varvec{\alpha }=(\alpha _1,\alpha _2,\ldots ,\alpha _m)\) of roots is admissible for a pair \(({\textbf{x}},{\textbf{y}})\in {\mathbb {R}}^N\times {\mathbb {R}}^N\) if \(n(\textbf{x},\sigma _{\varvec{\alpha }}(\textbf{y}))=0\). In other words, the composition \(\sigma _{\alpha _m}\circ \sigma _{\alpha _{m-1}}\circ \ldots \circ \sigma _{\alpha _1}\) of the reflections \(\sigma _{\alpha _j}\) maps \({\textbf{y}}\) to a Weyl chamber of \({\textbf{x}}\). The set of the all admissible sequences \(\varvec{\alpha }\) for the pair \(({\textbf{x}},{\textbf{y}})\) will be denoted by \({\mathcal {A}}({\textbf{x}},{\textbf{y}})\). Note that if \(n({\textbf{x}},{\textbf{y}})=0\), then \(\varvec{\alpha }=\emptyset \in {\mathcal {A}}(\textbf{x},\textbf{y})\).
Let us define
The following upper and lower bounds for \(h_t({\textbf{x}},{\textbf{y}})\) were proved in [12].
Theorem 2.2
([12, 13] Theorem 1.1 and Remark 2.3) Assume that \(0<c_{u}<1/4\) and \(c_l>1/4\). There are constants \(C_{u},C_{l}>0\) such that for all \(\textbf{x},\textbf{y} \in \mathbb {R}^N\) and \(t>0\) we have
We also have the following regularity estimate for \(h_t(\textbf{x},\textbf{y})\) ( [12, Theorem 6.1]).
Lemma 2.3
Let \(\varepsilon _1 \in (0,1]\). There is a constant \(C>0\) such that for all \(\textbf{x},\textbf{y},\textbf{y}' \in \mathbb {R}^N\) and \(t>0\) we have
As an application of Theorem 2.2 and (2.22) it is possible to describe a behavior of the measure \(\mu _{{\textbf{x}}}\) near the points \(\sigma (\textbf{x})\) for \(\sigma \in G\) (see also [17, Theorem 2.1]). The behavior, stated in Theorem 2.4, gives another proof of the theorem of Gallardo and Rejeb (see [14, Theorem A 3)]), which says that all the points \(\sigma ({\textbf{x}})\), \(\sigma \in G\), belong to the support of the measure \(\mu _{{\textbf{x}}}\).
Theorem 2.4
([12]) For \(\textbf{x} \in \mathbb {R}^N\) and \(t>0\) we set
There is a constant \(C>0\) such that for all \(\textbf{x} \in \mathbb {R}^N\), \(t>0\), and \(\sigma \in G\) we have
2.8 Kernel of the Dunkl–Bessel Potential
For an even positive integer s, we set
It can be easily checked that for \(\textbf{x},\textbf{y} \in \mathbb {R}^N\) we have
Since \(\xi \longmapsto (1+\Vert \xi \Vert ^{2})^{-s/2}\) is radial, thanks to (2.7), for all \(1 \le j \le N\) we have
3 Some Formulas and Estimates for Dunkl Translations of Regular Enough Functions
In the present section we prove formulas and derive basic estimates for translations of certain functions. Then, in the next section, we shall use them for more advanced estimations.
We start by the following lemma, which is a consequence of the generalized heat kernel regularity estimates (2.36).
Lemma 3.1
Let \(\varepsilon _1 \in (0,1]\). There is a constant \(C>0\) such that for all \(t>0\) and \(\textbf{y},\textbf{y}' \in \mathbb {R}^N\), we have
Proof
We prove just (3.2), the proof of (3.1) is analogous (in fact, it was proved in [8, (3.6)]). By (2.28), the Plancherel’s equality (2.15), and (2.36) we get
\(\square \)
In order to estimate translations of non-radial functions we need further preparation. The following lemma and its proof, which is based on the fundamental theorem of calculus (see e.g. [11, pages 284-285]), will play a crucial role in our study. Recall that for a function f, \(f^{\{\alpha \}}\) is defined in (2.8).
Lemma 3.2
Let \(\ell \in \mathbb {N}_0\), \(M>0\). If \(f \in C^{\ell +1}(\mathbb {R}^N)\) is such that \(\partial _jf\) are bounded functions for all \(1 \le j \le N\), then \( f^{\{\alpha \}}\) belongs to \(C^{\ell }(\mathbb {R}^N)\) for all \(\alpha \in R\) and there is a constant \(C>0\) independent of f such that
Moreover, there is a constant \(C>0\) independent of \(\ell \) and f such that if
then \(|T^\beta f^{\{\alpha \}}(\textbf{x})| \le C(1+\Vert \textbf{x}\Vert )^{-\textbf{N}-M}\) for all \(|\beta |\le \ell \), \(\alpha \in R\), and \(\textbf{x} \in \mathbb {R}^N\).
Proposition 3.3
Let \(\phi \in \mathcal {S}(\mathbb {R}^N)\) and \(1 \le j \le N\). Then for all \(\textbf{x},\textbf{y} \in \mathbb {R}^N\) we have
where \(\phi _j\), \(\phi _\alpha \) are Schwartz class functions defined by
Moreover, if \(\phi \) is G-invariant, then
where \(\mathcal {F}\phi _j(\xi )=\partial _{j,\xi }\mathcal {F}\phi (\xi )=T_{j,\xi }\mathcal {F}\phi (\xi )\), i.e. \(\phi _j(\textbf{x})=-ix_j\phi (\textbf{x})\).
Proof
It is obvious, that \(\phi _j\) defined in (3.4) belong to \({\mathcal {S}}({\mathbb {R}}^N)\). Further, the functions
belong the Schwartz class (see Lemma 3.2). Hence, \(\phi _\alpha \in {\mathcal {S}}({\mathbb {R}}^N)\) for all \(\alpha \in R\). Thanks to the inversion formula and definition of Dunkl kernel (see (2.17) and (2.12)) we get
It follows from (2.13) that for fixed \(\textbf{x} \in \mathbb {R}^N\) we have \((E(-i\cdot ,\textbf{x})\mathcal {F}\phi (\cdot )) \in \mathcal {S}(\mathbb {R}^N)\). Hence, by the integration by parts formula (2.9) and the Leibniz-type rule (2.10) we get
Using (2.12) and inverse formula (2.17) we obtain
Therefore, (3.3) is a consequence of (3.6) and (3.7). The proof of (3.5) follows from (3.3) and (3.4), since \({\mathcal {F}}\phi \) is G-invariant, so \(\phi _\alpha \equiv 0\) and \(\partial _{j,\xi }{\mathcal {F}}\phi (\xi )=T_j{\mathcal {F}}\phi (\xi )\) in this case. \(\square \)
Let us note that Proposition 3.3 together with its proof can be generalized to \(\phi \) which not necessary belongs to \(\mathcal {S}(\mathbb {R}^N)\), but the quantities which appear in the proof make sense. One of such a possible generalization is presented in the proposition below, which will be used in the proof of Theorem 4.6.
Proposition 3.4
Let \(\delta >0\). Assume that \(f\in L^1(dw)\) is compactly supported and \(g\in L^1(dw)\) is G-invariant function such that \(|\mathcal {F}g(\xi )|\le (1+\Vert \xi \Vert )^{-{\textbf{N}}-\delta }\), \(\mathcal {F}g\in C^1({\mathbb {R}}^N)\), and \(|T_{j}\mathcal {F}g(\xi )|\le (1+\Vert \xi \Vert )^{-{\textbf{N}}-\delta }\) for all \(1\le j \le N\)and \(\xi \in \mathbb {R}^N\). Then
Proof
First, let us observe that for every multi index \(\nu \in \mathbb {N}_0^N\), a function \(f\in L^1(dw)\), \(\text {supp}\, f\subseteq B(0,r)\), and \(\xi \in \mathbb {R}^N\) one has
Indeed, by (2.13),
Similarly, by Lemma 3.2,
Consequently, all of the integrals in (3.8) can be interpreted as the Dunkl transforms of \(L^1(dw)\)-functions. Hence, in order to establish (3.8), it is enough to note that applying the Leibniz-type rule (2.10) twice: firstly to the functions: \(E(-i \cdot ,\textbf{y})\mathcal {F}f\) (not necessarily G-invariant) and \(\mathcal {F}g\) (G-invariant) and then to the functions \(E(-i\cdot ,\textbf{y})\) and \(\mathcal {F}f\), we obtain
and repeat the proof of Proposition 3.3. \(\square \)
Proposition 3.5
Let \(\delta >0\) and \(0<\varepsilon _1 \le 1\). Assume that \(f \in L^1(dw)\) and \(g\in L^1(dw)\) is such that \(|\mathcal {F}g(\xi )|\le (1+\Vert \xi \Vert )^{-{\textbf{N}}-\delta }\) for all \(\xi \in \mathbb {R}^N\). Then the following statements hold.
-
(a)
There is a constant \(C_1>0\) independent of f, g such that for all \(1 \le j \le N\) and \(\textbf{x},\textbf{y} \in \mathbb {R}^N\), one has
$$\begin{aligned} |(f*g)({\textbf{x}},{\textbf{y}})|\le C w(B({\textbf{x}},1))^{-1/2} w(B({\textbf{y}}, 1))^{-1/2}\Vert f\Vert _{L^1(dw)}. \end{aligned}$$(3.12) -
(b)
If additionally g is G-invariant, \( \mathcal {F}g\in C^1({\mathbb {R}}^N)\), and satisfies \(|T_{j}\mathcal {F}g(\xi )|\le (1+\Vert \xi \Vert )^{-{\textbf{N}}-\delta }\) for all \(\xi \in \mathbb {R}^N\), then there is a constant \(C_2>0\) independent of f, g such that for all \(f\in L^1(dw)\) such that \(\text {supp}\, f\subseteq B(0,r)\) and \(\textbf{x},\textbf{y} \in \mathbb {R}^N\), we have
$$\begin{aligned} |(x_j-y_j) (f*g)({\textbf{x}},{\textbf{y}})|\le C_2r w(B({\textbf{x}},1))^{-1/2} w(B({\textbf{y}}, 1))^{-1/2}\Vert f\Vert _{L^1(dw)}. \end{aligned}$$(3.13) -
(c)
Assume \(\delta >\varepsilon _1\). If g is G-invariant, \( \mathcal {F}g\in C^1({\mathbb {R}}^N)\), and \(|T_{j}\mathcal {F}g(\xi )|\le (1+\Vert \xi \Vert )^{-{\textbf{N}}-\delta }\) for all \(\xi \in \mathbb {R}^N\), then there is a constant \(C_3>0\) independent of f, g such that for all \(f\in L^1(dw)\) such that \(\text {supp}\, f\subseteq B(0,r)\) and \(\textbf{x},\textbf{y},\textbf{y}' \in \mathbb {R}^N\), we have
$$\begin{aligned} \begin{aligned}&|x_j-y_j|| (f*g)({\textbf{x}},{\textbf{y}})-(f*g)({\textbf{x}},{\textbf{y}}')|\\&\quad \le C_3r\Vert \textbf{y}-\textbf{y}'\Vert ^{\varepsilon _1} w(B({\textbf{x}},1))^{-1/2} w(B({\textbf{y}}, 1))^{-1/2}\Vert f\Vert _{L^1(dw)}\\&\qquad + C_3r\Vert \textbf{y}-\textbf{y}'\Vert ^{\varepsilon _1} w(B({\textbf{x}},1))^{-1/2} w(B({\textbf{y}}', 1))^{-1/2}\Vert f\Vert _{L^1(dw)}. \end{aligned} \end{aligned}$$(3.14)
Proof
Let \(U_0=B(0,1)\) and \(U_\ell =B(0,2^\ell ){\setminus } B(0,2^{\ell -1})\) for \(\ell \in \mathbb {N}\). In order to prove (3.12), we use the Cauchy–Schwarz inequality, (3.1), and (2.6) (cf. [8, Proposition 3.7]),
so (3.12) is proved. In order to prove (3.13), we use (3.8). We shall estimate the first component of the right-hand side of (3.8), the others are treated in the same way. Recall that \(\Vert \partial _j \mathcal {F}f\Vert _{L^{\infty }} \le { \textbf{c}_k^{-1}}r\Vert f\Vert _{L^1(dw)}\) (see (3.10)). Therefore, similarly as in (3.15), we obtain
We now turn to prove (3.14). We write
The required estimate for \(I_2\) follows from (3.12). To deal with \(I_1\), we use (3.8) and obtain
In order to estimate \(I_{1,1}\), we proceed similarly to (3.15) and (3.16). By the Cauchy–Schwarz inequality together with (3.1), (3.2), and (2.6) we have
The estimate for \(I_{1,3}\) goes identically. In order to deal with \(I_{1,2}\), we recall that
(see (3.11)). Moreover, \(\Vert \sigma _{\alpha }(\textbf{y})-\sigma _{\alpha }(\textbf{y}')\Vert =\Vert \textbf{y}-\textbf{y}'\Vert \) for all \(\textbf{y},\textbf{y}' \in \mathbb {R}^N\) and \(\alpha \in R\). Consequently, for \(I_{1,2}\) one can repeat the same proof as for \(I_{1,1}\). \(\square \)
Since any sufficiently regular function can be written as a convolution of a nice radial function with an \(L^1\)-function, as a consequence of Proposition 3.5 we obtain the following theorem.
Theorem 3.6
Let s be an even integer greater than \(\textbf{N}\). Then for any \(0 \le \varepsilon _1 < s-\textbf{N}\), \(\varepsilon _1 \le 1\), there is a constant \(C>0\) such that for all \(f \in C^{s}(\mathbb {R}^N)\) such that \(\text {supp }f \subseteq B(0,1)\), and for all \(\textbf{x},\textbf{y},\textbf{y}' \in \mathbb {R}^N\) we have
Proof
For \(\textbf{x},\textbf{y} \in \mathbb {R}^N\) we write
where \(J^{\{s\}}\) is defined in (2.39) and
Therefore, by (2.18) we have \(\widetilde{f}=(1-\Delta _k)^{s/2}f\). Consequently, by the assumption \(\text {supp }f \subseteq B(0,1)\) and Lemma 3.2, there is a constant \(C>0\) such that
Hence, applying Proposition 3.5 with \(\widetilde{f}\), \(J^{\{s\}}\) (which is G-invariant), \(\delta :=s-\textbf{N}\), and any \(0<\varepsilon _1 <\delta \) (the assumptions are satisfied thanks to the definition of \(J^{\{s\}}\) and (2.41)), we obtain (3.18) and (3.19). \(\square \)
4 Applications of Formulas and Estimates from Section 3
4.1 Estimates for Dunkl Translations of Schwartz-Class Functions
As a consequence of Theorem 3.6, we obtain the following theorem.
Theorem 4.1
Let s be an even integer greater than \(\textbf{N}\). Assume that for a certain \( \kappa \ge {-{\textbf{N}}/2-1}\) and a function \(g \in C^s(\mathbb {R}^N)\) one has
Then there is a constant \(C>0\) (independent of g) such that for all \(\textbf{x},\textbf{y} \in \mathbb {R}^N\) and \(t>0\) we have
where \( g_t({\textbf{x}})=t^{-{\textbf{N}}}g({\textbf{x}}/t)\).
Remark 4.2
Let us note that by (2.6), \( w(B({\textbf{x}}, t+d({\textbf{x}},{\textbf{y}})))^{-1} \le w(B({\textbf{x}}, {t}))^{-1} (1+d({\textbf{x}},{\textbf{y}})/t)^{ -N}\) hence, under assumptions of Theorem 4.1, we have
Proof of Theorem 4.1
By scaling it is enough to prove (4.2) for \(t=1\). Let \(\widetilde{\Psi }_0 \in C_c^{\infty }((-\frac{1}{2},\frac{1}{2}))\) and \(\widetilde{\Psi } \in C_c^{\infty }((\frac{1}{8},1))\) be such that
Then
where the convergence is in \(L^2(dw({\textbf{x}}))\). By continuity of the generalized translations on \(L^2(dw)\) for all \({\textbf{y}}\in {\mathbb {R}}^N\) we have
where the convergence is in \(L^2(dw({\textbf{x}}))\). We turn to prove that the series converges absolutely for all \({\textbf{x}},{\textbf{y}}\in {\mathbb {R}}^N\). Indeed, for fixed \(\ell \in \mathbb {N}_0\) we consider \(\widetilde{g}_\ell (\textbf{x})=g_\ell (2^\ell \textbf{x})\). Then \(\widetilde{g}_\ell \) is supported by B(0, 1) and it follows from (4.1) that there is a constant \(C>0\) such that for all \(\ell \in \mathbb {N}_0\) we have
Applying Theorem 3.6 we get
therefore, by scaling and (2.3),
Finally, by (2.6),
where in the last step we have used the fact that the quantities \(w(B(\textbf{x},d(\textbf{x},\textbf{y})+1))\) and \(w(B(\textbf{y},d(\textbf{x},\textbf{y})+1))\) are comparable. \(\square \)
Assume \(\varphi \in \mathcal {S}(\mathbb {R}^N)\). It follows from Theorem 4.1 that for any \(M>0\) there is a constant \(C_M>0\) such that for all \(\textbf{x},\textbf{y} \in \mathbb {R}^N\) we have
Moreover, if additionally a Schwartz class function \(\varphi \) is G-invariant, then
Let us remark that if g is radial then the bound for \(\tau _{{\textbf{x}}}(-{\textbf{y}})\) can be improved under a weaker assumption on g. This is stated in the following proposition.
Proposition 4.3
Assume that \(\kappa >2-N\) and \(\kappa >-{\textbf{N}}/2\). Then there is a constant \(C>0\) such that for all radial functions g satisfying \(|g({\textbf{x}})|\le (1+\Vert {\textbf{x}}\Vert )^{-{\textbf{N}}-\kappa }\) one has
Proof
The proof follows the same pattern as that of Theorem 4.1. To this end we note that from the estimates for the Dunkl heat kernel (1.7) and the fact that the generalized translation of a non-negative radial function is non-negative combined with Theorem 2.1 we have
where \(g_\ell \) are define as in (4.5). Summing up the estimates we arrive in the desired bound. \(\square \)
Now we provide the estimates for the Dunkl translations of the (non-necessarily radial) Schwartz-class functions \(\varphi \), which make use of the function \(\Lambda (\textbf{x},\textbf{y},1)\) (see (2.33)). The following lemma has an easy proof (see [12, 13]).
Lemma 4.4
For any sequence \(\{\sigma _j\}_{j=0}^{m}\) of elements of the group G, \(m \ge |G|^2+1\), satisfying the condition \(\sigma _0=\textrm{id}\) and
where \(g_{j+1} \in \{\textrm{id}\} \cup \{\sigma _{\alpha }\;:\; \alpha \in R\}\), and \(\textbf{x},\textbf{y} \in \mathbb {R}^N\), there is a sequence \(\varvec{\alpha } \in \mathcal {A}(\textbf{x},\textbf{y})\) of elements of R such that \(\ell (\varvec{\alpha }) \le |G|\) and for all \(t>0\) we have
Theorem 4.5
Let \(\varphi \in \mathcal {S}(\mathbb {R}^N)\) and \(M>0\). Let \(\varphi _t:=t^{-\textbf{N}}\varphi (\cdot /t)\) There is a constant \(C_{M,\varphi } >0\) such that for all \(\textbf{x},\textbf{y} \in \mathbb {R}^N\) and \(t>0\), we have
Proof
By scaling, without loss of generality, we may assume \(t=1\). It follows by (3.3) that there is a constant \(C>0\) independent of \(\textbf{x},\textbf{y} \in \mathbb {R}^N\) and \(\phi \in \mathcal {S}(\mathbb {R}^N)\) such that
where \(\phi _j\), \(\phi _{\alpha }\) are defined in (3.4).
Fix a function \(\varphi \) from the Schwartz class \({\mathcal {S}}({\mathbb {R}}^N)\). In the first step we estimate \(\varphi ({\textbf{x}},{\textbf{y}})\) by (4.14). In the second step we apply the formula (4.14) to \(\varphi _j\) and \(\varphi _{\alpha }\) obtaining
where \(\varphi _{j,j_1},\ \varphi _{j,\alpha '}, \ \varphi _{\alpha ,j_1}, \ \varphi _{\alpha ,\alpha '}\in {\mathcal {S}}({\mathbb {R}}^N)\). Then we continue this procedure with the use of (4.14) to estimate \(\varphi _{j,j_1},\ \varphi _{j,\alpha '}, \ \varphi _{\alpha ,j_1}, \ \varphi _{\alpha ,\alpha '}\) and so on. Set \(m=|G|^2\). Let \(\mathcal {B}\) be the set of all sequences \(\{\sigma _j\}_{j=0}^{m}\) of length \(m+1\) satisfying the assumptions of Lemma 4.4. Finally, after all together \((m+1)\)–steps described above, we get
where \(\psi _{g,\ell } \in \mathcal {S}(\mathbb {R}^N)\) and \(n=(N+|R|)^{m+1}\). Since \(d(\textbf{x},g(\textbf{y}))=d(\textbf{x},\textbf{y})\) (see (2.29)), by (4.7) we get
Moreover, by Lemma 4.4 we have
Hence, taking into account (4.15), (4.16), and (4.17) we obtain (4.13). \(\square \)
4.2 Singular Integral Operators
Basic examples of singular integral operators are Riesz transforms. The Riesz transforms
in the Dunkl setting were studied by Thangavelu and Xu [27] (in dimension 1 and in the product case) and by Amri and Sifi [2] (in higher dimensions) who proved the bounds on \(L^p(dw)\) spaces. Further, in [4] the Riesz transforms \({\mathcal {R}}_j\) were used for characterization of the Hardy space \(H^1_{\Delta _k}\).
Recently, some various approaches to the theory of singular integrals, which use the \(d(\textbf{x},\textbf{y})\), \(\Vert \textbf{x}-\textbf{y}\Vert \) and \(w(B(\textbf{x},1))\) were investigated. For instance, in [10], the convolution–type singular integrals \(f \mapsto K*f\) were studied under some assumptions on the kernel K (see (A), (D), and (L) in Sect. 4.2.1 below). On the other hand, in [25], the authors proposed certain assumptions on kernels of non-necessarily convolution–type singular integrals (see (CZ1), (CZ2), (CZ3) below) which are relevant for proving some harmonic analysis spirit results in the Dunkl setting. As an example, it was proved there that the kernels of Riesz transforms \({\mathcal {R}}_j\) have the expected properties. In this section, we will use the results of Sect. 3 to unify these two approaches and prove that the kernel estimates of [25] are satisfied for the Dunkl type convolution operators considered in [10]. Consequently, we obtain a large class of examples of operators satisfying the assumptions (CZ1), (CZ2), and (CZ3). Moreover, thanks to the results of [25], we obtain several Fourier analysis spirit theorems for the convolution type operators.
4.2.1 Assumptions of [10]
Let \(s_0\) be an even positive integer larger than \(\textbf{N}\), which will be fixed in this section. Consider a function \(K\in C^{s_0} ({\mathbb {R}}^N{\setminus } \{0\})\) such that
Set
where \(\phi \) is a fixed radial \(C^\infty \)-function supported by the unit ball B(0, 1) such that \(\phi ({\textbf{x}})=1\) for \(\Vert {\textbf{x}}\Vert <1/2\). It was proved in [10, Theorems 4.1 and 4.2] that under (A) and (D) the operators \(f \mapsto f*K^{\{t\}}\) are bounded on \(L^p(dw)\) for \(1<p<\infty \) and they are of weak–type (1, 1) with the bounds independent of \(t>0\). Further, assuming additionally (L), the limit \(\lim _{t\rightarrow 0} f*K^{\{t\}} ({\textbf{x}})\) exists and defines a bounded operator \({\textbf{T}}\) on \(L^p(dw)\) for \(1<p<\infty \), which is of weak-type (1,1) as well [10, Theorem 4.3 and Theorem 3.7]. Moreover, in this case, the maximal operator
is bounded on \(L^p(dw)\) for \(1<p<\infty \) and of weak-type (1, 1) (Theorem 5.1 of [10]).
4.2.2 Assumptions of [25]
In [25] (see also [15]) the following definition of Dunkl–Calderón–Zygmund singular integral operators was proposed. Let \(\eta >0\). Let \(\dot{C}_0^{\eta }(\mathbb {R}^N)\) denote the space of continuous functions f with compact support satisfying
We say that a sequence \(\{f_n\}_{n \in \mathbb {N}}\) converges to f in \( \dot{C}^{\eta }_0(\mathbb {R}^N) \), if the functions are supported in the same compact set in \({\mathbb {R}}^N\) and \(\lim _{n\rightarrow \infty } \Vert f_n-f\Vert _\eta =0\). Let \(\dot{C}^{\eta }_0(\mathbb {R}^N)'\) be its dual space endowed with weak-* topology. An operator \(\textbf{T}:\dot{C}^{\eta }_0(\mathbb {R}^N) \longmapsto \dot{C}^{\eta }_0(\mathbb {R}^N)'\) is said to be a Dunkl–Calderón-Zygmund singular integral operator associated with a kernel \(\mathcal {K}(\textbf{x},\textbf{y})\) (which is not necessary the Dunkl translation of some function) if the following estimates are satisfied: for some \(0 <\varepsilon \le 1\):
and, furthermore,
We finish this subsection by the remark that the conditions (CZ1), (CZ2), and (CZ3) imply the following Calderón-Zygmund integral bounds for \({\mathcal {K}}({\textbf{x}},{\textbf{y}})\) on the space of homogeneous type \(({\mathbb {R}}^N,\Vert {\textbf{x}}-{\textbf{y}}\Vert ,dw)\) (see [25]): there is a constant \(A>0\) such that for all \(r>0\) one has
4.2.3 Assumptions (CZ1), (CZ2), and (CZ3) for Convolution Kernels
Theorem 4.6
Assume that a kernel \(K\in C^{s_0} ({\mathbb {R}}^N{\setminus } \{0\})\) satisfies (D) for a certain even integer \(s_0>{\textbf{N}}\). Then the kernel defined by
for \(\textbf{x},\textbf{y} \in \mathbb {R}^N\), \(\textbf{x} \ne \textbf{y}\), satisfies the assumptions (CZ1), (CZ2), and (CZ3) with some \(0<\varepsilon <\min (1,s_0-\textbf{N})\). Moreover, if additionally (A) and (L) are satisfied, then \({\textbf{K}}({\textbf{x}},{\textbf{y}})\) is a kernel associated with the Dunkl-Calderón–Zygmund operator \({\textbf{T}}\).
Proof
Let \(0<\varepsilon <\min (1,s_0-\textbf{N})\). For any \(t>0\) let us denote
Then \(K^{\{t/2,t\}}\) is \(C^{s_0}({\mathbb {R}}^N)\)-function supported by \(B(0,t) \setminus B(0,t/4)\) (cf. [10, (3.1)]), hence \(\mathcal {F}K^{\{t/2,t\}}\in L^{1}(dw)\). Firstly, let us consider \(K^{\{t/2,t\}}\) for \(t=1\). By Theorem 3.6 applied with \(s=s_0\), \(\varepsilon _1=\varepsilon \), and assumption (D) there is a constant \({\widetilde{C}}>0\) such that
for all \(\textbf{x},\textbf{y},\textbf{y}' \in \mathbb {R}\). For the other \(t>0\), note that \(K_t({\textbf{x}})=t^{-{{\textbf{N}}}}K({\textbf{x}}/t)\) satisfies the assumption (D) with the same constants \(C_{\beta }\) as K. Hence, proceeding by scaling, for all \(\textbf{x},\textbf{y},\textbf{y}' \in \mathbb {R}^N\) we obtain
We now turn to prove that \({\textbf{K}}({\textbf{x}},{\textbf{y}})\) is well defined (see (4.21)). Since \(\text {supp }K^{\{t/2,t\}} \subseteq B(0,t)\), by Theorem 2.1 concerning the support of the Dunkl translated function, we have
For \({\textbf{x}},{\textbf{y}} \in \mathbb {R}^N\) such that \(d(\textbf{x}, \textbf{y})>0\), let us set
where the series converges absolutely. Indeed, thanks to (4.24) and then (2.6) we have
where we have used the fact that dw is G-invariant and doubling (see (2.5)), so the quantities \(w(B(\textbf{x},d(\textbf{x},\textbf{y})))\) and \(w(B(\textbf{y},d(\textbf{x},\textbf{y})))\) are comparable. Since \(\tau _{{\textbf{x}}}\) is a contraction on \(L^2(dw)\), we conclude that
for any fixed \({\textbf{x}} \in \mathbb {R}^N\) with convergence in \(L^2(dw({\textbf{y}}))\). Now, from (4.24) and (4.26) we deduce that for \(t<d({\textbf{x}},{\textbf{y}})/4\) we have
hence the limit in (4.21) exists and \({\mathcal {K}}({\textbf{x}},{\textbf{y}})=\textbf{K}({\textbf{x}},{\textbf{y}})\) for \(d({\textbf{x}},{\textbf{y}})>0\).
We now prove that \({\textbf{K}}({\textbf{x}},{\textbf{y}})\) is the kernel associated with the operator \({\textbf{T}}\). To this end let \(f,g\in L^2(dw)\) be such that g is compactly supported and \(\text {supp }g\cap \text {supp }f=\emptyset \). Then there is \(\eta >0\) such that \(\Vert {\textbf{x}}-{\textbf{y}}\Vert >\delta \) for \({\textbf{y}}\in \text {supp}\, f\) and \({\textbf{x}}\in \text {supp}\, g\). Thus, from the results stated in Sect. 4.2.1, we have
The functions \(K^{\{2^{-\ell \}}}({\textbf{x}},{\textbf{y}})f({\textbf{y}})g({\textbf{x}})\) converge pointwise to \({\mathcal {K}}({\textbf{x}},{\textbf{y}})f({\textbf{y}})g({\textbf{x}})\) and are dominated by the integrable function
since g has compact support. Hence, (4.18) holds, by the Lebesgue dominated convergence theorem.
The proof of (CZ2) is similar but it uses (4.25) instead of (4.24). Indeed, assume \(\Vert \textbf{y}-\textbf{y}'\Vert <\frac{d(\textbf{x},\textbf{y})}{2}\). Then \(\frac{1}{2}d(\textbf{x},\textbf{y}) \le d(\textbf{x},\textbf{y}')\) and, by Theorem 2.1,
Consequently, by (4.25),
where we have used the fact that thank to the assumption \(\Vert \textbf{y}-\textbf{y}'\Vert <\frac{d(\textbf{x},\textbf{y})}{2}\) the quantities \(w(B(\textbf{x},d(\textbf{x},\textbf{y})))\), \(w(B(\textbf{y},d(\textbf{x},\textbf{y})))\), and \(w(B(\textbf{y}',d(\textbf{x},\textbf{y})))\) are comparable. Finally, (CZ3) is a consequence of the fact \(K(\textbf{x},\textbf{y})=K( -\textbf{y}, -\textbf{x})\). \(\square \)
4.3 Dunkl Transform Multiplier Operators
Our aim of this subsection is to prove that for bounded functions m the Dunkl transform multiplier operators \(f\mapsto {\mathcal {F}}^{-1}(m(\xi )\mathcal {F}f(\xi ))\) admit associated kernels \(K({\textbf{x}},{\textbf{y}})\) satisfying (depending on the regularity of m) (CZ1)–(CZ3) or (4.19)–(4.20).
4.3.1 Multipliers-Pointwise Type Estimates
For an \(L^1(dw)\)-function f we set
Theorem 4.7
Assume n is a positive integer and \(0<\varepsilon \le 1\). There is a constants \(C>0\) such that for \(f \in C^{n}(\mathbb {R}^N)\) such that \(\text {supp }f \subseteq B(0,4)\) and for all \(\textbf{x},\textbf{y}, {\textbf{y}}' \in \mathbb {R}^N\), \(\Vert {\textbf{y}}-{\textbf{y}}'\Vert \le {1}\), we have
For the proof we need the following lemma.
Lemma 4.8
Let n be a non-negative integer. Then there is a constant \(C_{n}>0\) such that for \(f\in C^n(\mathbb {R}^N)\), \(\text {supp}\, f\subseteq B(0,4)\), and \(\textbf{x},\textbf{y} \in \mathbb {R}^N\) one has
Proof of Lemma 4.8
The proof goes by induction. If \(n =0\), then using the Cauchy-Schwarz inequality, (3.1), and (2.6) we get
Now assume that the inequality (4.32) holds for n. Let \(f\in C^{n+1}(\mathbb {R}^N)\), \(\text {supp}\, f\subseteq B(0,4)\). Then the functions \(f_j=\partial _j f\in C^{n}(\mathbb {R}^N)\) and \(f^{\{\alpha \}}\in C^n(\mathbb {R}^N)\) (see (2.8)) are supported in B(0, 4) and
(see Lemma 3.2). The same calculation as in the proof of Proposition 3.3 gives
Recall that by (2.2) and (2.29) for all \(\sigma \in G\) we have \(w(B(\sigma ({\textbf{y}}),1))=w(B({\textbf{y}}, 1))\) and \( d({\textbf{x}},\sigma ({\textbf{y}}))=d({\textbf{x}},{\textbf{y}})\). Using (4.35), (4.34), and the induction hypothesis we deduce that
\(\square \)
Proof of Theorem 4.7
We start by proving (4.30) first. Let \(f_j\), \(f^{\{\alpha \}}\) be as in the proof of Lemma 4.8. Then, by (4.35), (4.34), and Lemma 4.8 applied to \(f_j\), \(f^{\{\alpha \}}\) we get
so (4.30) is proved. Now let us prove (4.31). Fix \(0<\varepsilon \le 1.\) Consider \({\textbf{x}},{\textbf{y}},{\textbf{y}}' \in \mathbb {R}^N\), \(\Vert {\textbf{y}}-{\textbf{y}}'\Vert \le \frac{d(\textbf{x},\textbf{y})}{2}\). Let \({\tilde{f}}(\xi )=f(\xi )e^{\Vert \xi \Vert ^2}\). Then \(\text {supp}\, {\tilde{f}}\in B(0,4)\), \(\Vert \widetilde{f}\Vert _{C^n(\mathbb {R}^N)} \le C'_n\Vert f\Vert _{C^n(\mathbb {R}^N)}\), and
Applying (4.30) to \(\widetilde{f}\) and then (2.36), we obtain
Since \(\Vert {\textbf{y}}-{\textbf{y}}'\Vert \le 1\), for all \(\textbf{z} \in \mathbb {R}^N\) we have
It follows from the estimate on the heat kernel (see either (1.7) or Theorem 2.2) that
So we conclude the desired inequality (4.31) from (4.37) and (4.38), because \(w(B({\textbf{y}},1))\sim w(B({\textbf{y}}',1))\). \(\square \)
Corollary 4.9
Suppose that \(n \in \mathbb {N}\) is the smallest integer such that \(n>\textbf{N}\) and \(m \in C^{n}(\mathbb {R}^N {\setminus } \{0\})\) satisfies the following Mihlin–type condition: for all \(\beta \in \mathbb {N}_0^{N}\), \(|\beta | \le n\) there is a constant \(C_{\beta }>0\) such that
Then the integral kernel \(K(\textbf{x},\textbf{y})\) of the multiplier operator \(\mathcal {T}_mf=\mathcal {F}^{-1}((\mathcal {F}f)m)\) satisfies the conditions (CZ1), (CZ2), (CZ3).
Proof
Let \(\phi \) be a radial \(C^\infty ({\mathbb {R}}^N)\) function, \(\text {supp}\, \phi \subseteq B(0,4)\setminus B(0,1/4)\), which forms a resolution of the identity, that is,
We write
Then \(K(\textbf{x},\textbf{y})=\sum _{\ell \in \mathbb {Z}}K_{\ell }(\textbf{x},\textbf{y})\) and, by homogeneity,
Let us note that the functions \(m_{\ell }\) are supported by B(0, 4). Moreover, it follows from (4.39) that \(\sup _{\ell \in \mathbb {Z}}\Vert m_{\ell }\Vert _{C^{n}(\mathbb {R}^N)} \le C\). Therefore, by Theorem 4.7, (2.3), and (2.6),
Similarly, using (4.31) if \(\Vert 2^{\ell }\textbf{y}-2^{\ell }\textbf{y}'\Vert \le 1\), and (4.30) if \(\Vert 2^{\ell }\textbf{y}-2^{\ell }\textbf{y}'\Vert > 1\), we get
Finally, (CZ1) follows from (4.41). Indeed, fix \(0<\varepsilon \le 1\), \(\varepsilon <N\). Then
The proof of (CZ2) with \(\varepsilon \le n-{\textbf{N}}\), \(0<\varepsilon \le 1\), follows the pattern presented in (4.43) but it uses (4.42) instead of (4.41).
Finally, (CZ3) is a consequence of the fact that \(K(\textbf{x},\textbf{y})=K( -\textbf{y}, -\textbf{x})\). \(\square \)
4.3.2 Multipliers-Integral Type Estimates
Let m be a bounded function on \({\mathbb {R}}^N\) which for a certain \(s>{\textbf{N}}\) satisfies
where \(\psi \in C^\infty ({\mathbb {R}}^N)\) is a fixed radial function \(\text {supp}\, \psi \subseteq \{\xi \in \mathbb {R}^N\;:\; 1/4\le \Vert \xi \Vert \le 4\}\), \(\psi (\xi ) = 1\) for all \(\xi \in \mathbb {R}^N\) such that \(1/2\le \Vert \xi \Vert \le 2\), and
denotes the classical Sobolev norm of the classical Sobolev space \(W^s_2(\mathbb {R}^N,d\textbf{x})\). It was proved in [8, Theorem 1.2] that the Dunkl multiplier operator
originally defined on \(L^2(dw)\cap L^p(dw)\), has a unique extension to a bounded operator on \(L^p(dw)\) for \(1<p<\infty \). Moreover, \(\mathcal {T}_m\) is of weak-type (1,1) and bounded on the relevant Hardy space. In order to prove the results the authors considered the integral kernels (see [8, (5.3)]):
where \(\phi \) is a radial \(C^\infty ({\mathbb {R}}^N)\) function, \(\text {supp}\, \phi \subseteq B(0,4)\setminus B(0,1/4)\), which forms a resolution of the identity as in (4.40) and showed the following estimates with respect to \(d({\textbf{x}},{\textbf{y}})\) (see [8, formulas (5.8), (5.10), and (5.11)]): there are \(\delta >0\) and \(C>0\) such that for all \(\textbf{y},\textbf{y}' \in \mathbb {R}^N\) we have
The estimates imply that for every ball \(B=B({\textbf{x}}_0,r)\) one has
for all \({\textbf{y}},{\textbf{y}}'\in B\). Here \(B^*=B({\textbf{x}}_0, 2r)\) and \({\mathcal {O}}(B^*)=\{\sigma ({\textbf{x}}):\sigma \in G,\ {\textbf{x}}\in B^*\}\). The bounds (4.46)–(4.48) play crucial roles in proving the Hörmander’s multiplier theorem ([8, Theorem 1.2]).
In this subsection we will prove the following proposition.
Proposition 4.10
Suppose that m is as in [8, Theorem 1.2], that is, (4.44) holds for a certain \(s>{\textbf{N}}\). Let \(K_\ell \) be defined by (4.45). Then the integral kernel \(K({\textbf{x}},{\textbf{y}}):=\sum _{\ell \in \mathbb {Z}} K_\ell ({\textbf{x}},{\textbf{y}})\) associated with the multiplier \(\mathcal {T}_m\) satisfies the Calderón–Zygmund integral conditions (4.19) and (4.20).
In other words, \({\mathcal {T}}_m\) is a Calderón-Zygmund operator on the space of homogeneous type \(({\mathbb {R}}^N, \Vert {\textbf{x}} -{\textbf{y}}\Vert , dw)\).
Proof
Fix \(s_2>{\textbf{N}}+1\) (sufficiently large) and assume that \(\eta \in W^{s_2}_2({\mathbb {R}}^N, d{\textbf{x}})\), \( \text {supp}\, \eta \subseteq B(0,4)\). Then
(cf. Lemma 3.2). Applying the technique from the proof of Proposition 3.3, for all \(j \in \{1,2,\ldots ,N\}\) we have
Since \(s_2-1>{\textbf{N}}\), it follows from (5.10) of [8] (see (4.46)) that for all \(\textbf{y} \in \mathbb {R}^N\) we have
Consequently, from (4.51) and (4.52) we conclude
Further, if \(s_1>{\textbf{N}}\) and \(\eta \in W^{{s_1}}_2({\mathbb {R}}^N, d\textbf{x})\), \(\text {supp}\,\eta \subseteq B(0,4)\), then (5.10) of [8] (see also (4.46)) implies
Now, (4.53) and (4.54) together with the interpolation argument of Mauceri and Meda [18] (see also [3, Proposition 5.3]) give that if \(s>{\textbf{N}}\), then there are constants \(C>0\) and \(0<\theta <1\) such that for all \(\eta \in W^{s}_2({\mathbb {R}}^N,d\textbf{x})\) supported in B(0, 4), and for all \(\textbf{y} \in \mathbb {R}^N\) we have
Hence, by scaling, for all \(\ell \in \mathbb {Z}\) and \(\textbf{y} \in \mathbb {R}^N\) we have
Consequently,
Further, it follows from Lemma 3.1 (see Proposition 3.7 of [8]) that
By (2.5), \(w(B({\textbf{x}},2^{-\ell }))\sim w(B({\textbf{y}},2^{-\ell }))\), if \(\Vert {\textbf{x}}-{\textbf{y}}\Vert <2r\le 2^{\ell +1}.\) So applying (4.58) and (2.6), we get
Thus (4.19) is proved.
In order to prove (4.20) we observe that (4.48) together with (4.56) give
whenever \({\textbf{y}},{\textbf{y}}'\in B({\textbf{y}}_0,r)\). Finally (4.20) follows from (4.59). \(\square \)
4.4 Non-positivity of Dunkl Translation Operators
In this subsection, we will use Proposition 3.3 to prove that for any root system R and a multiplicity function \(k>0\) there is \(\textbf{x} \in \mathbb {R}^N\) such that \(\tau _{\textbf{x}}\) is not a positive operator (see Theorem 4.11 for details). If \(G=\mathbb {Z}_2\), the result follows from the explicit formula for \(\tau _{\textbf{x}}\) (see [19]). For G being symmetric group, the result was proved by Thangavelu and Xu (see [26, Proposition 3.10]).
Theorem 4.11
For any \(N \in \mathbb {N}\) there is a sequence of N non-negative functions \(\{\varphi _j\}_{j=1}^N\), \(\varphi _j \in C^{\infty }(\mathbb {R}^N)\), such that for any system of roots \(R \subset \mathbb {R}^N\) and any positive multiplicity function k, at least one \(\varphi _j\) satisfies the following property: there are \(\textbf{x},\textbf{y} \in \mathbb {R}^N\) such that \(\varphi _j(\textbf{x},\textbf{y})<0\).
Proof
Let \(\varphi \in C^{\infty }(\mathbb {R}^N)\) be a radial function (\(\varphi (\textbf{x})=\widetilde{\varphi }(\Vert \textbf{x}\Vert )\)) supported by B(0, 1/2) such that \(0 \le \varphi (\textbf{x}) \le 1\) for all \(\textbf{x} \in \mathbb {R}^N\) and \(\varphi \equiv 1\) on B(0, 1/4). For \(1 \le j \le N\) we set
Since \(\varphi \) is supported by B(0, 1/2), the functions \(\varphi _j\) are non-negative. Then, using (3.5), for all \(\textbf{x},\textbf{y} \in \mathbb {R}^N\) we have
Take any \(\alpha \in R\) and let \(1 \le j \le N\) be such that \(\langle \alpha ,e_j \rangle \ne 0\). Then, by (2.1), for any \(\textbf{x} \in \mathbb {R}^N\) we get
One the one hand, let us note that for all \(\textbf{x} \in \mathbb {R}^N\) we have
Indeed, thanks to (2.22), the fact that \(\varphi \equiv 1\) on B(0, 1/4), and Theorem 2.4 we get
On the other hand, for any \(\alpha \in \mathbb {R}^N\) such that \(\langle \alpha ,e_j \rangle \ne 0\) there is \(\textbf{x} \in \mathbb {R}^N\) such that
Consequently, for such a \(\textbf{x}\), from (4.61), (4.62), and (4.63), we obtain our claim. \(\square \)
Remark 4.12
The result that the generalized translations do not preserve positivity of some functions can be also obtained using the generalized heat kernel and Theorem 2.2. To this end let us observe that here is a constant \(C_1>0\) such that for all \(\textbf{x} \in \mathbb {R}^N\) we have
where \(h_t(\textbf{x})\) is defined in (2.27). We now set
Then, thanks to (4.64), the function \(\varphi _j\) is non-negative. Further, by (3.5) together with Theorem 2.2 (recall that \(d({\textbf{x}},\sigma _{\alpha }({\textbf{x}}))=0\)), we get
Finally, by (2.26), we have \(h_1(\textbf{x},\sigma _{\alpha }(\textbf{x}))>0\) and (if \(\langle \alpha ,e_j \rangle \ne 0\)) one can take \(\textbf{x} \in \mathbb {R}^N\) such that \(C_2+\langle \alpha ,e_j \rangle \langle \textbf{x},\alpha \rangle <0\). Consequently, \(\varphi _j(\textbf{x},\sigma _{\alpha }(\textbf{x}))<0\).
References
Amri, B., Hammi, A.: Dunkl–Schrödinger operators. Complex Anal. Oper. Theory 113, 1033–1058 (2019)
Amri, B., Sifi, M.: Riesz transforms for Dunkl transform. Ann. Math. Blaise Pascal 19(1), 247–262 (2012)
Anker, J.-Ph., Ben Salem, N., Dziubański, J., Jacek, Hamda, N.: The Hardy space H1 in the rational Dunkl setting. Constr. Approx. 42(1), 93–128 (2015)
Anker, J.-Ph., Dziubański, J., Hejna, A.: Harmonic functions, conjugate harmonic functions and the Hardy space \(H^1\) in the rational Dunkl setting. J. Fourier Anal. Appl. 25, 2356–2418 (2019)
de Jeu, M.F.E.: The Dunkl transform. Invent. Math. 113, 147–162 (1993)
Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Am. Math. 311(1), 167–183 (1989)
Dunkl, C.F.: Hankel transforms associated to finite reflection groups. In: Proceeding of the special session on hypergeometric functions on domains of positivity, Jack polynomials and applications, Proceedings, Tampa 1991, Contemp. Math. vol. 138, pp. 123–138 (1989)
Dziubański, J., Hejna, A.: Hörmander’s multiplier theorem for the Dunkl transform. J. Funct. Anal. 277, 2133–2159 (2019)
Dziubański, J., Hejna, A.: Remark on atomic decompositions for the Hardy space \(H^1\) in the rational Dunkl setting. Studia Math. 251(1), 89–110 (2020)
Dziubański, J., Hejna, A.: Singular integrals in the rational Dunkl setting. Revista Matemática Complutense 12, 1–27 (2021)
Dziubański, J., Hejna, A.: Upper and lower bounds for Littlewood-Paley square functions in the Dunkl setting. Studia Math. 262, 275–303 (2022)
Dziubański, J., Hejna, A.: Upper and lower bounds for Dunkl heat kernel, arXiv:2111.03513, to appear in Calculus of Variation and Partial Differential Equations
Dziubański, J., Hejna, A.: On Dunkl Schrödinger semigroups with Green bounded potentials, arXiv:2204.03443
Gallardo, L., Rejeb, Ch.: Support properties of the intertwining and the mean value operators in Dunkl theory. Proc. Am. Math. Soc. 146(1), 145–152 (2018)
Han, Y., Lee, M-Y., Li, J., Wick, B.D.: Riesz transform and commutators in the Dunkl setting, preprint arXiv:2105.11275
Hofmann, S., Lu, G.Z., Mitrea, D., Mitrea, M., Yan, L.X.: Hardy spaces associated with non-negative self-adjoint operators satisfying Davies–Gafney estimates. Mem. Am. Math. Soc. 1007, 214 (2011)
Jiu, J., Li, Z.: On the representing measures of Dunkl’s intertwining operator. J. Approx. Theory 269, 105605, 10 (2021)
Mauceri, G., Meda, S.: Vector-valued multipliers on stratified groups. Rev. Mat. Iberoam. 6(3–4), 141–154 (1990)
Rösler, M.: Bessel-type signed hypergroups on \(\mathbb{R} \). In: Heyer, H., Mukherjea, A. (eds.) Probability measures on groups and related structures XI, Proceeding of Oberwolfach, vol. 1994, pp. 292–304. World Scientific, Singapore (1995)
Rösler, M.: Generalized Hermite polynomials and the heat equation for Dunkl operators. Commun. Math. Phys. 192, 519–542 (1998)
Rösler, M.: Positivity of Dunkl’s intertwining operator. Duke Math. J. 98(3), 445–463 (1999)
Rösler, M.: A positive radial product formula for the Dunkl kernel. Trans. Am. Math. Soc. 355(6), 2413–2438 (2003)
Rösler, M.: Dunkl operators (theory and applications). In: Koelink, E., Van Assche, W. (eds.) Orthogonal polynomials and special functions (Leuven, 2002), 93–135. Lecture Notes in Mathematics 1817, Springer(2003)
Rösler, M., Voit, M.: Dunkl theory, convolution algebras, and related Markov processes. In: Graczyk, P., Rösler, M., Yor, M. (eds.) Harmonic and Stochastic Analysis of Dunkl Processes, Travaux en Cours, pp. 1–112. Hermann, Paris (2008)
Tan, Ch., Han, Y., Han, Y., Lee, M.-Y., Li, J.: Singular integral operators, \(T1\) theorem, Littlewood–Paley theory and Hardy spaces in Dunkl Setting, arXiv:2204.01886
Thangavelu, S., Xu, Y.: Convolution operator and maximal function for the Dunkl transform. J. Anal. Math. 97, 25–55 (2005)
Thangavelu, S., Xu, Y.: Riesz transform and Riesz potentials for Dunkl transform. J. Comput. Appl. Math. 199(1), 181–195 (2007)
Trimèche, K.: Paley–Wiener theorems for the Dunkl transform and Dunkl translation operators. Integral Transforms Spec. Funct. 13(1), 17–38 (2002)
Acknowledgements
The authors thank Suman Mukherjee for a careful reading of an earlier draft.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Veluma Thangavelu.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Dziubański, J., Hejna, A. Remarks on Dunkl Translations of Non-radial Kernels. J Fourier Anal Appl 29, 52 (2023). https://doi.org/10.1007/s00041-023-10034-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00041-023-10034-2
Keywords
- Rational Dunkl theory
- Dunkl transform
- Heat kernels
- Root systems
- Generalized translations
- Singular integrals
- Multipliers