---
abstract: 'On $\mathbb R^N$ equipped with a normalized root system $R$ and a multiplicity function $k\geq 0$ let us consider a (non-radial) kernel $K(\mathbf x)$ which has properties similar to those from the classical theory. We prove that a singular integral Dunkl convolution operator associated with the kernel $K$ is bounded on $L^p$ for $1<p<\infty$ and of weak-type (1,1). Further we study a maximal function related to the Dunkl convolutions with truncation of $K$.'
address: 'J. Dziubański and A. Hejna, Uniwersytet Wrocławski, Instytut Matematyczny, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland'
author:
- Jacek Dziubański and Agnieszka Hejna
title: Singular integrals in the rational Dunkl setting 
---

[^1]

Introduction
============

The aim of this note is to study singular integral convolution operators in the Dunkl setting. We fix a normalized root system $R$ in $\mathbb R^N$ and a multiplicity function $k\geq 0$. Let $dw(\mathbf x)$ denote the associated measure and $\mathbf N$ the homogeneous dimension (see Section \[sec:preliminaries\]). For a positive integer $s$ consider a kernel $K\in C^{s} (\mathbb R^N\setminus \{0\})$ such that $$\label{eq:uni_on_annulus}\tag{A} \sup_{0<a<b<\infty} \Big| \int_{a<\|\mathbf x\|<b} K(\mathbf x)\, dw(\mathbf x)\Big|<\infty,$$ $$\label{eq:assumption1}\tag{D} 
\Big|\frac{\partial^\beta}{\partial \mathbf x^\beta} K(\mathbf x)\Big|\leq C\|\mathbf x\|^{-\mathbf N-|\beta|} \quad \text{for all} \ |\beta |\leq s.$$ Set $$K^{\{t\}}(\mathbf x)=K(\mathbf x)(1-\phi(t^{-1} \mathbf x)),$$ where $\phi$ is a fixed radial $C^\infty$-function supported by the unit ball $B(0,1)$ such that $\phi (\mathbf x)=1$ for $\|\mathbf x\|<1/2$. We prove that if $s$ is sufficiently large, then there are constants $C_p>0$ independent of $t>0$ such that $$\begin{aligned}
     \| f*K^{\{t\}} \|_{L^p(dw)}\leq C_p\| f\|_{L^p(dw)} \quad \text{for} \ 1<p<\infty
 \end{aligned}$$ and $$\begin{aligned}
    w(\{\mathbf x\in\mathbb R^N:| f*K^{\{t\}}(\mathbf x)|>\lambda\} )\leq C_1\lambda^{-1}\| f\|_{L^1(dw)}\end{aligned}$$ (Theorems \[theorem:weak\_type\_truncated1\] and \[theorem:strong\_type\_truncated1\]), where the symbol $*$ denotes the Dunkl convolution. We also prove that under the additional assumption $$\label{eq:limitA}\tag{L} 
     \lim_{\varepsilon \to 0} \int_{\varepsilon <|\mathbf x|<1} K(\mathbf x)\, dw(\mathbf x)=L,$$ where $L\in \mathbb C$, the limit $\lim_{t\to 0} f*K^{\{t\}} (\mathbf x)$ exists and defines a bounded operator on $L^p(dw)$ for $1<p<\infty$, which is of weak-type (1,1) as well (Theorem \[theorem:weak\_type\_K\], see also Theorem \[theorem:main\_L2\]). Moreover, in this case, the maximal operator $$K^*f(\mathbf x)=\sup_{t>0} |f*K^{\{t\}}(\mathbf x)|$$ is bounded on $L^p(dw)$ for $1<p<\infty$ and of weak-type (1,1) (Theorem \[theorem:maximal\]).

If $k\equiv 0$, then $dw$ is the Lebesgue measure in $\mathbb R^N$ and the Dunkl convolution reduces to the classical one. So the the above results are well known and $s=1$ suffices in this case (see i.e. [@Duo Chapter 5], [@St1], [@St2]). However, in the general case of $R$ and $k$ the main difficulty which one faces trying to study singular integral operators in the Dunkl setting lies in the lack of knowledge about boundendess of the so called Dunkl translations $\tau_{\mathbf x}$ on $L^p(dw)$-spaces for $p\ne 2$. Consequently, it is not known if for a fixed non-radial $L^1$-function $f$ the Dunkl convolution operator $g\mapsto f*g$ is bounded on $L^p(dw)$. The recent observations made in [@DzH] allow us to obtain some knowledge for the functions $\tau_{\mathbf y}f(\mathbf x)$ provided $f$ satisfies certain regularity conditions in smoothness and decay. In the present paper we explore and extend these ideas of [@DzH] for proving boundedness of singular integral convolution operators provided $s=s_{0}$ in , where $s_0$ is the smallest even integer bigger than $\mathbf{N}/2$.

Preliminaries and notation {#sec:preliminaries}
==========================

The Dunkl theory is a generalization of the Euclidean Fourier analysis. It started with the seminal article [@Dunkl] and developed extensively afterwards (see e.g. [@RoeslerDeJeu], [@Dunkl0], [@Dunkl3], [@Dunkl2], [@GR], [@Roesler2], [@Roesle99], [@Roesler2003], [@ThangaveluXu], [@Trimeche2002]). In this section we present basic facts concerning the theory of the Dunkl operators. For details we refer the reader to [@Dunkl], [@Roesler3], and [@Roesler-Voit].

We consider the Euclidean space $\mathbb R^N$ with the scalar product $\langle\mathbf x,\mathbf y\rangle=\sum_{j=1}^N x_jy_j
$, $\mathbf x=(x_1,...,x_N)$, $\mathbf y=(y_1,...,y_N)$, and the norm $\| \mathbf x\|^2=\langle \mathbf x,\mathbf x\rangle$. For a nonzero vector $\alpha\in\mathbb R^N$, the reflection $\sigma_\alpha$ with respect to the hyperplane $\alpha^\perp$ orthogonal to $\alpha$ is given by $$\begin{aligned}
\sigma_\alpha (\mathbf x)=\mathbf x-2\frac{\langle \mathbf x,\alpha\rangle}{\| \alpha\| ^2}\alpha.\end{aligned}$$ In this paper we fix a normalized root system in $\mathbb R^N$, that is, a finite set $R\subset \mathbb R^N\setminus\{0\}$ such that $\sigma_\alpha (R)=R$ and $\|\alpha\|=\sqrt{2}$ for every $\alpha\in R$. The finite group $G$ generated by the reflections $\sigma_\alpha \in R$ is called the [*Weyl group*]{} ([*reflection group*]{}) of the root system. A [*multiplicity function*]{} is a $G$-invariant function $k:R\to\mathbb C$ which will be fixed and $\geq 0$ throughout this paper. Let $$\begin{aligned}
dw(\mathbf x)=\prod_{\alpha\in R}|\langle \mathbf x,\alpha\rangle|^{k(\alpha)}\, d\mathbf x\end{aligned}$$ be the associated measure in $\mathbb R^N$, where, here and subsequently, $d\mathbf x$ stands for the Lebesgue measure in $\mathbb R^N$. We denote by $\mathbf N=N+\sum_{\alpha \in R} k(\alpha)$ the homogeneous dimension of the system. Clearly, $$\begin{aligned}
 w(B(t\mathbf x, tr))=t^{\mathbf N}w(B(\mathbf x,r)) \ \ \text{\rm for all } \mathbf x\in\mathbb R^N, \ t,r>0 \end{aligned}$$ and $$\begin{aligned}
\int_{\mathbb R^N} f(\mathbf x)\, dw(\mathbf x)=\int_{\mathbb R^N} t^{-\mathbf N} f(\mathbf x\slash t)\, dw(\mathbf x)\ \ \text{for} \ f\in L^1(dw)  \   \text{\rm and} \  t>0.\end{aligned}$$ Observe that ([^2]) $$\begin{aligned}
 w(B(\mathbf x,r))\sim r^{N}\prod_{\alpha \in R} (|\langle \mathbf x,\alpha\rangle |+r)^{k(\alpha)},\end{aligned}$$ so $dw(\mathbf x)$ is doubling, that is, there is a constant $C>0$ such that $$\label{eq:doubling} w(B(\mathbf x,2r))\leq C w(B(\mathbf x,r)) \ \ \text{ for all } \mathbf x\in\mathbb R^N, \ r>0.$$

For $\xi \in \mathbb{R}^N$, the [*Dunkl operators*]{} $T_\xi$ are the following $k$-deformations of the directional derivatives $\partial_\xi$ by a difference operator: $$\begin{aligned}
     T_\xi f(\mathbf x)= \partial_\xi f(\mathbf x) + \sum_{\alpha\in R} \frac{k(\alpha)}{2}\langle\alpha ,\xi\rangle\frac{f(\mathbf x)-f(\sigma_\alpha \mathbf x)}{\langle \alpha,\mathbf x\rangle}.\end{aligned}$$ The Dunkl operators $T_{\xi}$, which were introduced in [@Dunkl], commute and are skew-symmetric with respect to the $G$-invariant measure $dw$. Suppose that $\xi\ne 0$, $f,g \in C^1(\mathbb{R}^N)$ and $g$ is radial. The following Leibniz rule can be confirmed by a direct calculation: $$\begin{aligned}
T_{\xi}(f g)=f(T_\xi g)+g(T_\xi f). \end{aligned}$$ For fixed $\mathbf y\in\mathbb R^N$ the [*Dunkl kernel*]{} $E(\mathbf x,\mathbf y)$ is the unique analytic solution to the system $$\label{eq:Dunkl_kernel_definition}
    T_\xi f=\langle \xi,\mathbf y\rangle f, \ \ f(0)=1.$$ The function $E(\mathbf x ,\mathbf y)$, which generalizes the exponential function $e^{\langle \mathbf x,\mathbf y\rangle}$, has the unique extension to a holomorphic function on $\mathbb C^N\times \mathbb C^N$. Moreover, it satisfies $E(\mathbf{x},\mathbf{y})=E(\mathbf{y},\mathbf{x})$ for all $\mathbf{x},\mathbf{y} \in \mathbb{C}^N$.

Let $\{e_j\}_{1 \leq j \leq N}$ denote the canonical orthonormal basis in $\mathbb R^N$ and let $T_j=T_{e_j}$. For multi-index $\beta=(\beta_1,\beta_2,\ldots,\beta_N)  \in\mathbb N_0^N$, we set $$|\beta|=\beta_1+\beta_2 +\ldots +\beta_N,$$ $$\partial^{\beta}=\partial_1^{\beta_1} \circ \partial_2^{\beta_2}\circ \ldots \circ \partial_N^{\beta_N},$$ $$T^\beta = T_1^{\beta_1}\circ T_2^{\beta_2}\circ \ldots \circ T_N^{\beta_N}.$$ In our further consideration we shall need the following lemma.

For all $\mathbf{x} \in \mathbb{R}^N$, $\mathbf{z} \in \mathbb{C}^N$ and $\nu \in \mathbb{N}_0^{N}$ we have $$|\partial^{\nu}_{\mathbf{z}}E(\mathbf{x},\mathbf{z})| \leq \|\mathbf{x}\|^{|\nu|}\exp(\|\mathbf{x}\|\|{\rm Re \;}\mathbf{z}\|).$$ In particular, $$\begin{aligned}
 | E(i\xi, \mathbf x)|\leq 1 \quad \text{ for all } \xi,\mathbf x\in \mathbb R^N.\end{aligned}$$

See [@Roesle99 Corollary 5.3].

\[coro:Roesler\] There is a constant $C>0$ such that for all $\mathbf{x},\xi \in \mathbb{R}^N$ we have $$|E(i\xi,\mathbf{x})-1| \leq C\|\mathbf{x}\|\|\xi\|.$$

The *Dunkl transform* $$\begin{aligned}
\mathcal F f(\xi)=c_k^{-1}\int_{\mathbb R^N} E(-i\xi, \mathbf x)f(\mathbf x)\, dw(\mathbf x),
  \end{aligned}$$ where $$c_k=\int_{\mathbb{R}^N}e^{-\frac{\|\mathbf{x}\|^2}{2}}\,dw(\mathbf{x})>0,$$ originally defined for $f\in L^1(dw)$, is an isometry on $L^2(dw)$, i.e., $$\label{eq:Plancherel}
       \|f\|_{L^2(dw)}=\|\mathcal{F}f\|_{L^2(dw)} \text{ for all }f \in L^2(dw),$$ and preserves the Schwartz class of functions $\mathcal S(\mathbb R^N)$ (see [@deJeu]). Its inverse $\mathcal F^{-1}$ has the form $$\begin{aligned}
 \mathcal F^{-1} g(x)=c_k^{-1}\int_{\mathbb R^N} E(i\xi, \mathbf x)g(\xi)\, dw(\xi).
  \end{aligned}$$ The [*Dunkl translation*]{} $\tau_{\mathbf{x}}f$ of a function $f\in\mathcal{S}(\mathbb{R}^N)$ by $\mathbf{x}\in\mathbb{R}^N$ is defined by $$\begin{aligned}
\tau_{\mathbf{x}} f(\mathbf{y})=c_k^{-1} \int_{\mathbb{R}^N}{E}(i\xi,\mathbf{x})\,{E}(i\xi,\mathbf{y})\,\mathcal{F}f(\xi)\,{dw}(\xi).\end{aligned}$$ It is a contraction on $L^2(dw)$, however it is an open problem if the Dunkl translations are bounded operators on $L^p(dw)$ for $p\ne 2$.

The following specific formula was obtained by Rösler [@Roesler2003] for the Dunkl translations of (reasonable) radial functions $f({\mathbf{x}})=\tilde{f}({\|\mathbf{x}\|})$: $$\label{eq:translation-radial}
\tau_{\mathbf{x}}f(-\mathbf{y})=\int_{\mathbb{R}^N}{(\tilde{f}\circ A)}(\mathbf{x},\mathbf{y},\eta)\,d\mu_{\mathbf{x}}(\eta)\text{ for all }\mathbf{x},\mathbf{y}\in\mathbb{R}^N.$$ Here $$A(\mathbf{x},\mathbf{y},\eta)=\sqrt{{\|}\mathbf{x}{\|}^2+{\|}\mathbf{y}{\|}^2-2\langle \mathbf{y},\eta\rangle}=\sqrt{{\|}\mathbf{x}{\|}^2-{\|}\eta{\|}^2+{\|}\mathbf{y}-\eta{\|}^2}$$ and $\mu_{\mathbf{x}}$ is a probability measure, which is supported in the set $\operatorname{conv}\mathcal{O}(\mathbf{x})$, where $\mathcal O(\mathbf x) =\{\sigma(\mathbf x): \sigma \in G\}$ is the orbit of $\mathbf x$. Formula  implies that for all radial $f \in L^1(dw)$ and $\mathbf{x} \in \mathbb{R}^N$ we have $$\begin{aligned}
    \|\tau_{\mathbf{x}}f(\mathbf{y})\|_{L^1(dw(\mathbf{y}))} \leq \|f(\mathbf{y})\|_{L^1(dw(\mathbf{y}))}.\end{aligned}$$

[The *Dunkl convolution* $f*g$ of two reasonable functions (for instance Schwartz functions) is defined by $$(f*g)(\mathbf{x})=c_k\,\mathcal{F}^{-1}[(\mathcal{F}f)(\mathcal{F}g)](\mathbf{x})=\int_{\mathbb{R}^N}(\mathcal{F}f)(\xi)\,(\mathcal{F}g)(\xi)\,E(\mathbf{x},i\xi)\,dw(\xi) \text{ for }\mathbf{x}\in\mathbb{R}^N,$$ or, equivalently, by]{} $$\begin{aligned}
  {(f{*}g)(\mathbf{x})=\int_{\mathbb{R}^N}f(\mathbf{y})\,\tau_{\mathbf{x}}g(-\mathbf{y})\,{dw}(\mathbf{y})=\int_{\mathbb R^N} f(\mathbf y)g(\mathbf x,\mathbf y) \,dw(\mathbf{y}) \text{ for all } \mathbf{x}\in\mathbb{R}^N},  \end{aligned}$$ where, here and subsequently, $g(\mathbf x,\mathbf y)=\tau_{\mathbf x}g(-\mathbf y)$.

By an interpolation argument, if $1\leq p\leq 2$ and $q=2p/(2-p)$, then $$\begin{aligned}
  \|f*g\|_{L^q(dw)}\leq \| f\|_{L^2(dw)}\| g\|_{L^p(dw)}.  \end{aligned}$$

The [*Dunkl Laplacian*]{} associated with $R$ and $k$ is the differential-difference operator $\Delta=\sum_{j=1}^N T_{j}^2$, which acts on $C^2(\mathbb{R}^N)$-functions by

$$\begin{aligned}
    \Delta f(\mathbf x)=\Delta_{\rm eucl} f(\mathbf x)+\sum_{\alpha\in R} k(\alpha) \delta_\alpha f(\mathbf x),\end{aligned}$$

$$\begin{aligned}
    \delta_\alpha f(\mathbf x)=\frac{\partial_\alpha f(\mathbf x)}{\langle \alpha , \mathbf x\rangle} - \frac{\|\alpha\|^2}{2} \frac{f(\mathbf x)-f(\sigma_\alpha \mathbf x)}{\langle \alpha, \mathbf x\rangle^2}.\end{aligned}$$

Obviously, $\mathcal F(\Delta f)(\xi)=-\| \xi\|^2\mathcal Ff(\xi)$. The operator $\Delta$ is essentially self-adjoint on $L^2(dw)$ (see for instance [@AH Theorem3.1]) and generates the semigroup $e^{t\Delta}$ of linear self-adjoint contractions on $L^2(dw)$. The semigroup has the form $$\begin{aligned}
  e^{t\Delta} f(\mathbf x)=\mathcal F^{-1}(e^{-t\|\xi\|^2}\mathcal Ff(\xi))(\mathbf x)=\int_{\mathbb R^N} h_t(\mathbf x,\mathbf y)f(\mathbf y)\, dw(\mathbf y),
  \end{aligned}$$ where the heat kernel $$\label{eq:heat_def}
      h_t(\mathbf x,\mathbf y)=\tau_{\mathbf x}h_t(-\mathbf y), \ \ h_t(\mathbf x)=\mathcal F^{-1} (e^{-t\|\xi\|^2})(\mathbf x)=c_k^{-1} (2t)^{-\mathbf N\slash 2}e^{-\| \mathbf x\|^2\slash (4t)}$$ is a $C^\infty$-function of all variables $\mathbf x,\mathbf y \in \mathbb{R}^N$, $t>0$, and satisfies $$\begin{aligned}
 0<h_t(\mathbf x,\mathbf y)=h_t(\mathbf y,\mathbf x),
  \end{aligned}$$ $$\begin{aligned}
 \int_{\mathbb R^N} h_t(\mathbf x,\mathbf y)\, dw(\mathbf y)=1.
 \end{aligned}$$ Set $$V(\mathbf x,\mathbf y,t)=\max (w(B(\mathbf x,t)),w(B(\mathbf y, t))).$$ Let $$d(\mathbf x,\mathbf y)=\min_{\sigma\in G}\| \sigma(\mathbf x)-\mathbf y\|$$ be the distance of the orbit of $\mathbf x$ to the orbit of $\mathbf y$. The following theorem was proved in [@DzH1 Theorem 3.1] (see also [@ADzH Theorem 4.1]).

\[theorem:heat\] There are constants $C,c>0$ such that for all $\mathbf{x},\mathbf{y} \in \mathbb{R}^N$ and $t>0$ we have $$\label{eq:Gauss}
h_t(\mathbf{x},\mathbf{y}) \leq C\Big(1+\frac{\|\mathbf{x}-\mathbf{y}\|}{t}\Big)^{-2}\,V(\mathbf{x},\mathbf{y},\!\sqrt{t\,})^{-1}\,e^{-\hspace{.25mm}c\hspace{.5mm}d(\mathbf{x},\mathbf{y})^2\slash t}.$$ Moreover, if $\|\mathbf{y}-\mathbf{y}'\| \leq \sqrt{t}$, then $$\label{eq:Gauss-Lipschitz}
|h_t(\mathbf{x},\mathbf{y})-h_t(\mathbf{x},\mathbf{y}')| \leq C\frac{\|\mathbf{y}-\mathbf{y}'\|}{\sqrt{t}}\Big(1+\frac{\|\mathbf{x}-\mathbf{y}\|}{t}\Big)^{-2}\,V(\mathbf{x},\mathbf{y},\!\sqrt{t\,})^{-1}\,e^{-\hspace{.25mm}c\hspace{.5mm}d(\mathbf{x},\mathbf{y})^2\slash t}.$$

Theorem \[theorem:heat\] and  imply the following Lemma (see [@DzH1 Corollary 3.5]).

\[lem:nonradial\_estimation\] Suppose that $\varphi \in C_c^{\infty}(\mathbb{R}^N)$ is radial and supported by the unit ball. Then there is $C>0$ such that for all $\mathbf{x},\mathbf{y}\in \mathbb{R}^{N}$ and $t>0$ we have $$\begin{aligned}
    &|\varphi_t(\mathbf{x},\mathbf{y})| \leq C \Big(1+\frac{\|\mathbf{x}-\mathbf{y}\|}{t}\Big)^{-2}V(\mathbf{x},\mathbf{y},t)^{-1}\chi_{[0,1]}(d(\mathbf{x},\mathbf{y})/t).\end{aligned}$$

 L2 estimates
=============

In the present section we assume that $K(\mathbf x)$ satisfies and with $s=1$, that is, $$\label{eq:K_der1} \tag{$\text{D}(1)$}
    |\partial^{\beta}K(\mathbf{x})| \leq C_{\beta}\|\mathbf{x}\|^{-\mathbf{N}-|\beta|} \text{ for }|\beta| \leq 1.$$

Our aim is to we prove that convolution operators with the truncated kernels $K^{\{t\}}$ are uniformly bounded on $L^2(dw).$ Then we add the assumption and show the $L^2$-bound of the limiting operator.

We start by the easy observation that implies $$\label{eq:K_der} \tag{$\text{D'}(1)$}
    |T^{\beta}K(\mathbf{x})| \leq C_{\beta}\|\mathbf{x}\|^{-\mathbf{N}-|\beta|} \text{ for }|\beta| \leq 1.$$

Let $\phi $ be a fixed $C^\infty(\mathbb{R}^N)$ radial function supported in the unit ball such that $\phi(\mathbf x)=1$ for all $\mathbf{x} \in \mathbb{R}^N$ such that $ \| \mathbf x\|\leq 1/2$.

For $0<a<b<\infty$, let $$K_{a,\infty}(\mathbf x)=K(\mathbf x)\chi_{\{\mathbf y: a<\| \mathbf y\|\}}(\mathbf x) \quad \text{and} \quad  K_{a,b}(\mathbf x)=K(\mathbf x)\chi_{\{\mathbf y: a<\| \mathbf y\|<b\}}(\mathbf x),$$ $$K^{\{a\}}(\mathbf x)=K(\mathbf x)(1-\phi(a^{-1}\mathbf x)) \quad \text{and}\quad K^{\{a,b\}}(\mathbf x)=K^{\{a\}}(\mathbf x)-K^{\{b\}}(\mathbf x).$$ Let us list the following easily proved properties of the truncated kernels which follow from and : $$K_{a,b}, \, K^{\{a,b\}}\in L^1(dw)\cap L^2(dw), \quad  K_{a,\infty}, \, K^{\{a\}}\in L^2(dw),$$ $$\text{supp}\, K^{\{a,b\}}\subseteq \{\mathbf x\in\mathbb R^N: a/2\leq \| \mathbf x\|\leq b\},$$ $$\label{eq:L1compare} \sup_{0<a<b<\infty} \| K_{a,b}-K^{\{a,b\}}\|_{L^1(dw)}=C_0<\infty,$$ $$\lim_{b\to \infty} \| K_{a,b}-K_{a,\infty}\|_{L^2(dw)}=0, \quad \lim_{b\to \infty} \| K^{\{a,b\}}-K^{\{a\}}\|_{L^2(dw)}=0.$$ Consequently, by the Plancherel identity (see ), $$\label{eq:L^2limit}
\lim_{b \to \infty} \| \mathcal FK_{a,b}-\mathcal F K_{a,\infty}\|_{L^2(dw)}=0, \quad \lim_{b \to \infty} \| \mathcal FK^{\{a,b\}}-\mathcal F K^{\{a\}}\|_{L^2(dw)}=0.$$ Moreover, there are constants $A',C>0$ such that for all $0<a<b<\infty$ one has $$\begin{aligned}
    |K^{\{a,b\}}(\mathbf x)|\leq C\| \mathbf x\|^{-\mathbf N}, \quad  |K^{\{a\}}(\mathbf x)|\leq C\| \mathbf x\|^{-\mathbf N},\end{aligned}$$ $$\begin{aligned}
    |T_jK^{\{a,b\}}(\mathbf x)| \leq C\| \mathbf x\|^{-\mathbf N-1}, \quad  |T_jK^{\{a\}}(\mathbf x)| \leq C\| \mathbf x\|^{-\mathbf N-1},\end{aligned}$$ $$\begin{aligned}
\label{newA}    \Big|\int K^{\{a,b\}}(\mathbf x)\, dw(\mathbf x)\Big| \leq A'.\end{aligned}$$

\[lem:uniform\_boundedness0\] and  imply that there is a constant $C>0$ such that for all $0<a<b<\infty$ one has $|\mathcal{F}K_{a,b}(\xi)| \leq C$ and $|\mathcal{F}K^{\{a,b\}}(\xi)| \leq C$ .

Thanks to it suffices to prove the second inequality. Assume first that $\xi \in \mathbb{R}^N$ satisfies $a \leq \|\xi\|^{-1} \leq b$. Put $t=\|\xi\|^{-1}$. We have $K^{\{a,b\}}=K^{\{a,t\}}+K^{\{t,b\}}$ and, consequently, $$\label{eq:first_splitting}
\begin{split}
    \mathcal{F}K^{\{a,b\}}(\xi)=\mathcal{F}K^{\{a,t\}}(\xi)+\mathcal{F}K^{\{t,b\}}(\xi)
    &=:I_1+I_2.
\end{split}$$ In order to estimate $I_1$, we write $$\begin{aligned}
    I_1=c_k^{-1}\int K^{\{a,t\}} (\mathbf{x})(E(-i\xi,\mathbf{x})-1)\,dw(\mathbf{x})+c_k^{-1}\int K^{\{a,t\}}(\mathbf{x})\,dw(\mathbf{x})=:I_{1,1}+I_{1,2}.\end{aligned}$$ Clearly, by  we get $|I_{1,2}| \leq C$. For $I_{1,1}$, by Corollary \[coro:Roesler\] and , we obtain $$\begin{aligned}
    |I_{1,1}| \leq C\|\xi\|\int_{\|\mathbf{x}\| \leq t}\|\mathbf{x}\|^{-\mathbf{N}+1}\,dw(\mathbf{x}) \leq C.\end{aligned}$$ We now turn to estimate $I_2$. Choose $j \in \{1,2,\ldots,N\}$ such that $|\xi_j| \geq {N}^{-1/2}\|\xi\|$. Then, thanks to , we have $$\begin{aligned}
        |I_2|& \leq C\sqrt{N}\|\xi\|^{-1} \Big|\int_{\mathbb{R}^N} K^{\{t,b\}}(\mathbf x)T_jE(-i
        \xi ,\mathbf x )\, dw(\mathbf x)\Big|\\
        &\leq C\sqrt{N}\|\xi\|^{-1} \Big|\int_{\mathbb{R}^N} T_j K^{\{t,b\}}(\mathbf x)E(-i
        \xi ,\mathbf x )\, dw(\mathbf x)\Big|\\
        &\leq C\sqrt{N}\|\xi\|^{-1} \Big|\int_{\frac{1}{2} t\leq\|\mathbf x\| } \| \mathbf x\|^{-\mathbf N-1} \, dw(\mathbf x)\Big| \leq C'.\end{aligned}$$ The cases $\|\xi\|^{-1}< a $ or $\|\xi\|^{-1}>b$ can be treated similarly (we have to deal with just one integral in ).

From and Lemma \[lem:uniform\_boundedness0\] we easily deduce the following corollary.

\[coro:uniform\_K\_a\] and  imply that there is a constant $C>0$ such that for every $0<a<\infty$ we have $$\|\mathcal FK^{\{a\}}\|_{L^\infty} \leq C, \quad \text{and} \quad  \|\mathcal FK_{a,\infty}\|_{L^\infty} \leq C.$$

\[uniform\_L2\] There is a constant $C>0$ such that for all $0<a<b<\infty$ we have $$\| K^{\{a,b\}}*f\|_{L^2(dw)}+  \| K_{a,b}*f\|_{L^2(dw)}+ 
   \| K^{\{a\}}*f\|_{L^2(dw)}+ \| K_{a,\infty}*f\|_{L^2(dw)}  \leq C\| f\|_{L^2(dw)}.$$ Moreover, for every $a>0$ and $f\in L^2(dw)$ we have $$\label{eq:L2strong}
\lim_{b\to\infty}  \|K^{\{a,b\}} f-K^{\{a\}}f   \|_{L^2(dw)}=0.$$

The corollary follows directly from Lemma \[lem:uniform\_boundedness0\], Corollary \[coro:uniform\_K\_a\], the Plancherel identity , and the definition of the Dunkl convolution.

From now on to the end of the paper we assume additionally that is satisfied by $K$ as well.

\[lem:L\] Property  implies $$\begin{aligned}
     \lim_{a\to 0} \int_{\| \mathbf x\|<1} K^{\{a\}} (\mathbf x)\, dw(\mathbf x)=L.
 \end{aligned}$$

Let us define $\widetilde{\phi}(\|\mathbf{x}\|)=\phi(\mathbf{x})$. For $0< r\leq 1$ we set $$\label{eq:A_F}
     F(r)=\int_{r<\|\mathbf{x}\|<1}K(\mathbf{x})\,dw(\mathbf{x})=\int_r^{1}\int_{\mathbb{S}^{N-1}}r_1^{N-1}K(r_1\omega)\prod_{\alpha \in R}|\langle r_1\omega,\alpha \rangle|^{k(\alpha)}\,d{\boldsymbol\sigma}(\omega)\,dr_1,$$ where $\boldsymbol\sigma$ is the spherical measure. Note that $F$ is differentiable and satisfies $$\begin{aligned}
     F'(r)=-\int_{\mathbb{S}^{N-1}}r^{N-1}K(r\omega)\prod_{\alpha \in R}|\langle r\omega,\alpha \rangle|^{k(\alpha)}\,d{\boldsymbol\sigma}(\omega).
 \end{aligned}$$ Consequently, by the definition of $K^{\{a\}}$ and integration by parts we have $$\begin{aligned}
     \int_{\| \mathbf x\|<1} K^{\{a\}} (\mathbf x)\, dw(\mathbf x)&=\int_0^1 \big(1- \widetilde{\phi}(a^{-1}r)\big) \int_{\mathbb{S}^{N-1}}r^{N-1}K(r\omega)\prod_{\alpha \in R}|\langle r\omega,\alpha \rangle|^{k(\alpha)}\,d{\boldsymbol\sigma}(\omega)\,dr\\&=\int_0^1 \big(1- \widetilde{\phi}(a^{-1}r)\big)(-F'(r))\,dr\\
     &=-\int_0^1 (\widetilde{\phi}(a^{-1}r))'F(r)\,dr.
 \end{aligned}$$ We write $$\label{eq:A_split}
 \begin{split}
    - \int_0^1 (\widetilde{\phi}(a^{-1}r))'F(r)\,dr&=-\int_0^1 (\widetilde{\phi}(a^{-1}r))'(F(r)-L)\,dr-L\int_0^1 (\widetilde{\phi}(a^{-1}r))'\,dr\\&=:S_1(a)+S_2(a).
 \end{split}$$ Clearly, for all $a<1/4$, we have $$\label{eq:lim_A_2}
     S_2(a)=L.$$ Note that $\text{supp}\, (\widetilde{\phi}(a^{-1}\cdot))'\subseteq [a/2,a]$, hence $$\begin{aligned}
     |S_1(a)| &\leq \int_0^1| (\widetilde{\phi}(a^{-1}r))'||F(r)-L| \,dr
     =\int_{a/2}^{a}|(\widetilde{\phi}(a^{-1}r))'||F(r)-L| \,dr \\&\leq \max_{r \in [a/2,a]}|F(r)-L|\int_{a/2}^{a}|a^{-1}\widetilde{\phi}'(a^{-1}r)|\,dr \leq C\|(\widetilde{\phi})'\|_{L^{\infty}}\max_{r \in [0,a]}|F(r)-L|.
 \end{aligned}$$ Consequently, thanks to  and  we obtain $\lim_{a \to 0}S_1(a)=0$. Combining this fact with  and  we get the claim.

\[lem:K\_a\] Under , , and for almost every $\xi\in\mathbb R^N$, the limit $$\begin{aligned}
    \lim_{a\to 0} \mathcal FK_{a,\infty}(\xi)    \end{aligned}$$ exists and defines a bounded function denoted by $\mathcal FK(\xi)$.

According to Corollary  \[coro:uniform\_K\_a\], it suffices to show that $\mathcal FK_{a,\infty}(\xi)$ is a Cauchy sequence as $a\to 0$. To this end we write $$\begin{aligned}
|\mathcal FK_{a_1,\infty}(\xi)-\mathcal FK_{a_2,\infty}(\xi)|
&\leq c_k^{-1}\Big| \int_{a_1\leq \|\mathbf x\| \leq a_2} K(\mathbf x) \Big(E(-i\xi, \mathbf x) -1\Big)\, dw(\xi) \Big|\\
&\quad + c_k^{-1}\Big|\int_{a_1\leq \|\mathbf x\| \leq a_2} K(\mathbf x)\, dw(\mathbf x)\Big|=:I_1+I_2.
 \end{aligned}$$ Thanks to Corollary \[coro:Roesler\], $$I_1\leq  C\|\xi\|  \int_{a_1\leq \|\mathbf x\| \leq a_2} \|\mathbf x\|^{-\mathbf N +1}  dw(\mathbf x)\to 0$$ as $a_1$ and $a_2$ tend to $0^+$. The convergence of $I_2$ to 0 is a consequence of .

As a consequence of Corollary \[coro:uniform\_K\_a\] and Lemma \[lem:K\_a\] we obtain the following theorem.

\[theorem:limiting\_operator\_L2\] Under , , and , for every $f\in L^2(dw)$ the limit $$\begin{aligned}
    \lim_{a\to 0} K_{a,\infty}*f\end{aligned}$$ exists in $L^2(dw)$ and defines a operator, which is bounded on $L^2(dw)$ and will be denoted by $Kf$. Moreover, $$\begin{aligned}
    Kf=\mathcal F^{-1} (\mathcal FK(\cdot)\mathcal Ff(\cdot)).\end{aligned}$$

\[theorem:main\_L2\] Under , , and for almost every $\xi\in\mathbb R^N$, $$\begin{aligned}
  \lim_{ a\to 0} \mathcal FK^{\{a\}}(\xi)  = \mathcal FK(\xi),\end{aligned}$$ where $\mathcal FK(\xi)$ is defined in Lemma \[lem:K\_a\]. Moreover, for every $f\in L^2(dw)$ the limit $$\begin{aligned}
    \lim_{a\to 0} K^{\{a\}}*f    \end{aligned}$$ exists in $L^2(dw)$ and is equal to $Kf$ (see Theorem \[theorem:limiting\_operator\_L2\]).

Let $J(a,\xi)=\mathcal F( K^{\{a,1\}})(\xi)-\mathcal F( K_{a,1})(\xi)$. It suffices to show that $\lim_{a\to 0^+}J(a,\xi)=0$ for all $\xi\in\mathbb R^N$. Similarly to the proof of Lemma \[lem:K\_a\], we write $$\begin{aligned}
J(a,\xi)=&c_k^{-1}\int_{\mathbb{R}^N} \Big(K^{\{a,1\}}(\mathbf x)-K_{a,1}(\mathbf x)\Big)\Big( E(-i\xi,\mathbf x)-1\Big)\, dw(\mathbf x)\\
&+ c_k^{-1}\int_{\mathbb{R}^N} \Big(K^{\{a,1\}}(\mathbf x)-K_{a,1}(\mathbf x)\Big)\, dw(\mathbf x)=:J_1(a,\xi)+J_2(a,\xi).\end{aligned}$$ By Lemma \[lem:L\] we have $\lim_{a\to 0^+}J_2(a,\xi)=0$. To deal with $J_1(a,\xi)$ we note that $$\text{supp}\,(K^{\{a,1\}}-K_{a,1})\subseteq \{\mathbf x: a/2\leq \| \mathbf x\|\leq a\} \quad \text{and}\quad  |K^{\{a,1\}}(\mathbf x)-K_{a,1}(\mathbf x)|\leq C\| \mathbf x\|^{-\mathbf N}.$$ Hence, using Corollary \[coro:Roesler\], we get $$J_1\leq C\int_{a/2\leq \| \mathbf x\|\leq a} \|\xi\| \|\mathbf x\|^{-\mathbf N+1}\, dw(\mathbf x)\to 0\quad \text{as} \quad a\to 0.$$

 Lp estimates
=============

The purpose of this section is to study singular integrals operators on $L^p(dw)$. For this purpose we need to make the following stronger assumption on the kernel $K$, namely that  holds for $|\beta|\leq s_0$, where $s_0$ is the smallest even positive integer bigger than $\mathbf N\slash 2$, that is, $$\label{eq:K_der2}\tag{$\text{D}(s_0)$}
    |\partial^{\beta}K(\mathbf{x})| \leq C_{\beta}\|\mathbf{x}\|^{-\mathbf{N}-|\beta|} \text{ for all }|\beta| \leq s_0.$$ Clearly,  implies $$\label{eq:K_der2_Dunkl}
    |T^{\beta}K(\mathbf{x})| \leq C_{\beta}\|\mathbf{x}\|^{-\mathbf{N}-|\beta|} \text{ for all }|\beta| \leq s_0.$$

The goal of this section is to prove the following theorems.

\[theorem:weak\_type\_truncated1\] Under  and , there is a constant $C>0$ such that for all $0<a < b<\infty $, $\lambda>0$, and $f \in L^1(dw)\cap L^2(dw)$ we have $$\label{eq:Psi_nm_weak}
    w(\{\mathbf{x} \in \mathbb{R}^N\,:\,|K^{\{a,b\}}f(\mathbf{x})|>\lambda\}) \leq C\lambda^{-1}\|f\|_{L^1(dw)},$$ $$\label{eq:Psi_n_weak}
     w(\{\mathbf{x} \in \mathbb{R}^N\,:\,|K^{\{a\}}f(\mathbf{x})|>\lambda\}) \leq C\lambda^{-1}\|f\|_{L^1(dw)}.$$

\[theorem:strong\_type\_truncated1\] Let $1<p<\infty$. Under  and , there is a constant $C=C_p>0$ such that for all $0<a < b<\infty$, and $f \in L^p(dw)\cap L^2(dw)$ we have $$\label{eq:Psi_nm_strong}
    \|K^{\{a,b\}} f\|_{L^p(dw)} \leq C\|f\|_{L^p(dw)},$$ $$\label{eq:Psi_n_strong}
    \|K^{\{a\}} f\|_{L^p(dw)} \leq C\|f\|_{L^p(dw)}.$$ Moreover, the operators $K^{\{a,b\}}$ converge strongly to $K^{\{a\}}$ in $L^p(dw)$ as $b \to \infty$.

As the consequence of Theorems \[theorem:weak\_type\_truncated1\] and \[theorem:strong\_type\_truncated1\] we obtain the following theorem for the operator $K$ defined in Theorem \[theorem:limiting\_operator\_L2\].

\[theorem:weak\_type\_K\] Let $1 \leq p <\infty$. Under , , and there is a constant $C=C_p>0$ such that for all $\lambda>0$, and $f \in L^p(dw)\cap L^2(dw)$ we have $$\label{eq:K_weak}
    w(\{\mathbf{x} \in \mathbb{R}^N\,:\,|Kf(\mathbf{x})|>\lambda\}) \leq C\lambda^{-1}\|f\|_{L^1(dw)} \ \ \text{ if }p=1,$$ $$\label{eq:K_strong}
    \|K f\|_{L^p(dw)} \leq C\|f\|_{L^p(dw)} \ \ \text{ if } 1<p<\infty.$$ Moreover, if $1<p<\infty$, the operators $K^{\{a\}}$ converge strongly to $K$ in $L^p(dw)$ as $a \to 0$.

Bessel potential
----------------

For $s>0$ we set $$\begin{aligned}
    J^{\{s\}}(\mathbf{x})=\mathcal{F}^{-1}((1+\|\cdot\|^2)^{-s\slash 2})(\mathbf{x}).\end{aligned}$$ By gamma function identity we have $$\begin{aligned}
    (1+\|\xi\|^2)^{-s\slash 2}=\int_0^{\infty}e^{-t}e^{-t\|\xi\|^2}t^{s\slash 2}\,\frac{dt}{t},\end{aligned}$$ which leads us to $$\label{eq:subordination}
    J^{\{s\}}(\mathbf{x})=\int_0^{\infty}e^{-t}h_t(\mathbf{x})t^{s\slash 2}\,\frac{dt}{t}$$ (see ). The function $J^{\{s\}}$ is radial, positive and belongs $L^1(dw)$. As a consequence of  (see i.e. [@Grafakos]), we get the following proposition.

\[propo:bessel\_estimate\] Let $M>0$. There is a constant $C=C_{s,M}>0$ such that $$\begin{aligned}
    0<J^{\{s\}}(\mathbf{x}) \leq C \begin{cases}
    \|\mathbf{x}\|^{s-\mathbf{N}} & \text{ if }\|\mathbf{x}\| \leq 1\slash 2, \quad 0<s<\mathbf N,\\
    -\ln \|\mathbf x\| & \text{ if }\|\mathbf{x}\| \leq 1\slash 2, \quad s=\mathbf N,\\
    1 & \text{ if } \| \mathbf x\|\leq 1/2, \quad s>\mathbf N,\\
    (1+\|\mathbf{x}\|^2)^{-M} & \text{ if }\|\mathbf{x}\|>1/2,  \quad 0<s.
    \end{cases}\end{aligned}$$

Let $J^{\{s\}}(\mathbf x,\mathbf y)=\tau_{\mathbf x}(J^{\{s\}})(-\mathbf y)$. Clearly, by , $0<J^{\{s\}}(\mathbf x,\mathbf y)$ for all $\mathbf{x},\mathbf{y} \in \mathbb{R}^N$ and $$\begin{aligned}
    \int_{\mathbb{R}^N} J^{\{s\}}(\mathbf x,\mathbf y)\, dw(\mathbf x)=\int_{\mathbb{R}^N} J^{\{s\}}(\mathbf x,\mathbf y)\, dw(\mathbf y)=\int_{\mathbb{R}^N} J^{\{s\}}(\mathbf x)\, dw(\mathbf x).\end{aligned}$$

\[lem:Lipschitz\] Let $0<\delta < s$ and $0<\delta \leq 1$. Then there is a constant $C>0$ such that $$\begin{aligned}
    \int_{\mathbb{R}^N} |J^{\{s\}} (\mathbf x,\mathbf y)-J^{\{s\}} (\mathbf x,\mathbf y')|\, dw(\mathbf x)\leq C\min (1, \| \mathbf y-\mathbf y'\|^\delta)  \ \ \text{for all }\mathbf{y},\mathbf{y}' \in \mathbb{R}^N.\end{aligned}$$

By  we have $$\begin{aligned}
\label{eq:diff_bessel}
    J^{\{s\}}(\mathbf{x},\mathbf{y})-J^{\{s\}}(\mathbf{x},\mathbf{y}')=\int_0^{\infty}e^{-t}\big(h_t(\mathbf{x},\mathbf{y})-h_t(\mathbf{x},\mathbf{y}')\big)t^{s/2}\frac{dt}{t}.\end{aligned}$$ Moreover, by  we get $$\label{eq:Holder_heat}
    \int_{\mathbb{R}^N} |h_t(\mathbf x,\mathbf y)-h_t(\mathbf x,\mathbf y')|\,dw(\mathbf x)\leq C\min \Big(1, \big(\| \mathbf y-\mathbf y'\|\slash \sqrt{t}\big)^\delta\Big).$$ The lemma is a direct consequence of and .

Auxiliary estimates on Kj
-------------------------

The following theorem and proposition were proved in [@DzH Theorem 1.7] and [@DzH Proposition 4.4] respectively.

\[teo:support\] Let $f \in L^2(dw)$, $\text{\rm supp}\, f \subseteq B(0,r)$, and $\mathbf{x} \in \mathbb{R}^N$. Then $$\label{eq:inclusion_support}
{\text{\rm supp}\,}\tau_{\mathbf{x}}f(-\, \cdot) \subseteq \mathcal{O}(B(\mathbf x,r)),$$ where $\mathcal O(B(\mathbf x, r))=\bigcup_{\sigma \in G} B(\sigma (\mathbf x), r))$ is the orbit of the Euclidean closed ball $B(\mathbf x,r)=\{\mathbf y \in \mathbb{R}^N: \| \mathbf x-\mathbf y\|\leq r\}$.

\[propo:compact\_supports\] There is a constant $C>0$ such that for any $r_1,r_2>0$, any $f\in L^1(dw)$ such that ${\rm supp\,} f \subseteq B(0,r_2)$, any continuous radial function $\phi$ such that $\text{\rm supp}\, \phi \subseteq B(0,r_1)$, and for all $\mathbf{y} \in \mathbb{R}^N$ we have $$\begin{aligned}
\|\tau_{\mathbf{y}}(f * \phi)\|_{L^1(dw)} \leq C (r_1(r_1+r_2))^{\frac{\mathbf{N}}{2}}
\| \phi\|_{L^{\infty}}\|f\|_{L^1(dw)}.\end{aligned}$$

\[propo:j\_small\] Let $0 \leq \delta\leq 1$ be such that $\mathbf N\slash 2+\delta<s_0$. Suppose that the function $K$ satisfies . There is a constant $C>0$ such that for all $j \in \mathbb{Z}$ we have $$\label{eq:K_j_bound}
    \int_{\mathbb{R}^N}\sup_{2^{j-1}\leq a\leq b<2^{j+1}}|K^{\{a,b\}}(\mathbf{x},\mathbf{y})|d(\mathbf{x},\mathbf{y})^{\delta}\,dw(\mathbf{y}) \leq C2^{j\delta} \quad \text{for all } \mathbf{x} \in \mathbb{R}^N,$$ $$\label{eq:K_j_Holder}
    \int_{\mathbb{R}^N}\sup_{2^{j-1}\leq a\leq b<2^{j+1}}|K^{\{a,b\}}(\mathbf{x},\mathbf{y})-K^{\{a,b\}}(\mathbf{x},\mathbf{y}')|\,dw(\mathbf{x}) \leq C2^{-j\delta}\|\mathbf{y}-\mathbf{y}'\|^{\delta} \quad \text{for all } \mathbf{y}, \mathbf y' \in \mathbb{R}^N ,$$ $$\label{eq:K_j_Holder_22}
    \int_{\mathbb{R}^N}\sup_{2^{j-1}\leq a\leq b<2^{j+1}}|K^{\{a,b\}}(\mathbf{y},\mathbf{x})-K^{\{a,b\}}(\mathbf{y'},\mathbf{x})|\,dw(\mathbf{x}) \leq C2^{-j\delta}\|\mathbf{y}-\mathbf{y}'\|^{\delta} \quad \text{for all } \mathbf{y}, \mathbf y' \in \mathbb{R}^N ,$$ $$\label{eq:K^n_Holder_22}
    \int_{\mathbb{R}^N}\sup_{2^{j-1}\leq a \leq 2^{j+1}}|K^{\{a\}}(\mathbf{y},\mathbf{x})-K^{\{a\}}(\mathbf{y'},\mathbf{x})|\,dw(\mathbf{x}) \leq C2^{-j\delta}\|\mathbf{y}-\mathbf{y}'\|^{\delta} \quad \text{for all } \mathbf{y}, \mathbf y' \in \mathbb{R}^N .$$

It suffices to prove the proposition for $j=0$ and then use scaling. Let $\widetilde \phi:[0,\infty)\to \mathbb R$ be defined by relation $\widetilde\phi(\| \mathbf y\|)=\phi (\mathbf y)$. Set $\Phi(t,\mathbf y)=  -K(\mathbf y)t^{-2} \| \mathbf y\|\widetilde\phi '(t^{-1}\| \mathbf y\|)$. Assume that $2^{-1} \leq a\leq b \leq 2$, then $$\begin{aligned}
        K^{\{a,b\}} (\mathbf y)&=K(\mathbf y)\big(\widetilde \phi (b^{-1}\| \mathbf y\|)-\widetilde\phi (a^{-1}\|\mathbf y\|)\big)=\int_a^b K(\mathbf y)\frac{d}{dt} \{\widetilde \phi(t^{-1}\| \mathbf y\|)\}\, dt= \int_a^b \Phi(t,\mathbf y)\, dt,\end{aligned}$$ where the integral converges in $L^2(dw)$, because $t\mapsto \Phi(t,\cdot)$ is a continuous function from $(0,\infty)$ to $L^2(dw)$. Let us denote $\Phi(t,\mathbf x,\mathbf y)=\tau_{\mathbf x}(\Phi (t,\cdot))(-\mathbf y)$. Since the Dunkl translation $\tau_{\mathbf{x}}$ is continuous on $L^2(dw)$, for fixed $\mathbf x\in\mathbb R^N$ we have $$K^{\{a,b\}}(\mathbf x,\mathbf y)=\int_a^b \Phi(t,\mathbf x,\mathbf y)\, dt,$$ where the integral converges in $L^2(dw(\mathbf y))$.

Note that $\text{supp}\, \Phi(t,\cdot)\subseteq \{\mathbf y: t/2\leq \| \mathbf y\| \leq t\}\subseteq B(0,2)$, because $1/2 \leq a\leq t\leq b \leq 2$. Let $\mathbf N/2<s_1\leq s_0$. Set $$\begin{aligned}
    F(t,\mathbf{y})={\widetilde{\Phi}(t,\cdot)}*(J^{\{s_1\}})(\mathbf{y}), 
\quad \text{where} \quad 
    \widetilde{\Phi}(t,\mathbf{y})=(I-\Delta)^{s_0/2}\Phi(t,\mathbf{y}).\end{aligned}$$ By the assumption  (see also ), $\text{supp}\, \widetilde{\Phi}(t,\cdot)\subseteq B(0,2)$ and $|\widetilde{\Phi} (t,\mathbf x)|\leq C'$ for $1/2\leq t\leq 2$, where the constant $C'$ depends only on the constants $C_\beta$ in and the (fixed) function $\phi$. Consequently, $\widetilde{\Phi}(t,\cdot) \in L^p(dw)$ for $1\leq p\leq \infty$. In particular there is $C''>0$ independent of $j$ such that $$\label{eq:assumption_usage}
   \sup_{1/2\leq t\leq 2} \|\widetilde{\Phi}(t,\cdot)\|_{L^1(dw)} \leq C'' .$$ Let $\psi(\mathbf y)=\phi(\mathbf y)-\phi(2\mathbf y)$. We write $$\begin{aligned}
\begin{split}
    J^{\{s_1\}}(\mathbf{y})&=\sum_{\ell \in \mathbb{Z}}\psi(2^{-\ell}\mathbf{y})(J^{\{s_1\}})(\mathbf{y})=:\sum_{\ell \in \mathbb{Z}}\psi^{\{\ell \}}(\mathbf{y}),
\end{split}\end{aligned}$$ where the convergence is pointwise and in $L^1(dw)$. Recall that $\psi^{\{\ell\}}$ are radial functions. Hence, $$\begin{aligned}
    F(t,\cdot)=\sum_{\ell \in \mathbb{Z}} \widetilde{\Phi}(t,\cdot)*\psi^{\{\ell\}}\end{aligned}$$ with convergence in $L^2(dw)$. Therefore for all $\mathbf x\in\mathbb R^N$, $$\begin{aligned}
    F(t,\mathbf x,\mathbf y):=\tau_{\mathbf x} F(t,-\mathbf y)=\sum_{\ell \in \mathbb{Z}} \tau_{\mathbf x}\big( \widetilde{\Phi}(t,\cdot)*\psi^{\{\ell\}}\big)(-\mathbf y),\end{aligned}$$ where the series converges in $L^2(dw)$. We shall show that the convergence is in $L^1(dw)$ as well and the $L^1$-norm of the sum is independent of $t\in [1/2,2]$ and $\mathbf x \in \mathbb{R}^N$.

To this end we apply together with Proposition \[propo:compact\_supports\] and Proposition \[propo:bessel\_estimate\] (with $M>\mathbf{N}$) and obtain $$\begin{aligned}
     \sum_{\ell \in \mathbb{Z}} \|\tau_{\mathbf{x}}(\widetilde{\Phi}(t,\cdot)*\psi^{\{\ell\}})(\cdot)\|_{L^1(dw)}
      & \leq C \sum_{\ell \in \mathbb{Z}} 2^{\ell  \mathbf{N}/2}(2+2^{\ell})^{\mathbf{N}/2}\|\widetilde{\Phi}(t,\cdot)\|_{L^1(dw)}\|\psi^{\{\ell\}}\|_{L^{\infty}}\\
      &\leq C'\sum_{\ell \leq 0}  2^{\ell \mathbf{N}/2}2^{\ell(s_0-\mathbf{N})} + C'\sum_{\ell >0} 2^{\ell \mathbf N}2^{-\ell M}<\infty.\end{aligned}$$ Thus, $$\label{eq:F_0_bound}  \sup_{1/2\leq t\leq 2}\int_{\mathbb{R}^N} |F(t,\mathbf x,\mathbf y)|dw(\mathbf y)\leq C''<\infty,$$ where $C''$ depends on the constants in . By the same arguments, $$\label{eq:F_t_bound}  \sup_{1/2\leq t\leq 2}\int_{\mathbb{R}^N} |F(t,\mathbf x,\mathbf y)|dw(\mathbf x)\leq C''<\infty.$$ Note that if $s_1=s_0$ then $F(t,\cdot)=\Phi(t,\cdot)$ and $$\sup_{1/2\leq a\leq b\leq 2}  | K^{\{a,b\}}(\mathbf x,\mathbf y)|=  \sup_{1/2\leq a\leq b\leq 2}\Big|\int_a^b \Phi(t,\mathbf x,\mathbf y)\, dt\Big|\leq \int_{1/2}^2 |\Phi(t,\mathbf x,\mathbf y)|\, dt.$$ Consequently, applying with $F(t,\cdot)=\Phi(t,\cdot)$, we obtain $$\label{eq:K^ab}
 \int \sup_{1/2\leq a\leq b\leq 2}  | K^{\{a,b\}}(\mathbf x,\mathbf y)|\, dw(\mathbf y)\leq \int_{\mathbb R^N}\int_{1/2}^2 |\Phi(t,\mathbf x,\mathbf y)|\, dt\, dw(\mathbf y)\leq C'''.$$ Now for $j=0$ follows from , since, thanks to Theorem \[teo:support\], $K^{\{a,b\}}(\mathbf x,\mathbf y)=0$ if $d(\mathbf x,\mathbf y)>2$.

In order to prove we take $s_1,s>0$ such that $\mathbf N/2 <s_1$ and $s_0=s_1+s$. Then $$\begin{aligned}
    \Phi(t,\cdot)= \widetilde{\Phi}(t,\cdot)* J^{\{s_1\}}*J^{\{s\}}=F(t,\cdot)*J^{\{s\}}.\end{aligned}$$ Thus, $$\label{eq:supab}
\begin{split}
      & \sup_{1/2\leq a\leq b\leq 2} |K^{\{a,b\}}(\mathbf x,\mathbf y)-K^{\{a,b\}}(\mathbf x,\mathbf y')| =  \sup_{1/2\leq a\leq b\leq 2}\Big| \int_a^b \Big(\Phi(t,\mathbf x,\mathbf y)-\Phi(t,\mathbf x,\mathbf y')\Big)\, dt \Big| \\
      &\leq   \sup_{1/2\leq a\leq b\leq 2} \int_{a}^b     \int | F(t,\mathbf x,\mathbf z)| \Big|J^{\{s\}}(\mathbf z,\mathbf y)- J^{\{s\}}(\mathbf z,\mathbf y')\Big|\, dw(\mathbf z)\, dt \\
      &\leq \int_{1/2}^2     \int | F(t,\mathbf x,\mathbf z)| \Big|J^{\{s\}}(\mathbf z,\mathbf y)- J^{\{s\}}(\mathbf z,\mathbf y')\Big|\, dw(\mathbf z)\, dt .\\
\end{split}$$ Integrating  with respect to $dw(\mathbf x)$ and using  together with Lemma \[lem:Lipschitz\], we obtain for $j=0$. The proof of for $j=0$ is identical.

In order to prove , , and for arbitrary $j\in \mathbb Z$ we use scaling. To this end we fix $j\in\mathbb Z$ and write $G_j(\mathbf x)=2^{j\mathbf N}K(2^j\mathbf x)$. Then $G_j$ satisfies with the same constants $C_\beta$. Moreover, one can easily check that $$K^{\{a,b\}}(\mathbf x)=2^{-j\mathbf N} G_j^{\{2^{-j}a,2^{-j}b\}}(2^{-j}\mathbf x),$$ and, consequently, $$K^{\{a,b\}}(\mathbf x,\mathbf y)=2^{-j\mathbf N} G_j^{\{2^{-j}a,2^{-j}b\}}(2^{-j}\mathbf x,2^{-j}\mathbf y),$$ Now if $2^{j-1}\leq a\leq b\leq 2^{j+1}$, then $1/2\leq 2^{-j}a\leq 2^{-j}b\leq 2$ and we apply already proved results to $G_j$ and obtain the desired inequalities.

We now turn to prove . If $2^{j-1}\leq a\leq 2^{j+1}$, then we write $$K^{\{a\}}= K^{\{a,2^{j+1}\}}+\sum_{\ell=j+1}^\infty K^{\{2^{\ell},2^{\ell+1}\}},$$ where the convergence is in $L^2(dw)$. Now application of gives .

For a cube $Q\subset \mathbb R^N$, let $c_Q$ be its center and ${\text{\rm diam}}(Q)$ be the length of its diameter. Let $Q^{*}$ denote the cube with the same center $c_Q$ such that ${\text{\rm diam}}(Q^{*})=2{\text{\rm diam}}(Q)$. Let us remind that $$\begin{aligned}
    \mathcal{O}(Q^{*})=\{\sigma(\mathbf{x})\,:\,x\in Q^{*}, \, \sigma \in G\}.\end{aligned}$$ The following corollary is a direct consequence of Propositions \[propo:j\_small\].

\[coro:j\_small\_and\_large\] There are constants $C, \delta>0$ such that for any $j \in \mathbb{Z}$ and any every cube $Q\subset\mathbb R^N$, and $\mathbf y,\mathbf y'\in Q$ we have $$\begin{aligned}
    \int_{\mathbb{R}^N \setminus \mathcal{O}(Q^{*})}\sup_{2^{j-1}\leq a\leq b\leq 2^{j+1}}|K^{\{a,b\}}(\mathbf{x},\mathbf{y})&-K^{\{a,b\}}(\mathbf{x},\mathbf{y}')|\,dw(\mathbf{x}) \\
    &\leq C\min(2^{-j\delta}{\text{\rm diam}}(Q)^{\delta},2^{\delta j}({\text{\rm diam}}(Q))^{-\delta}).\end{aligned}$$

Proofs of Theorems \[theorem:weak\_type\_truncated1\], \[theorem:strong\_type\_truncated1\], and \[theorem:weak\_type\_K\]
--------------------------------------------------------------------------------------------------------------------------

We will prove  first. Take any $0<a\leq b<\infty$. There is a constant $C_1>1$, which depends on the doubling constant in  and $N$, such that $w(Q)\leq C_1 w(Q')$, where $Q'$ is any sub-cube of $Q$ such that $\text{\rm diam} (Q')=\text{\rm diam} (Q)\slash 2$.

Let $f\in L^1(dw)\cap L^2(dw)$. Fix $\lambda>0$. We denote by $\mathcal Q_\lambda$ the collection of all maximal (disjoint) dyadic cubes $Q_\ell$ in $\mathbb R^N$ satisfying $$\lambda< \frac{1}{w(Q_\ell)}\int_{Q_\ell}|f(\mathbf{x})|\,dw(\mathbf{x}).$$ Then $$\frac{1}{w(Q_\ell)} \int_{Q_\ell}|f(\mathbf{x})|\,dw(\mathbf{x})\leq C_1\lambda.$$ Set $\Omega=\bigcup_{Q_\ell \in\mathcal Q_\lambda} Q_\ell$. Then $w(\Omega)\leq \lambda^{-1}\| f\|_{L^1(dw)}$. Form the corresponding Calderón–Zygmund decomposition of $f$, namely, $f=\mathbf{g}+\mathbf{b}$, where

$$\label{eq:good}
\mathbf{g}(\mathbf{x})=f\chi_{\Omega^c}(\mathbf{x})+\sum_{\ell}w(Q_\ell)^{-1}\Big(\int_{Q_\ell}f(\mathbf{y})\,dw(\mathbf{y})\Big)\chi_{Q_\ell}(\mathbf{x}),$$

$$\label{eq:bad}
\mathbf{b}(\mathbf{x})=\sum_{\ell}\mathbf{b}_\ell(\mathbf{x}), \text{ where }\mathbf{b}_\ell(\mathbf{x})=\left(f(\mathbf{x})-w(Q_\ell)^{-1}\int_{Q_\ell}f(\mathbf{y})\,dw(\mathbf{y})\right)\chi_{Q_\ell}(\mathbf{x}).$$

Clearly, $\mathbf{g}(\mathbf x),\mathbf{b}(\mathbf x)\in L^1(dw(\mathbf x))\cap L^2(dw(\mathbf x))$, $|\mathbf{g}(\mathbf x)|\leq C_1\lambda$, $\| \mathbf{g}\|_{L^2(dw)}^2 \leq C\lambda \| f\|_{L^1}$, $\sum_\ell \| \mathbf{b}_\ell\|_{L^1(dw)}\leq C\| f\|_{L^1(dw)}$. Further, $$\begin{aligned}
w(\{\mathbf{x} \in \mathbb{R}^N\,:\,|K^{\{a,b\}}f(\mathbf{x})|> \lambda\}) &\leq w(\{\mathbf{x} \in \mathbb{R}^N\,:\,|K^{\{a,b\}}\mathbf{g}(\mathbf{x})|>\lambda/2\})\\&+w(\{\mathbf{x} \in \mathbb{R}^N\,:\,|K^{\{a,b\}}\mathbf{b}(\mathbf{x})|>\lambda/2\}).\end{aligned}$$ Since $\|K^{\{a,b\}}\|_{L^2(dw) \to L^2(dw)} \leq C_2$, where $C_2>0$ is independent of $0<a,b<\infty$ (see Corollary \[uniform\_L2\]), we obtain $$\begin{aligned}
w(\{\mathbf{x} \in \mathbb{R}^N\,:\,|K^{\{a,b\}}\mathbf{g}(\mathbf{x})|>\lambda/2\}) \leq \frac{4}{\lambda^{2}}C_2^2\|\mathbf{g}\|_{L^2(dw)}^2\leq \frac{C_3}{\lambda}\|f\|_{L^1(dw)}.\end{aligned}$$ Define $\Omega^{*}=\mathcal{O}\Big(\bigcup_{Q_\ell\in\mathcal Q_\lambda}Q_\ell^{*}\Big)$. There is a constant $C_2>1$, which depends on the Weyl group, doubling constant, and $N$ such that $$\begin{aligned}
w(\Omega^{*}) \leq C_2w(\Omega) \leq C_2\lambda^{-1}\|f\|_{L^1(dw)}.\end{aligned}$$ Thus it suffices to estimate $K^{\{a,b\}} \mathbf{b}(\mathbf x)$ on $\mathbb R^N\setminus \Omega^*$. Let $n_0, n_1 \in \mathbb{Z}$ be such that $2^{n_0-1}<a\leq 2^{n_0}$, $2^{n_1}\leq b<2^{n_1+1}$. If $n_0<n_1$, we write $$\label{K-decomp} K^{\{a,b\}}=K^{\{a,2^{n_0}\}}+\Big(\sum_{j=n_0}^{n_1-1}K^{\{2^{j},2^{j+1}\}}\Big)+K^{\{2^{n_1},b\}},$$ otherwise $2^{n_0-1}\leq a\leq b\leq 2^{n_0+1}$ and we consider just the single kernel $K^{\{a,b\}}$.

Since $\sum_{\ell}\mathbf{b}_\ell$ converges to $\mathbf{b}$ in $L^2(dw)$, $K^{\{a,b\}}\mathbf{b}=\sum_{\ell} K^{\{a,b\}}\mathbf{b}_\ell$ with convergence in $L^2(dw)$. So we have $$\begin{aligned}
|K^{\{a,b\}}\mathbf{b}(\mathbf{x})| \leq \sum_{\ell}\Big( |K^{\{a,2^{n_0}\}}\mathbf{b}_\ell(\mathbf x)|+\Big(\sum_{j=n_0}^{n_1-1}|K^{\{2^{j},2^{j+1}\}}\mathbf{b}_\ell(\mathbf x)|\Big)+|K^{\{2^{n_1},b\}}\mathbf{b}_\ell(\mathbf x)|\Big)  .\end{aligned}$$ By the fact that ${\text{\rm supp}\,}\mathbf{b}_\ell \subseteq Q_\ell$ and $\int_{\mathbb{R}^N} \mathbf{b}_\ell(\mathbf{y})\,dw(\mathbf{y})=0$, we get $$\label{eq:int_ell_j}\begin{split}
&\int_{\mathbb{R}^N \setminus \Omega^{*}}|K^{\{2^{j},2^{j+1}\}}\mathbf{b}_\ell(\mathbf{x})|\,dw(\mathbf{x})
 =\int_{\mathbb{R}^N \setminus \Omega^{*}}\left|\int_{Q_\ell} K^{\{2^{j}, 2^{j+1}\}}(\mathbf{x},\mathbf{y})\mathbf{b}_{\ell}(\mathbf{y})\,dw(\mathbf{y})\right|\,dw(\mathbf{x})\\
&\leq \int_{\mathbb{R}^N \setminus \mathcal O(Q_\ell^{*})}\left|\int_{Q_\ell}
 \Big(K^{\{2^j, 2^{j+1}\}}(\mathbf{x},\mathbf{y})-K^{\{2^j,2^{j+1}\}} (\mathbf x,c_{Q_\ell})\Big)\mathbf{b}_{\ell}(\mathbf{y})\,dw(\mathbf{y})\right|\,dw(\mathbf{x})\\
& \leq C \min \Big(2^{-j\delta}{\text{\rm diam}}(Q_\ell)^{\delta},2^{\delta j}{\text{\rm diam}}(Q_{\ell})^{-\delta}\Big)\| \mathbf{b}_\ell\|_{L^1(dw)},
\end{split}$$ where in the last inequality we have used Corollary \[coro:j\_small\_and\_large\] with $\delta>0$ small enough. Similarly, $$\int_{\mathbb{R}^N\setminus \Omega^{*}}|K^{\{a,2^{n_0}\}}\mathbf{b}_\ell(\mathbf{x})|\,dw(\mathbf{x})\leq C \min \Big(2^{-n_0\delta}{\text{\rm diam}}(Q_\ell)^{\delta},2^{n_0\delta }{\text{\rm diam}}(Q_{\ell})^{-\delta}\Big)\| \mathbf{b}_\ell\|_{L^1(dw)},$$ $$\label{eq:end}
 \int_{\mathbb{R}^N\setminus \Omega^{*}}|K^{\{2^{n_1},b\}}\mathbf{b}_\ell(\mathbf{x})|\,dw(\mathbf{x})\leq C \min \Big(2^{-n_1\delta}{\text{\rm diam}}(Q_\ell)^{\delta},2^{n_1\delta }{\text{\rm diam}}(Q_{\ell})^{-\delta}\Big)\| \mathbf{b}_\ell\|_{L^1(dw)}.$$ Summing up the inequalities – over $\ell \in \mathbb{Z}$ we end up with $$\int_{\mathbb R^N\setminus \Omega^*} | K^{\{a,b\}}\mathbf{b}(\mathbf x)|\,dw(\mathbf x)\leq C \sum_{\ell} \| \mathbf{b}_\ell\|_{L^1}\leq C\| f\|_{L^1(dw)},$$ with $C$ independent of $0<a\leq b<\infty$. Consequently, by the Chebyshev inequality, this completes the proof of weak type $(1,1)$ of the operator $K^{\{a,b\}}$.

In order to prove , let us note that for any $f\in L^2(dw)\cap L^1(dw)$ and any $a>0$ there is an increasing sequence $m_j\to\infty$, $m_j>a$, such that $K^{\{a\}} f=\lim_{j\to\infty}  K^{\{a,m_j\}}f $ with convergence in $L^2(dw)$ and almost everywhere. Therefore, up to a set of measure zero we have $$\label{eq:cup_cap_cup}
    \{\mathbf{x} \in \mathbb{R}^N\,:\,|K^{\{a\}}f(\mathbf{x})|>\lambda\}= 
    \bigcup_{n=1}^\infty \bigcap_{j \geq n}\{\mathbf{x} \in \mathbb{R}^N\,:\,|K^{\{a,m_j\}}f(\mathbf{x})|>\lambda\}.$$ So, thanks to  and , we have $$\begin{aligned}
    w(\{\mathbf{x} \in \mathbb{R}^N\,:\,|K^{\{a\}}f(\mathbf{x})|>\lambda\}) &\leq \sup_{a \leq b}w(\{\mathbf{x} \in \mathbb{R}^N\,:\,|K^{\{a,b\}}f(\mathbf{x})|>\lambda\}) \leq \frac{C}{\lambda}\|f\|_{L^1(dw)}.\end{aligned}$$

First we note that by and with $\delta=0$ we have $$\int |K^{\{a,b\}}(\mathbf x,\mathbf y)|\, dw(\mathbf x)\leq C_{a,b}, \quad  \int |K^{\{a,b\}}(\mathbf x,\mathbf y)|\, dw(\mathbf y)\leq C_{a,b}.$$ Thus the operators $K^{\{a,b\}}$ are bounded on $L^p(dw)$. Moreover, $(K^{\{a,b\}})^*=(K^*)^{\{a,b\}}$, where $K^*(\mathbf x)=\overline{K(-\mathbf x)}$. Now for fixed $1<p<\infty$ the uniform bound of the operators $K^{\{a,b\}}$ on $L^p(dw)$ follows from interpolation, Corollary \[uniform\_L2\], Theorem \[theorem:weak\_type\_truncated1\], and duality.

Furthermore,  and the strong convergence of $K^{\{a,b\}}$ to $K^{\{a\}}$ on the space $L^p(dw)$ for $1<p<2$ as ($b\to\infty$) also follows from the Marcinkiewicz interpolation theorem and Corollary \[eq:L2strong\]. In order to prove  for $p>2$, let us show first that for $f \in \mathcal{S}(\mathbb{R}^N)$ the function $(a,\infty)\ni b\mapsto K^{\{a,b\}}f\in L^p(dw)$ satisfies the Cauchy condition as $b\to\infty$. Clearly, $$\begin{aligned}
    &\|K^{\{a,b_1\}}f-K^{\{a,b_2\}}f\|_{L^p(dw)} \\&\leq \|K^{\{a,b_1\}}f-K^{\{a,b_2\}}f\|_{L^2(dw)}^{1/(p-1)}\|K^{\{a,b_1\}}f-K^{\{a,b_2\}}f\|_{L^{2p}(dw)}^{1-1/(p-1)}.\end{aligned}$$ Thanks to we have $$\begin{aligned}
    \lim_{b_1,b_2 \to \infty}\|K^{\{a,b_1\}}f-K^{\{a,b_2\}}f\|_{L^2(dw)}^{1/(p-1)}=0,\end{aligned}$$ moreover, by  , we get $$\begin{aligned}
    \|K^{\{a,b_1\}}f-K^{\{a,b_2\}}f\|_{L^{2p}(dw)}^{1-1/(p-1)} \leq C\|f\|_{L^{2p}(dw)}^{1-1/(p-1)}<\infty.\end{aligned}$$ Consequently, $$\label{eq:convergence_on_schwartz}
    \lim_{b_1,b_2 \to \infty}\|K^{\{a,b_1\}}f-K^{\{a,b_2\}}f\|_{L^p(dw)} =0$$ for $f \in \mathcal{S}(\mathbb{R}^N)$. In order to prove  for $f \in L^p(dw)$, it is enough to take a sequence $\mathcal{S}(\mathbb{R}^N) \ni f_{\ell} \to f$ in $L^p(dw)$ and write $$\begin{aligned}
    \|K^{\{a,b_1\}}f-K^{\{a,b_2\}}f\|_{L^p(dw)} &\leq \|K^{\{a,b_1\}}(f-f_\ell)\|_{L^p(dw)}+\|K^{\{a,b_1\}}f_\ell-K^{\{a,b_2\}}f_\ell\|_{L^p(dw)}\\&+\|K^{\{a,b_2\}}(f_{\ell}-f)\|_{L^p(dw)},\end{aligned}$$ then use  for the first and third summand and  for the second one.

In order to prove , we use the same argument as in the second part of the proof of Theorem \[theorem:weak\_type\_truncated1\] (see ). In order to prove , see the proof of Theorem \[theorem:strong\_type\_truncated1\].

Maximal function associated with singular integral
==================================================

Let $$\label{eq:maximal}
    K^{*}f(\mathbf{x})=\sup_{a>0}|K^{\{a\}}f(\mathbf{x})|.$$ The goal of this section is to prove the following theorem.

\[theorem:maximal\] Under , , and , the operator $K^{*}$ is of weak type $(1,1)$ and it is bounded on $L^p(dw)$ for $1<p<\infty$.

Cotlar type inequality
----------------------

Let $$\begin{aligned}
\mathcal{M}_{\rm HL}f(\mathbf{x})=\sup_{\mathbf{x} \in B}\frac{1}{w(B)}\int_{B}|f(\mathbf{y})|\,dw(\mathbf{y}),\end{aligned}$$ where the supremum is taken over all Euclidean balls $B$ which contain $\mathbf{x}$, be the non-centered Hardy-Littlewood maximal function defined on the space of homogeneous type $(\mathbb R^N, \|\mathbf x-\mathbf y\|, dw)$. The following lemma is in the spirit of Cotlar’s inequality (cf. [@Duo Lemma 5.15]).

Let $p \in [1,\infty)$. There is a constant $C>0$ such that for all $f \in L^p(dw) \cap L^{\infty}$ and $\mathbf{x} \in \mathbb{R}^N$ we have $$\label{eq:Cotlar}
    K^*f(\mathbf{x}) \leq C\Big(\sum_{\sigma \in G}(\mathcal{M}_{\rm HL}Kf)(\sigma(\mathbf{x}))+\|f\|_{L^{\infty}}\Big).$$

We assume additionally that $f\in L^2(dw)$. Then this assumption can be easily relaxed by a density argument. Let $\varphi \in C_c^{\infty}(\mathbb{R}^N)$ be a radial function such that ${\text{\rm supp}\,}\varphi \subseteq B(0,1)$ and $\int_{\mathbb{R}^N} \varphi\, dw=1$. Fix $a>0$ and set $\widetilde{K^{\{a\}}}=K-K^{\{a\}}$. Let as remind that $\varphi_a(\mathbf x)=a^{-\mathbf N} \varphi(\mathbf x /a)$. Then $$\begin{split}
        K^{\{a\}}f&= (K^{\{a\}}-\varphi_a*\varphi_a*K^{\{a\}})*f+\varphi_a*\varphi_a*(Kf)-((\varphi_a*(\widetilde{K^{\{a\}}}\varphi_a))*f\\&=:J_1+J_2-J_3.
    \end{split}$$ Clearly, by Lemma \[lem:nonradial\_estimation\], $|J_2|\leq C \sum_{\sigma\in G} ( \mathcal M_{\rm HL}Kf)(\sigma \cdot)$. In order to estimate $J_3$ we note that $\text{supp}\, \widetilde{K^{\{a\}}}\varphi_a \subset B(0,2a)$, so by Cauchy–Schwarz inequality and Theorem \[theorem:strong\_type\_truncated1\] together with Lemma \[lem:nonradial\_estimation\] we have $$\label{eq:CS_app}
    \|\widetilde{K^{\{a\}}}\varphi_a \|_{L^1(dw)}\leq C w(B(0,2a))^{1\slash 2} \| \varphi_a\|_{L^2(dw)}\leq C.$$ Applying Proposition \[propo:compact\_supports\] we get $$\int_{\mathbb{R}^N} |(\varphi_a*(\widetilde{K^{\{a\}}}\varphi_a))(\mathbf x,\mathbf y)|\, dw(\mathbf y)\leq Ca^{\mathbf N} \|\varphi_a\|_{L^\infty} \cdot \|\widetilde{K^{\{a\}}}\varphi_a \|_{L^1(dw)}\leq C,$$ which, together with , implies $|J_3|\leq C\| f\|_{L^\infty}.$ Finally, in order to evaluate $J_1$, we consider the integral kernel of the operator $(K^{\{a\}}-\varphi_a*\varphi_a*K^{\{a\}})$. Using  and Lemma \[lem:nonradial\_estimation\] we get $$\begin{aligned}
        \int_{\mathbb{R}^N}| & K^{\{a\}}(\mathbf x,\mathbf y) -(\varphi_a*\varphi_a*K^{\{a\}})(\mathbf x,\mathbf y)|\, dw(\mathbf y)\\
        &= \int_{\mathbb{R}^N} \Big|\int_{\mathbb{R}^N} (\varphi_a*\varphi_a)(\mathbf x,\mathbf z) (K^{\{a\}}(\mathbf x,\mathbf y)-K^{\{a\}}(\mathbf z,\mathbf y))\, dw(\mathbf z)\Big|\, dw(\mathbf y)\\
        &\leq \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} |(\varphi_a*\varphi_a)(\mathbf x,\mathbf z)||K^{\{a\}} (\mathbf x,\mathbf y)-K^{\{a\}}(\mathbf z,\mathbf y)|\, dw(\mathbf y)\, dw(\mathbf z)\\
        &\leq \int_{\mathbb{R}^N} |(\varphi_a*\varphi_a)(\mathbf x,\mathbf z)|  a^{-\delta} \| \mathbf x-\mathbf z\|^{\delta} \, dw(\mathbf z) \\
        &\leq \int_{\mathcal{O}(B(\mathbf{x},4a))} w(B(\mathbf x,2a))^{-1} \Big(1+\frac{\| \mathbf x-\mathbf z\|}{a}\Big)^{-2} a^{-\delta} \| \mathbf x-\mathbf z\|^{\delta } \, dw(\mathbf z) \leq C,  \end{aligned}$$ which gives $|J_1|\leq C\| f\|_{L^\infty}$.

Proof of Theorem \[theorem:maximal\]
------------------------------------

It is enough to prove that $ K^{*}$ is of weak type $(p,p)$ for $1 \leq p <\infty$. Let $f\in L^p(dw)\cap L^2(dw)$. Fix $\lambda>0$. We denote by $\mathcal Q_\lambda$ the collection of all maximal (disjoint) dyadic cubes $Q_\ell$ in $\mathbb R^N$ satisfying $$\label{eq:lambda_upper}
    \lambda^p< \frac{1}{w(Q_\ell)}\int_{Q_\ell}|f(\mathbf{x})|^p\,dw(\mathbf{x}).$$ Then $$\label{eq:lambda_lower}
\frac{1}{w(Q_\ell)} \int_{Q_\ell}|f(\mathbf{x})|^p\,dw(\mathbf{x})\leq C_1\lambda^p.$$ Set $\Omega=\bigcup_{Q_\ell \in\mathcal Q_\lambda} Q_\ell$. Thanks to  we have $$\label{eq:sum_Q_j}
\sum_{\ell}w(Q_\ell)=w(\Omega)\leq \lambda^{-p}\| f\|_{L^p(dw)}^p.$$ Form the corresponding Calderón–Zygmund decomposition of $f$, namely, $f=\mathbf{g}+\mathbf{b}$, where

$$\label{eq:good_max}
\mathbf{g}(\mathbf{x})=f\chi_{\Omega^c}(\mathbf{x})+\sum_{\ell}w(Q_\ell)^{-1}\Big(\int_{Q_\ell}f(\mathbf{y})\,dw(\mathbf{y})\Big)\chi_{Q_\ell}(\mathbf{x}),$$

$$\label{eq:bad_max}
\mathbf{b}(\mathbf{x})=\sum_{\ell}\mathbf{b}_\ell(\mathbf{x}), \text{ where }\mathbf{b}_\ell(\mathbf{x})=\left(f(\mathbf{x})-w(Q_\ell)^{-1}\int_{Q_\ell}f(\mathbf{y})\,dw(\mathbf{y})\right)\chi_{Q_\ell}(\mathbf{x}).$$

Clearly, $\|\mathbf{g}\|_{L^p(dw)}+\|\mathbf{b}\|_{L^p(dw)} \leq C\| f\|_{L^p(dw)}$, $\|\mathbf{g}\|_{L^2(dw)}+\|\mathbf{b}\|_{L^2(dw)} \leq C\| f\|_{L^2(dw)}$, and $|\mathbf{g}(\mathbf x)|\leq C_1^{ 1/p}\lambda$. Further, $$\begin{aligned}
w(\{\mathbf{x} \in \mathbb{R}^N\,:\,|K^{*}f(\mathbf{x})|> \lambda) &\leq w(\{\mathbf{x} \in \mathbb{R}^N\,:\,|K^{*}\mathbf{b}(\mathbf{x})|>\lambda/2\})\\&+w(\{\mathbf{x} \in \mathbb{R}^N\,:\,|K^{*}\mathbf{g}(\mathbf{x})|>\lambda/2\})=:S_1+S_2.\end{aligned}$$

Let us estimate $S_1$ first. Define $\Omega^{*}=\mathcal{O}\Big(\bigcup_{Q_\ell\in\mathcal Q_\lambda}Q_\ell^{*}\Big)$. There is a constant $C_2>1$, which depends on the Weyl group, doubling constant, and $N$ such that $$\begin{aligned}
\label{eq:OmegaStar}
w(\Omega^{*}) \leq C_2w(\Omega) \leq C_2\lambda^{-p}\|f\|^p_{L^p(dw)}.\end{aligned}$$ Thus it suffices to estimate $K^{*}\mathbf{b}(\mathbf x)$ on $\mathbb R^N\setminus \Omega^*$. Note that $\sum_{\ell}\mathbf{b}_\ell$ converges to $\mathbf{b}$ in $L^2(dw)$. Let us remind that $c_{Q_\ell}$ is the center of $Q_j$. We write $$\label{eq:on_bad_computation}
    \begin{split}
        &\|K^{*}\mathbf{b}\|_{L^1(\mathbb{R}^N \setminus \Omega^{*},dw)}=\int_{\mathbb{R}^N \setminus \Omega^{*}}\sup_{a>0}\Big|\sum_{\ell}\int_{Q_{\ell}}K^{\{a\}}(\mathbf{x},\mathbf{y})\mathbf{b}_{\ell}(\mathbf{y})\,dw(\mathbf{y})\Big|\,dw(\mathbf{x}) \\&\leq \int_{\mathbb{R}^N \setminus \Omega^{*}}\sup_{a>0}\Big|\sum_{\ell}\int_{Q_{\ell}}(K^{\{a\}}(\mathbf{x},\mathbf{y})-K^{\{a\}}(\mathbf{x},c_{Q_{\ell}}))\mathbf{b}_{\ell}(\mathbf{y})\,dw(\mathbf{y})\Big|\,dw(\mathbf{x}) \\& \leq \sum_{\ell}\sum_{j \in \mathbb{Z}}\int_{Q_{\ell}}|\mathbf{b}_{\ell}(\mathbf{y})|\int_{\mathbb{R}^N \setminus {\mathcal O(Q_{\ell}^{*})}}\sup_{2^{j-1} \leq a' \leq b' \leq 2^{j+1}}|K^{\{a',b'\}}(\mathbf{x},\mathbf{y})-K^{\{a',b'\}}(\mathbf{x},c_{Q_{\ell}})|\,dw(\mathbf{x})\,dw(\mathbf{y})\\
& \leq C \sum_{\ell}\sum_{j \in \mathbb{Z}}\min \Big(2^{-j\delta}{\text{\rm diam}}(Q_\ell)^{\delta},2^{\delta j}{\text{\rm diam}}(Q_{\ell})^{-\delta}\Big)\| \mathbf{b}_\ell\|_{L^1(dw)},
\end{split}$$ where in the last inequality we have used Corollary \[coro:j\_small\_and\_large\] with $\delta>0$ small enough. By Hölder’s inequality and  we have $$\label{eq:sum_bad}
    \sum_{\ell}\|\mathbf{b}_{\ell}\|_{L^1(dw)} \leq \sum_{\ell}w(Q_j)\left(w(Q_{\ell})^{-1}\int_{Q_\ell}|\mathbf{b}_{\ell}(\mathbf{x})|^p\,dw(\mathbf{x})\right)^{1/p} \leq C\lambda\sum_{\ell}w(Q_{\ell}).$$ Combining , , and  we get $$\label{eq:K_star_L_1}
    \|K^{*}\mathbf{b}\|_{L^1(\mathbb{R}^N \setminus \Omega^{*},\,dw)} \leq C \lambda^{-p+1}\|f\|_{L^p}^p.$$ Chebyshev’s inequality applied to   together with imply $S_1\leq C \lambda^{-p}\| f\|_{L^p(dw)}^p$.

In order to estimate $S_2$, let us note that it is enough to obtain a proper bound for $$w(\{\mathbf{x} \in \mathbb{R}^N\,:\,|K^{*}\mathbf{g}(\mathbf{x})|>2C_1^{1/p}C_3\lambda\}),$$ where $C_3$ is the constant from . Thanks to the fact that $\|\mathbf{g}\|_{L^{\infty}} \leq C_1^{1/p}\lambda$ and  we have $$w(\{\mathbf{x} \in \mathbb{R}^N\,:\,\big|K^{*}\mathbf{g}(\mathbf{x})\big|>2C_1^{1/p}C_3\lambda\}) \leq w(\{\mathbf{x} \in \mathbb{R}^N\,:\,|\sum_{\sigma \in G}\mathcal{M}_{\rm HL}K\mathbf{g}(\sigma(\mathbf{x}))|>C_1^{1/p}\lambda\}).$$ If $p>1$, then let us remind that $\mathcal{M}_{\rm HL}$ and $K$ are bounded operators on $L^p(dw)$ (see Theorem \[theorem:weak\_type\_K\]), so $S_2\leq C  \lambda^{-p}\| f\|_{L^p(dw)}^p$. If $p=1$, then $\|\mathbf{g}\|_{L^2(dw)} \leq C\lambda \| f\|_{L^1(dw)}$ and, consequently, $$\begin{aligned}
   w(\{\mathbf{x} \in \mathbb{R}^N\,:\,|\sum_{\sigma \in G}\mathcal{M}_{\rm HL}K\mathbf{g}(\sigma(\mathbf{x}))|>C_1^{1/p}\lambda\}) \leq C\lambda^{-2}\|\mathbf{g}\|_{L^{2}(dw)}^{2} \leq C\lambda^{-1}\|f\|_{L^1(dw)}.\end{aligned}$$

[99]{}

J.-Ph. Anker, J. Dziubański, A. Hejna, *Harmonic functions, conjugate harmonic functions and the Hardy space $H^1$ in the rational Dunkl setting*, J. Fourier Anal. Appl. 25 (2019), 2356–2418.

B. Amri, A. Hammi, *Dunkl-Schrödinger operators*, [Complex Anal. Oper. Theory (2018).]{}

M.F.E. de Jeu, *The Dunkl transform*, Invent. Math. 113 (1993), 147–162.

M. de Jeu, M. Rösler, *Asymptotic analysis for the Dunkl kernel*, J. Approx. Theory 119 (2002), no. 1, 110–126.

C.F. Dunkl, *Reflection groups and orthogonal polynomials on the sphere*, Math. Z. 197 (1988), no. 1, 33–60.

C.F. Dunkl, *Differential-difference operators associated to reflection groups*, Trans. [Amer]{}. Math. 311 [(1989), no. 1,]{} 167–183[.]{}

C.F. Dunkl, *Hankel transforms associated to finite reflection groups*, in: [*Proc. of the special session on hypergeometric functions on domains of positivity, Jack polynomials and applications,*]{} Proceedings, Tampa 1991, Contemp. Math. 138 (1989), 123–138.

C.F. Dunkl, *Integral kernels with reflection group invariance*, Canad. J. Math. 43 (1991), no. 6, 1213–1227.

J. Duoandikoetxea, *Fourier Analysis*, Graduate Studies in Mathematics, 29. American Mathematical Society, Providence, RI, 2001. xviii+222 pp. ISBN: 0-8218-2172-542-01.

J. Dziubański, A. Hejna, *Remark on atomic decompositions for Hardy space $H^1$ in the rational Dunkl setting*, [\[doi:10.4064/sm180618-25-11\]](https://doi.org/10.4064/sm180618-25-11), to appear in Studia Math.

J. Dziubański and A. Hejna, *Hörmander’s multiplier theorem for the Dunkl transform*, Journal of Functional Analysis 277 (2019), 2133-2159.

L. Gallardo, C. Rejeb, *A new mean value property for harmonic functions relative to the Dunkl-Laplacian operator and applications*, Trans. Amer. Math. Soc. 368 (2015), [no. 5,]{} 3727–3753.

L. Grafakos, *Modern Fourier Analysis*, 3rd edition, Graduate Texts in Mathematics, 250. Springer, New York, 2014.

M. Rösler, *Generalized Hermite polynomials and the heat equation for Dunkl operators*, Comm. Math. Phys. [192]{} (1998), 519–542.

M. Rösler, *Positivity of Dunkl’s intertwining operator*, Duke Math. J. 98 (1999), no. 3, 445–463.

M. Rösler, *A positive radial product formula for the Dunkl kernel*, Trans. Amer.Math. Soc. 355 (2003), no. 6, 2413–2438[.]{}

M. Rösler: *Dunkl operators (theory and applications). In: Koelink, E., Van Assche, W. (eds.) Orthogonal polynomials and special functions* (Leuven, 2002), 93–135. Lect. Notes Math. 1817, Springer-Verlag (2003).

M. Rösler, M. Voit, *Dunkl theory, convolution algebras, and related Markov processes*, in *Harmonic and stochastic analysis of Dunkl processes*, [P. Graczyk, M. Rösler, M. Yor (eds.), 1–112, Travaux en cours 71,]{} Hermann, Paris, 2008.

E.M. Stein, *Singular integral and differentiability properties of functions*, Princeton [Math.]{} Series 30, Princeton Univ. Press, 1970.

E.M. Stein, *Harmonic analysis* [(*real variable methods, orthogonality and oscillatory integrals*)]{}, [Princeton Math. Series 43]{}, Princeton Univ. Press, 1993.

S. Thangavelu, Y. Xu, *Convolution operator and maximal function for the Dunkl transform*, J. Anal. Math. 97 (2005), 25–55.

K. Triméche, *Paley-Wiener theorems for the Dunkl transform and Dunkl translation operators*, Integral Transforms Spec. Funct. 13 (2002), no. 1, 17–38.

[^1]: Research supported by the National Science Centre, Poland (Narodowe Centrum Nauki), Grant 2017/25/B/ST1/00599.

[^2]: The symbol $\sim$ between two positive expressions means that their ratio remains between two positive constants.
