On semigroups generated by sums of even powers of Dunkl operators

TL;DR

This paper studies a class of differential-difference operators built from Dunkl operators on Euclidean space, showing they generate semigroups with kernels exhibiting specific exponential decay properties related to the operators' structure.

Contribution

The authors establish the self-adjointness of sums of even powers of Dunkl operators and derive precise exponential decay estimates for the associated semigroup kernels.

Findings

The semigroup generated by the operator is a contraction on L^2(dw).

The kernel q(x) exhibits exponential decay of order rac{2\u221e}{2\u221e-1}.

Decay estimates extend to the Dunkl translation and the metric d(x,y).

Abstract

On the Euclidean space $\mathbb R^N$ equipped with a normalized root system $R$, a multiplicity function $k\geq 0$, and the associated measure $dw(\mathbf x)=\prod_{\alpha\in R} |\langle \mathbf x,\alpha\rangle|^{k(\alpha)}d\mathbf x$ we consider the differential-difference operator $$L=(-1)^{\ell+1} \sum_{j=1}^m T_{\zeta_j}^{2\ell},$$ where $\zeta_1,...,\zeta_m$ are nonzero vectors in $\mathbb R^N$, which span $\mathbb R^N$, and $T_{\zeta_j}$ are the Dunkl operators. The operator $L$ is essentially self-adjoint on $L^2(dw)$ and generates a semigroup $\{S_t\}_{t \geq 0}$ of linear self-adjoint contractions, which has the form $S_tf(\mathbf x)=f*q_t(\mathbf{x})$, $q_t(\mathbf x)=t^{-\mathbf N/ (2\ell)}q(\mathbf x/ t^{1/ (2\ell)})$, where $q(\mathbf x)$ is the Dunkl transform of the function $ \exp(-\sum_{j=1}^m \langle \zeta_j,\xi\rangle^{2\ell})$. We prove that $q(\mathbf x)$ satisfies the following exponential decay: $$ |q(\mathbf x)| \lesssim \exp(-c \| \mathbf x\|^{2\ell/ (2\ell-1)})$$ for a certain constant $c>0$. Moreover, if $q(\mathbf x,\mathbf y)=\tau_{\mathbf x}q(-\mathbf y)$, then $|q(\mathbf x,\mathbf y)|\lesssim w(B(\mathbf x,1))^{-1} \exp(-c d(\mathbf x,\mathbf y)^{2\ell / (2\ell-1)})$, where $d(\mathbf x,\mathbf y)=\min_{\sigma\in G}\| \mathbf x- \sigma(\mathbf y)\| $, $G$~is the reflection group for $R$, and $\tau_{\mathbf x}$ denotes the Dunkl translation.

Equations

$$L=(-1{)}^{\ell +1}\sum _{j=1}^m{T}_{{\zeta }_j}^{2\ell },$$

$$|q(\mathbf{x})|\lesssim \exp (-c\Vert \mathbf{x}{\Vert }^{2\ell /(2\ell -1)})$$

$$L=(-1{)}^{\ell +1}\sum _{j=1}^m{T}_{{\zeta }_j}^{2\ell },$$

$$S_{t}f(\mathbf{x})=f\ast q_t(\mathbf{x}),$$

$$q_{t}(\mathbf{x})=t^{-\mathbf{N}/(2\ell)}q\Big{(}\frac{\mathbf{x}}{t^{1/(2\ell)}}\Big{)}.$$

$$|q(\mathbf{x})|\le C\exp (-c\Vert \mathbf{x}{\Vert }^{2\ell /(2\ell -1)}).$$

$$S_{t}f(\mathbf{x})={\int }_{{\mathbb{R}}^N}{q}_t(\mathbf{x},\mathbf{y})f(\mathbf{y})dw(\mathbf{y}).$$

$$d(\mathbf{x},\mathbf{y})=\underset{\sigma \in G}{\min }\Vert \sigma (\mathbf{x})-\mathbf{y}\Vert$$

$$|q(\mathbf{x},\mathbf{y})|\le C(\max \{w(B(\mathbf{x},1)),w(B(\mathbf{y},1)\}{)}^{-1}\exp (-cd(\mathbf{x},\mathbf{y}{)}^{2\ell /(2\ell -1)}).$$

$$|q_t(\mathbf{x},\mathbf{y})|\le C(\max \{w(B(\mathbf{x},t^{1/(2\ell )})),w(B(\mathbf{y},t^{1/(2\ell )})\}{)}^{-1}\exp \left(-c\frac{d(\mathbf{x},\mathbf{y}{)}^{2\ell /(2\ell -1)}}{t^{1/(2\ell -1)}}\right).$$

$${\sigma }_{\alpha }(\mathbf{x})=\mathbf{x}-2\frac{⟨\mathbf{x},\alpha ⟩}{\Vert \alpha {\Vert }^2}\alpha .$$

$$dw(\mathbf{x})=\prod _{\alpha \in R}|⟨\mathbf{x},\alpha ⟩{|}^{k(\alpha )}d\mathbf{x}$$

$$w(B(t\mathbf{x},tr))=t^{\mathbf{N}}w(B(\mathbf{x},r))\text{for all }\mathbf{x}\in {\mathbb{R}}^N,t,r>0$$

$${\int }_{{\mathbb{R}}^N}f(\mathbf{x})dw(\mathbf{x})={\int }_{{\mathbb{R}}^N}{t}^{-\mathbf{N}}f(\mathbf{x}/t)dw(\mathbf{x})\text{for}f\in L^1(dw)\text{and}t>0.$$

$$w(B(\mathbf{x},r))\sim r^N\prod _{\alpha \in R}(|⟨\mathbf{x},\alpha ⟩|+r{)}^{k(\alpha )},$$

$$w(B(\mathbf{x},2r))\le Cw(B(\mathbf{x},r))\text{ for all }\mathbf{x}\in {\mathbb{R}}^N,r>0.$$

$$T_{\xi }f(\mathbf{x})={\partial }_{\xi }f(\mathbf{x})+\sum _{\alpha \in R}\frac{k(\alpha )}{2}⟨\alpha ,\xi ⟩\frac{f(\mathbf{x})-f({\sigma }_{\alpha }\mathbf{x})}{⟨\alpha ,\mathbf{x}⟩}.$$

$$T_{\xi }(fg)=f(T_{\xi }g)+g(T_{\xi }f).$$

$$T_{\xi }f=⟨\xi ,\mathbf{y}⟩f,f(0)=1.$$

$$|\beta |={\beta }_1+{\beta }_2+\dots +{\beta }_N,$$

$${\partial }^{\beta }={\partial }_1^{{\beta }_1}\circ {\partial }_2^{{\beta }_2}\circ \dots \circ {\partial }_N^{{\beta }_N},$$

$$T^{\beta }=T_1^{{\beta }_1}\circ T_2^{{\beta }_2}\circ \dots \circ T_N^{{\beta }_N}.$$

$$|{\partial }_{\mathbf{z}}^{\nu }E(\mathbf{x},\mathbf{z})|\le \Vert \mathbf{x}{\Vert }^{|\nu |}\exp (\Vert \mathbf{x}\Vert \Vert \mathrm{Re}\mathbf{z}\Vert ).$$

$$|E(i\xi ,\mathbf{x})|\le 1\text{ for all }\xi ,\mathbf{x}\in {\mathbb{R}}^N.$$

$$\mathcal{F}f(\xi )=c_k^{-1}{\int }_{{\mathbb{R}}^N}E(-i\xi ,\mathbf{x})f(\mathbf{x})dw(\mathbf{x}),$$

$$c_k={\int }_{{\mathbb{R}}^N}{e}^{-\frac{\Vert \mathbf{x}{\Vert }^2}{2}}dw(\mathbf{x})>0,$$

$$\Vert f{\Vert }_{L^2(dw)}=\Vert \mathcal{F}f{\Vert }_{L^2(dw)}\text{ for all }f\in L^2(dw),$$

$${\mathcal{F}}^{-1}g(x)=c_k^{-1}{\int }_{{\mathbb{R}}^N}E(i\xi ,\mathbf{x})g(\xi )dw(\xi ).$$

$$\mathcal{F}(T_{\zeta }f)(\xi )=-i⟨\zeta ,\xi ⟩\mathcal{F}f(\xi )\text{ for all }\xi ,\zeta \in {\mathbb{R}}^N,$$

$$\mathcal{F}(Lf)(\xi)=-\Big{(}\sum_{j=1}^{m}\langle\zeta_{j},\xi\rangle^{2\ell}\Big{)}\mathcal{F}f(\xi)\text{ for all }\xi\in\mathbb{R}^{N}.$$

Full text

[TABLE]

where $\zeta_{1},...,\zeta_{m}$ are nonzero vectors in $\mathbb{R}^{N}$, which span $\mathbb{R}^{N}$, and $T_{\zeta_{j}}$ are the Dunkl operators. The operator $L$ is essentially self-adjoint on $L^{2}(dw)$ and generates a semigroup $\{S_{t}\}_{t\geq 0}$ of linear self-adjoint contractions, which has the form $S_{t}f(\mathbf{x})=f*q_{t}(\mathbf{x})$, $q_{t}(\mathbf{x})=t^{-\mathbf{N}/(2\ell)}q(\mathbf{x}/t^{1/(2\ell)})$, where $q(\mathbf{x})$ is the Dunkl transform of the function $\exp(-\sum_{j=1}^{m}\langle\zeta_{j},\xi\rangle^{2\ell})$. We prove that $q(\mathbf{x})$ satisfies the following exponential decay:

[TABLE]

for a certain constant $c>0$. Moreover, if $q(\mathbf{x},\mathbf{y})=\tau_{\mathbf{x}}q(-\mathbf{y})$, then $|q(\mathbf{x},\mathbf{y})|\lesssim w(B(\mathbf{x},1))^{-1}\exp(-cd(\mathbf{x},\mathbf{y})^{2\ell/(2\ell-1)})$, where $d(\mathbf{x},\mathbf{y})=\min_{\sigma\in G}\|\mathbf{x}-\sigma(\mathbf{y})\|$, $G$ is the reflection group for $R$, and $\tau_{\mathbf{x}}$ denotes the Dunkl translation.

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

1. Introduction

Let $\zeta_{1},...,\zeta_{m}\in\mathbb{R}^{N}$ be non-zero vectors which span $\mathbb{R}^{N}$. For $\ell\in\mathbb{N}$ (which will be fixed throughout the paper) we consider the symmetric differential-difference operator

[TABLE]

where $T_{\zeta_{j}}$ are Dunkl operators associated with a normalized system of roots $R$ and a multiplicity function $k\geq 0$ (see Section 2 for details). Let $dw$ denote the related measure (see (2.2)). The operator $L$ is essentially self-adjoint on $L^{2}(dw)$ and its closure generates a semigroup of self-adjoint linear contractions $\{S_{t}\}_{t\geq 0}$ on $L^{2}(dw)$. The semigroup has the form

[TABLE]

where $q_{t}(\mathbf{x})=\mathcal{F}^{-1}(\exp(-t\sum_{j=1}^{m}\langle\zeta_{j},\cdot\rangle^{2\ell})(\mathbf{x}).$ Here and subsequently, $*$ denotes the Dunkl convolution, while $\mathcal{F}$ and $\mathcal{F}^{-1}$ stand for the Dunkl transform and its inverse respectively (see (2.10)). Clearly, $q_{t}\in\mathcal{S}(\mathbb{R}^{N})$, and if we set $q(\mathbf{x})=q_{1}(\mathbf{x})$, then, by homogeneity,

[TABLE]

Our first result is to prove that the decay of $q(\mathbf{x})$ is exponential. This is stated in the following theorem.

Theorem 1.1**.**

There are constants $C,c>0$ such that for all $\mathbf{x}\in\mathbb{R}^{N}$ we have

[TABLE]

Let $\tau_{\mathbf{x}}$ denote the Dunkl translation (see (2.15)). Then $q_{t}(\mathbf{x},\mathbf{y})=\tau_{\mathbf{x}}q_{t}(-\mathbf{y})$ are the integral kernels of the operators $S_{t}$ with respect to the measure $dw$, that is,

[TABLE]

Let

[TABLE]

be the distance of the orbit of $\mathbf{x}$ to the orbit of $\mathbf{y}$, where $G$ denotes the Weyl group associated with $R$ (see Section 2). We denote by $B(\mathbf{x},r)$ the (closed) Euclidean ball centered at $\mathbf{x}\in\mathbb{R}^{N}$ and radius $r$. Our second result expresses the decay of $q_{t}(\mathbf{x},\mathbf{y})$ by means of the distance $d(\mathbf{x},\mathbf{y})$.

Theorem 1.2**.**

There are constants $C,c>0$ such that for all $\mathbf{x},\mathbf{y}\in\mathbb{R}^{N}$ we have

[TABLE]

Remark 1.3**.**

By a scaling argument applied to (1.3) (see (1.2) and (2.3)) we obtain that there are $C,c>0$ such that for all $\mathbf{x},\mathbf{y}\in\mathbb{R}^{N}$ and $t>0$ we have

[TABLE]

To prove the first theorem we borrow ideas of [9] and [10]. We first introduce a family of weighted $L^{2}$-spaces with weights of exponential growth and prove that (1.1) defines strongly continuous semigroups of linear operators on these spaces. This is done by proving Gårding inequalities for associated weighted linear forms and applying a theorem of J.-L. Lions (see Theorem 5.1). We expect that if a convolution operator preserves weighted $L^{2}$-spaces with weights of exponential growth and has some smoothness properties, then its convolution kernel should have some fast decay, and in fact it has.

Let us note that the function $q(\mathbf{x})$ is not radial. Therefore in the proof of Theorem 1.2 we cannot apply the formula of Rösler (see (2.16)) for translations of radial functions. In order to prove Theorem 1.2 we use methods developed in [11] based on the description of the support the Dunkl translations of compactly supported functions combined with the observation that any sufficiently regular fast decaying function can be written as a convolution of two functions such that one of them is radial (see [11]). Let us emphasis difficulties we have to face when we apply the method of exponential weights. The first one is that the Dunkl operators do not satisfy the Leibniz rule. The second one concerns the lack of knowledge about boundedness of the Dunkl translations on $L^{p}(dw)$ spaces and the fact that the translations do not form a group of operators as it is in the case of Lie groups.

2. Preliminaries and notation

The Dunkl theory is a generalization of the Euclidean Fourier analysis. It started with the seminal article [6] and developed extensively afterwards (see e.g. [4], [5], [7], [8], [12], [15], [16], [17], [20], [21]). In this section we present basic facts concerning the theory of the Dunkl operators. For details we refer the reader to [6], [18], and [19].

We consider the Euclidean space $\mathbb{R}^{N}$ with the scalar product $\langle\mathbf{x},\mathbf{y}\rangle=\sum_{j=1}^{N}x_{j}y_{j}$, $\mathbf{x}=(x_{1},...,x_{N})$, $\mathbf{y}=(y_{1},...,y_{N})$, 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

[TABLE]

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

[TABLE]

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,

[TABLE]

and

[TABLE]

Observe that (111The symbol $\sim$ between two positive expressions means that their ratio remains between two positive constants.)

[TABLE]

so $dw(\mathbf{x})$ is doubling, that is, there is a constant $C>0$ such that

[TABLE]

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:

[TABLE]

The Dunkl operators $T_{\xi}$, which were introduced in [6], commute and are skew-symmetric with respect to the $G$-invariant measure $dw$. Suppose that $\xi\neq 0$, $f,g\in C^{1}(\mathbb{R}^{N})$ and $g$ is radial. The following Leibniz rule can be confirmed by a direct calculation:

[TABLE]

For fixed $\mathbf{y}\in\mathbb{R}^{N}$ the Dunkl kernel $E(\mathbf{x},\mathbf{y})$ is the unique analytic solution to the system

[TABLE]

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 $\mathbb{C}^{N}\times\mathbb{C}^{N}$. 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

[TABLE]

[TABLE]

[TABLE]

In our further consideration we shall need the following lemma.

Lemma 2.1**.**

For all $\mathbf{x}\in\mathbb{R}^{N}$, $\mathbf{z}\in\mathbb{C}^{N}$ and $\nu\in\mathbb{N}_{0}^{N}$ we have

[TABLE]

In particular,

[TABLE]

Proof.

See [16, Corollary 5.3]. ∎

The Dunkl transform

[TABLE]

where

[TABLE]

originally defined for $f\in L^{1}(dw)$, is an isometry on $L^{2}(dw)$, i.e.,

[TABLE]

and preserves the Schwartz class of functions $\mathcal{S}(\mathbb{R}^{N})$ (see [3]). Its inverse $\mathcal{F}^{-1}$ has the form

[TABLE]

Obviously, for all $f\in\mathcal{S}(\mathbb{R}^{N})$, we have

[TABLE]

and, consequently,

[TABLE]

The Dunkl transform $\mathcal{F}$ is an analogue of the classical Fourier transform.

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

[TABLE]

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 $p\neq 2$.

The following specific formula was obtained by Rösler [17] for the Dunkl translations of (reasonable) radial functions $f({\mathbf{x}})=\tilde{f}({\|\mathbf{x}\|})$:

[TABLE]

Here

[TABLE]

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 (2.16) implies that for all radial $f\in L^{1}(dw)$ and $\mathbf{x}\in\mathbb{R}^{N}$ we have

[TABLE]

The Dunkl convolution $f*g$ of two reasonable functions (for instance Schwartz functions) is defined by

[TABLE]

or, equivalently, by

[TABLE]

where, here and subsequently, $g(\mathbf{x},\mathbf{y})=\tau_{\mathbf{x}}g(-\mathbf{y})$.

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

[TABLE]

[TABLE]

Obviously, $\mathcal{F}(\Delta f)(\xi)=-\|\xi\|^{2}\mathcal{F}f(\xi)$. The operator $\Delta$ is essentially self-adjoint on $L^{2}(dw)$ (see for instance [2, Theorem 3.1]) and generates the semigroup $e^{t\Delta}$ of linear self-adjoint contractions on $L^{2}(dw)$. The semigroup has the form

[TABLE]

where the heat kernel

[TABLE]

is a $C^{\infty}$-function of all variables $\mathbf{x},\mathbf{y}\in\mathbb{R}^{N}$, $t>0$ and satisfies

[TABLE]

[TABLE]

Set

[TABLE]

The following theorem was proved in [1, Theorem 4.1].

Theorem 2.2**.**

There are constants $C,c>0$ such that for all $\mathbf{x},\mathbf{y}\in\mathbb{R}^{N}$ and $t>0$ we have

[TABLE]

3. Weighted Hilbert spaces and bilinear forms

3.1. Definition and properties of exponential weight functions

For any $s>0$ and $\mathbf{x}\in\mathbb{R}^{N}$ let us define

[TABLE]

Clearly,

[TABLE]

Lemma 3.1**.**

For every $\beta\in\mathbb{N}_{0}^{N}$ there is a constant $C_{\beta}>0$ such that for all $\mathbf{x}\in\mathbb{R}^{N}$ and $s>0$ we have

[TABLE]

where, here and subsequently, $\partial_{\mathbf{x}}^{\beta}$ denotes the partial derivative with respect to the variable $\mathbf{x}$.

Proof.

The proof is straightforward. ∎

Lemma 3.2**.**

Suppose that $\phi:\mathbb{R}^{N}\times(0,\infty)\to\mathbb{R}$ is a $C^{\infty}(\mathbb{R}^{N})$-function such that for any $\beta\in\mathbb{N}_{0}^{N}$ there is $C_{\beta}>0$ such that

[TABLE]

Then for every $\zeta\neq 0$ and every $\alpha\in R$ the functions

[TABLE]

satisfy (3.4).

Proof.

Thanks to Lemma 3.1 and (2.7) it is enough to check the claim for $\psi_{\alpha}$ for all $\alpha\in R$. Note that

[TABLE]

therefore

[TABLE]

so the claim is a consequence of (3.4) for $\phi$. ∎

Lemma 3.3**.**

Suppose that $C^{\infty}\ni\phi:\mathbb{R}^{N}\times(0,\infty)\to\mathbb{R}$ satisfies (3.4). Then for every $\zeta\neq 0$ there are $\phi_{\zeta},\phi_{\alpha,\zeta}\in C^{\infty}(\mathbb{R}^{N}\times(0,\infty))$, $\alpha\in R$, which satisfy (3.4), such that for all $f\in C^{1}(\mathbb{R}^{N})$, $\mathbf{x}\in\mathbb{R}^{N}$, and $s>1/4$ we have

[TABLE]

Proof.

By (2.7) we have

[TABLE]

Setting

[TABLE]

and using Lemma 3.2 we get the claim. ∎

Lemma 3.4**.**

Suppose that $C^{\infty}\ni\phi:\mathbb{R}^{N}\times(0,\infty)\to\mathbb{R}$ is a function such that $\phi(\mathbf{x},s)=\phi(\mathbf{x}^{\prime},s)$ for all $\|\mathbf{x}\|=\mathbf{\|}\mathbf{x}^{\prime}\|$ and it satisfies (3.4). Then for every $\zeta\neq 0$ there is $\phi_{\zeta}\in C^{\infty}(\mathbb{R}^{N}\times(0,\infty))$ which satisfies (3.4) such that for all $f\in C^{1}(\mathbb{R}^{N})$, $\mathbf{x}\in\mathbb{R}^{N}$, and $s>1/4$ we have

[TABLE]

Proof.

The claim follows directly by (2.8) and (3.3). ∎

For $\sigma\in G$ let $f^{\sigma}(\mathbf{x})=f(\sigma(\mathbf{x}))$. It is easy to check that for all $\zeta\neq 0$ we have

[TABLE]

Iteration of Lemma 3.3 together with (3.7) and Lemma 3.4 gives the following proposition.

Proposition 3.5**.**

For every $\beta\in\mathbb{N}_{0}^{N}$ there are functions $\phi_{\beta,\beta^{\prime},\sigma}(\mathbf{x},s)$ which satisfy (3.4) such that for all $f\in C_{c}^{\infty}(\mathbb{R}^{N})$, $\mathbf{x}\in\mathbb{R}^{N}$, and $s>1/4$ we have

[TABLE]

3.2. Weighted Hilbert spaces

We define a family $\{\mathcal{H}_{s}\}_{s>0}$ of weighted $L^{2}$-spaces by

[TABLE]

To unify our notation we write

[TABLE]

Clearly, for $s_{1}\leq s_{2}$ we have

[TABLE]

Let us note that for all $\mathbf{x}\in\mathbb{R}^{N}$ and $s>0$ we have

[TABLE]

Therefore,

[TABLE]

Let us recall that $\eta(\mathbf{x})=\eta(\mathbf{x},1)$. The following corollary in a consequence of (3.8) and (3.11).

Corollary 3.6**.**

For every $\beta\in\mathbb{N}_{0}^{N}$ there is a constant $C_{\beta}>0$ such that for every $f\in C_{c}^{\infty}(\mathbb{R}^{N})$ we have

[TABLE]

[TABLE]

Proposition 3.7**.**

For every $\delta>0$ and $\ell_{1}\in\mathbb{N}$ (in particular, for $\ell_{1}=\ell$) there is a constant $C_{\delta,\ell_{1}}>0$ such that for all $f\in C^{\infty}_{c}(\mathbb{R}^{N})$ we have

[TABLE]

Proof.

Thanks to  (2.11), (2.13),  (2.14), and the fact that $\zeta_{1},...,\zeta_{m}$ span $\mathbb{R}^{N}$, we get that for every $\beta\in\mathbb{N}_{0}^{N}$ there is a constant $C_{\beta}>0$ such that

[TABLE]

Moreover, for every $\ell_{1}\in\mathbb{N}_{0}$, $\beta\in\mathbb{N}_{0}^{N}$ such that $|\beta|<\ell_{1}$, and every $\delta>0$ there is a constant $C_{\beta,\delta}>0$ such that

[TABLE]

The proof of (3.14) is by induction on $\ell_{1}$. Assume that (3.14) holds for $\ell_{1}$. Using (3.12) we have

[TABLE]

Then, by (3.16) (for the first summand) and induction hypothesis (3.14) (for the second summand) for any $\varepsilon>0$ we get

[TABLE]

Finally, joining (3.17) and (3.18) and applying (3.13) we get

[TABLE]

The proof is finished by taking $\varepsilon=\frac{1}{4}\min\{\delta,1\}(C_{\ell_{1}}+C)^{-1}$. ∎

Proposition 3.8**.**

Let $\beta\in\mathbb{N}_{0}^{N}$. There is a constant $C_{\beta}>0$ such that for all $f\in C^{\infty}_{c}(\mathbb{R}^{N})$ we have

[TABLE]

Proof.

Thanks to (3.12) and Proposition 3.7 with $\delta=1$ we get

[TABLE]

In order to estimate $\|T^{\beta}(f\eta)\|^{2}_{L^{2}(dw)}$, we use (3.15), then (3.13), which lead to

[TABLE]

The claim follows by Proposition 3.7 with $\delta=1$ applied to $\sum_{|\beta^{\prime}|<|\beta|}\|T^{\beta^{\prime}}f\|^{2}_{L^{2}(\eta^{2}dw)}$. ∎

Corollary 3.9**.**

Let $n<\ell$ be a positive integer. For every $\delta>0$ there is a constant $C=C_{\delta}>0$ such that for all $f\in C_{c}^{\infty}(\mathbb{R}^{N})$ and for all $s>1/4$ we have

[TABLE]

[TABLE]

Proof.

Let us apply (3.14) to $f_{\{s\}}(\mathbf{x})=\frac{1}{s^{\mathbf{N}/2}}f(\mathbf{x}/s)$. Then (3.20) follows from the fact that

[TABLE]

Finally, (3.21) is a consequence of (3.20) and (3.11). ∎

Corollary 3.10**.**

There is a constant $C>0$ such that for all $f\in C^{\infty}_{c}(\mathbb{R}^{N})$ and $s>1/4$ we have

[TABLE]

Proof.

The proof is the same as the proof of Corollary 3.9, but instead of (3.14) we use (3.19). ∎

3.3. Weighted Sobolev spaces

For $s>0$ we define the weighted Sobolev space $V_{\ell,s}$ as the completion of $C_{c}^{\infty}(\mathbb{R}^{N})$-functions in the norm

[TABLE]

Clearly, $V_{\ell,s}\subset\mathcal{H}_{s}$. Moreover, $V_{\ell,s}$ is a dense subspace of $\mathcal{H}_{s}$.

Proposition 3.11**.**

Assume that $f\in\mathcal{H}_{s}$. Then the following statements are equivalent:

1. (a)

$f\in V_{\ell,s}$; 2. (b)

for any $\beta\in\mathbb{N}_{0}^{N}$ such that $|\beta|\leq\ell$ there is a function $f_{\beta,s}\in\mathcal{H}_{s}$ such that for every $\varphi\in C^{\infty}_{c}(\mathbb{R}^{N})$ we have

[TABLE]

Proof.

See Appendix A. ∎

Remark 3.12**.**

If $0<s_{1}<s_{2}$ and $f\in V_{\ell,s_{2}}$, then $f\in V_{\ell,s_{1}}$ and the functions $f_{\beta,s_{1}}$ and $f_{\beta,s_{2}}$ from Proposition 3.11 coincide. They will be denoted by $T^{\beta}f$.

3.4. Bilinear forms

Definition 3.13**.**

For $s>1/4$ we define the bilinear form $a_{s}(\cdot,\cdot)$ with the domain $V_{\ell,s}$ by

[TABLE]

Proposition 3.14**.**

The form $a_{s}(f,g)$ is bounded on $V_{\ell,s}$. More precisely, there is a constant $C>0$ such that for every $s>1/4$ and every $f,g\in V_{\ell,s}$ we have

[TABLE]

Proof.

By Proposition 3.5 there are functions $\phi_{j,\beta^{\prime},\sigma}(\mathbf{x},s)$, $\beta^{\prime}\in\mathbb{N}_{0}^{N}$ and $\sigma\in G$, such that

[TABLE]

and

[TABLE]

Hence, using the Cauchy-Schwarz inequality we obtain

[TABLE]

Now, applying  (3.21) we get

[TABLE]

The proposition is a direct consequence of (3.25). ∎

Proposition 3.15** (Gårding inequality).**

There are constants $\alpha,C_{\alpha}>0$ such that for all $s>1/4$ and $f\in V_{\ell,s}$ we have

[TABLE]

Proof.

Similarly to the proof of Proposition 3.14, applying Proposition 3.5, we have

[TABLE]

Using (3.21) and the Cauchy-Schwarz inequality for any $\delta>0$ there is a constant $C_{\delta}^{\prime}>0$ such that for any $\varepsilon>0$ we have

[TABLE]

Taking $\delta,\varepsilon>0$ small enough such that $C\delta m+CC_{\delta}^{\prime}\varepsilon<1/2$ we conclude the proposition from (3.26) and (3.27). ∎

4. Perturbations of the bilinear form

For $\varepsilon\geq 0$ and $s>1/4$ we consider the following bilinear form

[TABLE]

with the domain $V_{\ell,s}$. Let us note that $b_{s,0}(f,g)=a_{s}(f,g)$.

Proposition 4.1**.**

For every $\varepsilon\geq 0$ and $s>1/4$ the form $b_{s,\varepsilon}$ is bounded on $V_{\ell,s}$.

Proof.

Thanks to (3.21) and (3.22) there is a constant $C>0$ such for all $s>1/4$ we have

[TABLE]

Hence, using Lemma 3.4 and then either (3.21) or (3.22), we obtain

[TABLE]

Now Proposition 4.1 follows from (4.1) and Proposition 3.14. ∎

Proposition 4.2** (Gårding inequality for the perturbed bilinear form).**

There are $\varepsilon_{0}>0$ and $\alpha,C_{\alpha}>0$ such that for all $0\leq\varepsilon\leq\varepsilon_{0}$, $f\in V_{\ell,s}$, and every $s>1/4$ we have

[TABLE]

Proof.

It suffices to take $\varepsilon_{0}>0$ small enough and apply Proposition 3.15 together with (4.1). ∎

The number $\varepsilon_{0}$ from Proposition 4.2 will be fixed throughout the remaining part of the paper.

5. Semigroups of operators and Lions theorem

5.1. Lions theorem

The following theorem is essentially due to J.-L. Lions [13]. Its proof, which includes holomorphy of the semigroup under consideration, and which is a combination of a number of propositions from [13] and [14], can be found in [9, Proposition (1.1)].

Theorem 5.1**.**

Let $\mathcal{H}$ be a Hilbert space and $V$ be a dense subspace of $\mathcal{H}$ such that $V$ is a Hilbert space with the inner product $\langle\cdot,\cdot\rangle_{V}$ and the norm $\|\cdot\|_{V}$, and for some constant $c>0$ we have $\|f\|_{\mathcal{H}}\leq c\|f\|_{V}$ for all $f\in V$. Let $b(\cdot,\cdot)$ be a bounded bilinear form on $V$. It defines an operator $A:D(A)\mapsto\mathcal{H}$ as follows

[TABLE]

Suppose that for some $\alpha>0$ and $\lambda_{0}\in\mathbb{R}$ we have

[TABLE]

Then $A$ is the infinitesimal generator of a strongly continuous semigroup $\{T_{t}\}_{t\geq 0}$ of operators on $\mathcal{H}$ which is holomorphic in a sector

[TABLE]

for some $\kappa>0$. Moreover,

[TABLE]

5.2. Semigroup $\{S_{t}\}_{t\geq 0}$ of operators on $L^{2}(dw)$

For $\varepsilon\in\{0,\varepsilon_{0}\}$ let us define the symmetric bilinear form

[TABLE]

with the domain $V_{\ell,0}=\{f\in L^{2}(dw):(1+\|\xi\|)^{\ell}\mathcal{F}f(\xi)\in L^{2}(dw)\}$ and the norm

[TABLE]

The form can be written by means of the Dunkl transform as

[TABLE]

Proposition 5.2**.**

Let $\varepsilon\in\{0,\varepsilon_{0}\}$. The form $b_{0,\varepsilon}$ is bounded on $V_{\ell,0}$. Moreover, it satisfies the following Gårding inequality: there are $\lambda_{0},\alpha>0$ such that

[TABLE]

Proof.

By the Cauchy–Schwarz inequality and (2.11) we have

[TABLE]

which implies that the form $b_{0,\varepsilon}$ is bounded on $V_{\ell,0}$. The Gårding inequality can be verified by the same way. ∎

As the consequence of the boundedness of $b_{0,\varepsilon}$, we conclude that it defines a self-adjoint linear operator $A^{(\varepsilon)}$, which, thanks to Theorem 5.1 and the Gårding inequality (5.3), generates a strongly continuous semigroup $\{S_{t}^{(\varepsilon)}\}_{t\geq 0}$ of bounded self-adjoint linear operators on $\mathcal{H}_{0}=L^{2}(dw)$, which has the form

[TABLE]

where

[TABLE]

Let us also remark (see Proposition 5.5) that the operator $A^{(\varepsilon)}$ is the closure in the space $\mathcal{H}_{0}$ of

[TABLE]

initially defined on $C^{\infty}_{c}(\mathbb{R}^{N})$ (for the proof see Appendix C with $s=0$).

5.3. Semigroups on weighted Hilbert spaces

We are in a position to apply Theorem 5.3 to the weighted bilinear forms $b_{s,\varepsilon}$, where $\varepsilon\in\{0,\varepsilon_{0}\}$ and $s>1/4$. Let us remind that the forms $b_{s,\varepsilon}$ are bounded (see Propositions 3.14 and 4.1). Let $A_{s}^{(\varepsilon)}$ be the operator associated with the form $b_{s,\varepsilon}$ with its domain $D(A_{s}^{(\varepsilon)})\subset V_{\ell,s}\subset\mathcal{H}_{s}$. The following theorem is a direct consequence of Propositions 4.1, 4.2, and Theorem 5.1.

Theorem 5.3**.**

Let $\varepsilon\in\{0,\varepsilon_{0}\}$. There are constants $c_{0},\kappa>0$ such that for all $s>1/4$ the operator $A_{s}^{(\varepsilon)}$ is the infinitesimal generator of a strongly continuous semigroup $\{S^{\{\varepsilon,s\}}_{t}\}_{t\geq 0}$ of operators on $\mathcal{H}_{s}$ which is holomorphic is a sector

[TABLE]

which for all $f\in\mathcal{H}_{s}$ satisfies

[TABLE]

for all $t\geq 0$ and for all $f\in\mathcal{H}_{s}$.

Clearly, $\mathcal{H}_{s_{1}}\subset\mathcal{H}_{s_{2}}\subset\mathcal{H}_{0}$ and $V_{\ell,s_{1}}\subset V_{\ell,s_{2}}\subset V_{\ell,0}$ for $s_{1}\geq s_{2}\geq 0$. The next theorem asserts that the semigroups $\{S_{t}^{\{\varepsilon,s\}}\}_{t\geq 0}$ can be thought as the semigroup $\{S^{(\varepsilon)}_{t}\}_{t\geq 0}$ acting on $\mathcal{H}_{s}$.

Theorem 5.4**.**

Let $\varepsilon\in\{0,\varepsilon_{0}\}$. For all $s>1/4$ and $f\in\mathcal{H}_{s}\subset L^{2}(dw)$ we have

[TABLE]

Proof.

See Appendix B. ∎

Proposition 5.5**.**

Let $\varepsilon\in\{0,\varepsilon_{0}\}$. For $s>1/4$ let $\lambda>c_{0}s^{2\ell}$, where $c_{0}>0$ is the constant from (5.6). Then for every $n\in\mathbb{N}$ the space $C_{c}^{\infty}(\mathbb{R}^{N})$ is a core for $(\lambda I-A_{s}^{(\varepsilon)})^{n}$.

Proof.

See Appendix C. ∎

6. Pointwise estimates for integral kernel of $S_{t}$

We define the sequence $\{d(n)\}_{n\in\mathbb{N}}$ inductively by

[TABLE]

Lemma 6.1**.**

For every $\beta\in\mathbb{N}_{0}^{N}$ there is a constant $C>0$ such that for every $s>1/4$ and every $f\in C_{c}^{\infty}(\mathbb{R}^{N})$ we have

[TABLE]

Proof.

The proof goes by induction on $|\beta|$. First, let us note that $\eta(\mathbf{x},s)=\eta(\mathbf{x}^{\prime},s)$ for $\|\mathbf{x}\|=\|\mathbf{x}^{\prime}\|$, so for any function $f\in C^{\infty}_{c}(\mathbb{R}^{N})$ integration by parts (see (2.8)) gives

[TABLE]

The claim for $|\beta|=1$ follows from (6.2), the Cauchy–Schwarz inequality, and Lemma 3.1, because

[TABLE]

[TABLE]

Assume that (6.1) is satisfied for $\beta\in\mathbb{N}_{0}^{N}$ such that $|\beta|=n$. Consider multi-index $\beta+e_{j}$, where $|\beta|=n$. Then by (6.1) with $f$ replaced by $T_{j}f$ we get

[TABLE]

Using again the Cauchy-Schwarz inequality together with Lemma 3.1, we obtain

[TABLE]

and

[TABLE]

Hence, repeating the calculation presented in (6.2) we get

[TABLE]

Now (6.4) together with (6.3) completes the proof. ∎

Lemma 6.2**.**

Let $\varepsilon\in\{0,\varepsilon_{0}\}$ and $\beta\in\mathbb{N}_{0}^{N}$. There are constants $C,\lambda_{0}>0$, and $M\in\mathbb{N}$ such that for all $\lambda>\lambda_{0}$ and $f\in C_{c}^{\infty}(\mathbb{R}^{N})$, we have

[TABLE]

Proof.

Let us recall that $(\lambda I-A^{(\varepsilon)})^{M}f=(\lambda I-L^{(\varepsilon)})^{M}f$ for $f\in C^{\infty}_{c}(\mathbb{R}^{N})$. The lemma is a consequence of (2.11) and (2.13). ∎

Combination of Lemma 6.1 and Lemma 6.2 leads to the following corollary.

Corollary 6.3**.**

Let $\varepsilon\in\{0,\varepsilon_{0}\}$ and $\beta\in\mathbb{N}_{0}^{N}$. There are constants $C,\lambda_{0}>0$, and $M=M_{\beta}\in\mathbb{N}$ such that for all $f\in C_{c}^{\infty}(\mathbb{R}^{N})$, $\lambda\geq\lambda_{0}$, and $s>1/4$, we have

[TABLE]

Lemma 6.4**.**

Let $\varepsilon\in\{0,\varepsilon_{0}\}$ and $\beta\in\mathbb{N}_{0}^{N}$. There are constants $C,\lambda_{0}>0$, and $M=M_{\beta}>0$ such that for all $s>1/4$, $\lambda\geq\lambda_{0}$, and $f\in C_{c}^{\infty}(\mathbb{R}^{N})$ we have

[TABLE]

Proof.

By Proposition 3.5 we get

[TABLE]

Then, applying Corollary 6.3 to each term of the sum we obtain the claim. ∎

Lemma 6.5**.**

Let $\varepsilon\in\{0,\varepsilon_{0}\}$ and $\beta\in\mathbb{N}_{0}^{N}$. There are constants $C,c>0$ such that for all $s>1/4$, $f\in\mathcal{H}_{2^{|\beta|+1}s}$, and $1/2<t<2$ we have

[TABLE]

Proof.

Let $M=M_{\beta}$ be as in Lemma 6.4 and let $c_{0}$ be the constant from (5.6). We claim that Lemma 6.4 is satisfied if $f\in D((A^{(\varepsilon)}_{s_{1}})^{M})$, where $s_{1}=2^{|\beta|+1}s$, and $\lambda>\max(\lambda_{0},c_{0}s_{1}^{2\ell})$. Indeed, since $C_{c}^{\infty}(\mathbb{R}^{N})$ is a core for $(\lambda I-A^{(\varepsilon)}_{s_{1}})^{M}$ (see Proposition 5.5) and $D((A_{s_{1}}^{(\varepsilon)})^{M})\subset D((\lambda I-A^{(\varepsilon)}_{s_{1}})^{M})$, there are $f_{n}\in C_{c}^{\infty}(\mathbb{R}^{N})$ such that

[TABLE]

Consequently, by (3.9), (3.11), and Corollary 7.3 in Appendix B we have

[TABLE]

Now the claim follows, because $T^{\beta}$ is closed on $L^{2}(dw)$.

Set $\lambda=\max(\lambda_{0},2c_{0}s_{1}^{2\ell})$. If $f\in\mathcal{H}_{s_{1}}$, then $S^{(\varepsilon)}_{t}f\in D((A_{s_{1}}^{(\varepsilon)})^{M})$, because $\{S^{(\varepsilon)}_{t}\}_{t\geq 0}$ is analytic. Hence, by Lemma 6.4, we get

[TABLE]

By Proposition 5.2 and Theorem 5.1 we have that $A^{(\varepsilon)}$ is the generator of the semigroup $\{S^{(\varepsilon)}_{t}\}_{t\geq 0}$ of self-adjoint linear operators on $L^{2}(dw)$. Therefore, since $1/2<t<2$, by the spectral theorem (or Cauchy integral formula) we obtain

[TABLE]

Moreover, by Theorem 5.3 and the fact that $1/2<t<2$ we have

[TABLE]

which completes the proof. ∎

6.1. Pointwise estimate for convolution kernels of semigroups

Corollary 6.6**.**

Let $\varepsilon\in\{0,\varepsilon_{0}\}$. There are constants $C,c>0$ and $M\in\mathbb{N}$ such that for all $s>1/4$, $f\in\mathcal{H}_{2^{2M+1}s}$, $\mathbf{x}\in\mathbb{R}^{N}$, and $1/2<t<2$ we have

[TABLE]

Proof.

By (2.12), the Cauchy-Schwarz inequality, and (2.11), for $M\in\mathbb{N}$ such that $M>\mathbf{N}/2$ and for any function $g\in D(\Delta^{M})$, we have

[TABLE]

Therefore, if for $f\in\mathcal{H}_{2^{2M+1}s}$ we plug $g(\mathbf{x})=\eta(\mathbf{x},s)S_{t}^{(\varepsilon)}f(\mathbf{x})$ in (6.7) and use Lemma 6.5, we obtain the claim, because $\exp(s\|\mathbf{x}\|)\leq\eta(\mathbf{x},s)$ for all $\mathbf{x}\in\mathbb{R}^{N}$ and $s>1/4$. ∎

Lemma 6.7**.**

There is a constant $C>0$ such that for all $\mathbf{x},\xi\in\mathbb{R}^{N}$ we have

[TABLE]

Proof.

For all $\mathbf{x},\xi\in\mathbb{R}^{N}$ we have

[TABLE]

Therefore, by Cauchy-Schwarz inequality and Lemma 2.1 we get

[TABLE]

∎

Recall that the kernel $q_{t}^{(\varepsilon)}(\mathbf{x})$ is given by (5.4). Our goal is to obtain pointwise estimates of $q_{t}^{(\varepsilon)}$ for $t=1$.

Lemma 6.8**.**

Let $\varepsilon\in\{0,\varepsilon_{0}\}$. There is a constant $C>0$ such that for all $\mathbf{x}\in\mathbb{R}^{N}$ we have

[TABLE]

Proof.

For any $\mathbf{y}\in\mathbb{R}^{N}$ we have

[TABLE]

Therefore, the claim is a consequence of Lemma 6.7, Lemma 2.1, and the fact that $q_{1}^{(\varepsilon)}\in\mathcal{S}(\mathbb{R}^{N})$, so $\mathcal{F}(q_{1}^{(\varepsilon)})\in\mathcal{S}(\mathbb{R}^{N})$ as well. ∎

Theorem 1.1 is a special case (for $\varepsilon=0$) of the theorem below.

Theorem 6.9**.**

Let $\varepsilon\in\{0,\varepsilon_{0}\}$. There are constants $C,c>0$ such that for all $\mathbf{x}\in\mathbb{R}^{N}$ we have

[TABLE]

Proof.

Since $q_{1}^{(\varepsilon)}\in\mathcal{S}(\mathbb{R}^{N})$, it suffices to prove (6.9) for large $\|\mathbf{x}\|$. For any $\mathbf{x}\in\mathbb{R}^{N}$, $s>1$ and $r>0$ we write

[TABLE]

By Lemma 6.8 we have

[TABLE]

Furthermore, it follows by the definition of the Dunkl translation that

[TABLE]

Therefore, by Corollary 6.6 and (3.2) we get that there is $M\in\mathbb{N}$ such that

[TABLE]

where in the last inequality we have used  (2.5). Therefore, taking into account (6.10) and (6.11) we obtain

[TABLE]

Set

[TABLE]

then (6.12) reduces to

[TABLE]

Finally, setting $s=\delta\|\mathbf{x}\|^{1/(2\ell-1)}$ for $\delta>0$ small enough we obtain the claim. ∎

6.2. Pointwise estimations for the integral kernel of the semigroup

The following proposition was proved in [11, Proposition 4.4].

Proposition 6.10**.**

There is a constant $C>0$ such that for any $r_{1},r_{2}>0$, any $f\in L^{1}(dw)$ such that $\text{supp }f\subseteq B(0,r_{2})$, 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

[TABLE]

The lemma below is a suitable adaptation of  [11, Proposition 4.10].

Lemma 6.11**.**

Let $a,b>1$ and $f,g$ be measurable functions such that $g$ is radial and continuous, and there are constants $C,c>0$ such that

[TABLE]

Then there are constants $C^{\prime},c^{\prime}>0$ such that for all $\mathbf{y}\in\mathbb{R}^{N}$ we have

[TABLE]

Proof.

Let $\Psi_{0}\in C^{\infty}(-\frac{1}{2},\frac{1}{2})$ and $\Psi\in C^{\infty}(\frac{1}{8},1)$ be such that

[TABLE]

Set $f_{n}(\mathbf{x})=f(\mathbf{x})\Psi_{n}(\|\mathbf{x}\|)$ and $g_{j}(\mathbf{x})=g(\mathbf{x})\Psi_{j}(\|\mathbf{x}\|)$, where $n,j\geq 0$. Clearly, $\tau_{\mathbf{y}}(f*g)=\sum_{n,j=0}^{\infty}\tau_{\mathbf{y}}(f_{n}*g_{j})$ (see [11, Proposition 4.10] for details). Since ${\rm supp}\,f_{n}\subseteq B(0,2^{n})$ and ${\rm supp}\,g_{j}\subseteq B(0,2^{j})$, we have

[TABLE]

By Proposition 6.10 we obtain

[TABLE]

By (6.14) we have $\|f_{n}\|_{L^{1}(dw)}\leq C\exp(-2^{na}c/2)$ and $\|g_{j}\|_{L^{\infty}}\leq C\exp(-2^{jb}c)$, so (6.15) leads to

[TABLE]

Finally, we see that if $c^{\prime}>0$ is small enough, then the double series above is convergent, so we are done. ∎

Proof of Theorem 1.2.

We write

[TABLE]

where $h_{\varepsilon_{0}/2}$ is the Dunkl heat kernel (see (2.20)). This gives

[TABLE]

By Theorem 2.2 applied to $h_{\varepsilon_{0}/2}(\mathbf{x},\mathbf{z})$, we have

[TABLE]

where in the last inequality we have used the fact that the measure $dw$ is doubling (see (2.6)). The functions $f=q_{1}^{(\varepsilon_{0})}$ and $g=h_{\varepsilon_{0}/2}$ satisfy the assumptions of Lemma 6.11 with $a=\frac{2\ell}{2\ell-1}$ and $b=2$ respectively (see Theorems 6.9 and 2.2), so the last integral is bounded by a constant.

Thanks to the inequality $|h_{\varepsilon_{0}/2}(\mathbf{x},\mathbf{z})|\leq Cw(B(\mathbf{x},\varepsilon_{0}))^{-1}$ (see Theorem 2.2), $|J_{2}|$ is less than

[TABLE]

Since the functions $f=q_{1}^{(\varepsilon_{0})}$ and $g=h_{\varepsilon_{0}/2}$ satisfy the assumptions of Lemma 6.11 with $a=\frac{2\ell}{2\ell-1}$ and $b=2$ respectively, the last integral is bounded by a constant independent of $\mathbf{y}$, provided $c>0$ is small enough. The proof is complete. ∎

7. Appendix

A. Proof of Proposition 3.11

Lemma 7.1**.**

Let $s>1/4$ and let $\Phi$ be a radial $C^{\infty}_{c}(\mathbb{R}^{N})$-function such that $\int\Phi\,dw=1$ and ${\rm supp}\,\Phi\subset B(0,1)$. There is a constant $C=C_{\Phi}>0$ such that for all $f\in\mathcal{H}_{s}$ we have

[TABLE]

Moreover,

[TABLE]

Here and subsequently, $\Phi_{1/n}(\mathbf{x})=n^{\mathbf{N}}\Phi(n\mathbf{x})$.

Proof.

Let us note that by the definition of $\eta(\mathbf{x},s)$ (see (3.1)), there is a constant $C>0$ such that for all $\mathbf{x},\mathbf{y}\in\mathbb{R}^{N}$ and $s>1/4$ we have

[TABLE]

therefore, by the Cauchy–Schwarz inequality,

[TABLE]

Since $\Phi$ is radial, by (2.17) (see also (2.4)) we have

[TABLE]

Consequently, combining (7.3) and (7.4) we get

[TABLE]

Because ${\rm supp}\,\Phi_{1/n}\subseteq B(0,1)$ for all $n\in\mathbb{N}$ and $\Phi_{1/n}$ is radial, (2.16) implies that ${\rm supp}\,\Phi_{1/n}(\cdot,\mathbf{y})\subset\mathcal{O}(B(\mathbf{y},1))$ for all $\mathbf{y}\in\mathbb{R}^{N}$. Therefore, $d(\mathbf{x},\mathbf{y})\leq 1$ for all $\mathbf{x}\in{\rm supp}\,\Phi_{1/n}(\cdot,\mathbf{y})$, so applying (7.5) to (7.6) we get

[TABLE]

where in the last inequality we have used  the first inequality of (7.3).

To finish the proof it suffices to show that (7.2) holds for compactly supported $\mathcal{H}_{s}$-functions, because they form a dense set there. Fix $f\in\mathcal{H}_{s}$. Let $R>0$ be such that $\text{supp}\,f\subseteq B(0,R)$. Then ${\rm supp\,}f*\Phi_{1/n}\subset B(0,R+1)$. By (3.2) we get

[TABLE]

The right-hand side of the above inequality tends to zero, since one can easily prove (using the Dunkl transform) that $\Phi_{1/n}$ is an approximate of the identity on $L^{2}(dw)$. ∎

Proof of Proposition 3.11 (a)$\Rightarrow$(3.23).

Let $\boldsymbol{f}=\{f_{n}\}_{n\in\mathbb{N}}\subset C^{\infty}_{c}(\mathbb{R}^{N})$ be a Cauchy sequence in $V_{\ell,s}$. Clearly, by completeness of $\mathcal{H}_{s}$, there is $f\in\mathcal{H}_{s}\subset L^{2}(dw)$ such that $\lim_{n\to\infty}\|f_{n}-f\|_{\mathcal{H}_{s}}=0$. Let $|\beta|\leq\ell$, by Corollary 3.9 the sequence $\{T^{\beta}f_{n}\}_{n\in\mathbb{N}}$ is a Cauchy sequence in $\mathcal{H}_{s}$, thus it converges to a function $f_{\beta,s}$ in $\mathcal{H}_{s}$ and in $L^{2}(dw)$ as well. Let $\varphi\in C^{\infty}_{c}(\mathbb{R}^{N})$. Integrating by parts we obtain

[TABLE]

Assume now that ${\boldsymbol{g}}=\{g_{n}\}_{n\in\mathbb{N}}$ is another Cauchy sequence in $V_{\ell,s}$, such that $\{g_{n}\}_{n\in\mathbb{N}}$ converge to the $f$ in $\mathcal{H}_{s}$. Then (7.8) implies that $g_{\beta,s}=f_{\beta,s}$ thus $\{g_{n}\}_{n\in\mathbb{N}}$ corresponds to the same element in $V_{\ell,s}$. Hence we have proved that for every element $\boldsymbol{f}$ in $V_{\ell,s}$ we can find a unique element in $f\in\mathcal{H}_{s}$ which satisfies (3.23). ∎

Proof of Proposition 3.11 (3.23)$\Rightarrow$(a).

Let $\Phi$ be a radial $C^{\infty}_{c}(\mathbb{R}^{N})$-function such that $\int\Phi\,dw=1$ and ${\rm supp}\,\Phi\subset B(0,1)$. Let $\Psi$ be a radial $C^{\infty}_{c}(\mathbb{R}^{N})$-function such that $\Psi\equiv 1$ on $B(0,1)$ and $0\leq\Psi\leq 1$. For $n\in\mathbb{N}$ we set

[TABLE]

Since $f\in\mathcal{H}_{s}$, we have $f_{n}\in C^{\infty}_{c}(\mathbb{R}^{N})$ for all $n\in\mathbb{N}$. By iteration of (3.6), for all $\beta\in\mathbb{N}_{0}^{N}$ such that $|\beta|\leq\ell$, there are functions $\Psi_{\beta,\beta^{\prime},\sigma}\in C^{\infty}_{c}(\mathbb{R}^{N})$ such that

[TABLE]

(see also (3.8)). Therefore, by the definition of $f_{\beta^{\prime},s}$, we get

[TABLE]

It follows from (7.9) and Lemma 7.1 that

[TABLE]

which completes the proof of the proposition.

∎

B. Proof of Theorem 5.4.

We remark that Theorem 5.4 is the part (c) of Corollary 7.3. The operator $L^{(\varepsilon)}=(-1)^{\ell+1}\sum_{j=1}^{m}T_{\zeta j}^{2\ell}-\varepsilon\Delta$ is understood as a differential-difference operator acting on $C^{\infty}(\mathbb{R}^{N})$-functions. We define its action on all $L^{2}(dw)$-functions by means of distributions, that is,

[TABLE]

Lemma 7.2**.**

Let $f\in V_{\ell,s}$. Then $f\in D(A^{(\varepsilon)}_{s})$ if and only if $L^{(\varepsilon)}f$ belongs to $\mathcal{H}_{s}$ in the sense of distributions (cf. (7.10)).

Proof.

Assume that $f\in D(A^{(\varepsilon)}_{s})$. Set $g=A^{(\varepsilon)}_{s}f\in\mathcal{H}_{s}$. Fix $\varphi\in C_{c}^{\infty}(\mathbb{R}^{N})$. We may assume that $\varphi$ is real-valued. Define $\psi(\mathbf{x})=\varphi(\mathbf{x})\eta(\mathbf{x},s)^{-1}$. Then $\psi\in C_{c}^{\infty}(\mathbb{R}^{N})\subset V_{\ell,s}$. By the definition of $A^{(\varepsilon)}_{s}$ (see Subsection 5.3) we get

[TABLE]

which proves that $g=L^{(\varepsilon)}f$ in the weak sense.

Converselly, assume that $f\in V_{\ell,s}$ is such that $L^{(\varepsilon)}f\in\mathcal{H}_{s}$ in the weak sense. Set $g=L^{(\varepsilon)}f$. Take $\varphi\in C_{c}^{\infty}(\mathbb{R}^{N})$. Then $\varphi(\mathbf{x})\eta(\mathbf{x},s)\in C_{c}^{\infty}(\mathbb{R}^{N})$ and

[TABLE]

By a density argument (see Subsection 3.3), the formula  (7.11) holds for all $\varphi\in V_{\ell,s}$, which implies that $f\in D(A^{(\varepsilon)}_{s})$ and $A^{(\varepsilon)}_{s}f=g$. ∎

Corollary 7.3**.**

Let $\varepsilon\in\{0,\varepsilon_{0}\}$ and $c_{0}$ be the constant from (5.6). For every $s_{1}>s_{2}>1/4$ we have

1. (a)

$D(A^{(\varepsilon)}_{s_{1}})\subset D(A^{(\varepsilon)}_{s_{2}})\subset D(A^{(\varepsilon)})$* and $A^{(\varepsilon)}_{s_{1}}\subset A^{(\varepsilon)}_{s_{2}}\subset A^{(\varepsilon)}$;* 2. (b)

$R(\lambda;A^{(\varepsilon)}_{s_{1}})\subset R(\lambda;A^{(\varepsilon)}_{s_{2}})\subset R(\lambda;A^{(\varepsilon)})$* for all $\lambda>c_{0}s_{1}^{2\ell}$, where $R(\lambda;A^{(\varepsilon)}_{s})$ denotes the resolvent operator, that is, $R(\lambda;A^{(\varepsilon)}_{s_{j}})=(\lambda I-A^{(\varepsilon)}_{s_{j}})^{-1}$* 3. (c)

$S_{t}^{\{\varepsilon,s_{1}\}}\subset S_{t}^{\{\varepsilon,s_{2}\}}\subset S^{(\varepsilon)}_{t}$* for all $t>0$.*

Proof.

The statements  (a) and (b) are consequences of Lemma 7.2. To prove (c) we take $\omega>0$ sufficiently large. Then, by the Lions theorem (see Theorem 5.1), the operators $\widetilde{A^{(\varepsilon)}_{s_{1}}}=A^{(\varepsilon)}_{s_{1}}-\omega I$, $\widetilde{A^{(\varepsilon)}_{s_{2}}}=A^{(\varepsilon)}_{s_{2}}-\omega I$, and $\widetilde{A^{(\varepsilon)}}=A^{(\varepsilon)}-\omega I$, generate contraction semigroups $\{e^{-t\omega}S_{t}^{\{\varepsilon,s_{1}\}}\}_{t\geq 0}$, $\{e^{-t\omega}S_{t}^{\{\varepsilon,s_{2}\}}\}_{t\geq 0}$, and $\{e^{-t\omega}S^{(\varepsilon)}_{t}\}_{t\geq 0}$ respectively (each semigroup acts on its corresponding Hilbert space $\mathcal{H}_{s_{j}}$). It follows from the statements (a) and (b) that the Yosida approximations of $\widetilde{A^{(\varepsilon)}_{s_{j}}}$ (see [14, Section 3.1]) satisfy

[TABLE]

for $\lambda>0$, which implies (c), by the proof of the Hille-Yosida theorem (see [14]). ∎

C. Proof of Proposition 5.5

Since $\lambda>c_{0}s^{2\ell}$, the operator $\lambda I-A_{s}^{(\varepsilon)}$ is invertible on $\mathcal{H}_{s}$. Let $R(\lambda;A_{s}^{(\varepsilon)})$ denote its inverse. Since $R(\lambda;A^{(\varepsilon)}_{s})^{n}$ is bounded operator on $\mathcal{H}_{s}$, it suffices to prove that $(\lambda I-A^{(\varepsilon)}_{s})^{n}(C_{c}^{\infty}(\mathbb{R}^{N}))$ is a dense subspace in $\mathcal{H}_{s}$. For this purpose let

[TABLE]

We claim that $\mathcal{V}_{s}^{\infty}$ is a core for $(\lambda I-A^{(\varepsilon)}_{s})^{n}$, because for $f\in C_{c}^{\infty}(\mathbb{R}^{N})$ we have $T^{\beta}R(\lambda;A^{(\varepsilon)}_{s})^{n}f=R(\lambda;A^{(\varepsilon)}_{s})^{n}T^{\beta}f\in D((A^{(\varepsilon)}_{s})^{n})\subset\mathcal{H}_{s}$ and, consequently, $R(\lambda;A^{(\varepsilon)}_{s})^{n}f\in\mathcal{V}_{s}^{\infty}$. Therefore $C_{c}^{\infty}(\mathbb{R}^{N})\subset(\lambda I-A^{(\varepsilon)}_{s})^{n}(\mathcal{V}_{s}^{\infty})$, which proves the claim.

Let $\Psi$ be as in Appendix A and let $f\in\mathcal{V}_{s}^{\infty}$. Then $f_{j}(\mathbf{x})=\Psi(\mathbf{x}/j)f(\mathbf{x})\in C_{c}^{\infty}(\mathbb{R}^{N})$ for all $j\in\mathbb{N}$. It is not difficult to prove that $\lim_{j\to\infty}\|T^{\beta}f_{j}-T^{\beta}f\|_{\mathcal{H}_{s}}=0$ for every multi-index $\beta\in\mathbb{N}_{0}^{N}$, which finishes the proof of the proposition.