On Semigroups Generated by Sums of Even Powers of Dunkl Operators

Abstract

On the Euclidean space ${\mathbb {R}}^N$ equipped with a normalized root system R, a multiplicity function $k\ge 0$, 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

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

where $\zeta _1,\ldots ,\zeta _m$ 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 $\{S_t\}_{t \ge 0}$ 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 $\exp (-\sum _{j=1}^m \langle \zeta _j,\xi \rangle ^{2\ell })$ and $*$ stands for the Dunkl convolution. We prove that $q({\mathbf {x}})$ satisfies the following exponential decay:

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

for a certain constant $c>0$. Moreover, if $q({\mathbf {x}},{\mathbf {y}})=\tau _{{\mathbf {x}}}q(-{\mathbf {y}})$, then

$$\begin{aligned} |q({\mathbf {x}},{\mathbf {y}})|\lesssim w(B({\mathbf {x}},1))^{-1} \exp (-c d({\mathbf {x}},{\mathbf {y}})^{2\ell /(2\ell -1)}), \end{aligned}$$

where $d({\mathbf {x}},{\mathbf {y}})=\min _{\sigma \in G}\Vert {\mathbf {x}}- \sigma ({\mathbf {y}})\Vert$, G is the reflection group for R, and $\tau _{{\mathbf {x}}}$ denotes the Dunkl translation.

Appendices

7. Appendix

1.1 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 $\mathrm{supp}\,\Phi \subset B(0,1)$. There is a constant $C=C_{\Phi }>0$ such that for all $f \in {\mathcal {H}}_s$ we have

$$\begin{aligned} \Vert \Phi _{1/n}*f\Vert _{{\mathcal {H}}_s} \le C\Vert f\Vert _{{\mathcal {H}}_s}. \end{aligned}$$

(7.1)

Moreover,

$$\begin{aligned} \lim _{n\rightarrow \infty } \Vert f-\Phi _{1/n}*f\Vert _{{\mathcal {H}}_s} =0 \text { for all } f\in {\mathcal {H}}_s. \end{aligned}$$

(7.2)

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

$$\begin{aligned} e^{s\Vert {\mathbf {x}}\Vert } \le \eta ({\mathbf {x}},s) \le C e^{s\Vert {\mathbf {x}}\Vert } \le C e^{sd({\mathbf {x}},{\mathbf {y}})+s\Vert {\mathbf {y}}\Vert }, \end{aligned}$$

(7.3)

therefore, by the Cauchy–Schwarz inequality,

$$\begin{aligned} \begin{aligned}&\Vert \Phi _{1/n}*f\Vert ^2_{{\mathcal {H}}_s} \le C\int _{{\mathbb {R}}^{N}} \Big |\int _{{\mathbb {R}}^N}\Phi _{1/n}({\mathbf {x}},{\mathbf {y}})f({\mathbf {y}})\,dw({\mathbf {y}})\Big |^2e^{s\Vert {\mathbf {x}}\Vert }\,dw({\mathbf {x}}) \\ {}&\le C\int _{{\mathbb {R}}^{N}} \int _{{\mathbb {R}}^N}|\Phi _{1/n}({\mathbf {x}},{\mathbf {y}})|\,dw({\mathbf {y}})\int _{{\mathbb {R}}^N}|\Phi _{1/n}({\mathbf {x}},{\mathbf {y}})||f({\mathbf {y}})|^2\,dw({\mathbf {y}})e^{s\Vert {\mathbf {x}}\Vert }\,dw({\mathbf {x}}) . \end{aligned} \end{aligned}$$

(7.4)

Since $\Phi$ is radial, by (2.19) (see also (2.4)) we have

$$\begin{aligned} \int _{{\mathbb {R}}^{N}}|\Phi _{1/n}({\mathbf {x}},{\mathbf {y}})|\,dw({\mathbf {y}}) \le \int _{{\mathbb {R}}^{N}}|\Phi ({\mathbf {y}})|\,dw({\mathbf {y}}) \le C. \end{aligned}$$

(7.5)

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

$$\begin{aligned}&\Vert \Phi _{1/n}*f\Vert ^2_{{\mathcal {H}}_s}\nonumber \\&\quad \le C' \int _{{\mathbb {R}}^N}|f({\mathbf {y}})|^2e^{s\Vert {\mathbf {y}}\Vert }\int _{{\mathbb {R}}^{N}}|\Phi _{1/n}({\mathbf {x}},{\mathbf {y}})|e^{sd({\mathbf {x}},{\mathbf {y}})}\,dw({\mathbf {x}})\,dw({\mathbf {y}}). \end{aligned}$$

(7.6)

Because $\mathrm{supp}\,\Phi _{1/n} \subseteq B(0,1)$ for all $n \in {\mathbb {N}}$ and $\Phi _{1/n}$ is radial, (2.18) implies that $\mathrm{supp}\,\Phi _{1/n}(\cdot ,{\mathbf {y}}) \subset {\mathcal {O}}(B({\mathbf {y}},1))$ for all ${\mathbf {y}} \in {\mathbb {R}}^N$. Therefore, $d({\mathbf {x}},{\mathbf {y}}) \le 1$ for all ${\mathbf {x}} \in \mathrm{supp}\,\Phi _{1/n}(\cdot ,{\mathbf {y}})$, so applying (7.5) to (7.6) we get

$$\begin{aligned} \Vert \Phi _{1/n} * f \Vert ^2_{{\mathcal {H}}_s} \le C'e^{s}\int _{{\mathbb {R}}^{N}}|f({\mathbf {y}})|^2e^{s\Vert {\mathbf {y}}\Vert }\,dw({\mathbf {y}}) \le C''e^s\Vert f\Vert _{{\mathcal {H}}_s}^2, \end{aligned}$$

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 $\mathrm{supp\,} f*\Phi _{1/n}\subset B(0,R+1)$. By (3.2) we get

$$\begin{aligned} \Vert f*\Phi _{1/n} -f\Vert ^2_{{\mathcal {H}}_s} \le e^{(R+1)s +1}\Vert f*\Phi _{1/n} -f\Vert ^2_{L^2(dw)}. \end{aligned}$$

(7.7)

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 identity on $L^2(dw)$. $\square$

Proof

(Proof of Proposition 3.11 (a)$\Rightarrow$(b)) Let ${\varvec{f}}=\{f_n\}_{n \in {\mathbb {N}}} \subset C^{\infty }_c({\mathbb {R}}^{N})$ 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 $\lim _{n\rightarrow \infty } \Vert f_n-f\Vert _{{\mathcal {H}}_s}=0$. Let $|\beta | \le \ell$, 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

$$\begin{aligned} \begin{aligned} (-1)^{|\beta |} \int _{{\mathbb {R}}^N} f({\mathbf {x}})T^\beta \varphi ({\mathbf {x}})\, dw({\mathbf {x}})&=\lim _{n\rightarrow \infty } (-1)^{|\beta |} \int _{{\mathbb {R}}^N} f_n({\mathbf {x}})T^\beta \varphi ({\mathbf {x}})\, dw({\mathbf {x}})\\&=\lim _{n\rightarrow \infty }\int _{{\mathbb {R}}^N} T^{\beta }f_n({\mathbf {x}})\varphi ({\mathbf {x}})\, dw({\mathbf {x}})\\&=\int _{{\mathbb {R}}^N} f_{\beta , s}({\mathbf {x}})\varphi ({\mathbf {x}})\, dw({\mathbf {x}}). \end{aligned} \end{aligned}$$

(7.8)

Assume now that ${{\varvec{g}}}=\{ g_n\}_{n\in {\mathbb {N}}}$ 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 ${\varvec{f}}$ in $V_{\ell ,s}$ we can find a unique element in $f\in {\mathcal {H}}_s$ which satisfies (3.23). $\square$

Proof

(Proof of Proposition 3.11 (b)$\Rightarrow$(a)) Let $\Phi$ be a radial $C^{\infty }_c({\mathbb {R}}^{N})$-function such that $\int \Phi \,dw=1$ and $\mathrm{supp}\,\Phi \subset B(0,1)$. Let $\Psi$ be a radial $C^\infty _c({\mathbb {R}}^N)$-function such that $\Psi \equiv 1$ on B(0, 1) and $0 \le \Psi \le 1$. For $n \in {\mathbb {N}}$ we set

$$\begin{aligned} f_n({\mathbf {x}})=\Psi ({\mathbf {x}}/n)(\Phi _{1/n}*f)({\mathbf {x}}). \end{aligned}$$

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 (3.5), for all $\beta \in {\mathbb {N}}_0^{N}$ such that $|\beta | \le \ell$, there are functions $\Psi _{\beta ,\beta ',\sigma } \in C^{\infty }_c({\mathbb {R}}^N)$ such that

$$\begin{aligned} T^{\beta }f_n({\mathbf {x}})&=T^{\beta }(\Phi _{1/n}*f)({\mathbf {x}})\Psi ({\mathbf {x}}/n)\\ {}&\quad +\sum _{\sigma \in G}\sum _{\beta ' \in {\mathbb {N}}^{N}_0, \,|\beta '| < |\beta |} n^{|\beta '|-|\beta |} T^{\beta '}(\Phi _{1/n}*f)(\sigma ({\mathbf {x}}))\Psi _{\beta ,\beta ',\sigma } ({\mathbf {x}}/n). \end{aligned}$$

Therefore, by the definition of $f_{\beta ',s}$, we get

$$\begin{aligned} \begin{aligned} T^{\beta }f_n({\mathbf {x}})&=(\Phi _{1/n}*f_{\beta ,s})({\mathbf {x}})\Psi ({\mathbf {x}}/n)\\ {}&\quad +\sum _{\sigma \in G}\sum _{\beta ' \in {\mathbb {N}}^{N}_0, \,|\beta '| < |\beta |}n^{|\beta '|-|\beta |}(\Phi _{1/n}*f_{\beta ',s})(\sigma ({\mathbf {x}}))\Psi _{\beta ,\beta ',\sigma }({\mathbf {x}}/n). \end{aligned}\nonumber \\ \end{aligned}$$

(7.9)

It follows from (7.9) and Lemma 7.1 that

$$\begin{aligned} \lim _{n \rightarrow \infty }\Vert T^{\beta }f_n-f_{\beta ,s}\Vert _{{\mathcal {H}}_s}=0 \text { for all }|\beta | \le \ell , \end{aligned}$$

which completes the proof of the proposition. $\square$

B. Proof of Theorem 5.5.

We remark that Theorem 5.5 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,

$$\begin{aligned}&\int _{{\mathbb {R}}^N} (L^{(\varepsilon )}f)({\mathbf {x}})\varphi ({\mathbf {x}})\, dw({\mathbf {x}})\nonumber \\&\quad =\int _{{\mathbb {R}}^N} f({\mathbf {x}})(L^{(\varepsilon )}\varphi )({\mathbf {x}})\, dw({\mathbf {x}}) \text { for all } \varphi \in C_c^\infty ({\mathbb {R}}^N). \end{aligned}$$

(7.10)

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)). In this case, $A^{(\varepsilon )}_sf=L^{(\varepsilon )}f$.

Proof

Assume that $f\in D(A^{(\varepsilon )}_s)$. Set $g=A^{(\varepsilon )}_sf\in {\mathcal {H}}_s$. 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

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}^N} g({\mathbf {x}})\varphi ({\mathbf {x}})\, dw({\mathbf {x}})&= \int _{{\mathbb {R}}^N} g({\mathbf {x}})\psi ({\mathbf {x}})\eta ({\mathbf {x}},s)\, dw({\mathbf {x}})=b_{s,\varepsilon }(f,\psi )\\&=-\int _{{\mathbb {R}}^N} \sum _{j=1}^m T_{\zeta _j}^\ell f({\mathbf {x}})T_{\zeta _j}^\ell (\psi ({\mathbf {x}})\eta ({\mathbf {x}},s))\, dw({\mathbf {x}})\\&\quad +\varepsilon \int _{{\mathbb {R}}^N} \sum _{j=1}^N T_{j}f({\mathbf {x}})T_{j}(\psi ({\mathbf {x}})\eta ({\mathbf {x}},s))\, dw({\mathbf {x}})\\&=\int _{{\mathbb {R}}^N} f({\mathbf {x}}) L^{(\varepsilon )}\varphi ({\mathbf {x}})\, dw({\mathbf {x}}), \end{aligned} \end{aligned}$$

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

Conversely, 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

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}^N } g({\mathbf {x}})(\varphi ({\mathbf {x}})\eta ({\mathbf {x}}, s))\, dw({\mathbf {x}})&= \int _{{\mathbb {R}}^N } f({\mathbf {x}})L^{(\varepsilon )}(\varphi ({\mathbf {x}})\eta ({\mathbf {x}},s))\, dw({\mathbf {x}})\\&= b_{s,\varepsilon }(f,\varphi ). \end{aligned} \end{aligned}$$

(7.11)

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 $A^{(\varepsilon )}_sf=g$. $\square$

Corollary 7.3

Let $\varepsilon \in \{0,\varepsilon _0\}$ and $c_0$ be the constant from (5.7). For every $s_1>s_2>1/4$ we have

• (a) $D(A^{(\varepsilon )}_{s_1})\subset D(A^{(\varepsilon )}_{s_2}) \subset D(A^{(\varepsilon )})$ and $A^{(\varepsilon )}_{s_1}\subset A^{(\varepsilon )}_{s_2}\subset A^{(\varepsilon )}$;

• (b) $R(\lambda ; A^{(\varepsilon )}_{s_1})\subset R(\lambda ; A^{(\varepsilon )}_{s_2})\subset R(\lambda ; A^{(\varepsilon )})$ for all $\lambda >c_0s_1^{2\ell }$, where $R(\lambda ; A^{(\varepsilon )}_{s_j})$ denotes the resolvent operator, that is, $R(\lambda ; A^{(\varepsilon )}_{s_j})=(\lambda I-A^{(\varepsilon )}_{s_j})^{-1}$;

• (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 $\{e^{-t\omega }S_t^{\{\varepsilon ,s_1\}}\}_{t \ge 0}$, $\{e^{-t\omega }S_t^{\{\varepsilon ,s_2\}}\}_{t \ge 0}$, and $\{e^{-t\omega }S^{(\varepsilon )}_t\}_{t \ge 0}$ 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 [19, Section 3.1]) satisfy

$$\begin{aligned} \lambda ^2 R(\lambda ; \widetilde{A^{(\varepsilon )}_{s_1}})-\lambda I\subset \lambda ^2 R(\lambda ; \widetilde{ A^{(\varepsilon )}_{s_2}})-\lambda I\subset \lambda ^2 R(\lambda ; \widetilde{A^{(\varepsilon )}})-\lambda I, \end{aligned}$$

for $\lambda >0$, which implies (c), by the proof of the Hille-Yosida theorem (see [19]). $\square$

C. Proof of Proposition  5.6

The proof goes in two steps. Let

$$\begin{aligned} {\mathcal {V}}_s^\infty =\{ f\in C^\infty ({\mathbb {R}}^N): T^\beta f\in {\mathcal {H}}_s\quad \text { for every } \beta \in {\mathbb {N}}_0^N\}. \end{aligned}$$

In the first step we prove that ${\mathcal {V}}_s^\infty$ is a core for $(\lambda I-A_{s}^{(\varepsilon )})^n$. To this end we note that 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 a bounded operator on ${\mathcal {H}}_s$, it suffices to prove that $(\lambda I-A^{(\varepsilon )}_s)^n ({\mathcal {V}}_s^\infty )$ is a dense subspace in ${\mathcal {H}}_s$. 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 completes the proof of the first step.

In the second step we show that ${\mathcal {V}}_s^\infty$ is contained in the domain of the closure of the operator $(\lambda I-A_{s}^{(\varepsilon ))})^n$ from the space $C_c^\infty ({\mathbb {R}}^N)$. 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 using (3.5) and Lemmas 3.1 and 3.3 that $\lim _{j\rightarrow \infty } \Vert T^\beta f_j-T^\beta f\Vert _{{\mathcal {H}}_s}=0$ for every multi-index $\beta \in {\mathbb {N}}_0^N$. Consequently,

$$\begin{aligned} \lim _{j\rightarrow \infty } (\Vert f_j-f\Vert _{{\mathcal {H}}_s}+\Vert (\lambda I-A_{s}^{(\varepsilon )})^n f_j- (\lambda I-A_{s}^{(\varepsilon )})^nf\Vert _{{\mathcal {H}}_s})=0, \end{aligned}$$

which finishes the proof of the second step. Finally, the proof of the proposition is complete. $\square$