Uncertainty principles on weighted spheres, balls and simplexes (Q2788788)

Let \(G\subset O(d)\) be a finite reflection group on \(\mathbb{R}^d\). For \(v\in\mathbb{R}^d\backslash\{0\}\), \(\sigma_v\) is defined as follows NEWLINE\[NEWLINE\sigma_v x:=x-\frac{2\langle x,v\rangle}{\|v\|^2}v,\quad x\in\mathbb{R}^d,NEWLINE\]NEWLINE where \(\langle\cdot,\cdot\rangle\) denotes the Euclidean inner product on \(\mathbb{R}^d\) and \(\|x\|:=\sqrt{\langle x,x\rangle}\). Denote by \(\mathcal{R}\) the root system of \(G\), normalized so that \({\langle v,v\rangle}=2\) for all \(v\in\mathcal{R}\). Let \(\kappa:\mathcal{R}\rightarrow[0,\infty)\), \(v\mapsto\kappa_v=\kappa(v)\) be a nonnegative multiplicative function on \(\mathcal{R}\); that is, \(\kappa\) is a nonnegative \(G\)-invariant function on \(\mathcal{R}\). Let \(h_\kappa\) denote the weight function on \(\mathbb{R}^d\) defined by NEWLINE\[NEWLINEh_\kappa(x):=\prod_{v\in\mathcal{R}_+}|\langle x,v\rangle|^{\kappa_v}, \quad x\in\mathbb{R}^d,NEWLINE\]NEWLINE which is \(G\)-invariant.NEWLINENEWLINEThe Dunkl operators associated with \(G\) and \(\kappa\) are defined by NEWLINE\[NEWLINE\mathcal{D}_if(x):=\partial_if(x)+\sum_{v\in\mathcal{R}_+}\kappa \langle v,e_i\rangle\frac{f(x)-f(\sigma_vx)}{\langle x,v\rangle}, \quad i\in\{1,\ldots,d\},\quad f\in C^1(\mathbb{R}^d),NEWLINE\]NEWLINE where \(\partial_i=\frac{\partial}{\partial{x_i}}\) and \(\mathcal{R}_+\) is a fixed positive subsystem of \(\mathcal{R}\). By restricting to the unit sphere, the weighted analogue \(\Delta_{\kappa,0}\) of the Laplace-Beltrami operator \(\Delta_0\) and the analogue \(\nabla_{\kappa,0}\) of the tangential gradient \(\nabla_0\) are defined as follows: NEWLINE\[NEWLINE\Delta_{\kappa,0}f(x):=\Delta_{\kappa}F(z)|_{z=x},\quad x\in\mathbb{S}^{d-1}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\nabla_{\kappa,0}f(x):=\Delta_{\kappa}F(z)|_{z=x},\quad x\in\mathbb{S}^{d-1},NEWLINE\]NEWLINE where \(F(z)=f(\frac{z}{\|z\|})\), \(\mathbb{S}^{d-1}\) denotes the unit sphere of \(\mathbb{R}^d\), \(\Delta_\kappa:=\sum_{j=1}^d\mathcal{D}_j^2\) and \(\nabla_\kappa:=(\mathcal{D}_1,\cdots,\mathcal{D}_d)\).NEWLINENEWLINEIn this paper, the author shows that, if \(f\in C^1(\mathbb{S}^{d-1})\) is such that NEWLINE\[NEWLINE\int_{\mathbb{S}^{d-1}}f(x)h_{\kappa}^2(x)\,d\sigma(x)=0 \text{ and }\int_{\mathbb{S}^{d-1}}|f(x)|^2h_{\kappa}^2(x)\,d\sigma(x)=1,NEWLINE\]NEWLINE then NEWLINE\[NEWLINE\left[\min_{y\in\mathbb{S}^{d-1}}\int_{\mathbb{S}^{d-1}}(1-\langle x,y\rangle) |f(x)|^2h_{\kappa}^2(x)\,d\sigma(x)\right] \left[\int_{\mathbb{S}^{d-1}}|\sqrt{-\Delta_{\kappa,0}}f|^2 h_{\kappa}^2\,d\sigma(x)\right]NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\left[\min_{y\in\mathbb{S}^{d-1}}\int_{\mathbb{S}^{d-1}}(1-\langle x,y\rangle) |f(x)|^2h_{\kappa}^2(x)\,d\sigma(x)\right] \left[\int_{\mathbb{S}^{d-1}}|\nabla_{\kappa,0}f(x)|^2 h_{\kappa}^2(x)\,d\sigma(x)\right]NEWLINE\]NEWLINE both have positive lower bounds, which depend only on \(d\) and \(\kappa\).NEWLINENEWLINEAlso, the author establishes a similar result for the weighted orthogonal polynomial expansion (WOPEs) with respect to the weight function NEWLINE\[NEWLINEW_\kappa^B(x):=\left(\prod_{v\in\mathcal{R}_+}|\langle x,v\rangle|^{2\kappa_v}\right) \left(1-\|x\|^2\right)^{\mu-1/2},\quad \mu\geq0NEWLINE\]NEWLINE on the unit ball \(\mathbb{B}^d\), as well as for the WOPEs with respect to the weight function NEWLINE\[NEWLINEW_\kappa^T(x;\mathbb{Z}_2^d):=\left(\prod_{i=1}^dx_i^{\kappa_i-1/2}\right) \left(1-|x|\right)^{\kappa_{d+1}-1/2},\quad \min_{1\leq i\leq d+1}\kappa_i\geq0,NEWLINE\]NEWLINE or NEWLINE\[NEWLINEW_{\kappa,\mu}^T(x;H_d):=\prod_{i=1}^dx_i^{\kappa'-1/2}\prod_{1\leq i<j\leq d}|x_i-x_j|^\kappa(1-|x|)^{\mu-1/2},\quad \min\{\kappa',\kappa,\mu\}\geq0,NEWLINE\]NEWLINE on the simplex \(\mathbb{T}^d\).