Positive Cubature formulas and Marcinkiewicz-Zygmund inequalities on spherical caps

Let $\Pi_n^d$ denote the space of all spherical polynomials of degree at most $n$ on the unit sphere $\sph$ of $\mathbb{R}^{d+1}$, and let $d(x, y)$ denote the usual geodesic distance $\arccos x\cdot y$ between $x, y\in \sph$. Given a spherical cap $$B(e,\al)=\{x\in\sph: d(x, e) \leq \al\}, (e\in\sph, \text{$\al\in (0,\pi)$ is bounded away from $\pi$}),$$ we define the metric $$\rho(x,y):=\frac 1{\al} \sqrt{(d(x, y))^2+\al(\sqrt{\al-d(x, e)}-\sqrt{\al-d(y,e)})^2},$$ where $x, y\in B(e,\al)$. It is shown that given any $\be\ge 1$, $1\leq p<\infty$ and any finite subset $\Ld$ of $B(e,\al)$ satisfying the condition $\dmin_{\sub{\xi,\eta \in \Ld \xi\neq \eta}} \rho (\xi,\eta) \ge \f \da n$ with $\da\in (0,1]$, there exists a positive constant $C$, independent of $\al$, $n$, $\Ld$ and $\da$, such that, for any $f\in\Pi_{n}^d$, \begin{equation*} \sum_{\og\in \Ld} (\max_{x,y\in B_\rho (\og, \be\da/n)}|f(x)-f(y)|^p) |B_\rho(\og, \da/n)| \le (C \dz)^p \int_{B(e,\al)} |f(x)|^p d\sa(x),\end{equation*} where $d\sa(x)$ denotes the usual Lebesgue measure on $\sph$, $$B_\rho(x, r)=\Bl\{y\in B(e,\al): \rho(y,x)\leq r\Br\}, (r>0),$$ and $$\Bl|B_\rho(x, \f\da n)\Br|=\int_{B_{\rho}(x, \da/n)} d\sa(y) \sim \al ^{d}\Bl[ (\f{\da}n)^{d+1}+ (\f\da n)^{d} \sqrt{1-\f{d(x, e)}\al}\Br].$$ As a consequence, we establish positive cubature formulas and Marcinkiewicz-Zygmund inequalities on the spherical cap $B(e,\al)$.