Bulletin of the
Korean Mathematical Society BKMS

Let $R$ be a root system in $\mathbb{R}^d$ with Coxeter-Weyl group $W$ and let $k$ be a nonnegative multiplicity function on $R$. The generalized volume mean of a function $f\in L^1_{loc}(\mathbb{R}^d,m_k)$, with $m_k$ the measure given by $dm_k(x):=\omega_k(x)dx:=\prod_{\alpha\in R}|\mathop{\langle\alpha,x\rangle}|^{k(\alpha)}dx$, is defined by: $\forall\ x\in \mathbb{R}^d$, $\forall\ r>0$, $M_B^r(f)(x):=\frac{1}{m_k[B(0,r)]}\int_{\mathbb{R}^d}f(y)h_k(r,x,y)\omega_k(y)dy$, where $h_k(r,x,\cdot)$ is a compactly supported nonnegative explicit measurable function depending on $R$ and $k$. In this paper, we prove that for almost every $x\in\mathbb{R}^d$, $\lim_{r\rightarrow0}M_B^r(f)(x)=f(x)$.

Keywords: generalized volume mean value operator, harmonic kernel, Dunkl-Laplace operator, Dunkl transform

MSC numbers: Primary 42B25, 42B37, 43A32; Secondary 31B05, 33C52