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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 $x\in\mathbb{R}^d$, $\lim_{r\rightarrow0}M_B^r(f)(x)=f(x)$.
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참고문헌 (17)
참고문헌 신청
C. F. Dunkl / 1991 / Integral kernels with re ection group invariance / Canad. J. Math 43 (6) : 1213 ~ 1227
C. F. Dunkl / 1992 / Hankel transforms associated to finite re ection groups, Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL
C. F. Dunkl / 2001 / Orthogonal Polynomials of Several variables / Cambridge Univ. Press
E. M. Opdam / 1993 / Dunkl operators, Bessel functions and the discriminant of a finite Cox-eter group / Compositio Math 85 (3) : 333 ~ 373
E. M. Stein / 2005 / Real Analysis: Measure Theory, Integration and Hilbert Spaces / Princeton University Press
C. F. Dunkl / 1992 / Hankel transforms associated to finite re ection groups, Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL
C. F. Dunkl / 2001 / Orthogonal Polynomials of Several variables / Cambridge Univ. Press
E. M. Opdam / 1993 / Dunkl operators, Bessel functions and the discriminant of a finite Cox-eter group / Compositio Math 85 (3) : 333 ~ 373
E. M. Stein / 2005 / Real Analysis: Measure Theory, Integration and Hilbert Spaces / Princeton University Press
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