Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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VOLUME MEAN OPERATOR AND DIFFERENTIATION RESULTS ASSOCIATED TO ROOT SYSTEMS

  • Rejeb, Chaabane (Universite de Tunis El Manar Faculte des Sciences de Tunis Laboratoire d'Analyse Mathematiques et Applications LR11ES11)
  • Received : 2016.08.17
  • Accepted : 2017.07.14
  • Published : 2017.11.30
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Abstract

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 $dmk(x):={\omega}_k(x)dx:=\prod_{{\alpha}{\in}R}{\mid}{\langle}{\alpha},x{\rangle}{\mid}^{k({\alpha})}dx$, is defined by: ${\forall}x{\in}\mathbb{R}^d$, ${\forall}r$ > 0, $M^r_B(f)(x):=\frac{1}{m_k[B(0,r)]}\int_{\mathbb{R}^d}f(y)h_k(r,x,y){\omega}_k(y)dy$, 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 $x{\in}\mathbb{R}^d$, $lim_{r{\rightarrow}0}M^r_B(f)(x)= f(x)$.

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References

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