Volume mean operator and differentiation results associated to root systems

Chaabane Rejeb (1)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique

Let R be a root system in R d with Coxeter-Weyl group W and let k be a non-negative multiplicity function on R. The generalized volume mean of a function f ∈ L 1 loc (R d , m k), with m k the measure given by dm k (x) := ω k (x)dx := ∏ α∈R | ⟨α, x⟩ | k(α) dx, is defined by: ∀ x ∈ R d , ∀ r > 0, M r B (f)(x) := 1 m k [B(0,r)] ∫ R d f (y)h k (r, x, y)ω k (y)dy, where h k (r, x, .) is a compactly supported nonnegative explicit measurable function depending on R and k. In this paper, we prove that for almost every x ∈ R d , lim r→0 M r B (f)(x) = f (x). MSC (2010) primary: 42B25, 42B37, 43A32; secondary: 31B05, 33C52.

Mots clés: Generalized volume mean value operator; harmonic kernel; Dunkl-Laplace operator; Dunkl transform

Domaines: Mathématiques [math]

Chaabane Rejeb. Volume mean operator and differentiation results associated to root systems. 2017. ⟨hal-01436147⟩