A Mean Value Theorem on Bounded Symmetric Domains

Let Ω be a Cartan domain of rank r and genus p and Bν , ν > p−1, the Berezin transform on Ω; the number Bνf(z) can be interpreted as a certain invariant-mean-value of a function f around z. We show that a Lebesgue integrable function satisfying f = Bνf = Bν+1f = · · · = Bν+rf , ν ≥ p, must be M-harmonic. In a sense, this result is reminiscent of Delsarte’s two-radius mean-value theorem for ordinary harmonic functions on the complex n-space Cn, but with the role of radius r played by the quantity 1/ν. Let Ω = G/K be an irreducible bounded symmetric (Cartan) domain in its Harish-Chandra realization (i.e. a circular convex domain in C centered at the origin), dm the Lebesgue measure on Ω normalized so that m(Ω) = 1, and denote by K(z, w) the Bergman kernel of Ω with respect to dm and by p, r its genus and rank, respectively. For ν ∈ R, it is known [FK1] that the integral c(ν)−1 := ∫ Ω K(z, z)1−ν/p dm(z) is finite if and only if ν > p − 1; in that case, one can consider the weighted Bergman spaces Aν(Ω) of functions analytic on Ω and squareintegrable against the probability measure dμν := c(ν)K(z, z)1−ν/p dm(z). It can be shown that the point evaluations are continuous linear functionals on Aν and the corresponding reproducing kernels are [FK1] Kν(z, w) = K(z, w). (1) The Berezin transform Bν on Ω is the integral operator defined by Bνf(w) : = ∫ Ω f(z) |Kν(z, w)|2 Kν(w,w) dμν(z)