A mean value theorem on bounded symmetric domains

Abstract

LetΩ\Omegabe a Cartan domain of rankrrand genusppandBνB_\nu,ν>p−1\nu >p-1, the Berezin transform onΩ\Omega; the numberBνf(z)B_{\nu }f(z)can be interpreted as a certain invariant-mean-value of a functionffaround zz. We show that a Lebesgue integrable function satisfyingf=Bνf=Bν+1f=⋯=Bν+rff=B_\nu f=B_{\nu +1}f=\dots =B_{\nu +r}f,ν≥p\nu \ge p, must beM\mathcal {M}-harmonic. In a sense, this result is reminiscent of Delsarte’s two-radius mean-value theorem for ordinary harmonic functions on the complexnn-spaceCn\mathbf {C}^{n}, but with the role of radiusrrplayed by the quantity 1/ν1/\nu.