An F. and M. Riesz type theorem for the unit ball in complex $N$-space

Abstract:

Let ${M_B}$ denote Lebesgue measure on the open unit ball, $B$, in complex $N$-space and let $M(B)$ denote the space of Borel measures on $B$. The volume Poisson kernel $\chi :\overline B \times B \to (0,\infty )$ is defined and then we prove Theorem. If $\mu \in M(B)$ and if ${\mu ^\sharp }(w) = {\smallint _B}\chi (z,w)d\mu (z)$ is pluriharmonic in $B$, then ${\mu ^\sharp } \in L’({M_B})$ and $\mu = {\mu ^\sharp } \cdot {M_B}$.