Monge–Ampère operators and Jessen functions of holomorphic almost periodic mappings

For a holomorphic almost periodic mapping $f$ from a tube domain of ${\mathbf C}^n$ into ${\mathbf C}^q$, the properties of its Jessen function, i.e., the mean value of the function $\log|f|^2$, are studied. In particular, certain relations between the Jessen function and behavior of the mapping and its zero set are obtained. To this end certain operators $\Phi_l$ on plurisubharmonic functions are introduced in a way that for a smooth function $u$,
$$ (\Phi_l[u])^l\,(dd^c|z|^2)^n=(dd^cu)^l\wedge(dd^c|z|^2)^{n-l}. $$