Growth Estimates for Analytic Vector-Valued Functions in the Unit Ball Having Bounded $\mathbf{L}$-index in Joint Variables

Abstract

Our results concern growth estimates for vector-valued functions of $\mathbb{L}$-index in joint variables which are analytic in the unit ball.  There are deduced analogs of known growth estimates obtained early for functions analytic in the unit ball. Our estimates contain logarithm of $\sup$-norm instead of logarithm modulus of the function. They describe the behavior of logarithm of norm of analytic vector-valued function on a skeleton in a bidisc by behavior of the function $\mathbf{L}.$ These estimates are sharp in a general case.  The presented results are based on bidisc exhaustion of a unit ball.