The $H^ p$-corona theorem for the polydisc

Abstract:

Let ${H^p} = {H^p}({D^n})$ denote the usual Hardy spaces on the polydisc ${D^n}$. We prove in this paper the following theorem: Suppose ${f_1},{f_2}, \ldots ,{f_n} \in {H^\infty },{\left \| {{f_j}} \right \|_{{H^\infty }}} \leq 1$, and $\sum \nolimits _{j = 1}^m {|{f_j}(z)|} \geq \delta > 0$. Then for every $g$ in ${H^p}$, $1 < p < \infty$, there are ${H^p}$ functions $g,g, \ldots ,{g_m}$ such that $\sum \nolimits _{j = 1}^m {{f_j}(z){g_j}(z) = g(z)}$. Moreover, we have ${\left \| {{g_j}} \right \|_{{H^p}}} \leq c(m,n,\delta ,p){\left \| g \right \|_{{H^p}}}$. (When $p = 2,n = 1$, this theorem is known to be equivalent to Carleson’s corona theorem.)