Weighted $L^{2}$ estimates for $\overline{\partial }$ and the Corona problem of several complex variables

Abstract

In the paper, we apply Hörmander's weighted $L^{2}$ estimate for $\overline{\partial }$ to study the Corona problem on the unit ball $B_{n}$ in ${\mathbf{C}}^{n}$. We introduce a new holomorphic function space ${\mathcal S}(B_{n})$ which is slightly small than $H^{\infty}(B_{n})$. We can solve the Corona problems on ${\mathcal S}(B_{n})$ instead of $H^{\infty}(B_{n})$. We also provide a new proof of $H^{\infty }\cdot BMOA$ solution for the Corona problem which was first obtained by Varopoulos [41].