  We prove an $H^2-$Corona theorem with estimate $C(\delta)=C\delta^{-1-q}|\log
\delta|$ for $\delta\ll 1$ on delta-regular domains, where $q=\min\{n,m-1\}$
and $m$ is the number of generators. This class of domains includes smooth
bounded domains with defining functions that are plurisubharmonic on boundaries
and pseudoconvex domains of D'Angelo finite type.
