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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 smoothbounded domains with defining functions that are plurisubharmonic on boundariesand pseudoconvex domains of D'Angelo finite type.

$H^2-$Corona problem on $\delta-$regular domains