On peak-interpolation manifolds for $\boldsymbol {A}\boldsymbol {(}\boldsymbol {\Omega }\boldsymbol {)}$ for convex domains in $\boldsymbol {\mathbb {C}}^{\boldsymbol {n}}$

Abstract:

Let $\Omega$ be a bounded, weakly convex domain in ${\mathbb {C}}^n$, $n\geq 2$, having real-analytic boundary. $A(\Omega )$ is the algebra of all functions holomorphic in $\Omega$ and continuous up to the boundary. A submanifold $\boldsymbol {M}\subset \partial \Omega$ is said to be complex-tangential if $T_p(\boldsymbol {M})$ lies in the maximal complex subspace of $T_p(\partial \Omega )$ for each $p\in \boldsymbol {M}$. We show that for real-analytic submanifolds $\boldsymbol {M}\subset \partial \Omega$, if $\boldsymbol {M}$ is complex-tangential, then every compact subset of $\boldsymbol {M}$ is a peak-interpolation set for $A(\Omega )$.