On linear extension for interpolating sequences
Volume 186 / 2008
Abstract
Let $A$ be a uniform algebra on $X$ and $\sigma $ a probability measure on $X$. We define the Hardy spaces $H^{p}(\sigma )$ and the $H^{p}(\sigma )$ interpolating sequences $S$ in the $p$-spectrum ${\mathcal{M}}_{p}$ of $\sigma $. We prove, under some structural hypotheses on $A$ and $\sigma $, that if $S$ is a “dual bounded” Carleson sequence, then $S$ is $ H^{s}(\sigma )$-interpolating with a linear extension operator for $ s< p$, provided that either $p=\infty $ or $p\leq 2$. In the case of the unit ball of ${\mathbb{C}}^{n}$ we find, for instance, that if $S$ is dual bounded in $H^{\infty }({\mathbb{B}})$ then $S$ is $ H^{p}({\mathbb{B}})$-interpolating with a linear extension operator for any $1\leq p< \infty $. Already in this case this is a new result.
