On linear extension for interpolating sequences

Abstract

Title: On linear extension for interpolating sequences. Author: Eric Amar Abstract: Let A be a uniform algebra on the compact space 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 $\sigma $ that "Carleson type" conditions on S imply that S is interpolating with a linear extension operator in $\displaystyle H^{s}(\sigma), s<p$ provided that either $p=\infty $ or $p\leq 2$. This gives new results on interpolating sequences for Hardy spaces of the ball and the polydisc. In particular in the case of the unit ball of ${\mathbb{C}}^{n}$ we get that if there is a sequence $\{\rho_{a}\}_{a\in S}$ bounded in $H^{\infty}({\mathbb{B}})$ such that $\forall a,b\in S, \rho _{a}(b)=\delta_{ab}$, then S is $H^{p}({\mathbb{B}})$-interpolating with a linear extension operator for any $1\leq p<\infty $.