On linear extension for interpolating sequences

Eric Amar¹

¹ Université Bordeaux I 351 Cours de la Libération 33405 Talence, France

Studia Mathematica, Tome 186 (2008) no. 3, pp. 251-265

Résumé

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.

Détail

DOI : 10.4064/sm186-3-4

Keywords: uniform algebra sigma probability measure define hardy spaces sigma sigma interpolating sequences p spectrum mathcal sigma prove under structural hypotheses sigma dual bounded carleson sequence sigma interpolating linear extension operator provided either infty leq the unit ball mathbb instance dual bounded infty mathbb mathbb interpolating linear extension operator leq infty already this result