Interpolating sequences and Carleson measures in the Hardy–Sobolev spaces of the ball in ${\mathbb {C}}^{n}$
Volume 241 / 2018
Abstract
We study interpolating sequences and Carleson measures for the Hardy–Sobolev spaces $ H_{s}^{p}$ in the ball ${\mathbb {B}}$ of $ {\mathbb {C}}^{n}$. Our main results for interpolating sequences $ S\subset {\mathbb {B}}$ of the multiplier algebra ${\mathcal {M}}_{s}^{p}$ of $ H_{s}^{p}$ are: (i) there is always a bounded linear extension operator $E: l^{\infty }\rightarrow {\mathcal {M}}_{s}^{p}$ realizing the interpolation; (ii) the union of two interpolating sequences $ S_{1}, S_{2}$ for ${\mathcal {M}}_{s}^{p}$ is interpolating for ${\mathcal {M}}_{s}^{p}$ if and only if $ S_{1}$ and $ S_{2}$ are completely separated, generalizing a theorem of Varopoulos. We also establish a link between dual boundedness and Carleson sequences with the interpolation property for the Hardy–Sobolev spaces $ H_{s}^{p}$.
