Extension domains for Hardy spaces
Volume 272 / 2023
Studia Mathematica 272 (2023), 139-158
MSC: Primary 42B30.
DOI: 10.4064/sm220726-30-5
Published online: 26 September 2023
Abstract
We show that a proper open subset $\Omega \subset \mathbb R^{n}$ is an extension domain for $H^p$ ($0 \lt p\le 1$) if and only if it satisfies a certain geometric condition. When $n(1/p-1)\in \mathbb N$, this condition is equivalent to the global Markov condition for $\Omega ^c$, for $p=1$ it is stronger, and when $n(1/p-1)\notin \mathbb N\cup \{0\}$, every proper open subset is an extension domain for $H^p$. We show that in each case a linear extension operator exists. We apply our results to study some complemented subspaces of BMO$(\mathbb R^{n})$.
