Extension domains for Hardy spaces

TL;DR

This paper characterizes when open subsets of Euclidean space are extension domains for Hardy spaces $H^p$, linking geometric conditions to the existence of linear extension operators, and explores implications for subspaces of BMO.

Contribution

It provides a complete geometric characterization of extension domains for $H^p$ spaces and establishes the existence of linear extension operators under these conditions.

Findings

Proper open subsets are extension domains iff they satisfy a specific geometric condition.

When $n(1/p - 1)$ is an integer, the condition aligns with the global Markov condition.

For certain $p$, all proper open subsets are extension domains.

Abstract

We show that a proper open subset $\Omega\subset \mathbb{R}^n$ is an extension domain for $H^p$ ($0<p\le1$), if and only if it satisfies a certain geometric condition. When $n(\frac{1}{p}-1)\in \mathbb{N}$ this condition is equivalent to the global Markov condition for $\Omega^c$, for $p=1$ it is stronger, and when $n(\frac{1}{p}-1)\notin \mathbb{N}\cup \{0\}$ every proper open subset is an extension domain for $H^p$. It is shown that in each case a linear extension operator exists. We apply our results to study some complemented subspaces of $BMO(\mathbb{R}^n)$.