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Extension domains for Hardy spaces (2208.06684v6)
Published 13 Aug 2022 in math.FA
Abstract: We show that a proper open subset $\Omega\subset \mathbb{R}n$ is an extension domain for $Hp$ ($0<p\le1$), 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 $\Omegac$, 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 $Hp$. It is shown that in each case a linear extension operator exists. We apply our results to study some complemented subspaces of $BMO(\mathbb{R}n)$.