Abstract:In this paper we prove ``$H=W$" in the context of a Banach function space $X(\Omega)$. Let $\Omega$ be a subset of ${\mathbb R}^n$ and denote by $W^1_X(\Omega)$ the collection of all those $f\in X(\Omega)$ whose distributional derivatives $\partial_jf$ are contained in $X(\Omega)$. Our main result provides a small collection of ``universal" hypotheses on $X(\Omega)$ that ensure $W^1_X(\Omega)$ is equal to $H^1_X(\Omega)$, the formal closure of ${Lip}(\Omega)\cap W^1_X(\Omega)$ with respect to the norm \[\|f\|_{W^1_X(\Omega)} = \|f\|_{X(\Omega)} + \|\nabla f\|_{X(\Omega)}.\] The main theorem has two corollaries. The first gives a slightly stronger set of hypotheses for ``$H=W$", and the second gives density of $C^\infty_c({\mathbb R}^n)$ in $W^1_X({\mathbb R}^n)$.
From: David Cruz-Uribe OFS [view email]
[v1]
Wed, 17 Jun 2026 19:24:38 UTC (22 KB)
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