On $H=W$ in Banach function spaces

In this paper we prove $H=W$" in the context of a Banach function space $X(Ω)$. Let $Ω$ be a subset of ${\mathbb R}^n$ and denote by $W^1_X(Ω)$ the collection of all those $f\in X(Ω)$ whose distributional derivatives $\partial_jf$ are contained in $X(Ω)$. Our main result provides a small collection of universal" hypotheses on $X(\mathrm{\Omega })$ that ensure $W_X^1(\mathrm{\Omega })$ is equal to $H_X^1(\mathrm{\Omega })$, the formal closure of $Lip(\mathrm{\Omega })\cap W_X^1(\mathrm{\Omega })$ with respect to the norm $$\mathrm{\parallel }f{\mathrm{\parallel }}_{W_X^1(\mathrm{\Omega })}=\mathrm{\parallel }f{\mathrm{\parallel }}_{X(\mathrm{\Omega })}+\mathrm{\parallel }\mathrm{\nabla }f{\mathrm{\parallel }}_{X(\mathrm{\Omega })}\mathrm{.}$$ The main theorem has two corollaries. The first gives a slightly stronger set of hypotheses for ``$H=W$", and the second gives density of $C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ in $W_X^1({\mathbb{R}}^n)$.