Approximation in the Zygmund Class

Abstract

We study the distance in the Zygmund class $\Lambda_{\ast}$ to the subspace $\operatorname{I}(\operatorname{BMO})$ of functions with distributional derivative with bounded mean oscillation. In particular, we describe the closure of $\operatorname{I}(\operatorname{BMO})$ in the Zygmund seminorm. We also generalise this result to Zygmund measures on $\mathbb{R}^d.$ Finally, we apply the techniques developed in the article to characterise the closure of the subspace of functions in $\Lambda_{\ast}$ that are also in the classical Sobolev space $W^{1,p},$ for $1 < p < \infty.$