Approximation in the Zygmund Class

Abstract

We study the distance in the Zygmund class ${\Lambda }_{\ast }$ to the subspace $I(\mathrm{BMO})$ of functions with distributional derivative with bounded mean oscillation. In particular, we describe the closure of $I(\mathrm{BMO})$ 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 $1<p<\infty .$

27 pages, 2 figures, comments welcome