A new function space and applications
This article is concerned with showing that certain functions with values in \(\mathbb{Z}\) are constant. The starting point are the known results that if a \(\mathbb{Z}\)-valued function is contained in any of the function spaces \(\mathrm {VMO}(\Omega)\), \(W^{1,1}(\Omega)\), or \(W^{1/p,p}(\Omega)\), then it is constant. The authors introduce a space \(B\) that contains the above three spaces and they further introduce a subspace \(B_0\) of \(B\). They show that if \(f\) is \(\mathbb{Z}\)-valued and in \(B_0\), then \(f\) is constant. Actually, for \(f\) being a characteristic function (and \(n\geq 2\)), they show a quantitative improvement of the result. This quantitative result is extended to \(\mathbb{Z}\)-valued functions in \(B\) (under some smallness condition on the function, which is also shown to be necessary).
scientific article; zbMATH DE number 6495629