A counterexample for improved Sobolev inequalities over the 2-adic group (Q2842346)

The improved Sobolev inequalities have the form \(\| f\|_{\dot W^{s,q}}\leq C\| f\|_{\dot W^{s+1,p}}^\theta\) \(\| f\|_{\dot B_\infty ^{-\beta ,\infty}}^{1-\theta}\) and concern functions \(f:\mathbb R^n\to\mathbb R\) belonging to certain Sobolev and Besov spaces. The author considers Sobolev and Besov spaces of functions \(f:\mathbb Z_2\to\mathbb R\) defined on the 2-adic group \(\mathbb Z_2=\{x\in\mathbb Q_2\mid | x|_2\leq 1\}\), and shows that, in certain cases, the counterpart of an improved Sobolev inequality is not a true relation. This means that the improved Sobolev inequalities depend on the group's structure. The presented results are obtained by using the Littlewood-Paley decomposition of a function \(f:\mathbb Z_2\to\mathbb R\).