A counterexample for Improved Sobolev Inequalities over the 2-adic group

TL;DR

This paper constructs a counterexample in the 2-adic group setting, showing that an improved Sobolev inequality involving BV and Besov norms does not hold universally.

Contribution

It demonstrates that the improved Sobolev inequality fails in the 2-adic group context by explicitly constructing a counterexample based on the properties of p-adic groups.

Findings

Counterexample disproves the improved Sobolev inequality

Identification of BV with a specific Besov space in p-adic groups

Highlights unique topological properties of p-adic groups

Abstract

On the framework of the 2-adic group Z_2, we study a Sobolev-like inequality where we estimate the L^2 norm by a geometric mean of the BV norm and the Besov space B(-1,\infty,\infty) norm. We first show, using the special topological properties of the p-adic groups, that the set of functions of bounded variations BV can be identified to the Besov space B(1,\infty,1). This identification lead us to the construction of a counterexample to the improved Sobolev inequality.