A counterexample for Improved Sobolev Inequalities over the 2-adic group
Abstract
On the framework of the 2-adic group Z2, we study a Sobolev-like inequality where we estimate the L2 norm by a geometric mean of the BV norm and the Besov space B(-1,∞,∞) 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,∞,1). This identification lead us to the construction of a counterexample to the improved Sobolev inequality.
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