Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2013; 28(2): 231-241

Printed April 1, 2013

https://doi.org/10.4134/CKMS.2013.28.2.231

Copyright © The Korean Mathematical Society.

A counterexample for improved Sobolev inequalities over the $2$-adic group

Diego Chamorro

23 Bd. de France

Abstract

On the framework of the $2$-adic group $\mathbb{Z}_2$, we study a Sobolev-like inequality where we estimate the $L^2$ norm by a geometric mean of the $BV$ norm and the $\dot{B}^{-1,\infty}_{\infty}$ 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 $\dot{B}^{1,\infty}_{1}$. This identification lead us to the construction of a counterexample to the improved Sobolev inequality.

Keywords: Sobolev inequalities, $p$-adic groups

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