# zbMATH — the first resource for mathematics

Sobolev constants for trees. (Constantes de Sobolev des arbres.) (French) Zbl 1152.58013
Summary: For $$p \in [1,+ \infty [$$ and for any tree $$T$$ of valency at least $$3$$, we study the Sobolev constant of exponent $$p$$ of $$T$$, that is the smallest constant $$\sigma _p$$ such that for every $$u \in \ell _p(T )$$, one has $$\| u\| ^p_p \leq \sigma _p\| du\| ^p_p$$. Our motivation comes from the search of finite graphs with small Poincaré constants of exponent $$p$$, in order to construct examples of groups which admit the fixed point property on $$L^p$$-spaces.

##### MSC:
 58E35 Variational inequalities (global problems) in infinite-dimensional spaces 31C45 Other generalizations (nonlinear potential theory, etc.)
##### Keywords:
Sobolev constants; Poincaré constants; trees; graphs
Full Text: