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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.)
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