Sobolev inequalities on homogeneous spaces

We consider a homogeneous spaceX=(X, d, m) of dimension 1 and a local regular Dirichlet forma inL² (X, m). We prove that if a Poincaré inequality of exponent 1p< holds on every pseudo-ballB(x, R) ofX, then Sobolev and Nash inequalities of any exponentq[p, ), as well as Poincaré inequalities of any exponentq[p, +), also hold onB(x, R).Lavoro eseguito nell'ambito del Contratto CNR Strutture variazionali irregolari.