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Sobolev inequalities on homogeneous spaces

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We consider a homogeneous spaceX=(X, d, m) of dimension ν≥1 and a local regular Dirichlet forma inL 2 (X, m). We prove that if a Poincaré inequality of exponent 1≤p<ν holds on every pseudo-ballB(x, R) ofX, then Sobolev and Nash inequalities of any exponentq∈[p, ν), as well as Poincaré inequalities of any exponentq∈[p, +∞), also hold onB(x, R).

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Lavoro eseguito nell'ambito del Contratto CNR “Strutture variazionali irregolari”.

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Biroli, M., Mosco, U. Sobolev inequalities on homogeneous spaces. Potential Anal 4, 311–324 (1995).

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