Abstract
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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Biroli M., Mosco U.,Sobolev inequalities for Dirichlet forms on homogeneous spaces, in “Boundary value problems for partial differential equations and applications”, C. Baiocchi and J.L. Lions Eds., Research Notes in Applied Mathematics, Masson, 1993.
Biroli M., Mosco U.,Sobolev and isoperimetric inequalities for Dirichlet forms on homogeneous spaces, Rend. Mat. Acc. Lincei (1994), Roma, to appear.
Coifman R.R., Weiss G.,Analyse harmonique sur certaines espaces homogènes, Lectures Notes in Math. 242, Springer V., Berlin-Heidelberg-New York, 1971.
Fukushima M.,Dirichlet forms and Markov processes, North Holland Math. Library, North Holland, Amsterdam, 1980.
Mosco U.,Composite media and asymptotic Dirichlet forms, J. Funct. Anal.123, 2 (1994), 368–421.
Stampacchia G., Le problème de Dirichlet pour les equations elliptiques du second ordre à coefficient discontinus, Ann. Inst. Fourier, 15(1965), 189–258.
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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). https://doi.org/10.1007/BF01053449
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DOI: https://doi.org/10.1007/BF01053449