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By Calvo M., Villarroya A., Oller J.M.

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1 we have uα Lr¯ (Bα ) ≥ C −1 µ2α log 1 µα 2/3 . Inserting into (64), we obtain αµ4α log 1 µα 4/3 ≤ C µ4α log 1 µα 4/3 + αµ4α log 1 . µα Once again we obtain α ≤ C, a contradiction. 1 is thus established in the remaining limit case n = 6. 2. We adapt some ideas from [6]. Let (M, g) be a smooth compact Riemannian manifold without boundary, n ≥ 3. 1. Suppose that there exist ε¯ > 0 and Aε¯ > 0 such that (65) u 2 L2∗ (M,g) ≤ K2 {|∇g u|2 + c(n)Rg u2 } dvg + Aε¯ u 2 L1 (M,g) , M for all u ∈ H 1 (M ) such that diamg (supp) u < ε¯.

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26] E. Hebey and M. Vaugon, Meilleures constantes dans le th´eor`eme d’inclusion de Sobolev, Ann. Inst. H. Poincar´e 13 No. 1 (1996), 57–93. [27] J. Lee and T. Parker, The Yamabe Problem, Bull. Amer. Math. Soc. 17 (1987), 37–91. Y. Li and M. Zhu, Sharp Sobolev Trace Inequalities on Riemannian Manifolds with Boundaries, Commun. Pure. Appl. Math. 50 (1997), 449– 487. Y. Li and M. Zhu, Sharp Sobolev inequalities involving boundary terms, Geom. Funct. Anal. 8 (1998), 59–87. [30] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann.

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A biplot method for multivariate normal populations with unequal covariance matrices by Calvo M., Villarroya A., Oller J.M.


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