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Adams inequality on metric measure spaces. (English) Zbl 1186.46031

The Adams inequality [D.R.Adams, Stud.Math.48, 99–105 (1973; Zbl 0237.46037)] says in \(\mathbb{R}^n\) that, for \(1 \leq p < q\) and \(p<n\), \(\| u \|_q \leq CM^{1/q} \| \nabla u \|_p\) for every \(u \in C^{\infty}_0(\mathbb{R}^n)\) whenever \(\nu\) is a Borel measure satisfying
\[ \nu(B(x,r)) \leq Mr^{\alpha}, \, \alpha = q(n-p)/p.\tag{\(*\)} \]
Here, \(\| u \|_q\) refers to the \(q\)-norm of \(u\) with respect to the measure \(\nu\). The author proves the corresponding inequality for Lipschitz functions with compact support in a complete metric measure space \((X, \mu)\) admitting a weak \((1, 1)\)-Poincaré inequality. The measure \(\mu\) is assumed to be a doubling Radon measure; for the assumptions on \(X\) and \(\mu\), see [P.Hajłasz and P.Koskela, “Sobolev met Poincaré” (Mem.Am.Math.Soc.688) (2000; Zbl 0954.46022)]. The proofs for \(p = 1\) and \(p >1\) require a different treatment. Inequality (\(*\)) is expressed in terms of the measure \(\mu\). The proof makes use of the isoperimetric inequality and the co-area formula in \(X\) [see L.Ambrosio, Set-Valued Anal.10, No.2–3, 111–128 (2002; Zbl 1037.28002) and M.Miranda, J. Math.Pures Appl.(9) 82, No.8, 975–1004 (2003; Zbl 1109.46030)], as well as the Marcinkiewicz interpolation theorem.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
31C15 Potentials and capacities on other spaces
26D10 Inequalities involving derivatives and differential and integral operators
30L99 Analysis on metric spaces
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References:

[1] Adams, D. R.: A trace inequality for generalized potentials. Studia Math. 48 (1973), 99-105. · Zbl 0237.46037
[2] Adams, D. and Hedberg, L.: Function spaces and potential theory. Grundlehren der Mathematischen Wissenschaften 314 . Springer-Verlag, Berlin, 1996. · Zbl 0834.46021
[3] Ambrosio, L.: Fine properties of sets of finite perimeter in doubling metric measure spaces. Set-Valued Anal. 10 (2002), no. 2-3, 111-128. · Zbl 1037.28002 · doi:10.1023/A:1016548402502
[4] Björn, A., Björn, J. and Shanmugalingam, N.: The Dirichlet problem for \(p\)-harmonic functions on metric spaces. J. Reine Angew. Math. 556 (2003), 173-203. · Zbl 1018.31004 · doi:10.1515/crll.2003.020
[5] Björn, A., Björn, J. and Shanmugalingam, N.: Sobolev extensions of Hölder continuous and characteristic functions on metric spaces Canad. J. Math. 59 (2007), no. 6, 1135-1153. · Zbl 1137.46018 · doi:10.4153/CJM-2007-049-7
[6] Björn, J., MacManus, P. and Shanmugalingam, N.: Fat sets and pointwise boundary estimates for \(p\)-harmonic functions in metric spaces. J. Anal. Math. 85 (2001), 339-369. · Zbl 1003.31004 · doi:10.1007/BF02788087
[7] Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9 (1999), no. 3, 428-517. · Zbl 0942.58018 · doi:10.1007/s000390050094
[8] Edmunds, D.E., Kokilashvili, V. and Meskhi, A.: Bounded and compact integral operators. Mathematics and its applications 543 . Kluwer Academic Publishers, Dordrecht, 2002. · Zbl 1023.42001
[9] Hajłasz, P.: Sobolev spaces on an arbitrary metric space. Potential Anal. 5 (1996), no. 4, 403-415. · Zbl 0859.46022
[10] Hajłasz, P. and Koskela, P.: Sobolev met Poincaré. Mem. Amer. Math. Soc. 145 (2000), no. 688. · Zbl 0954.46022
[11] Heinonen, J.: Lectures on analysis on metric spaces. Universititext, Springer-Verlag, New-York, 2001. · Zbl 0985.46008 · doi:10.1007/978-1-4613-0131-8
[12] Heinonen, J. and Koskela, P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181 (1998), no. 1, 1-61. · Zbl 0915.30018 · doi:10.1007/BF02392747
[13] Keith, S. and Zhong X.: The Poincare inequality is an open ended condition. Ann. of Math. (2) 167 (2008), no. 2, 575-599. · Zbl 1180.46025 · doi:10.4007/annals.2008.167.575
[14] Kilpeläinen, T., Kinnunen, J. and Martio, O.: Sobolev spaces with zero boundary values on metric spaces. Potential Anal. 12 (2000), no. 3, 233-247. · Zbl 0962.46021 · doi:10.1023/A:1008601220456
[15] Kinnunen, J. and Korte, R.: Characterizations of Sobolev inequalities on metric spaces. J. Math. Anal. Appl. 344 (2008), no. 2, 1093-1104. · Zbl 1154.46018 · doi:10.1016/j.jmaa.2008.03.070
[16] Kinnunen, J., Korte, R., Shanmugalingam, N. and Tuominen, H.: Lebesgue points and capacities via the boxing inequality in metric spaces. Indiana Univ. Math. J. 57 (2008), no. 1, 401-430. · Zbl 1146.46018 · doi:10.1512/iumj.2008.57.3168
[17] Kinnunen, J. and Martio, O.: Nonlinear potential theory on metric spaces. Illinois J. Math. 46 (2002), no. 3, 857-883. · Zbl 1030.46040
[18] Kinnunen, J. and Martio, O.: Potential theory of quasiminimizers. Ann. Acad. Sci. Fenn. Math. 28 (2003), no. 2, 459-490. · Zbl 1035.31007
[19] Kinnunen, J. and Shanmugalingam, N.: Regularity of quasi-minimizers on metric spaces. Manuscripta Math. 105 (2001), no. 3, 401-423. · Zbl 1006.49027 · doi:10.1007/s002290100193
[20] Koskela, P. and MacManus, P.: Quasiconformal mappings and Sobolev spaces. Studia Math. 131 (1998), no. 1, 1-17. · Zbl 0918.30011
[21] Maz’ja, V.G.: Sobolev spaces. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1985.
[22] Miranda Jr., M.: Functions of bounded variation on “good” metric spaces. J. Math. Pures Appl. (9) 82 (2003), no. 8, 975-1004. · Zbl 1109.46030 · doi:10.1016/S0021-7824(03)00036-9
[23] Muckenhoupt, B. and Wheeden, R.: Weighted norm inequalities for fractional integrals. Trans. Amer. Math. Soc. 192 (1974), 261-274. · Zbl 0289.26010 · doi:10.2307/1996833
[24] Shanmugalingam, N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoamericana 16 (2000), 243-279. · Zbl 0974.46038 · doi:10.4171/RMI/275
[25] Shanmugalingam, N.: Harmonic functions on metric spaces. Illinois J. Math. 45 (2001), no. 3, 1021-1050. · Zbl 0989.31003
[26] Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series 30 . Princeton University Press, Princeton, N.J., 1970. · Zbl 0207.13501
[27] Turesson, B.O.: Nonlinear potential theory and weighted Sobolev spaces. Lecture Notes in Mathematics, 1736 . Springer-Verlag, Berlin, 2000. · Zbl 0949.31006 · doi:10.1007/BFb0103908
[28] Ziemer, W.P.: Weakly differentiable functions. Graduate Texts in Mathematics 120 . Springer-Verlag, New York, 1989. · Zbl 0692.46022 · doi:10.1007/978-1-4612-1015-3
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