# zbMATH — the first resource for mathematics

Inequalities of John-Nirenberg type in doubling spaces. (English) Zbl 0990.46019
The author introduces the concept of an $$H$$-chain set $$\Omega$$ in a doubling space $$X$$; roughly speaking this means that there exists a “fairly short” chain of balls from any $$x\in\Omega$$ to a fixed $$x_0\in\Omega$$. $$H$$-chain sets generalize the notion of Hölder domains in Euclidean space but are not necessarily connected. It is shown that every $$H$$-chain set $$\Omega$$ is mean porous and that its outer layer has measure bounded by a power of its thickness. As a consequence the author shows that a John-Nirenberg type inequality holds on an open subset $$\Omega$$ of a doubling space $$X$$ if, and often only if, $$\Omega$$ is an $$H$$-chain set.

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 42B35 Function spaces arising in harmonic analysis
Full Text:
##### References:
 [1] [Bo] B. Bojarski,Remarks on Sobolev imbedding inequalities, Proc. of the conference on Complex Analysis, Joensuu 1987, Lecture Notes in Math.1351, Springer-Verlag, Berlin, 1989, pp. 52–68. [2] [B1] S. M. Buckley,Is the maximal function of a Lipschitz function continuous?, Ann. Acad. Sci. Fenn.24 (1999), 519–528. [3] [B2] S. M. Buckley,Strong doubling conditions, Math. Ineq. Appl.1 (1998), 533–542. · Zbl 0921.28001 [4] [BKL1] S. M. Buckley, P. Koskela and G. Lu,Subelliptic Poincaré inequalities: the case p<1, Publ. Mat.39 (1995), 313–334. · Zbl 0895.26005 [5] [BKL2] S. M. Buckley, P. Koskela and G. Lu,Boman equals John, Proc. XVIth Rolf Nevanlinna Colloquium, de Gruyter, Berlin, 1996, pp. 91–99. [6] [CW1] R. Coifman and G. Weiss,Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math.242, Springer-Verlag, Berlin, 1971. [7] [CW2] R. Coifman and G. Weiss,Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc.83 (1977), 569–645. · Zbl 0358.30023 [8] [GN] N. Garofalo and D.-M. Nhieu,Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and existence of minimal surfaces, Comm. Pure Appl. Math.49 (1996), 1081–1144. · Zbl 0880.35032 [9] [GM] F. W. Gehring and O. Martio,Lipschitz classes and quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I Math.10 (1985), 203–219. · Zbl 0584.30018 [10] [G1] Y. Gotoh,On global integrability of BMO functions on general domains, J. Analyse Math.75 (1998), 67–84. · Zbl 0916.30033 [11] [G2] Y. Gotoh,On domains with some growth conditions for quasihyperbolic metric, preprint. [12] [GS] J. Graczyk and S. Smirnov,Collett, Eckmann and Hölder, Invent. Math.133 (1998), 69–96. · Zbl 0916.30023 [13] [HK] P. Hajłasz and P. Koskela,Sobolev met Poincaré, to appear in Mem. Amer. Math. Soc. [14] [HKM] J. Heinonen, T. Kilpeläinen and O. Martio,Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Univ. Press, Oxford, 1993. [15] [HS] I. Holopainen and P. M. Soardi,A strong Liouville theorem for p-harmonic functions on graphs, Ann. Acad. Sci. Fenn. Ser. A I Math.22 (1997), 205–226. · Zbl 0874.31008 [16] [H] R. Hurri-Syrjänen,The John-Nirenberg inequality and a Sobolev inequality for general domains, J. Math. Anal. Appl.175 (1993), 579–587. · Zbl 0779.30018 [17] [JN] F. John and L. Nirenberg,On functions of bounded mean oscillation, Comm. Pure Appl. Math.14 (1961), 415–426. · Zbl 0102.04302 [18] [J] P. W. Jones,Extension theorems for BMO, Indiana Univ. Math. J.29 (1980), 41–66. · Zbl 0432.42017 [19] [JM] P. W. Jones and N. G. Makarov,Density properties of harmonic measure, Ann. of Math. (2)142 (1995), 427–455. · Zbl 0842.31001 [20] [K] P. Koskela,Old and new on the quasihyperbolic metric, inQuasiconformal Mappings and Analysis (Ann Arbor, MI, 1995), Springer, New York, 1998, pp. 205–219. · Zbl 0888.30019 [21] [KR] P. Koskela and S. Rohde,Hausdorff dimension and mean porosity, Math. Ann.309 (1997), 593–609. · Zbl 0890.30013 [22] [NSW] A. Nagel, E. M. Stein and S. Waigner,Balls and metrics defined by vector fields I: basic properties, Acta Math.155 (1985), 103–147. · Zbl 0578.32044 [23] [RR] H. M. Reimann and T. Rychener,Funktionen beschränkter mittelerer Oszillation, Lecture Notes in Math.489, Springer, Berlin, 1975. · Zbl 0324.46030 [24] [RL] L. Ruilin and Y. Lo,BMO functions in spaces of homogeneous type, Scientia Sinica (Series A)27 (1984), 695–708. · Zbl 0572.42011 [25] [SS1] W. Smith and D. A. Stegenga,Hölder domains and Poincaré domains, Trans. Amer. Math. Soc.319 (1990), 67–100. · Zbl 0707.46028 [26] [SS2] W. Smith and D. A. Stegenga,Exponential integrability of the quasihyperbolic metric in Hölder domains, Ann. Acad. Sci. Fenn. Ser. A I Math.16 (1991), 345–360. · Zbl 0725.46024 [27] [S] S. Staples,L p -averaging domains and the Poincaré inequality, Ann. Acad. Sci. Fenn. Ser. A I Math.14 (1989), 103–127. · Zbl 0706.26010 [28] [VSC] N. Th. Varopoulos, L. Saloff-Coste and T. Coulhon,Analysis and Geometry on Groups, Cambridge Tracts in Math. 100, Cambridge Univ. Press, Cambridge, 1992. [29] [VG] S. K. Vodop’yanov and A. V. Greshnov,On extension of functions of bounded mean oscillation from domains in a space of homogeneous type with intrinsic metric, Siberian Math. J.36 (1995), 873–901. · Zbl 0865.30029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.