Advertisement

Journal d’Analyse Mathématique

, Volume 79, Issue 1, pp 215–240 | Cite as

Inequalities of John—Nirenberg type in doubling spaces

  • Stephen M. Buckley
Article

Abstract

The concept of an H-chain set in a doubling spaceX, which generalizes that of a Hölder domain in Euclidean space, is defined and investigated. We show that every H-chain set is mean porous and that its outer layer has measure bounded by a power of its thickness. As a consequence, we show that a John-Nirenberg type inequality holds on an open subset Ω ofX if, and often only if, Ω is an H-chain set.

Keywords

Length Space Doubling Condition Doubling Measure Exponential Integrability Uniform Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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.Google Scholar
  2. [B1] S. M. Buckley,Is the maximal function of a Lipschitz function continuous?, Ann. Acad. Sci. Fenn.24 (1999), 519–528.MathSciNetGoogle Scholar
  3. [B2] S. M. Buckley,Strong doubling conditions, Math. Ineq. Appl.1 (1998), 533–542.zbMATHMathSciNetGoogle Scholar
  4. [BKL1] S. M. Buckley, P. Koskela and G. Lu,Subelliptic Poincaré inequalities: the case p<1, Publ. Mat.39 (1995), 313–334.zbMATHMathSciNetGoogle Scholar
  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.Google Scholar
  6. [CW1] R. Coifman and G. Weiss,Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math.242, Springer-Verlag, Berlin, 1971.zbMATHGoogle Scholar
  7. [CW2] R. Coifman and G. Weiss,Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc.83 (1977), 569–645.zbMATHMathSciNetGoogle Scholar
  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.zbMATHCrossRefMathSciNetGoogle Scholar
  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.zbMATHMathSciNetGoogle Scholar
  10. [G1] Y. Gotoh,On global integrability of BMO functions on general domains, J. Analyse Math.75 (1998), 67–84.zbMATHMathSciNetCrossRefGoogle Scholar
  11. [G2] Y. Gotoh,On domains with some growth conditions for quasihyperbolic metric, preprint.Google Scholar
  12. [GS] J. Graczyk and S. Smirnov,Collett, Eckmann and Hölder, Invent. Math.133 (1998), 69–96.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [HK] P. Hajłasz and P. Koskela,Sobolev met Poincaré, to appear in Mem. Amer. Math. Soc.Google Scholar
  14. [HKM] J. Heinonen, T. Kilpeläinen and O. Martio,Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Univ. Press, Oxford, 1993.zbMATHGoogle Scholar
  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.zbMATHMathSciNetGoogle Scholar
  16. [H] R. Hurri-Syrjänen,The John-Nirenberg inequality and a Sobolev inequality for general domains, J. Math. Anal. Appl.175 (1993), 579–587.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [JN] F. John and L. Nirenberg,On functions of bounded mean oscillation, Comm. Pure Appl. Math.14 (1961), 415–426.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [J] P. W. Jones,Extension theorems for BMO, Indiana Univ. Math. J.29 (1980), 41–66.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [JM] P. W. Jones and N. G. Makarov,Density properties of harmonic measure, Ann. of Math. (2)142 (1995), 427–455.zbMATHCrossRefMathSciNetGoogle Scholar
  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.Google Scholar
  21. [KR] P. Koskela and S. Rohde,Hausdorff dimension and mean porosity, Math. Ann.309 (1997), 593–609.zbMATHCrossRefMathSciNetGoogle Scholar
  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.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [RR] H. M. Reimann and T. Rychener,Funktionen beschränkter mittelerer Oszillation, Lecture Notes in Math.489, Springer, Berlin, 1975.Google Scholar
  24. [RL] L. Ruilin and Y. Lo,BMO functions in spaces of homogeneous type, Scientia Sinica (Series A)27 (1984), 695–708.zbMATHMathSciNetGoogle Scholar
  25. [SS1] W. Smith and D. A. Stegenga,Hölder domains and Poincaré domains, Trans. Amer. Math. Soc.319 (1990), 67–100.zbMATHCrossRefMathSciNetGoogle Scholar
  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.MathSciNetGoogle Scholar
  27. [S] S. Staples,L p -averaging domains and the Poincaré inequality, Ann. Acad. Sci. Fenn. Ser. A I Math.14 (1989), 103–127.zbMATHMathSciNetGoogle Scholar
  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.Google Scholar
  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.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Magnes Press 1999

Authors and Affiliations

  • Stephen M. Buckley
    • 1
  1. 1.Department of MathematicsNational University of IrelandMaynoothIreland

Personalised recommendations