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.
Reviewer: Walter Farkas (München)
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 42B35 | Function spaces arising in harmonic analysis |
References:
| [1] | Bojarski, B., Remarks on Sobolev imbedding inequalities, 52-68 (1989), Berlin: Springer-Verlag, Berlin |
| [2] | Buckley, S. M., Is the maximal function of a Lipschitz function continuous?, Ann. Acad. Sci. Fenn., 24, 519-528 (1999) · Zbl 0935.42010 |
| [3] | Buckley, S. M., Strong doubling conditions, Math. Ineq. Appl., 1, 533-542 (1998) · Zbl 0921.28001 |
| [4] | Buckley, S. M.; Koskela, P.; Lu, G., Subelliptic Poincaré inequalities: the case p<1, Publ. Mat., 39, 313-334 (1995) · Zbl 0895.26005 |
| [5] | Buckley, S. M.; Koskela, P.; Lu, G., Boman equals John, 91-99 (1996), Berlin: de Gruyter, Berlin · Zbl 0861.43007 |
| [6] | Coifman, R.; Weiss, G., Analyse harmonique non-commutative sur certains espaces homogènes (1971), Berlin: Springer-Verlag, Berlin · Zbl 0224.43006 |
| [7] | Coifman, R.; Weiss, G., Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83, 569-645 (1977) · Zbl 0358.30023 |
| [8] | Garofalo, N.; Nhieu, D.-M., Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and existence of minimal surfaces, Comm. Pure Appl. Math., 49, 1081-1144 (1996) · Zbl 0880.35032 · doi:10.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-A |
| [9] | Gehring, F. W.; Martio, O., Lipschitz classes and quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I Math., 10, 203-219 (1985) · Zbl 0584.30018 |
| [10] | Gotoh, Y., On global integrability of BMO functions on general domains, J. Analyse Math., 75, 67-84 (1998) · Zbl 0916.30033 · doi:10.1007/BF02788692 |
| [11] | [G2] Y. Gotoh,On domains with some growth conditions for quasihyperbolic metric, preprint. |
| [12] | Graczyk, J.; Smirnov, S., Collett, Eckmann and Hölder, Invent. Math., 133, 69-96 (1998) · Zbl 0916.30023 · doi:10.1007/s002220050239 |
| [13] | [HK] P. Hajłasz and P. Koskela,Sobolev met Poincaré, to appear in Mem. Amer. Math. Soc. |
| [14] | Heinonen, J.; Kilpeläinen, T.; Martio, O., Nonlinear Potential Theory of Degenerate Elliptic Equations (1993), Oxford: Oxford Univ. Press, Oxford · Zbl 0780.31001 |
| [15] | Holopainen, I.; Soardi, P. M., A strong Liouville theorem for p-harmonic functions on graphs, Ann. Acad. Sci. Fenn. Ser. A I Math., 22, 205-226 (1997) · Zbl 0874.31008 |
| [16] | Hurri-Syrjänen, R., The John-Nirenberg inequality and a Sobolev inequality for general domains, J. Math. Anal. Appl., 175, 579-587 (1993) · Zbl 0779.30018 · doi:10.1006/jmaa.1993.1191 |
| [17] | John, F.; Nirenberg, L., On functions of bounded mean oscillation, Comm. Pure Appl. Math., 14, 415-426 (1961) · Zbl 0102.04302 · doi:10.1002/cpa.3160140317 |
| [18] | Jones, P. W., Extension theorems for BMO, Indiana Univ. Math. J., 29, 41-66 (1980) · Zbl 0432.42017 · doi:10.1512/iumj.1980.29.29005 |
| [19] | Jones, P. W.; Makarov, N. G., Density properties of harmonic measure, Ann. of Math., 142, 2, 427-455 (1995) · Zbl 0842.31001 · doi:10.2307/2118551 |
| [20] | Koskela, P., Old and new on the quasihyperbolic metric, Quasiconformal Mappings and Analysis (Ann Arbor, MI, 1995), 205-219 (1998), New York: Springer, New York · Zbl 0888.30019 |
| [21] | Koskela, P.; Rohde, S., Hausdorff dimension and mean porosity, Math. Ann., 309, 593-609 (1997) · Zbl 0890.30013 · doi:10.1007/s002080050129 |
| [22] | Nagel, A.; Stein, E. M.; Waigner, S., Balls and metrics defined by vector fields I: basic properties, Acta Math., 155, 103-147 (1985) · Zbl 0578.32044 · doi:10.1007/BF02392539 |
| [23] | Reimann, H. M.; Rychener, T., Funktionen beschränkter mittelerer Oszillation (1975), Berlin: Springer, Berlin · Zbl 0324.46030 |
| [24] | Ruilin, L.; Lo, Y., BMO functions in spaces of homogeneous type, Scientia Sinica (Series A), 27, 695-708 (1984) · Zbl 0572.42011 |
| [25] | Smith, W.; Stegenga, D. A., Hölder domains and Poincaré domains, Trans. Amer. Math. Soc., 319, 67-100 (1990) · Zbl 0707.46028 · doi:10.2307/2001337 |
| [26] | Smith, W.; Stegenga, D. A., Exponential integrability of the quasihyperbolic metric in Hölder domains, Ann. Acad. Sci. Fenn. Ser. A I Math., 16, 345-360 (1991) · Zbl 0725.46024 |
| [27] | Staples, S., L^p-averaging domains and the Poincaré inequality, Ann. Acad. Sci. Fenn. Ser. A I Math., 14, 103-127 (1989) · Zbl 0706.26010 |
| [28] | Varopoulos, N. Th.; Saloff-Coste, L.; Coulhon, T., Analysis and Geometry on Groups (1992), Cambridge: Cambridge Univ. Press, Cambridge · Zbl 0813.22003 |
| [29] | Vodop’yanov, S. K.; Greshnov, A. V., On extension of functions of bounded mean oscillation from domains in a space of homogeneous type with intrinsic metric, Siberian Math. J., 36, 873-901 (1995) · Zbl 0865.30029 · doi:10.1007/BF02112531 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
