Locally biHölder continuous maps and their induced embeddings between Besov spaces. (English) Zbl 1541.46021
Summary: We introduce a class of homeomorphisms between metric spaces, which are locally biHölder continuous maps. Then an embedding result between Besov spaces induced by locally biHölder continuous maps between Ahlfors regular spaces is established, which extends the corresponding result of Björn, Björn, Gill, and Shanmugalingam [A. Björn et al., J. Reine Angew. Math. 725, 63–114 (2017; Zbl 1376.46025)]. Furthermore, an example is constructed to show that our embedding result is more general. We also introduce a geometric condition, named as uniform boundedness, to characterize when a quasisymmetric map between uniformly perfect spaces is locally biHölder continuous.
MSC:
| 46E36 | Sobolev (and similar kinds of) spaces of functions on metric spaces; analysis on metric spaces |
| 30L10 | Quasiconformal mappings in metric spaces |
Keywords:
locally biHölder continuous map; (power) quasisymmetric map; Besov space; Ahlfors regular metric space; embedding; uniform boundednessCitations:
Zbl 1376.46025References:
| [1] | A. Björn and J. Björn, Nonlinear potential theory on metric spaces, EMS Tracts Math. 17. European Mathematical Society (EMS), Zürich, 2011. xii+403 pp. · Zbl 1231.31001 |
| [2] | A. Björn, J. Björn and N. Shanmugalingam, The Dirichlet problem for p-harmonic functions on metric spaces, J. Reine Angew. Math. 556 (2003), 173-203. · Zbl 1018.31004 |
| [3] | A. Björn, J. Björn and N. Shanmugalingam, Extension and trace results for dou-bling metric measure spaces and their hyperbolic fillings, J. Math. Pures Appl. 159 (2022), 196-249. · Zbl 1486.31022 |
| [4] | A. Björn, J. Björn, T. Gill and N. Shanmugalingam, Geometric analysis on Can-tor sets and trees, J. Reine Angew. Math. 725 (2017), 63-114. · Zbl 1376.46025 |
| [5] | G. Bourdaud and W. Sickel, Changes of variable in Besov spaces, Math. Nachr. 198 (1999), 19-39. · Zbl 0930.46026 |
| [6] | M. Bourdon and H. Pajot, Cohomologie l p et espaces de Besov, J. Reine Angew. Math. 558 (2003), 85-108. · Zbl 1044.20026 |
| [7] | F. W. Gehring, The definitions and exceptional sets for quasiconformal maps, Ann. Acad. Sci. Fenn. Math. 281 (1960), 1-28. · Zbl 0090.05303 |
| [8] | A. Gogatishvili, P. Koskela and N. Shanmugalingam, Interpolation properties of Besov spaces defined on metric spaces, Math. Nachr. 283 (2010), 215-231. · Zbl 1195.46033 |
| [9] | A. Gogatishvili, P. Koskela and Y. Zhou, Characterizations of Besov and Triebel-Lizorkin spaces on metric measure spaces, Forum Math. 25 (2013), 787-819. · Zbl 1278.42029 |
| [10] | P. Hajłasz, Sobolev spaces on an arbitrary metric space, Potential Anal. 5 (1996), 403-415. · Zbl 0859.46022 |
| [11] | P. Hajłasz, Sobolev space on metric-measure spaces, in Heat kernels and analysis on manifolds, graphs and metric spaces (Paris 2002), Contemp. Math. 338 (2003), 173-218. · Zbl 1048.46033 |
| [12] | P. Hajłasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 668 (2000), x+101 pp. · Zbl 0954.46022 |
| [13] | J. Heinonen, Lectures on analysis on metric spaces, Universitext. Springer-Verlag, New York, 2001. x+140 pp. · Zbl 0985.46008 |
| [14] | J. Heinonen, T. Kilpeläinen and O. Martio, Harmonic morphisms in nonlinear potential theory, Nagoya Math. J. 125 (1992), 115-140. · Zbl 0776.31007 |
| [15] | J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear potential theory of degen-erate elliptic equations, Dover Publications, Inc., Mineola, NY, 2006. · Zbl 1115.31001 |
| [16] | J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with con-trolled geometry, Acta Math. 181 (1998), 1-61. · Zbl 0915.30018 |
| [17] | J. Heinonen, P. Koskela, N. Shanmugalingam and J. Tyson, Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients. Cambridge University Press, Cambridge, 2015. · Zbl 1332.46001 |
| [18] | S. Hencl and L. Kleprlík, Composition of q-quasiconformal maps and functions in Orlicz-Sobolev spaces, Illinois J. Math. 56 (2012), 931-955. · Zbl 1291.30133 |
| [19] | P. Koskela, J. Xiao, Y. Zhang and Y. Zhou, A quasiconformal composition prob-lem for the Q-spaces, J. Eur. Math. Soc. 19 (2017), 1159-1187. · Zbl 1372.42017 |
| [20] | P. Koskela, D. Yang and Y. Zhou, A characterization of Hajłasz-Sobolev and Triebel-Lizorkin spaces via grand Littlewood-Paley functions, J. Funct. Anal. 258 (2010), 2637-2661. · Zbl 1195.46034 |
| [21] | P. Koskela, D. Yang and Y. Zhou, Pointwise characterizations of Besov and Triebel-Lizorkin spaces and quasiconformal maps, Adv. Math. 226 (2011), 3579-3621. · Zbl 1217.46019 |
| [22] | H. M. Riemann, Functions of bounded mean oscillation and quasiconformal maps, Comment. Math. Helv. 49 (1974), 260-276. · Zbl 0289.30027 |
| [23] | N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoam. 16 (2000), 243-279. · Zbl 0974.46038 |
| [24] | P. Tukia and J. Väisälä, Quasi-symmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), 97-114. · Zbl 0403.54005 |
| [25] | J. Väisälä, Lectures on n-dimensional quasiconformal maps, Lect. Notes Math. 229, Springer-Verlag, Berlin-New York, 1971. · Zbl 0221.30031 |
| [26] | S. K. Vodopyanov, Maps of homogeneous groups and embeddings of function spaces, Sibirsk. Mat. Zh. 30 (1989), 25-41. · Zbl 0731.46019 |
| [27] | MOE-LCSM, School of Mathematics and Statistics Hunan Normal University Changsha, Hunan 410081, China. emails : mzhuang@hunnu.edu.cn, xtwang@hunnu.edu.cn, zwang@hunnu.edu.cn, zhxu@hunnu.edu.cn |
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.
