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Fractional Sobolev extension and imbedding


Author: Yuan Zhou
Journal: Trans. Amer. Math. Soc. 367 (2015), 959-979
MSC (2010): Primary 46E35; Secondary 42B35
DOI: https://doi.org/10.1090/S0002-9947-2014-06088-1
Published electronically: June 30, 2014
MathSciNet review: 3280034
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Abstract: Let $ \Omega $ be a domain of $ \mathbb{R}^n$ with $ n\ge 2$ and denote by $ W^{s,\,p}(\Omega )$ the fractional Sobolev space for $ s\in (0,\,1)$ and $ p\in (0, \,\infty )$. We prove that the following are equivalent:

(i) there exists a constant $ C_1>0$ such that for all $ x\in \Omega $ and $ r\in (0,\,1]$,

$\displaystyle \vert B(x,\,r)\cap \Omega \vert\ge C_1 r^n;$      

(ii) $ \Omega $ is a $ W^{s,\,p}$-extension domain for all $ s\in (0,\,1)$ and all $ p\in (0,\,\infty )$;

(iii) $ \Omega $ is a $ W^{s,\,p}$-extension domain for some $ s\in (0,\,1)$ and some $ p\in (0,\,\infty )$;

(iv) $ \Omega $ is a $ W^{s,\,p}$-imbedding domain for all $ s\in (0,\,1)$ and all $ p\in (0,\,\infty )$;

(v) $ \Omega $ is a $ W^{s,\,p}$-imbedding domain for some $ s\in (0,\,1)$ and some $ p\in (0,\,\infty )$.


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  • [1] N. Aronszajn, Boundary values of functions with finite Dirichlet integral, Techn. Report of Univ. of Kansas 14 (1955), 77-94.
  • [2] S. Buckley and P. Koskela, Sobolev-Poincaré implies John, Math. Res. Lett. 2 (1995), no. 5, 577-593. MR 1359964 (96i:46035)
  • [3] Stephen M. Buckley and Pekka Koskela, Criteria for imbeddings of Sobolev-Poincaré type, Internat. Math. Res. Notices 18 (1996), 881-901. MR 1420554 (98g:46041), https://doi.org/10.1155/S1073792896000542
  • [4] Ronald A. DeVore and Robert C. Sharpley, Besov spaces on domains in $ {\bf R}^d$, Trans. Amer. Math. Soc. 335 (1993), no. 2, 843-864. MR 1152321 (93d:46051), https://doi.org/10.2307/2154408
  • [5] Nobuhiko Fujii, A condition for a two-weight norm inequality for singular integral operators, Studia Math. 98 (1991), no. 3, 175-190. MR 1115188 (92k:42022)
  • [6] Emilio Gagliardo, Proprietà di alcune classi di funzioni in più variabili, Ricerche Mat. 7 (1958), 102-137 (Italian). MR 0102740 (21 #1526)
  • [7] F. W. Gehring and O. Martio, Lipschitz classes and quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 203-219. MR 802481 (87b:30029)
  • [8] F. W. Gehring and O. Martio, Quasiextremal distance domains and extension of quasiconformal mappings, J. Analyse Math. 45 (1985), 181-206. MR 833411 (87j:30043), https://doi.org/10.1007/BF02792549
  • [9] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364 (2001k:35004)
  • [10] A. Gogatishvili, P. Koskela and Y. Zhou, Characterizations of Besov and Triebel-Lizorkin spaces on metric measure spaces, Forum Math. 25 (2013), no. 4, 787-819. DOI 10.1515/FORM.2011.135. MR 3089750
  • [11] Vladimir Mikailovitch Goldšteĭn and Sergei Konstantinovitch Vodopjanov, Prolongement des fonctions de classe $ {\mathcal {L}}^{1}_{p}$ et applications quasi conformes, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 10, A453-A456 (French, with English summary). MR 571380 (81k:30024)
  • [12] Vladimir Goldstein and Serge Vodopanov, Prolongement de fonctions différentiables hors de domaines plans, C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), no. 12, 581-584 (French, with English summary). MR 647686 (83a:46043)
  • [13] Piotr Hajłasz, Pekka Koskela, and Heli Tuominen, Sobolev embeddings, extensions and measure density condition, J. Funct. Anal. 254 (2008), no. 5, 1217-1234. MR 2386936 (2009b:46070), https://doi.org/10.1016/j.jfa.2007.11.020
  • [14] Piotr Hajłasz, Pekka Koskela, and Heli Tuominen, Measure density and extendability of Sobolev functions, Rev. Mat. Iberoam. 24 (2008), no. 2, 645-669. MR 2459208 (2009i:46065), https://doi.org/10.4171/RMI/551
  • [15] Juha Heinonen, Lectures on Lipschitz analysis, Report. University of Jyväskylä Department of Mathematics and Statistics, vol. 100, University of Jyväskylä, Jyväskylä, 2005. MR 2177410 (2006k:49111)
  • [16] Alf Jonsson and Hans Wallin, Function spaces on subsets of $ {\bf R}^n$, Math. Rep. 2 (1984), no. 1, xiv+221. MR 820626 (87f:46056)
  • [17] A. Jonsson and H. Wallin, A Whitney extension theorem in $ L_{p}$ and Besov spaces, Ann. Inst. Fourier (Grenoble) 28 (1978), no. 1, vi, 139-192 (English, with French summary). MR 500920 (81c:46024)
  • [18] Peter W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math. 147 (1981), no. 1-2, 71-88. MR 631089 (83i:30014), https://doi.org/10.1007/BF02392869
  • [19] P. Koskela, Extensions and imbeddings, J. Funct. Anal. 159 (1998), no. 2, 369-383. MR 1658090 (99k:46056), https://doi.org/10.1006/jfan.1998.3331
  • [20] Pekka Koskela, Dachun Yang, and Yuan Zhou, Pointwise characterizations of Besov and Triebel-Lizorkin spaces and quasiconformal mappings, Adv. Math. 226 (2011), no. 4, 3579-3621. MR 2764899 (2011k:46050), https://doi.org/10.1016/j.aim.2010.10.020
  • [21] Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521-573. MR 2944369, https://doi.org/10.1016/j.bulsci.2011.12.004
  • [22] Jaak Peetre, New thoughts on Besov spaces, Mathematics Department, Duke University, Durham, N.C., 1976. Duke University Mathematics Series, No. 1. MR 0461123 (57 #1108)
  • [23] Vyacheslav S. Rychkov, On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains, J. London Math. Soc. (2) 60 (1999), no. 1, 237-257. MR 1721827 (2000m:46077), https://doi.org/10.1112/S0024610799007723
  • [24] Pavel Shvartsman, Local approximations and intrinsic characterization of spaces of smooth functions on regular subsets of $ {\mathbb{R}}^n$, Math. Nachr. 279 (2006), no. 11, 1212-1241. MR 2247585 (2007m:46051), https://doi.org/10.1002/mana.200510418
  • [25] Pavel Shvartsman, On Sobolev extension domains in $ \mathbf {R}^n$, J. Funct. Anal. 258 (2010), no. 7, 2205-2245. MR 2584745 (2011d:46073), https://doi.org/10.1016/j.jfa.2010.01.002
  • [26] L. N. Slobodeckiĭ, Generalized Sobolev spaces and their application to boundary problems for partial differential equations, Leningrad. Gos. Ped. Inst. Učen. Zap. 197 (1958), 54-112. MR 0203222 (34 #3075)
  • [27] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095 (44 #7280)
  • [28] Hans Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983. MR 781540 (86j:46026)
  • [29] Hans Triebel, Function spaces in Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise multipliers, Rev. Mat. Complut. 15 (2002), no. 2, 475-524. MR 1951822 (2003m:46059)
  • [30] Hans Triebel, Function spaces and wavelets on domains, EMS Tracts in Mathematics, vol. 7, European Mathematical Society (EMS), Zürich, 2008. MR 2455724 (2010b:46078)
  • [31] S. K. Vodopjanov, V. M. Goldšteĭn, and T. G. Latfullin, A criterion for the extension of functions of the class $ L_{2}^{1}$ from unbounded plane domains, Sibirsk. Mat. Zh. 20 (1979), no. 2, 416-419, 464 (Russian). MR 530508 (80j:46061)
  • [32] Y. Zhou, Criteria for optimal global integrability of Hajłasz-Sobolev functions, Ill. J. Math. 231 (2011), 1083-1103. MR 3069296

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Additional Information

Yuan Zhou
Affiliation: Department of Mathematics, Beihang University, Haidian District Xueyuan Road 37#, Beijing 100191, People’s Republic of China
Email: yuanzhou@buaa.edu.cn

DOI: https://doi.org/10.1090/S0002-9947-2014-06088-1
Received by editor(s): September 7, 2011
Received by editor(s) in revised form: December 2, 2012
Published electronically: June 30, 2014
Additional Notes: The author was supported by Program for New Century Excellent Talents in University of Ministry of Education of China, New Teachers’ Fund for Doctor Stations of Ministry of Education of China (#20121102120031), and National Natural Science Foundation of China (#11201015).
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