Fractional Sobolev extension and imbedding
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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 $r\in (0, 1]$, \begin{eqnarray*} |B(x, r)\cap \Omega |\ge C_1 r^n; \end{eqnarray*}
(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 $p\in (0, \infty )$.
References
- N. Aronszajn, Boundary values of functions with finite Dirichlet integral, Techn. Report of Univ. of Kansas 14 (1955), 77-94.
- S. Buckley and P. Koskela, Sobolev-Poincaré implies John, Math. Res. Lett. 2 (1995), no. 5, 577–593. MR 1359964, DOI 10.4310/MRL.1995.v2.n5.a5
- Stephen M. Buckley and Pekka Koskela, Criteria for imbeddings of Sobolev-Poincaré type, Internat. Math. Res. Notices 18 (1996), 881–901. MR 1420554, DOI 10.1155/S1073792896000542
- Ronald A. DeVore and Robert C. Sharpley, Besov spaces on domains in $\textbf {R}^d$, Trans. Amer. Math. Soc. 335 (1993), no. 2, 843–864. MR 1152321, DOI 10.1090/S0002-9947-1993-1152321-6
- Nobuhiko Fujii, A condition for a two-weight norm inequality for singular integral operators, Studia Math. 98 (1991), no. 3, 175–190. MR 1115188, DOI 10.4064/sm-98-3-175-190
- Emilio Gagliardo, Proprietà di alcune classi di funzioni in più variabili, Ricerche Mat. 7 (1958), 102–137 (Italian). MR 102740
- 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, DOI 10.5186/aasfm.1985.1022
- F. W. Gehring and O. Martio, Quasiextremal distance domains and extension of quasiconformal mappings, J. Analyse Math. 45 (1985), 181–206. MR 833411, DOI 10.1007/BF02792549
- 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
- Amiran Gogatishvili, Pekka Koskela, and Yuan Zhou, Characterizations of Besov and Triebel-Lizorkin spaces on metric measure spaces, Forum Math. 25 (2013), no. 4, 787–819. MR 3089750, DOI 10.1515/form.2011.135
- Vladimir Mikailovitch Gol′dšteĭn and Sergei Konstantinovitch Vodop′janov, Prolongement des fonctions de classe ${\cal L}^{1}_{p}$ et applications quasi conformes, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 10, A453–A456 (French, with English summary). MR 571380
- Vladimir Gol′dstein and Serge Vodop′anov, 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
- 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, DOI 10.1016/j.jfa.2007.11.020
- 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, DOI 10.4171/RMI/551
- Juha Heinonen, Lectures on Lipschitz analysis, Report. University of Jyväskylä Department of Mathematics and Statistics, vol. 100, University of Jyväskylä, Jyväskylä, 2005. MR 2177410
- Alf Jonsson and Hans Wallin, Function spaces on subsets of $\textbf {R}^n$, Math. Rep. 2 (1984), no. 1, xiv+221. MR 820626
- 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
- Peter W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math. 147 (1981), no. 1-2, 71–88. MR 631089, DOI 10.1007/BF02392869
- P. Koskela, Extensions and imbeddings, J. Funct. Anal. 159 (1998), no. 2, 369–383. MR 1658090, DOI 10.1006/jfan.1998.3331
- 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, DOI 10.1016/j.aim.2010.10.020
- 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, DOI 10.1016/j.bulsci.2011.12.004
- Jaak Peetre, New thoughts on Besov spaces, Duke University Mathematics Series, No. 1, Duke University, Mathematics Department, Durham, N.C., 1976. MR 0461123
- 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, DOI 10.1112/S0024610799007723
- Pavel Shvartsman, Local approximations and intrinsic characterization of spaces of smooth functions on regular subsets of ${\Bbb R}^n$, Math. Nachr. 279 (2006), no. 11, 1212–1241. MR 2247585, DOI 10.1002/mana.200510418
- Pavel Shvartsman, On Sobolev extension domains in $\mathbf R^n$, J. Funct. Anal. 258 (2010), no. 7, 2205–2245. MR 2584745, DOI 10.1016/j.jfa.2010.01.002
- 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
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- Hans Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983. MR 781540, DOI 10.1007/978-3-0346-0416-1
- 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, DOI 10.5209/rev_{R}EMA.2002.v15.n2.16910
- Hans Triebel, Function spaces and wavelets on domains, EMS Tracts in Mathematics, vol. 7, European Mathematical Society (EMS), Zürich, 2008. MR 2455724, DOI 10.4171/019
- S. K. Vodop′janov, V. M. Gol′dšteĭn, 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
- Yuan Zhou, Criteria for optimal global integrability of Hajłasz-Sobolev functions, Illinois J. Math. 55 (2011), no. 3, 1083–1103 (2013). MR 3069296
Additional Information
- Yuan Zhou
- Affiliation: Department of Mathematics, Beihang University, Haidian District Xueyuan Road 37#, Beijing 100191, People’s Republic of China
- MR Author ID: 792720
- Email: yuanzhou@buaa.edu.cn
- 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).
- © Copyright 2014 American Mathematical Society
- 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
- MathSciNet review: 3280034