Bulletin of the American Mathematical Society

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Authors: Alexander Brudnyi and Yuri Brudnyi
Title: Methods of geometric analysis in extension and trace problems. Volume 1
Additional book information: Monographs in Mathematics, Vol. 102, Birkh\"auser/Springer Basel AG, Basel, 2012, xxiv+560 pp., ISBN 978-3-0348-0208-6, US$129

Authors: Alexander Brudnyi and Yuri Brudnyi
Title: Methods of geometric analysis in extension and trace problems. Volume 2
Additional book information: Monographs in Mathematics, Vol. 103, Birkh\"auser/Springer Basel AG, Basel, 2012, xx+414 pp., ISBN 978-3-0348-0211-6, US$129

K. Ball, Markov chains, Riesz transforms and Lipschitz maps, Geom. Funct. Anal. 2 (1992), no. 2, 137–172. MR 1159828, DOI 10.1007/BF01896971

Edward Bierstone, Pierre D. Milman, and Wiesław Pawłucki, Differentiable functions defined in closed sets. A problem of Whitney, Invent. Math. 151 (2003), no. 2, 329–352. MR 1953261, DOI 10.1007/s00222-002-0255-6

L. E. J. Brouwer, Über die Erweiterung des Definitionsbereichs einer stetigen Funktion, Math. Ann. 79 (1918), no. 3, 209–211 (German). MR 1511921, DOI 10.1007/BF01458203

Yu. A. Brudnyĭ and P. A. Shvartsman, A linear extension operator for a space of smooth functions defined on a closed subset in $\textbf {R}^n$, Dokl. Akad. Nauk SSSR 280 (1985), no. 2, 268–272 (Russian). MR 775048

Ju. A. Brudnyi and P. A. Shvartsman, The traces of differentiable functions to closed subsets of $\textbf {R}^n$, Function spaces (Poznań, 1989) Teubner-Texte Math., vol. 120, Teubner, Stuttgart, 1991, pp. 206–210. MR 1155176

Yuri Brudnyi and Pavel Shvartsman, Generalizations of Whitney’s extension theorem, Internat. Math. Res. Notices 3 (1994), 129 ff., approx. 11. MR 1266108, DOI 10.1155/S1073792894000140

Yuri Brudnyi and Pavel Shvartsman, The Whitney problem of existence of a linear extension operator, J. Geom. Anal. 7 (1997), no. 4, 515–574. MR 1669235, DOI 10.1007/BF02921632

Yuri Brudnyi and Pavel Shvartsman, The trace of jet space $J^k\Lambda ^\omega$ to an arbitrary closed subset of $\mathbf R^n$, Trans. Amer. Math. Soc. 350 (1998), no. 4, 1519–1553. MR 1407483, DOI 10.1090/S0002-9947-98-01872-8

Yuri Brudnyi and Pavel Shvartsman, Whitney’s extension problem for multivariate $C^{1,\omega }$-functions, Trans. Amer. Math. Soc. 353 (2001), no. 6, 2487–2512. MR 1814079, DOI 10.1090/S0002-9947-01-02756-8

Yu. Brudnyi and P. Shvartsman, Stability of the Lipschitz extension property under metric transforms, Geom. Funct. Anal. 12 (2002), no. 1, 73–79. MR 1904557, DOI 10.1007/s00039-002-8237-9

Charles L. Fefferman, A sharp form of Whitney’s extension theorem, Ann. of Math. (2) 161 (2005), no. 1, 509–577. MR 2150391, DOI 10.4007/annals.2005.161.509

Charles Fefferman, Interpolation and extrapolation of smooth functions by linear operators, Rev. Mat. Iberoamericana 21 (2005), no. 1, 313–348. MR 2155023, DOI 10.4171/RMI/424

Charles Fefferman, The structure of linear extension operators for $C^m$, Rev. Mat. Iberoam. 23 (2007), no. 1, 269–280. MR 2351135, DOI 10.4171/RMI/495

Charles Fefferman, The $C^m$ norm of a function with prescribed jets. II, Rev. Mat. Iberoam. 25 (2009), no. 1, 275–421. MR 2514339, DOI 10.4171/RMI/570

Charles Fefferman, Extension of $\mathsf C^{m,\omega }$-smooth functions by linear operators, Rev. Mat. Iberoam. 25 (2009), no. 1, 1–48. MR 2514337, DOI 10.4171/RMI/568

Charles Fefferman, The $C^m$ norm of a function with prescribed jets I, Rev. Mat. Iberoam. 26 (2010), no. 3, 1075–1098. MR 2789377, DOI 10.4171/RMI/628

Charles Fefferman, Nearly optimal interpolation of data in $\textrm {C}^2(\Bbb R^2)$. Part I, Rev. Mat. Iberoam. 28 (2012), no. 2, 415–533. MR 2916966, DOI 10.4171/rmi/68

Charles Fefferman and Bo’az Klartag, Fitting a $C^m$-smooth function to data. I, Ann. of Math. (2) 169 (2009), no. 1, 315–346. MR 2480607, DOI 10.4007/annals.2009.169.315

Charles Fefferman and Bo’az Klartag, Fitting a $C^m$-smooth function to data. II, Rev. Mat. Iberoam. 25 (2009), no. 1, 49–273. MR 2514338, DOI 10.4171/RMI/569

Charles Fefferman, Arie Israel, and Garving K. Luli, The structure of Sobolev extension operators, Rev. Mat. Iberoam. 30 (2014), no. 2, 419–429. MR 3231204, DOI 10.4171/RMI/787

Charles L. Fefferman, Arie Israel, and Garving K. Luli, Sobolev extension by linear operators, J. Amer. Math. Soc. 27 (2014), no. 1, 69–145. MR 3110796, DOI 10.1090/S0894-0347-2013-00763-8

Charles Fefferman, Arie Israel, and Garving K. Luli, Fitting a Sobolev function to data, arXiv:1411.1786, 2014.

Georges Glaeser, Étude de quelques algèbres tayloriennes, J. Analyse Math. 6 (1958), 1–124; erratum, insert to 6 (1958), no. 2 (French). MR 101294, DOI 10.1007/BF02790231

Arie Israel, A bounded linear extension operator for $L^{2,p}(\Bbb R^2)$, Ann. of Math. (2) 178 (2013), no. 1, 183–230. MR 3043580, DOI 10.4007/annals.2013.178.1.3

William B. Johnson and Joram Lindenstrauss, Extensions of Lipschitz mappings into a Hilbert space, Conference in modern analysis and probability (New Haven, Conn., 1982) Contemp. Math., vol. 26, Amer. Math. Soc., Providence, RI, 1984, pp. 189–206. MR 737400, DOI 10.1090/conm/026/737400

Mojźesz David Kirszbraun, Über die zusammenziehende und lipschitzsche transformationen, Fund. Math., 22:77–108, 1934.

U. Lang and V. Schroeder, Kirszbraun’s theorem and metric spaces of bounded curvature, Geom. Funct. Anal. 7 (1997), no. 3, 535–560. MR 1466337, DOI 10.1007/s000390050018

Urs Lang and Thilo Schlichenmaier, Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions, Int. Math. Res. Not. 58 (2005), 3625–3655. MR 2200122, DOI 10.1155/IMRN.2005.3625

Erwan Le Gruyer, Minimal Lipschitz extensions to differentiable functions defined on a Hilbert space, Geom. Funct. Anal. 19 (2009), no. 4, 1101–1118. MR 2570317, DOI 10.1007/s00039-009-0027-1

Henri Lebesgue, Sur le probléme de Dirichlet, Rend. Circ. Mat. Palermo, 24:371–402, 1907.

E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), no. 12, 837–842. MR 1562984, DOI 10.1090/S0002-9904-1934-05978-0

Assaf Naor, Yuval Peres, Oded Schramm, and Scott Sheffield, Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces, Duke Math. J. 134 (2006), no. 1, 165–197. MR 2239346, DOI 10.1215/S0012-7094-06-13415-4

P. Shvartsman, Sobolev $W^1_p$-spaces on closed subsets of $\textbf {R}^n$, Adv. Math. 220 (2009), no. 6, 1842–1922. MR 2493183, DOI 10.1016/j.aim.2008.09.020

Heinrich Tietze, Über funktionen, die auf einer abgeschlossenen menge stetig sind, J. Reine Angew. Math., 145:9–14, 1915.

Paul Urysohn, Über die Mächtigkeit der zusammenhängenden Mengen, Math. Ann. 94 (1925), no. 1, 262–295 (German). MR 1512258, DOI 10.1007/BF01208659

F. A. Valentine, On the extension of a vector function so as to preserve a Lipschitz condition, Bull. Amer. Math. Soc. 49 (1943), 100–108. MR 8251, DOI 10.1090/S0002-9904-1943-07859-7

F. A. Valentine, A Lipschitz condition preserving extension for a vector function, Amer. J. Math. 67 (1945), 83–93. MR 11702, DOI 10.2307/2371917

Hassler Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), no. 1, 63–89. MR 1501735, DOI 10.1090/S0002-9947-1934-1501735-3

Hassler Whitney, Differentiable functions defined in closed sets. I, Trans. Amer. Math. Soc. 36 (1934), no. 2, 369–387. MR 1501749, DOI 10.1090/S0002-9947-1934-1501749-3

• Keith Ball, Markov chains, Riesz transforms and Lipschitz maps, Geom. Funct. Anal. 2 (1992), no. 2, 137–172. MR 1159828 (93b:46025), DOI 10.1007/BF01896971
• Edward Bierstone, Pierre D. Milman, and Wiesław Pawłucki, Differentiable functions defined in closed sets. A problem of Whitney, Invent. Math. 151 (2003), no. 2, 329–352. MR 1953261 (2004h:58009), DOI 10.1007/s00222-002-0255-6
• L. E. J. Brouwer, Über die Erweiterung des Definitionsbereichs einer stetigen Funktion, Math. Ann. 79 (1918), no. 3, 209–211 (German). MR 1511921, DOI 10.1007/BF01458203
• Yuri A. Brudnyĭ and Pavel A. Shvartsman, A linear extension operator for a space of smooth functions defined on a closed subset in $\textbf {R}^n$, Dokl. Akad. Nauk SSSR 280 (1985), no. 2, 268–272 (Russian). MR 775048 (86f:46031)
• Yuri A. Brudnyi and Pavel A. Shvartsman, The traces of differentiable functions to closed subsets of $\textbf {R}^n$, Function spaces (Poznań, 1989) Teubner-Texte Math., vol. 120, Teubner, Stuttgart, 1991, pp. 206–210. MR 1155176
• Yuri Brudnyi and Pavel Shvartsman, Generalizations of Whitney’s extension theorem, Internat. Math. Res. Notices 3 (1994), 129 ff., approx. 11 pp. (electronic). MR 1266108 (95c:58018), DOI 10.1155/S1073792894000140
• Yuri Brudnyi and Pavel Shvartsman, The Whitney problem of existence of a linear extension operator, J. Geom. Anal. 7 (1997), no. 4, 515–574. MR 1669235 (2000a:46051), DOI 10.1007/BF02921632
• Yuri Brudnyi and Pavel Shvartsman, The trace of jet space $J^k\Lambda ^\omega$ to an arbitrary closed subset of $\mathbf {R}^n$, Trans. Amer. Math. Soc. 350 (1998), no. 4, 1519–1553. MR 1407483 (98i:58010), DOI 10.1090/S0002-9947-98-01872-8
• Yuri Brudnyi and Pavel Shvartsman, Whitney’s extension problem for multivariate $C^{1,\omega }$-functions, Trans. Amer. Math. Soc. 353 (2001), no. 6, 2487–2512 (electronic). MR 1814079 (2002b:46052), DOI 10.1090/S0002-9947-01-02756-8
• Yuri Brudnyi and Pavel Shvartsman, Stability of the Lipschitz extension property under metric transforms, Geom. Funct. Anal. 12 (2002), no. 1, 73–79. MR 1904557 (2003c:54027), DOI 10.1007/s00039-002-8237-9
• Charles L. Fefferman, A sharp form of Whitney’s extension theorem, Ann. of Math. (2) 161 (2005), no. 1, 509–577. MR 2150391 (2006h:58008), DOI 10.4007/annals.2005.161.509
• Charles Fefferman, Interpolation and extrapolation of smooth functions by linear operators, Rev. Mat. Iberoamericana 21 (2005), no. 1, 313–348. MR 2155023 (2006h:58009), DOI 10.4171/RMI/424
• Charles Fefferman, The structure of linear extension operators for $C^m$, Rev. Mat. Iberoam. 23 (2007), no. 1, 269–280. MR 2351135 (2009c:41003), DOI 10.4171/RMI/495
• Charles Fefferman, The $C^m$ norm of a function with prescribed jets. II, Rev. Mat. Iberoam. 25 (2009), no. 1, 275–421. MR 2514339 (2011d:46050), DOI 10.4171/RMI/570
• Charles Fefferman, Extension of $\mathsf {C}^{m,\omega }$-smooth functions by linear operators, Rev. Mat. Iberoam. 25 (2009), no. 1, 1–48. MR 2514337 (2010k:46029), DOI 10.4171/RMI/568
• Charles Fefferman, The $C^m$ norm of a function with prescribed jets I, Rev. Mat. Iberoam. 26 (2010), no. 3, 1075–1098. MR 2789377 (2012b:46050), DOI 10.4171/RMI/628
• Charles Fefferman, Nearly optimal interpolation of data in $\textrm {C}^2(\mathbb {R}^2)$. Part I, Rev. Mat. Iberoam. 28 (2012), no. 2, 415–533. MR 2916966
• Charles Fefferman and Bo’az Klartag, Fitting a $C^m$-smooth function to data. I, Ann. of Math. (2) 169 (2009), no. 1, 315–346. MR 2480607 (2011g:58011), DOI 10.4007/annals.2009.169.315
• Charles Fefferman and Bo’az Klartag, Fitting a $C^m$-smooth function to data. II, Rev. Mat. Iberoam. 25 (2009), no. 1, 49–273. MR 2514338 (2011f:58020), DOI 10.4171/RMI/569
• Charles Fefferman, Arie Israel, and Garving K. Luli, The structure of Sobolev extension operators, Rev. Mat. Iberoam. 30 (2014), no. 2, 419–429. MR 3231204, DOI 10.4171/RMI/787
• Charles L. Fefferman, Arie Israel, and Garving K. Luli, Sobolev extension by linear operators, J. Amer. Math. Soc. 27 (2014), no. 1, 69–145. MR 3110796, DOI 10.1090/S0894-0347-2013-00763-8
• Charles Fefferman, Arie Israel, and Garving K. Luli, Fitting a Sobolev function to data, arXiv:1411.1786, 2014.
• Georges Glaeser, Étude de quelques algèbres tayloriennes, J. Analyse Math. 6 (1958), 1–124; erratum, insert to 6 (1958), no. 2 (French). MR 0101294 (21 \#107)
• Arie Israel, A bounded linear extension operator for $L^{2,p}(\mathbb {R}^2)$, Ann. of Math. (2) 178 (2013), no. 1, 183–230. MR 3043580, DOI 10.4007/annals.2013.178.1.3
• William B. Johnson and Joram Lindenstrauss, Extensions of Lipschitz mappings into a Hilbert space, Conference in modern analysis and probability (New Haven, Conn., 1982), Contemp. Math., vol. 26, Amer. Math. Soc., Providence, RI, 1984, pp. 189–206. MR 737400 (86a:46018), DOI 10.1090/conm/026/737400
• Mojźesz David Kirszbraun, Über die zusammenziehende und lipschitzsche transformationen, Fund. Math., 22:77–108, 1934.
• Urs Lang and Viktor Schroeder, Kirszbraun’s theorem and metric spaces of bounded curvature, Geom. Funct. Anal. 7 (1997), no. 3, 535–560. MR 1466337 (98d:53062), DOI 10.1007/s000390050018
• Urs Lang and Thilo Schlichenmaier, Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions, Int. Math. Res. Not. 58 (2005), 3625–3655. MR 2200122 (2006m:53061), DOI 10.1155/IMRN.2005.3625
• Erwan Le Gruyer, Minimal Lipschitz extensions to differentiable functions defined on a Hilbert space, Geom. Funct. Anal. 19 (2009), no. 4, 1101–1118. MR 2570317 (2011c:46163), DOI 10.1007/s00039-009-0027-1
• Henri Lebesgue, Sur le probléme de Dirichlet, Rend. Circ. Mat. Palermo, 24:371–402, 1907.
• E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), no. 12, 837–842. MR 1562984, DOI 10.1090/S0002-9904-1934-05978-0
• Assaf Naor, Yuval Peres, Oded Schramm, and Scott Sheffield, Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces, Duke Math. J. 134 (2006), no. 1, 165–197. MR 2239346 (2007k:46017), DOI 10.1215/S0012-7094-06-13415-4
• Pavel Shvartsman, Sobolev $W^1_p$-spaces on closed subsets of $\textbf {R}^n$, Adv. Math. 220 (2009), no. 6, 1842–1922. MR 2493183 (2010b:46077), DOI 10.1016/j.aim.2008.09.020
• Heinrich Tietze, Über funktionen, die auf einer abgeschlossenen menge stetig sind, J. Reine Angew. Math., 145:9–14, 1915.
• Paul Urysohn, Über die Mächtigkeit der zusammenhängenden Mengen, Math. Ann. 94 (1925), no. 1, 262–295 (German). MR 1512258, DOI 10.1007/BF01208659
• Frederick A. Valentine, On the extension of a vector function so as to preserve a Lipschitz condition, Bull. Amer. Math. Soc. 49 (1943), 100–108. MR 0008251 (4,269d)
• Frederick A. Valentine, A Lipschitz condition preserving extension for a vector function, Amer. J. Math. 67 (1945), 83–93. MR 0011702 (6,203e)
• Hassler Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), no. 1, 63–89. MR 1501735, DOI 10.2307/1989708
• Hassler Whitney, Differentiable functions defined in closed sets. I, Trans. Amer. Math. Soc. 36 (1934), no. 2, 369–387. MR 1501749, DOI 10.2307/1989844