A bounded linear extension operator for $L^{2,p}(\mathbb{R}^2)$

Arie Israel

doi:https://doi.org/10.4007/annals.2013.178.1.3

Abstract

Let $L^{2,p}(\mathbb{R}^2)$ be the Sobolev space of real-valued functions on the plane whose Hessian belongs to $L^p$. For any finite subset $E \subset \Bbb{R}^2$ and $p>2$, let $L^{2,p}(\Bbb{R}^2)|_E$ be the space of real-valued functions on $E$, equipped with the trace seminorm. In this paper we construct a bounded linear extension operator $T : L^{2,p}(\mathbb{R}^2)|_E \rightarrow L^{2,p}(\mathbb{R}^2)$. We also provide an explicit formula that approximates the $L^{2,p}(\mathbb{R}^2)|_E$ trace seminorm.

Show/hide bibliography for this article

[BS3] Y. A. Brudnyui and P. A. Shvartsman, "A linear extension operator for a space of smooth functions defined on a closed subset in ${\bf R}^n$," Dokl. Akad. Nauk SSSR, vol. 280, iss. 2, pp. 268-272, 1985.

Show bibtex

@article {BS3, MRKEY = {0775048}, AUTHOR = {Brudny{\u\i}, Yu. A. and Shvartsman, P. A.}, TITLE = {A linear extension operator for a space of smooth functions defined on a closed subset in {${\bf R}\sp n$}}, JOURNAL = {Dokl. Akad. Nauk SSSR}, FJOURNAL = {Doklady Akademii Nauk SSSR}, VOLUME = {280}, YEAR = {1985}, NUMBER = {2}, PAGES = {268--272}, ISSN = {0002-3264}, MRCLASS = {46E35 (26A16)}, MRNUMBER = {0775048}, MRREVIEWER = {J. Musielak}, ZBLNUMBER = {0597.46027}, NOTE={English translation in \emph{Soviet Math. Dokl.} {\bf 31} (1985), 48--51}, }
[BS1] $Go to document$ Y. Brudnyi and P. Shvartsman, "Generalizations of Whitney’s extension theorem," Internat. Math. Res. Notices, iss. 3, p. 129, 1994.

Show bibtex

@article {BS1, MRKEY = {1266108}, AUTHOR = {Brudnyi, Yuri and Shvartsman, Pavel}, TITLE = {Generalizations of {W}hitney's extension theorem}, JOURNAL = {Internat. Math. Res. Notices}, FJOURNAL = {International Mathematics Research Notices}, YEAR = {1994}, NUMBER = {3}, PAGES = {129 ff., approx. 11 pp.}, ISSN = {1073-7928}, MRCLASS = {58C25 (41A10 46E35)}, MRNUMBER = {1266108}, MRREVIEWER = {H. Wallin}, DOI = {10.1155/S1073792894000140}, ZBLNUMBER = {0845.57022}, }
[BS4] $Go to document$ Y. Brudnyi and P. Shvartsman, "The Whitney problem of existence of a linear extension operator," J. Geom. Anal., vol. 7, iss. 4, pp. 515-574, 1997.

Show bibtex

@article {BS4, MRKEY = {1669235}, AUTHOR = {Brudnyi, Yuri and Shvartsman, Pavel}, TITLE = {The {W}hitney problem of existence of a linear extension operator}, JOURNAL = {J. Geom. Anal.}, FJOURNAL = {The Journal of Geometric Analysis}, VOLUME = {7}, YEAR = {1997}, NUMBER = {4}, PAGES = {515--574}, ISSN = {1050-6926}, MRCLASS = {46E35 (58C25)}, MRNUMBER = {1669235}, MRREVIEWER = {P. Szeptycki}, DOI = {10.1007/BF02921632}, ZBLNUMBER = {0937.58007}, }
[BS2] $Go to document$ Y. Brudnyi and P. Shvartsman, "Whitney’s extension problem for multivariate $C^{1,\omega}$-functions," Trans. Amer. Math. Soc., vol. 353, iss. 6, pp. 2487-2512, 2001.

Show bibtex

@article {BS2, MRKEY = {1814079}, AUTHOR = {Brudnyi, Yuri and Shvartsman, Pavel}, TITLE = {Whitney's extension problem for multivariate {$C\sp {1,\omega}$}-functions}, JOURNAL = {Trans. Amer. Math. Soc.}, FJOURNAL = {Transactions of the American Mathematical Society}, VOLUME = {353}, YEAR = {2001}, NUMBER = {6}, PAGES = {2487--2512}, ISSN = {0002-9947}, CODEN = {TAMTAM}, MRCLASS = {46E35 (58C25)}, MRNUMBER = {1814079}, MRREVIEWER = {Lubo{š} Pick}, DOI = {10.1090/S0002-9947-01-02756-8}, ZBLNUMBER = {0973.46025}, }
[Evans] L. C. Evans, Partial Differential Equations, Providence, RI: Amer. Math. Soc., 1998, vol. 19.

Show bibtex

@book {Evans, MRKEY = {1625845}, AUTHOR = {Evans, Lawrence C.}, TITLE = {Partial Differential Equations}, SERIES = {Grad. Studies in Math.}, VOLUME = {19}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, RI}, YEAR = {1998}, PAGES = {xviii+662}, ISBN = {0-8218-0772-2}, MRCLASS = {35-01}, MRNUMBER = {1625845}, MRREVIEWER = {Luigi Rodino}, ZBLNUMBER = {0902.35002}, }
[F1] $Go to document$ C. Fefferman, "A sharp form of Whitney’s extension theorem," Ann. of Math., vol. 161, iss. 1, pp. 509-577, 2005.

Show bibtex

@article {F1, MRKEY = {2150391}, AUTHOR = {Fefferman, Charles}, TITLE = {A sharp form of {W}hitney's extension theorem}, JOURNAL = {Ann. of Math.}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {161}, YEAR = {2005}, NUMBER = {1}, PAGES = {509--577}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {58C25 (26B35)}, MRNUMBER = {2150391}, MRREVIEWER = {M. Laczkovich}, DOI = {10.4007/annals.2005.161.509}, ZBLNUMBER = {1102.58005}, }
[F5] $Go to document$ C. Fefferman, "Interpolation and extrapolation of smooth functions by linear operators," Rev. Mat. Iberoamericana, vol. 21, iss. 1, pp. 313-348, 2005.

Show bibtex

@article {F5, MRKEY = {2155023}, AUTHOR = {Fefferman, Charles}, TITLE = {Interpolation and extrapolation of smooth functions by linear operators}, JOURNAL = {Rev. Mat. Iberoamericana}, FJOURNAL = {Revista Matemática Iberoamericana}, VOLUME = {21}, YEAR = {2005}, NUMBER = {1}, PAGES = {313--348}, ISSN = {0213-2230}, MRCLASS = {58C25 (30E05 41A65 52A35)}, MRNUMBER = {2155023}, MRREVIEWER = {Yu. A. Brudny{\u\i}}, DOI = {10.4171/RMI/424}, ZBLNUMBER = {1084.58003}, }
[F3] $Go to document$ C. Fefferman, "Whitney’s extension problem for $C^m$," Ann. of Math., vol. 164, iss. 1, pp. 313-359, 2006.

Show bibtex

@article {F3, MRKEY = {2233850}, AUTHOR = {Fefferman, Charles}, TITLE = {Whitney's extension problem for {$C\sp m$}}, JOURNAL = {Ann. of Math.}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {164}, YEAR = {2006}, NUMBER = {1}, PAGES = {313--359}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {58C25 (26B35 46E35)}, MRNUMBER = {2233850}, MRREVIEWER = {Pavel A. Shvartsman}, DOI = {10.4007/annals.2006.164.313}, ZBLNUMBER = {1109.58016}, }
[F4] $Go to document$ C. Fefferman, "$C^m$ extension by linear operators," Ann. of Math., vol. 166, iss. 3, pp. 779-835, 2007.

Show bibtex

@article {F4, MRKEY = {2373373}, AUTHOR = {Fefferman, Charles}, TITLE = {{$C\sp m$} extension by linear operators}, JOURNAL = {Ann. of Math.}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {166}, YEAR = {2007}, NUMBER = {3}, PAGES = {779--835}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {46E35 (26B35 58C25)}, MRNUMBER = {2373373}, MRREVIEWER = {Pavel A. Shvartsman}, DOI = {10.4007/annals.2007.166.779}, ZBLNUMBER = {1161.46013}, }
[F2] $Go to document$ C. Fefferman, "Extension of ${C}^{m,\omega}$-smooth functions by linear operators," Rev. Mat. Iberoam., vol. 25, iss. 1, pp. 1-48, 2009.

Show bibtex

@article {F2, MRKEY = {2514337}, AUTHOR = {Fefferman, Charles}, TITLE = {Extension of {${C}\sp {m,\omega}$}-smooth functions by linear operators}, JOURNAL = {Rev. Mat. Iberoam.}, FJOURNAL = {Revista Mathemática Iberoamericana}, VOLUME = {25}, YEAR = {2009}, NUMBER = {1}, PAGES = {1--48}, ISSN = {0213-2230}, MRCLASS = {46E15 (26E10 41A05)}, MRNUMBER = {2514337}, MRREVIEWER = {M. Laczkovich}, DOI = {10.4171/RMI/568}, ZBLNUMBER = {1173.46014}, }
[FK1] $Go to document$ C. Fefferman and B. Klartag, "Fitting a $C^m$-smooth function to data. I," Ann. of Math., vol. 169, iss. 1, pp. 315-346, 2009.

Show bibtex

@article {FK1, MRKEY = {2480607}, AUTHOR = {Fefferman, Charles and Klartag, Bo'az}, TITLE = {Fitting a {$C\sp m$}-smooth function to data. {I}}, JOURNAL = {Ann. of Math.}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {169}, YEAR = {2009}, NUMBER = {1}, PAGES = {315--346}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {58C25 (26B05 65D10)}, MRNUMBER = {2480607}, MRREVIEWER = {M. Laczkovich}, DOI = {10.4007/annals.2009.169.315}, ZBLNUMBER = {1175.41001}, }
[FIL1] C. Fefferman, A. Israel, and G. K. Luli, The structure of Sobolev extension operators, 2012.

Show bibtex

@misc{FIL1, author={Fefferman, Charles and Israel, A. and Luli, G. K.}, TITLE={The structure of {S}obolev extension operators}, NOTE={to appear in \emph{Revista Matematica Iberoamericana}}, YEAR={2012}, }
[FIL2] $Go to document$ C. Fefferman, A. Israel, and G. K. Luli, Sobolev extension by linear operators.

Show bibtex

@misc{FIL2, SORTYEAR={2011}, AUTHOR={Fefferman, Charles and Israel, A. and Luli, G. K.}, TITLE={Sobolev extension by linear operators}, NOTE={posted on February 28, 2013 (to appear in print)}, DOI = {10.1090/S0894-0347-2013-00763-8}, }
[L1] $Go to document$ G. K. Luli, Whitney Extension for ${W}^{k,p}({E})$ in One Dimension.

Show bibtex

@misc{L1, author={Luli, Garving K.}, TITLE={Whitney Extension for ${W}^{k,p}({E})$ in One Dimension}, SORTYEAR={2013}, URL={http://users.math.yale.edu/~gkl6/papers/notes.pdf}, }
[L2] $Go to document$ G. K. Luli, "$C^{m,\omega}$ extension by bounded-depth linear operators," Adv. Math., vol. 224, iss. 5, pp. 1927-2021, 2010.

Show bibtex

@article {L2, MRKEY = {2646115}, AUTHOR = {Luli, Garving K.}, TITLE = {{$C\sp {m,\omega}$} extension by bounded-depth linear operators}, JOURNAL = {Adv. Math.}, FJOURNAL = {Advances in Mathematics}, VOLUME = {224}, YEAR = {2010}, NUMBER = {5}, PAGES = {1927--2021}, ISSN = {0001-8708}, CODEN = {ADMTA4}, MRCLASS = {58C25}, MRNUMBER = {2646115}, MRREVIEWER = {M. Laczkovich}, DOI = {10.1016/j.aim.2010.01.027}, ZBLNUMBER = {1195.47005}, }
[S2] P. A. Shvartsman, The traces of functions of two variables satisfying to the Zygmund condition, 1982.

Show bibtex

@misc{S2, author = {Shvartsman, P. A.}, TITLE = {The traces of functions of two variables satisfying to the {Z}ygmund condition}, NOTE={Studies in the Theory of Functions of Several Real Variables, Yaroslav. State Univ., Yaroslavl, pp. 145--168}, YEAR={1982}, }
[S3] P. A. Shvartsman, "Lipschitz selections of multivalued mappings and the traces of the Zygmund class functions to an arbitrary compact," Dokl. Akad. Nauk SSSR, vol. 276, iss. 3, pp. 559-562, 1984.

Show bibtex

@article {S3, author = {Shvartsman, P. A.}, TITLE = {Lipschitz selections of multivalued mappings and the traces of the {Z}ygmund class functions to an arbitrary compact}, JOURNAL = {Dokl. Akad. Nauk SSSR}, FJOURNAL = {Doklady Akademii Nauk SSSR}, NOTE = {In Russian; translated in {\it Soviet. Math. Dokl.} {\bf 29} (1984), 565--568}, VOLUME = {276}, YEAR = {1984}, NUMBER = {3}, PAGES = {559--562}, ISSN = {0002-3264}, MRCLASS = {46E35 (42B30)}, MRREVIEWER = {J. Musielak}, ZBLNUMBER = {0598.46026}, }
[S4] P. A. Shvartsman, "Traces of functions of Zygmund class," Sibirsk. Mat. Zh., vol. 28, iss. 5, pp. 203-215, 1987.

Show bibtex

@article {S4, MRKEY = {0924998}, AUTHOR = {Shvartsman, P. A.}, TITLE = {Traces of functions of {Z}ygmund class}, JOURNAL = {Sibirsk. Mat. Zh.}, FJOURNAL = {Akademiya Nauk SSSR. Sibirskoe Otdelenie. SibirskiĭMatematicheskiĭZhurnal}, VOLUME = {28}, YEAR = {1987}, NUMBER = {5}, PAGES = {203--215}, ISSN = {0037-4474}, MRCLASS = {46E99}, MRNUMBER = {0924998}, MRREVIEWER = {Boris I. Botvinnik}, NOTE = {In Russian; translated in \emph{Siberian Math. J.} {\bf 28} (1987), 853--863}, ZBLNUMBER = {0648.46030}, }
[S5] $Go to document$ P. Shvartsman, "Sobolev $W^1_p$-spaces on closed subsets of ${\bf R}^n$," Adv. Math., vol. 220, iss. 6, pp. 1842-1922, 2009.

Show bibtex

@article {S5, MRKEY = {2493183}, AUTHOR = {Shvartsman, P. }, TITLE = {Sobolev {$W\sp 1\sb p$}-spaces on closed subsets of {${\bf R}\sp n$}}, JOURNAL = {Adv. Math.}, FJOURNAL = {Advances in Mathematics}, VOLUME = {220}, YEAR = {2009}, NUMBER = {6}, PAGES = {1842--1922}, ISSN = {0001-8708}, CODEN = {ADMTA4}, MRCLASS = {46E35}, MRNUMBER = {2493183}, MRREVIEWER = {Olli Martio}, DOI = {10.1016/j.aim.2008.09.020}, ZBLNUMBER = {1176.46040}, }
[S6] P. Shvartsman, Lipschitz spaces generated by the Sobolev-Poincaré inequality and extensions of Sobolev functions, 2011.

Show bibtex

@misc{S6, author={Shvartsman, P.}, TITLE = {Lipschitz spaces generated by the {S}obolev-{P}oincaré inequality and extensions of {S}obolev functions}, YEAR={2011}, NOTE={preprint}, }
[S7] P. Shvartsman, Sobolev ${L}^2_p$-functions on closed subsets of $\mathbb{R}^2$, 2012.

Show bibtex

@misc{S7, author={Shvartsman, P.}, TITLE = {Sobolev ${L}^2_p$-functions on closed subsets of {$\mathbb{R}^2$}}, YEAR={2012}, ARXIV={1210.0590}, }
[St1] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton, N.J.: Princeton Univ. Press, 1970, vol. 30.

Show bibtex

@book {St1, MRKEY = {0290095}, AUTHOR = {Stein, Elias M.}, TITLE = {Singular Integrals and Differentiability Properties of Functions}, SERIES = {Princeton Math. Series}, VOLUME={30}, PUBLISHER = {Princeton Univ. Press}, ADDRESS = {Princeton, N.J.}, YEAR = {1970}, PAGES = {xiv+290}, MRCLASS = {46.38 (26.00)}, MRNUMBER = {0290095}, MRREVIEWER = {R. E. Edwards}, ZBLNUMBER = {0207.13501}, }
[St2] E. M. Stein, "The characterization of functions arising as potentials. II," Bull. Amer. Math. Soc., vol. 68, pp. 577-582, 1962.

Show bibtex

@article {St2, MRKEY = {0142980}, AUTHOR = {Stein, E. M.}, TITLE = {The characterization of functions arising as potentials. {II}}, JOURNAL = {Bull. Amer. Math. Soc.}, FJOURNAL = {Bulletin of the American Mathematical Society}, VOLUME = {68}, YEAR = {1962}, PAGES = {577--582}, ISSN = {0002-9904}, MRCLASS = {31.11}, MRNUMBER = {0142980}, MRREVIEWER = {I. I. Hirschman, Jr.}, ZBLNUMBER = {0127.32002}, }
[T1] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Second ed., Heidelberg: Johann Ambrosius Barth, 1995.

Show bibtex

@book {T1, MRKEY = {1328645}, AUTHOR = {Triebel, Hans}, TITLE = {Interpolation Theory, Function Spaces, Differential Operators}, EDITION = {Second}, PUBLISHER = {Johann Ambrosius Barth}, ADDRESS = {Heidelberg}, YEAR = {1995}, PAGES = {532}, ISBN = {3-335-00420-5}, MRCLASS = {46-02 (35Jxx 46E10 46E35 46M35)}, MRNUMBER = {1328645}, ZBLNUMBER = {0830.46028}, }
[T2] H. Triebel, Theory of Function Spaces. III, Basel: Birkhäuser, 2006.

Show bibtex

@book {T2, MRKEY = {2250142}, AUTHOR = {Triebel, Hans}, TITLE = {Theory of Function Spaces. {III}}, SERIES = {Monogr. Math.}, NUMBER = {100}, PUBLISHER = {Birkhäuser}, ADDRESS = {Basel}, YEAR = {2006}, PAGES = {xii+426}, ISBN = {978-3-7643-7581-2; 3-7643-7581-7}, MRCLASS = {46E35 (46-02)}, MRNUMBER = {2250142}, MRREVIEWER = {Lubo{š} Pick}, ZBLNUMBER = {1104.46001}, }
[W1] $Go to document$ H. Whitney, "Differentiable functions defined in closed sets. I," Trans. Amer. Math. Soc., vol. 36, iss. 2, pp. 369-387, 1934.

Show bibtex

@article {W1, MRKEY = {1501749}, AUTHOR = {Whitney, Hassler}, TITLE = {Differentiable functions defined in closed sets. {I}}, JOURNAL = {Trans. Amer. Math. Soc.}, FJOURNAL = {Transactions of the American Mathematical Society}, VOLUME = {36}, YEAR = {1934}, NUMBER = {2}, PAGES = {369--387}, ISSN = {0002-9947}, CODEN = {TAMTAM}, MRCLASS = {58C25}, MRNUMBER = {1501749}, DOI = {10.2307/1989844}, ZBLNUMBER = {0009.20803}, }
[W2] $Go to document$ H. Whitney, "Analytic extensions of differentiable functions defined in closed sets," Trans. Amer. Math. Soc., vol. 36, iss. 1, pp. 63-89, 1934.

Show bibtex

@article {W2, MRKEY = {1501735}, AUTHOR = {Whitney, Hassler}, TITLE = {Analytic extensions of differentiable functions defined in closed sets}, JOURNAL = {Trans. Amer. Math. Soc.}, FJOURNAL = {Transactions of the American Mathematical Society}, VOLUME = {36}, YEAR = {1934}, NUMBER = {1}, PAGES = {63--89}, ISSN = {0002-9947}, CODEN = {TAMTAM}, MRCLASS = {26B05}, MRNUMBER = {1501735}, DOI = {10.2307/1989708}, ZBLNUMBER = {0008.24902}, }

A bounded linear extension operator for $L^{2,p}(\mathbb{R}^2)$

Abstract

Authors