Geodesic
Sobolev $W^1_p$-spaces on $d$-thick closed subsets of $\mathbb R^n$
Sbornik. Mathematics, Tome 211 (2020) no. 6, pp. 786-837 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Let $S\subset\mathbb R^n$ be a nonempty closed set such that for some $d\in[0,n]$ and $\varepsilon>0$ the $d$-Hausdorff content $\mathscr H^d_\infty(S\cap Q(x,r))\geqslant\varepsilon r^d$ for all cubes $Q(x,r)$ with centre $x\in S$ and edge length $2r\in(0,2]$. For each $p>\max\{1,n-d\}$ we give an intrinsic characterization of the trace space $W_p^1(\mathbb R^n)|_S$ of the Sobolev space $W_p^1(\mathbb R^n)$ to the set $S$. Furthermore, we prove the existence of a bounded linear operator $\operatorname{Ext}\colon W_p^1(\mathbb R^n)|_S\to W_p^1(\mathbb R^n)$ such that $\operatorname{Ext}$ is the right inverse to the standard trace operator. Our results extend those available in the case $p\in(1,n]$ for Ahlfors-regular sets $S$. Bibliography: 36 titles.
Keywords: Whitney problem, extension operators.
Mots-clés : Sobolev spaces, traces 
@article{SM_2020_211_6_a1,
     author = {S. K. Vodopyanov and A. I. Tyulenev},
     title = {Sobolev $W^1_p$-spaces on~$d$-thick closed subsets of $\mathbb R^n$},
     journal = {Sbornik. Mathematics},
     pages = {786--837},
     year = {2020},
     volume = {211},
     number = {6},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2020_211_6_a1/}
}
TY  - JOUR
AU  - S. K. Vodopyanov
AU  - A. I. Tyulenev
TI  - Sobolev $W^1_p$-spaces on $d$-thick closed subsets of $\mathbb R^n$
JO  - Sbornik. Mathematics
PY  - 2020
SP  - 786
EP  - 837
VL  - 211
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/SM_2020_211_6_a1/
LA  - en
ID  - SM_2020_211_6_a1
ER  - 
%0 Journal Article
%A S. K. Vodopyanov
%A A. I. Tyulenev
%T Sobolev $W^1_p$-spaces on $d$-thick closed subsets of $\mathbb R^n$
%J Sbornik. Mathematics
%D 2020
%P 786-837
%V 211
%N 6
%U http://geodesic.mathdoc.fr/item/SM_2020_211_6_a1/
%G en
%F SM_2020_211_6_a1
S. K. Vodopyanov; A. I. Tyulenev. Sobolev $W^1_p$-spaces on $d$-thick closed subsets of $\mathbb R^n$. Sbornik. Mathematics, Tome 211 (2020) no. 6, pp. 786-837. http://geodesic.mathdoc.fr/item/SM_2020_211_6_a1/

[1] H. Whitney, “Analytic extensions of differentiable functions defined in closed sets”, Trans. Amer. Math. Soc., 36:1 (1934), 63–89 | DOI | MR | Zbl

[2] H. Whitney, “Differentiable functions defined in closed sets. I”, Trans. Amer. Math. Soc., 36:2 (1934), 369–387 | DOI | MR | Zbl

[3] G. Glaeser, “Étude de quelques algèbres tayloriennes”, J. Analyse Math., 6 (1958), 1–124 | DOI | MR | Zbl

[4] Yu. Brudnyi, P. Shvartsman, “The Whitney problem of existence of a linear extension operator”, J. Geom. Anal., 7:4 (1997), 515–574 | DOI | MR | Zbl

[5] E. Bierstone, P. D. Milman, W. Pawłucki, “Differentiable functions defined in closed sets. A problem of Whitney”, Invent. Math., 151:2 (2003), 329–352 | DOI | MR | Zbl

[6] C. Fefferman, “A sharp form of Whitney's extension theorem”, Ann. of Math. (2), 161:1 (2005), 509–577 | DOI | MR | Zbl

[7] C. Fefferman, “A generalized sharp Whitney theorem for jets”, Rev. Mat. Iberoam., 21:2 (2005), 577–688 | DOI | MR | Zbl

[8] C. Fefferman, “Whitney's extension problem for $C^m$”, Ann. of Math. (2), 164:1 (2006), 313–359 | DOI | MR | Zbl

[9] C. Fefferman, “$C^m$ extension by linear operators”, Ann. of Math. (2), 166:3 (2007), 779–835 | DOI | MR | Zbl

[10] V. G. Maz'ya, S. V. Poborchi, Differentiable functions on bad domains, World Sci. Publ., River Edge, NJ, 1997, xx+481 pp. | DOI | MR | Zbl

[11] C. L. Fefferman, A. Israel, G. K. Luli, “Sobolev extension by linear operators”, J. Amer. Math. Soc., 27:1 (2014), 69–145 | DOI | MR | Zbl

[12] C. Fefferman, A. Israel, G. K. Luli, “The structure of Sobolev extension operators”, Rev. Mat. Iberoam., 30:2 (2014), 419–429 | DOI | MR | Zbl

[13] C. Fefferman, A. Israel, G. K. Luli, “Fitting a Sobolev function to data I”, Rev. Mat. Iberoam., 32:1 (2016), 275–376 | DOI | MR | Zbl

[14] C. Fefferman, A. Israel, G. K. Luli, “Fitting a Sobolev function to data II”, Rev. Mat. Iberoam., 32:2 (2016), 649–750 | DOI | MR | Zbl

[15] A. Israel, “A bounded linear extension operator for $L^{2,p}(\mathbb{R}^{2})$”, Ann. of Math. (2), 178:1 (2013), 183–230 | DOI | MR | Zbl

[16] P. Shvartsman, “Sobolev $W^{1}_{p}$-spaces on closed subsets of $\mathbf{R}^{n}$”, Adv. Math., 220:6 (2009), 1842–1922 | DOI | MR | Zbl

[17] P. Shvartsman, “Sobolev $L^{2}_{p}$-functions on closed subsets of $\mathbf{R}^{2}$”, Adv. Math., 252 (2014), 22–113 | DOI | MR | Zbl

[18] P. Shvartsman, Extension criteria for homogeneous Sobolev space of functions of one variable, arXiv: 1812.00817v2

[19] P. Shvartsman, Sobolev functions on closed subsets of the real line: long version, arXiv: 1808.01467v2

[20] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Math. Ser., 30, Princeton Univ. Press, Princeton, N.J., 1970, xiv+290 pp. | MR | MR | Zbl | Zbl

[21] L. Ihnatsyeva, A. V. Vähäkangas, “Characterization of traces of smooth functions on Ahlfors regular sets”, J. Funct. Anal., 265:9 (2013), 1870–1915 | DOI | MR | Zbl

[22] A. Jonsson, H. Wallin, Function spaces on subsets of $\mathbb{R}^{n}$, Math. Rep., 2, no. 1, Harwood Acad. Publ., London, 1984, xiv+221 pp. | MR | Zbl

[23] P. Shvartsman, “Local approximations and intrinsic characterization of spaces of smooth functions on regular subsets of $\mathbb{R}^{n}$”, Math. Nachr., 279:11 (2006), 1212–1241 | DOI | MR | Zbl

[24] G. A. Kalyabin, “The intrinsic norming of the retractions of Sobolev spaces onto plain domains with the points of sharpness”, Abstracts of conference on functional spaces, approximation theory, nonlinear analysis in honor of S. M. Nikolskij, Moscow, 1995, 330

[25] V. S. Rychkov, “Linear extension operators for restrictions of function spaces to irregular open sets”, Studia Math., 140:2 (2000), 141–162 | DOI | MR | Zbl

[26] D. R. Adams, L. I. Hedberg, Function spaces and potential theory, Grundlehren Math. Wiss., 314, Springer-Verlag, Berlin, 1996, xii+366 pp. | DOI | MR | Zbl

[27] P. W. Jones, “Quasiconformal mappings and extendability of functions in Sobolev spaces”, Acta Math., 147:1-2 (1981), 71–88 | DOI | MR | Zbl

[28] A. P. Calderón, “Estimates for singular integral operators in terms of maximal functions”, Studia Math., 44:6 (1972), 563–582 | DOI | MR | Zbl

[29] H. Triebel, The structure of functions, Monogr. Math., 97, Birkhäuser Verlag, Basel, 2001, xii+425 pp. | DOI | MR | Zbl

[30] E. T. Sawyer, “A characterization of a two-weight norm inequality for maximal operators”, Studia Math., 75:1 (1982), 1–11 | DOI | MR | Zbl

[31] L. C. Evans, R. F. Gariepy, Measure theory and fine properties of functions, Stud. Adv. Math., CRC Press, Boca Raton, FL, 1992, viii+268 pp. | MR | Zbl

[32] C. Cascante, J. M. Ortega, I. E. Verbitsky, “On $L_{p}$-$L_{q}$ trace inequalities”, J. London Math. Soc. (2), 74:2 (2006), 497–511 | DOI | MR | Zbl

[33] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren Math. Wiss., 224, 2nd ed., Springer-Verlag, Berlin, 1983, xiii+513 pp. | DOI | MR | MR | Zbl | Zbl

[34] P. Hajłasz, P. Koskella, Sobolev met Poincaré, Mem. Amer. Math. Soc., 145, no. 688, Amer. Math. Soc., Providence, RI, 2000, x+101 pp. | DOI | MR | Zbl

[35] P. Hajłasz, “Sobolev spaces on an arbitrary metric space”, Potential Anal., 5:4 (1996), 403–415 | MR | Zbl

[36] S. K. Vodop'yanov, “Monotone functions and quasiconformal mappings on Carnot groups”, Siberian Math. J., 37:6 (1996), 1113–1136 | DOI | MR | Zbl