Sobolev $W^1_p$-spaces on $d$-thick closed subsets of $\mathbb R^n$

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

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 $\mathscr H^d_\infty(S\cap Q(x,r))\geq\varepsilon r^d$ 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.

About the authors

Sergei Konstantinovich Vodopyanov

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Email: vodopis@math.nsc.ru
Doctor of physico-mathematical sciences, Professor

Alexander Ivanovich Tyulenev

Steklov Mathematical Institute of Russian Academy of Sciences

Email: tyulenev-math@yandex.ru
Candidate of physico-mathematical sciences, Associate professor

References

  1. H. Whitney, “Analytic extensions of differentiable functions defined in closed sets”, Trans. Amer. Math. Soc., 36:1 (1934), 63–89
  2. H. Whitney, “Differentiable functions defined in closed sets. I”, Trans. Amer. Math. Soc., 36:2 (1934), 369–387
  3. G. Glaeser, “Etude de quelques algèbres tayloriennes”, J. Analyse Math., 6 (1958), 1–124
  4. Yu. Brudnyi, P. Shvartsman, “The Whitney problem of existence of a linear extension operator”, J. Geom. Anal., 7:4 (1997), 515–574
  5. E. Bierstone, P. D. Milman, W. Pawlucki, “Differentiable functions defined in closed sets. A problem of Whitney”, Invent. Math., 151:2 (2003), 329–352
  6. C. Fefferman, “A sharp form of Whitney's extension theorem”, Ann. of Math. (2), 161:1 (2005), 509–577
  7. C. Fefferman, “A generalized sharp Whitney theorem for jets”, Rev. Mat. Iberoam., 21:2 (2005), 577–688
  8. C. Fefferman, “Whitney's extension problem for $C^m$”, Ann. of Math. (2), 164:1 (2006), 313–359
  9. C. Fefferman, “$C^m$ extension by linear operators”, Ann. of Math. (2), 166:3 (2007), 779–835
  10. V. G. Maz'ya, S. V. Poborchi, Differentiable functions on bad domains, World Sci. Publ., River Edge, NJ, 1997, xx+481 pp.
  11. C. L. Fefferman, A. Israel, G. K. Luli, “Sobolev extension by linear operators”, J. Amer. Math. Soc., 27:1 (2014), 69–145
  12. C. Fefferman, A. Israel, G. K. Luli, “The structure of Sobolev extension operators”, Rev. Mat. Iberoam., 30:2 (2014), 419–429
  13. C. Fefferman, A. Israel, G. K. Luli, “Fitting a Sobolev function to data I”, Rev. Mat. Iberoam., 32:1 (2016), 275–376
  14. C. Fefferman, A. Israel, G. K. Luli, “Fitting a Sobolev function to data II”, Rev. Mat. Iberoam., 32:2 (2016), 649–750
  15. A. Israel, “A bounded linear extension operator for $L^{2,p}(mathbb{R}^{2})$”, Ann. of Math. (2), 178:1 (2013), 183–230
  16. P. Shvartsman, “Sobolev $W^{1}_{p}$-spaces on closed subsets of $mathbf{R}^{n}$”, Adv. Math., 220:6 (2009), 1842–1922
  17. P. Shvartsman, “Sobolev $L^{2}_{p}$-functions on closed subsets of $mathbf{R}^{2}$”, Adv. Math., 252 (2014), 22–113
  18. P. Shvartsman, Extension criteria for homogeneous Sobolev space of functions of one variable
  19. P. Shvartsman, Sobolev functions on closed subsets of the real line: long version
  20. И. Стейн, Сингулярные интегралы и дифференциальные свойства функций, Мир, М., 1973, 342 с.
  21. L. Ihnatsyeva, A. V. Vähäkangas, “Characterization of traces of smooth functions on Ahlfors regular sets”, J. Funct. Anal., 265:9 (2013), 1870–1915
  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.
  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
  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
  26. D. R. Adams, L. I. Hedberg, Function spaces and potential theory, Grundlehren Math. Wiss., 314, Springer-Verlag, Berlin, 1996, xii+366 pp.
  27. P. W. Jones, “Quasiconformal mappings and extendability of functions in Sobolev spaces”, Acta Math., 147:1-2 (1981), 71–88
  28. A. P. Calderon, “Estimates for singular integral operators in terms of maximal functions”, Studia Math., 44:6 (1972), 563–582
  29. H. Triebel, The structure of functions, Monogr. Math., 97, Birkhäuser Verlag, Basel, 2001, xii+425 pp.
  30. E. T. Sawyer, “A characterization of a two-weight norm inequality for maximal operators”, Studia Math., 75:1 (1982), 1–11
  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.
  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
  33. Д. Гилбарг, Н. Трудингер, Эллиптические дифференциальные уравнения с частными производными второго порядка, Наука, М., 1989, 464 с.
  34. P. Hajlasz, P. Koskella, Sobolev met Poincare, Mem. Amer. Math. Soc., 145, no. 688, Amer. Math. Soc., Providence, RI, 2000, x+101 pp.
  35. P. Hajlasz, “Sobolev spaces on an arbitrary metric space”, Potential Anal., 5:4 (1996), 403–415
  36. С. К. Водопьянов, “Монотонные функции и квазиконформные отображения на группах Карно”, Сиб. матем. журн., 37:6 (1996), 1269–1295

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2020 Водопьянов С.K., Тюленев А.I.

Согласие на обработку персональных данных

 

Используя сайт https://journals.rcsi.science, я (далее – «Пользователь» или «Субъект персональных данных») даю согласие на обработку персональных данных на этом сайте (текст Согласия) и на обработку персональных данных с помощью сервиса «Яндекс.Метрика» (текст Согласия).