Traces of Sobolev Spaces on Piecewise Ahlfors--David Regular Sets

A. I. Tyulenev

Matematičeskie zametki, Tome 114 (2023) no. 3, pp. 404-434.

Résumé

Let $(\operatorname{X},\operatorname{d},\mu)$ be a metric measure space with uniformly locally doubling measure $\mu$. Given $p \in (1,\infty)$, assume that $(\operatorname{X},\operatorname{d},\mu)$ supports a weak local $(1,p)$-Poincaré inequality. We characterize trace spaces of the first-order Sobolev $W^{1}_{p}(\operatorname{X})$-spaces to subsets $S$ of $\operatorname{X}$ that can be represented as a finite union $\bigcup_{i=1}^{N}S^{i}$, $N \in \mathbb{N}$, of Ahlfors–David regular subsets $S^{i} \subset \operatorname{X}$, $i \in \{1,\dots,N\}$, of different codimensions. Furthermore, we explicitly compute the corresponding trace norms up to some universal constants.

Détail

Mots-clés : Sobolev spaces
Keywords: metric measure spaces, lower content regular sets, Frostman-type measures.

Bibliographie

[1] M. Garcia-Bravo, T. Ikonen, Z. Zhu, Extensions and Approximations of Banach-valued Sobolev functions, arXiv: 2208.12594

[2] E. Saksman, T. Soto, “Traces of Besov, Triebel–Lizorkin and Sobolev spaces on metric spaces”, Anal. Geom. Metr. Spaces, 5:1 (2017), 98–115 | DOI | MR

[3] P. Shvartsman, “On extensions of Sobolev functions defined on regular subsets of metric measure spaces”, J. Approx. Theory, 144:2 (2007), 139–161 | DOI | MR

[4] S. K. Vodopyanov, A. I. Tyulenev, “Prostranstva Soboleva $W^{1}_{p}$ na $d$-tolstykh zamknutykh podmnozhestvakh $\mathbb{R}^{n}$”, Matem. sb., 211:6 (2020), 40–94 | DOI | MR

[5] S. K. Vodopyanov, A. I. Tyulenev, “O probleme Uitni dlya vesovykh prostranstv Soboleva”, Dokl. AN, 472:6 (2017), 634–638 | DOI | MR

[6] A. I. Tyulenev, “O pochti tochnom opisanii sledov prostranstv Soboleva na kompaktakh”, Matem. zametki, 110:6 (2021), 948–953 | DOI | MR

[7] R. Gibara, R. Korte, N. Shanmugalingam, Solving a Dirichlet Problem on Unbounded Domains via a Conformal Transformation, arXiv: 2209.09773

[8] R. Gibara, N. Shanmugalingam, Trace and Extension Theorems for Homogeneous Sobolev and Besov Spaces for Unbounded Uniform Domains in Metric Measure Spaces, arXiv: 2211.12708

[9] L. Maly, Trace and Extension Theorems for Sobolev-type Functions in Metric Spaces, arXiv: abs/1704.06344

[10] A. I. Tyulenev, “Restrictions of Sobolev $W_{p}^{1}(\mathbb{R}^{2})$-spaces to planar rectifiable curves”, Ann. Fenn. Math., 47:1 (2022), 507–531 | DOI | MR

[11] A. I. Tyulenev, “Sledy prostranstv Soboleva na neregulyarnykh podmnozhestvakh metricheskikh prostranstv s meroi”, Matem. sb., 214:9 (2023), 58–143

[12] J. Heinonen, P. Koskela, N. Shanmugalingam, J. Tyson, Sobolev Spaces on Metric Measure Spaces. An Approach Based on Upper Gradients, New Math. Monographs, 27, Cambridge Univ. Press, Cambridge, 2015 | MR

[13] J. Martín, W. A. Ortiz, “A Sobolev type embedding theorem for Besov spaces defined on doubling metric spaces”, J. Math. Anal. Appl., 479:2 (2019), 2302–2337 | DOI | MR

[14] L. Maly, N. Shanmugalingam, M. Snipes, “Trace and extension theorems for functions of bounded variation”, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 18:1 (2018), 313–341 | MR

[15] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability, Cambridge Stud. Adv. Math., 44, Cambridge Univ. Press, Cambridge, 1995 | MR

[16] P. Shmerkin, Porosity, Dimension, and Local Entropies: a Survey, arXiv: 1110.5682

[17] E. Järvenpää, M. Järvenpää, A. Käenmäki, T. Rajala, S. Rogovin, V. Suomala, “Packing dimension and Ahlfors regularity of porous sets in metric spaces”, Math. Z., 266:1 (2010), 83–105 | DOI | MR

[18] J. Cheeger, “Differentiability of Lipschitz functions on metric measure spaces”, Geom. Funct. Anal., 9:3 (1999), 428–517 | DOI | MR

[19] A. Björn, J. Björn, Nonlinear Potential Theory on Metric Spaces, EMS Tracts in Mathematics, 17, European Math. Soc., Zürich, 2011 | MR

[20] R. Alvarado, F. Wang, D. Yang, W. Yuan, “Pointwise characterization of Besov and Triebel–Lizorkin spaces on spaces of homogeneous type”, Studia Math., 268:2 (2023), 121–166 | DOI | MR

[21] T. Bruno, M. M. Peloso, M. Vallarino, “Besov and Triebel–Lizorkin spaces on Lie groups”, Math. Ann., 377:1–2 (2020), 335–377 | DOI | MR

[22] A. Gogatishvili, P. Koskela, N. Shanmugalingam, “Interpolation properties of Besov spaces defined on metric spaces”, Math. Nachr., 283:2 (2010), 215–231 | DOI | MR