Traces of Sobolev spaces to irregular subsets of metric measure spaces

A. I. Tyulenev

Sbornik. Mathematics, Tome 214 (2023) no. 9, pp. 1241-1320

Résumé

Given $p \in (1,\infty)$, let $(\operatorname{X},\operatorname{d},\mu)$ be a metric measure space with uniformly locally doubling measure $\mu$ supporting a weak local $(1,p)$-Poincaré inequality. For each $\theta \in [0,p)$ we characterize the trace space of the Sobolev $W^{1}_{p}(\operatorname{X})$-space to lower $\theta$-codimensional content regular closed sets $S \subset \operatorname{X}$. In particular, if the space $(\operatorname{X},\operatorname{d},\mu)$ is Ahlfors $Q$-regular for some $Q \geq 1$ and $p \in (Q,\infty)$, then we obtain an intrinsic description of the trace-space of the Sobolev space $W^{1}_{p}(\operatorname{X})$ to arbitrary closed nonempty sets $S \subset \operatorname{X}$. Bibliography: 43 titles.

Détail

Keywords: extensions.
Mots-clés : Sobolev spaces, traces

Bibliographie

[1] J. Heinonen, P. Koskela, N. Shanmugalingam and J. T. Tyson, Sobolev spaces on metric measure spaces. An approach based on upper gradients, New Math. Monogr., 27, Cambridge Univ. Press, Cambridge, 2015, xii+434 pp. | DOI | MR | Zbl

[2] N. Gigli and E. Pasqualetto, Lectures on nonsmooth differential geometry, SISSA Springer Ser., 2, Springer, Cham, 2020, xi+204 pp. | DOI | MR | Zbl

[3] L. Ambrosio, N. Gigli and G. Savaré, “Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces”, Rev. Mat. Iberoam., 29:3 (2013), 969–996 | DOI | MR | Zbl

[4] L. Ambrosio, N. Gigli and G. Savaré, “Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below”, Invent. Math., 195:2 (2014), 289–391 | DOI | MR | Zbl

[5] N. J. Korevaar and R. M. Schoen, “Sobolev spaces and harmonic maps for metric space targets”, Comm. Anal. Geom., 1:3–4 (1993), 561–659 | DOI | MR | Zbl

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

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

[8] N. Shanmugalingam, “Newtonian spaces: an extension of Sobolev spaces to metric measure spaces”, Rev. Mat. Iberoam., 16:2 (2000), 243–279 | DOI | MR | Zbl

[9] N. Gigli and A. Tyulenev, “Korevaar-Schoen's energy on strongly rectifiable spaces”, Calc. Var. Partial Differential Equations, 60:6 (2021), 235, 54 pp. | DOI | MR | Zbl

[10] A. Björn and J. Björn, Nonlinear potential theory on metric spaces, EMS Tracts Math., 17, Eur. Math. Soc. (EMS), Zürich, 2011, xii+403 pp. | DOI | MR | Zbl

[11] R. Gibara, R. Korte and N. Shanmugalingam, Solving a Dirichlet problem for unbounded domains via a conformal transformation, arXiv: 2209.09773

[12] R. Gibara and N. Shanmugalingam, “Trace and extension theorems for homogeneous Sobolev and Besov spaces for unbounded uniform domains in metric measure spaces”, Tr. Mat. Inst. Steklova, 323 (to appear) ; arXiv: 2211.12708

[13] A. Jonsson and 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

[14] L. Malý, Trace and extension theorems for Sobolev-type functions in metric spaces, arXiv: 1704.06344

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

[16] 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

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

[18] 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 | Zbl

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

[20] P. Shvartsman, “Whitney-type extension theorems for jets generated by Sobolev functions”, Adv. Math., 313 (2017), 379–469 | DOI | MR | Zbl

[21] S. K. Vodopyanov and A. I. Tyulenev, “Sobolev $W^1_p$-spaces on $d$-thick closed subsets of $\mathbb R^n$”, Sb. Math., 211:6 (2020), 786–837 | DOI | MR | Zbl

[22] A. I. Tyulenev, “Almost sharp descriptions of traces of Sobolev spaces on compacta”, Math. Notes, 110:6 (2021), 976–980 | DOI | MR | Zbl

[23] A. I. Tyulenev, “Almost sharp descriptions of traces of Sobolev $W_{p}^{1}(\mathbb{R}^{n})$-spaces to arbitrary compact subsets of $\mathbb{R}^{n}$. The case $p \in (1,n]$”, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 26 (2025) (to appear)

[24] M. García-Bravo, T. Ikonen and Zheng Zhu, Extensions and approximations of Banach-valued Sobolev functions, arXiv: 2208.12594

[25] J. Azzam and R. Schul, “An analyst's traveling salesman theorem for sets of dimension larger than one”, Math. Ann., 370:3–4 (2018), 1389–1476 | DOI | MR | Zbl

[26] J. Azzam and M. Villa, “Quantitative comparisons of multiscale geometric properties”, Anal. PDE, 14:6 (2021), 1873–1904 | DOI | MR | Zbl

[27] A. I. Tyulenev, “Some porosity-type properties of sets related to the $d$-Hausdorff content”, Proc. Steklov Inst. Math., 319 (2022), 283–306 | DOI | MR | Zbl

[28] E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Math. Ser., 43, Princeton Univ. Press, Princeton, NJ, 1993, xiv+695 pp. | MR | Zbl

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

[30] Yu. A. Brudnyĭ, “Spaces defined by means of local approximations”, Trans. Moscow Math. Soc., 24(1971) (1974), 73–139 | MR | Zbl

[31] P. Shmerkin, Porosity, dimension, and local entropies: a survey, arXiv: 1110.5682

[32] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Math. Monogr., The Clarendon Press, Oxford Univ. Press, New York, 2000, xviii+434 pp. | MR | Zbl

[33] M. Christ, “A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral”, Colloq. Math., 60/61:2 (1990), 601–628 | DOI | MR | Zbl

[34] J. Martín and 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 | Zbl

[35] P. Mattila, Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability, Cambridge Stud. Adv. Math., 44, Cambridge Univ. Press, Cambridge, 1995, xii+343 pp. | DOI | MR | Zbl

[36] T. J. Laakso, “Ahlfors $Q$-regular spaces with arbitrary $Q > 1$ admitting weak Poincaré inequality”, Geom. Funct. Anal., 10:1 (2000), 111–123 | DOI | MR | Zbl

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

[38] L. Ambrosio, M. Colombo and S. Di Marino, “Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope”, Variational methods for evolving objects, Adv. Stud. Pure Math., 67, Math. Soc. Japan, Tokyo, 2015, 1–58 | DOI | MR | Zbl

[39] L. Ambrosio, “Calculus, heat flow and curvature-dimension bounds in metric measure spaces”, Proceedings of the international congress of mathematicians (Rio de Janeiro 2018), v. 1, World Sci. Publ., Hackensack, NJ, 2018, 301–340 | DOI | MR | Zbl

[40] S. Di Marino and G. Speight, “The $p$-weak gradient depends on $p$”, Proc. Amer. Math. Soc., 143:12 (2015), 5239–5252 | DOI | MR | Zbl

[41] X. Tolsa, “$BMO$, $H^{1}$ and Calderón-Zygmund operators for non doubling measures”, Math. Ann., 319:1 (2001), 89–149 | DOI | MR | Zbl

[42] 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 | Zbl

[43] C. Cascante, J. M. Ortega and I. E. Verbitsky, “Nonlinear potentials and two weight trace inequalities for general dyadic and radial kernels”, Indiana Univ. Math. J., 53:3 (2004), 845–882 | DOI | MR | Zbl