Sobolev and bounded variation functions on metric measure spaces.

*(English)*Zbl 1362.53002
Barilari, Davide (ed.) et al., Geometry, analysis and dynamics on sub-Riemannian manifolds. Volume II. Lecture notes from the IHP Trimester held at the Institut Henri Poincaré, Paris, France, and from the CIRM summer school “Sub-Riemannian manifolds: from geodesics to hypoelliptic diffusion”, Luminy, France, Fall 2014. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-163-7/pbk; 978-3-03719-663-2/ebook). EMS Series of Lectures in Mathematics, 211-273 (2016).

These notes are devoted to analysis on metric spaces, especially to Sobolev theory. They arose from lectures of the first author on work of the first author et al. [Boll. Unione Mat. Ital. (9) 5, No. 3, 575–629 (2012; Zbl 1288.58016); Rev. Mat. Iberoam. 29, No. 3, 969–996 (2013; Zbl 1287.46027); Invent. Math. 195, No. 2, 289–391 (2014; Zbl 1312.53056); Duke Math. J. 163, No. 7, 1405–1490 (2014; Zbl 1304.35310); Ann. Probab. 43, No. 1, 339–404 (2015; Zbl 1307.49044)], as well as the first author et al. [Adv. Stud. Pure Math. 67, 1–58 (2015; Zbl 1370.46018)].

On Euclidean space one has (at least) three equivalent definitions of Sobolev spaces \(W^{1,p}({\mathbb R}^n)\): the W-definition via integration by parts, the H-definition as the closure of \({\mathcal C}_c^\infty\) with respect to the Sobolev norm, and the BL-definition based on absolute continuity along almost every line parallel to every coordinate axis. The goal of the lectures is to extend these definitions to metric measure spaces and to prove their equivalence in this general context.

For the entire collection see [Zbl 1351.53003].

On Euclidean space one has (at least) three equivalent definitions of Sobolev spaces \(W^{1,p}({\mathbb R}^n)\): the W-definition via integration by parts, the H-definition as the closure of \({\mathcal C}_c^\infty\) with respect to the Sobolev norm, and the BL-definition based on absolute continuity along almost every line parallel to every coordinate axis. The goal of the lectures is to extend these definitions to metric measure spaces and to prove their equivalence in this general context.

For the entire collection see [Zbl 1351.53003].

Reviewer: Thilo Kuessner (Seoul)

##### MSC:

53-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry |

53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |

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\textit{L. Ambrosio} and \textit{R. Ghezzi}, in: Geometry, analysis and dynamics on sub-Riemannian manifolds. Volume II. Lecture notes from the IHP Trimester held at the Institut Henri Poincaré, Paris, France, and from the CIRM summer school ``Sub-Riemannian manifolds: from geodesics to hypoelliptic diffusion'', Luminy, France, Fall 2014. Zürich: European Mathematical Society (EMS). 211--273 (2016; Zbl 1362.53002)