By Heinonen J.
We survey contemporary advances in research and geometry, the place first order differential research has been prolonged past its classical gentle settings. Such reviews have purposes to geometric pressure questions, yet also are of intrinsic curiosity. The transition from gentle areas to singular areas the place calculus is feasible parallels the classical improvement from delicate features to features with susceptible or generalized derivatives. in addition, there's a new approach of the classical geometric idea of Sobolev capabilities that's beneficial in additional normal contexts.
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Poincar´ e inequality and Gromov-Hausdorff convergence. We first require the concept of a measured Gromov-Hausdorff convergence. Let X = (X, d, µ, x) be a locally compact doubling p-Poincar´e space with a base point x ∈ X. 4) µ (E) := µ(B (x, 1))−1 µ(E) for E ⊂ X. Here and later B refers to a ball in the metric −1 d. 4). 6 and if, in addition, the following holds: for every sequence (yn ) of points such that yn ∈ X n and that yn → y ∈ X∞ , and for every r > 0, we have that µ n (B n (yn , r)) → µ∞ (B∞ (y, r)) as n → ∞, where B∞ refers to a ball in X∞ .
Surfaces of bounded curvature in the sense of Alexandrov also support (locally) a 1-Poincar´e inequality. This follows from the Bonk-Lang parametrization theorem cited earlier. Carnot-Carath´eodory spaces typically support a Poincar´e inequality. See [96], [184], [79, Section 11]. The validity of a Poincar´e inequality carries over to Gromov-Hausdorff limits of metric measure spaces, where a convergence of measures has to be incorporated in the definition. 6. 8 and the fact that every complete Riemannian manifold NONSMOOTH CALCULUS 53 with nonnegative Ricci curvature supports a 1-Poincar´e inequality [41].
1) holds for p-almost every curve γ in X. 6 that every Borel representative ρ of |∇u| is a p-weak upper gradient of u. In fact, it is necessary to take the quasicontinuous representative here, for every locally integrable function in Rn that possesses a p-integrable p-weak upper gradient is p-quasicontinuous. We will see this momentarily. Functions that have p-integrable p-weak upper gradients are the analogs of Sobolev functions on metric measure spaces. The precise definition, presented momentarily, requires a little care because of issues related to the pointwise definition.