Lectures on analysis in metric spaces

By Luigi Ambrosio, Francesco Serra Cassano. 2013

894 citations recorded by the source corpus.

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Citations in the corpus

Geometric Group Theory (2018, methodology): “[Hei01]) that for Ω = Ω′ = R, the two concepts are equivalent.”
Assouad Dimension and Fractal Geometry (2020, background): “Perhaps most notably, it plays an important role in quasi-conformal geometry, see [130, 192, 195], and embedding theory, see [222, 221, 240, 6, 7].”
Local Poincaré inequalities from stable curvature conditions on metric spaces (2011, methodology): “The local Poincaré inequality together with a doubling measure constitute the very foundation of analysis on metric spaces which was pioneered by Heinonen and Koskela [12] and Cheeger [4].”
On the Assouad dimension of self-similar sets with overlaps (2014, background): “With Hs denoting the s-dimensional Hausdorff measure, recall that a set F ⊆ Rd is called Ahlfors regular if there exist constants a, b > 0 such that a rs ≤ Hs ( Br(x) ∩ F ) ≤ b rs for s = dimH F , all x ∈ F and all 0 r < 1, see Heinonen [12, Chapter 8].”
Comparison of Metric Spectral Gaps (2013, background): “[44]) shows that c∞(X) = 1 for every separablemetric space X.”
A note on the dimensions of Assouad and Aikawa (2013, methodology): “[6], [7], [13], and [15].”
Approximation by finitely supported measures (2010, background): “Note that if μ is Ahlfors regular of dimension s, then s is the Hausdorff dimension of suppμ (and therefore s d), see [11], Section 8.”
John–Nirenberg lemmas for a doubling measure (2009, background): “1), the Vitali covering theorem and further information on metric spaces, see, for example, [10] or [4].”
Thomas Bayes’ walk on manifolds (2012, background): “[16]) if there exist positive c1, c2, d such that for all x ∈M, for all 0 < r ≤ 1, c1r ≤ |B(x, r)| ≤ c2r.”
An Overview of the Proof of the Splitting Theorem in Spaces with Non-Negative Ricci Curvature (2014, background): “x is a Lebesgue point for f (see for instance [31]), so that ∫ f dμx → f (x) for m̃-a.”
Existence of doubling measures via generalised nested cubes (2010, background): “A slightly simpler proof in the compact case can be found in Wu [18] (see also Heinonen [6]).”
Approximation properties and Schauder decompositions in Lipschitz-free spaces (2012, methodology): “For that purpose, we use a particular technique for interpolating Lipschitz functions on hypercubes of RN .”