•Journal•ISSN: 2299-3274
Analysis and Geometry in Metric Spaces
De Gruyter Open
About: Analysis and Geometry in Metric Spaces is an academic journal published by De Gruyter Open. The journal publishes majorly in the area(s): Metric space & Metric (mathematics). It has an ISSN identifier of 2299-3274. It is also open access. Over the lifetime, 135 publications have been published receiving 1365 citations.
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TL;DR: In this article, the authors present the basic theory of Carnot groups and discuss the regularity of isometries in the general case of Carathéodory spaces and of nilpotent metric Lie groups.
Abstract: Carnot groups are distinguished spaces that are rich of structure: they are those Lie groups equipped with a path distance that is invariant by left-translations of the group and admit automorphisms that are dilations with respect to the distance. We present the basic theory of Carnot groups together with several remarks.We consider them as special cases of graded groups and as homogeneousmetric spaces.We discuss the regularity of isometries in the general case of Carnot-Carathéodory spaces and of nilpotent metric Lie groups.
102 citations
TL;DR: Gigi et al. as discussed by the authors presented an overview of the Proof of the Splitting Theorem in Spaces with Non-Negative Ricci Curvature and analyzed the geometry in metric spaces.
Abstract: 3.0 License. Anal. Geom. Metr. Spaces 2014; 2:169–213 Analysis and Geometry in Metric Spaces Open Access Survey Paper Nicola Gigli* An Overview of the Proof of the Splitting Theorem in Spaces with Non-Negative Ricci Curvature DOI 10.2478/agms-2014-0006 Received May 21, 2013; accepted April 18, 2014
74 citations
TL;DR: For an equiregular sub-Riemannian manifold M, Popp's volume is a smooth volume which is canonically associated with the Riemannians structure as discussed by the authors.
Abstract: For an equiregular sub-Riemannian manifold M, Popp's volume is a smooth volume which is canonically associated with the sub-Riemannian structure, and it is a natural generalization of the Riemannian one. In this paper we prove a general formula for Popp's volume, written in terms of a frame adapted to the sub-Riemannian distribution. As a first application of this result, we prove an explicit formula for the canonical sub-Laplacian, namely the one associated with Popp's volume. Finally, we discuss sub-Riemannian isometries, and we prove that they preserve Popp's volume. We also show that, under some hypotheses on the action of the isometry group of M, Popp's volume is essentially the unique volume with such a property.
70 citations
TL;DR: In this paper, the authors discuss asymmetric length structures and asymmetric metric spaces and discuss newly found aspects of the theory: they identify three interesting classes of paths, and compare them; they note that a geodesic segment (as defined by Busemann) is not necessarily continuous in their setting; hence they present three different notions of intrinsic metric space.
Abstract: In this paper we discuss asymmetric length structures and asymmetric metric spaces. A length structure induces a (semi)distance function; by using the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempotent. As a typical application, we consider the length of paths defined by a Finslerian functional in Calculus of Variations. In the second part we generalize the setting of General metric spaces of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a geodesic segment (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of intrinsic metric space.
55 citations
TL;DR: In this paper, the authors introduced a Musielak-Orlicz-Hardy space Hφ, L(X), via the Lusin area function associated with L, and established its molecular characterization.
Abstract: Let X be a metric space with doubling measure and L a oneto-one operator of type ω having a bounded H∞-functional calculus in L2(X) satisfying the reinforced (pL, qL) off-diagonal estimates on balls, where pL ∈ [1, 2) and qL ∈ (2,∞]. Let φ : X × [0,∞) → [0,∞) be a function such that φ(x, ·) is an Orlicz function, φ(·, t) ∈ A∞(X) (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index I(φ) ∈ (0, 1] and φ(·, t) satisfies the uniformly reverse Holder inequality of order (qL/I(φ))′, where (qL/I(φ))′ denotes the conjugate exponent of qL/I(φ). In this paper, the authors introduce a Musielak-Orlicz-Hardy space Hφ, L(X), via the Lusin-area function associated with L, and establish its molecular characterization. In particular, when L is nonnegative self-adjoint and satisfies the Davies-Gaffney estimates, the atomic characterization of Hφ, L(X) is also obtained. Furthermore, a sufficient condition for the equivalence between Hφ, L(Rn) and the classical Musielak-Orlicz-Hardy spaceHφ(Rn) is given. Moreover, for the Musielak-Orlicz-Hardy spaceHφ, L(Rn) associated with the second order elliptic operator in divergence form on Rn or the Schrodinger operator L := −∆ + V with 0 ≤ V ∈ L1 loc(Rn), the authors further obtain its several equivalent characterizations in terms of various non-tangential and radial maximal functions; finally, the authors show that the Riesz transform∇L−1/2 is bounded from Hφ, L(Rn) to the Musielak-Orlicz space Lφ(Rn) when i(φ) ∈ (0, 1], from Hφ, L(Rn) to Hφ(Rn) when i(φ) ∈ ( n n+1 , 1], and fromHφ, L(Rn) to the weak MusielakOrlicz-Hardy space WHφ(Rn) when i(φ) = n n+1 is attainable and φ(·, t) ∈ A1(X), where i(φ) denotes the uniformly critical lower type index of φ.
50 citations