scholarly journals Optimal maps in essentially non-branching spaces

2017 ◽  
Vol 19 (06) ◽  
pp. 1750007 ◽  
Author(s):  
Fabio Cavalletti ◽  
Andrea Mondino
Keyword(s):  
Measure Space ◽  
Metric Measure Space ◽  
Absolutely Continuous ◽  
Contraction Property ◽  
Transport Plan ◽  
Measure Contraction Property

In this paper we prove that in a metric measure space [Formula: see text] verifying the measure contraction property with parameters [Formula: see text] and [Formula: see text], any optimal transference plan between two marginal measures is induced by an optimal map, provided the first marginal is absolutely continuous with respect to [Formula: see text] and the space itself is essentially non-branching. In particular this shows that there exists a unique transport plan and it is induced by a map.

scholarly journals Inscribed Radius Bounds for Lower Ricci Bounded Metric Measure Spaces with Mean Convex Boundary

2020 ◽  
Author(s):  
Annegret Burtscher ◽  
◽  
Christian Ketterer ◽  
Robert J. McCann ◽  
Eric Woolgar ◽  
...  
Keyword(s):  
Lower Bound ◽  
Mean Curvature ◽  
Measure Space ◽  
Riemannian Curvature ◽  
Metric Measure Spaces ◽  
Contraction Property ◽  
Measure Spaces ◽  
Generalized Mean Curvature ◽  
Measure Contraction Property ◽  
Mean Convex

Consider an essentially nonbranching metric measure space with the measure contraction property of Ohta and Sturm, or with a Ricci curvature lower bound in the sense of Lott, Sturm and Villani. We prove a sharp upper bound on the inscribed radius of any subset whose boundary has a suitably signed lower bound on its generalized mean curvature. This provides a nonsmooth analog to a result of Kasue (1983) and Li (2014). We prove a stability statement concerning such bounds and - in the Riemannian curvature-dimension (RCD) setting - characterize the cases of equality.

scholarly journals Generalized Lebesgue Points for Hajłasz Functions

2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Toni Heikkinen
Keyword(s):  
Function Space ◽  
Measure Space ◽  
Variable Exponent ◽  
Banach Function Space ◽  
Lebesgue Point ◽  
Metric Measure Space ◽  
Absolutely Continuous ◽  
Lebesgue Points ◽  
Boyd Index ◽  
Generalized Lebesgue Point

Let X be a quasi-Banach function space over a doubling metric measure space P. Denote by αX the generalized upper Boyd index of X. We show that if αX<∞ and X has absolutely continuous quasinorm, then quasievery point is a generalized Lebesgue point of a quasicontinuous Hajłasz function u∈M˙s,X. Moreover, if αX<(Q+s)/Q, then quasievery point is a Lebesgue point of u. As an application we obtain Lebesgue type theorems for Lorentz–Hajłasz, Orlicz–Hajłasz, and variable exponent Hajłasz functions.

scholarly journals Distortion Coefficients of the $$\alpha $$-Grushin Plane

2022 ◽  
Vol 32 (3) ◽  
Author(s):  
Samuël Borza
Keyword(s):  
Trigonometric Functions ◽  
Contraction Property ◽  
Grushin Plane ◽  
Measure Contraction Property ◽  
Property Condition

AbstractWe compute the distortion coefficients of the $$\alpha $$ α -Grushin plane. They are expressed in terms of generalised trigonometric functions. Estimates for the distortion coefficients are then obtained and a conjecture of a measure contraction property condition for the generalised Grushin planes is suggested.

scholarly journals Diffusion, Optimal Transport and Ricci Curvature for Metric Measure Space

2017 ◽  
Vol 2017-3 (103) ◽  
pp. 19-28
Author(s):  
Luigi Ambrosio ◽  
Nicola Gigli ◽  
Giuseppe Savaré
Keyword(s):  
Ricci Curvature ◽  
Measure Space ◽  
Optimal Transport ◽  
Metric Measure Space

scholarly journals Characterizations of Orlicz-Sobolev Spaces by Means of Generalized Orlicz-Poincaré Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Toni Heikkinen
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Measure Space ◽  
Poincaré Inequality ◽  
Metric Measure Space ◽  
Poincare Inequality ◽  
Poincaré Inequalities ◽  
Space Setting ◽  
Poincare Inequalities

Let Φ be anN-function. We show that a functionu∈LΦ(ℝn)belongs to the Orlicz-Sobolev spaceW1,Φ(ℝn)if and only if it satisfies the (generalized) Φ-Poincaré inequality. Under more restrictive assumptions on Φ, an analog of the result holds in a general metric measure space setting.

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