scholarly journals First order Poincaré inequalities in metric measure spaces

2013 ◽  
Vol 38 ◽  
pp. 287-308 ◽  
Author(s):  
Estibalitz Durand-Cartagena ◽  
Jesús A. Jaramillo ◽  
Nageswari Shanmugalingam
Keyword(s):  
Metric Measure Spaces ◽  
Poincaré Inequalities ◽  
First Order ◽  
Poincare Inequalities ◽  
Measure Spaces

scholarly journals Lusin-type Theorems for Cheeger Derivatives on Metric Measure Spaces

2015 ◽  
Vol 3 (1) ◽  
Keyword(s):  
Continuous Function ◽  
Large Class ◽  
Borel Function ◽  
Metric Measure Spaces ◽  
Famous Theorem ◽  
Poincaré Inequalities ◽  
Almost Everywhere ◽  
Poincare Inequalities ◽  
Measure Spaces ◽  
Doubling Measures

Abstract A theorem of Lusin states that every Borel function onRis equal almost everywhere to the derivative of a continuous function. This result was later generalized to Rn in works of Alberti and Moonens-Pfeffer. In this note, we prove direct analogs of these results on a large class of metric measure spaces, those with doubling measures and Poincaré inequalities, which admit a form of differentiation by a famous theorem of Cheeger.

scholarly journals Resistance conditions, Poincaré inequalities, the Lip-lip condition and Hardy’s inequalities

2016 ◽  
Vol 49 (1) ◽  
Author(s):  
Juha Kinnunen ◽  
Pilar Silvestre
Keyword(s):  
Poincaré Inequality ◽  
Metric Measure Spaces ◽  
Poincare Inequality ◽  
Hardy Inequalities ◽  
Poincaré Inequalities ◽  
Poincare Inequalities ◽  
Measure Spaces ◽  
Lipschitz Constants ◽  
Hardy’S Inequalities ◽  
Weaker Conditions

AbstractThis note investigates weaker conditions than a Poincaré inequality in analysis on metric measure spaces. We discuss two resistance conditions which are stated in terms of capacities. We show that these conditions can be characterized by versions of Sobolev–Poincaré inequalities. As a consequence, we obtain so-called Lip-lip condition related to pointwise Lipschitz constants. Moreover, we show that the pointwise Hardy inequalities and uniform fatness conditions are equivalent under an appropriate resistance condition.

scholarly journals A remark on Poincaré inequalities on metric measure spaces

2004 ◽  
Vol 95 (2) ◽  
pp. 299 ◽  
Author(s):  
Stephen Keith ◽  
Kai Rajala
Keyword(s):  
Measure Space ◽  
Poincaré Inequality ◽  
Metric Measure Space ◽  
Metric Measure Spaces ◽  
Regular Measure ◽  
Poincare Inequality ◽  
Poincaré Inequalities ◽  
Poincare Inequalities ◽  
Measure Spaces ◽  
Lipschitz Constants

We show that, in a complete metric measure space equipped with a doubling Borel regular measure, the Poincaré inequality with upper gradients introduced by Heinonen and Koskela [3] is equivalent to the Poincaré inequality with "approximate Lipschitz constants" used by Semmes in [9].

Orlicz-Poincaré inequalities, maximal functions and AΦ-conditions

2010 ◽  
Vol 140 (1) ◽  
pp. 31-48 ◽  
Author(s):  
Jana Björn
Keyword(s):  
Measure Space ◽  
General Setting ◽  
Maximal Functions ◽  
The Other ◽  
Metric Measure Spaces ◽  
Young Function ◽  
Poincaré Inequalities ◽  
Poincare Inequalities ◽  
Measure Spaces ◽  
Littlewood Maximal Operator

For a measure μ on ℝn (or on a doubling metric measure space) and a Young function Φ, we define two versions of Orlicz–Poincaré inequalities as generalizations of the usual p-Poincaré inequality. It is shown that, on ℝ, one of them is equivalent to the boundedness of the Hardy–Littlewood maximal operator from LΦ(ℝ,μ) to LΦ(ℝ,μ), while the other is equivalent to a generalization of the Muckenhoupt Ap-condition. While one direction in these equivalences is valid only on ℝ, the other holds in the general setting of doubling metric measure spaces. We also characterize both Orlicz–Poincaré inequalities on metric measure spaces by means of pointwise inequalities involving maximal functions of the gradient.

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