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1
Content available remote

Poincaré Inequalities for Mutually Singular Measures

100%
Schioppa A.
Analysis and Geometry in Metric Spaces
|
2015
|
tom 3
|
nr 1
EN
Using an inverse system of metric graphs as in [3], we provide a simple example of a metric space X that admits Poincaré inequalities for a continuum of mutually singular measures.
2
Content available remote

The Poincaré Inequality Does Not Improve with Blow-Up

100%
Schioppa A.
Analysis and Geometry in Metric Spaces
|
2016
|
tom 4
|
nr 1
EN
For each β > 1 we construct a family Fβ of metric measure spaces which is closed under the operation of taking weak-tangents (i.e. blow-ups), and such that each element of Fβ admits a (1, P)-Poincaré inequality if and only if P > β.
3
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How to get rid of one of the weights in a two-weight Poincaré inequality?

80%
Franchi B., Hajłasz P.
Annales Polonici Mathematici
|
2000
|
tom 74
|
nr 1
97-103
EN
We prove that if a Poincaré inequality with two different weights holds on every ball, then a Poincaré inequality with the same weight on both sides holds as well.
4
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Dimension Distortion by Sobolev Mappings in Foliated Metric Spaces

70%
Balogh Z. M., Tyson J. T., Wildrick K.
Analysis and Geometry in Metric Spaces
|
2013
|
tom 1
232-254
EN
We quantify the extent to which a supercritical Sobolev mapping can increase the dimension of subsets of its domain, in the setting of metric measure spaces supporting a Poincaré inequality. We show that the set of mappings that distort the dimensions of sets by the maximum possible amount is a prevalent subset of the relevant function space. For foliations of a metric space X defined by a David–Semmes regular mapping Π : X → W, we quantitatively estimate, in terms of Hausdorff dimension in W, the size of the set of leaves of the foliation that are mapped onto sets of higher dimension. We discuss key examples of such foliations, including foliations of the Heisenberg group by left and right cosets of horizontal subgroups.
5
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Lusin-type Theorems for Cheeger Derivatives on Metric Measure Spaces

70%
David G. C. D.
Analysis and Geometry in Metric Spaces
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2015
|
tom 3
|
nr 1
EN
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.
6
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Resistance Conditions and Applications

61%
Kinnunen J., Silvestre P.
Analysis and Geometry in Metric Spaces
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2013
|
tom 1
276-294
EN
This paper studies analytic aspects of so-called resistance conditions on metric measure spaces with a doubling measure. These conditions are weaker than the usually assumed Poincaré inequality, but however, they are sufficiently strong to imply several useful results in analysis on metric measure spaces. We show that under a perimeter resistance condition, the capacity of order one and the Hausdorff content of codimension one are comparable. Moreover, we have connections to the Sobolev inequality for compactly supported Lipschitz functions on balls as well as capacitary strong type estimates for the Hardy-Littlewood maximal function. We also consider extensions to Sobolev type inequalities with two different measures and Lorentz type estimates.
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