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INEQUALITIES OF JOHNNIRENBERG TYPE IN DOUBLING SPACES
"... Abstract. The concept of an Hchain set in a doubling space X, which generalizes that of a Hölder domain in Euclidean space, is defined and investigated. We show that every Hchain set is mean porous, and that its outer layer has measure bounded by a power of its thickness. As a consequence, we show ..."
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Abstract. The concept of an Hchain set in a doubling space X, which generalizes that of a Hölder domain in Euclidean space, is defined and investigated. We show that every Hchain set is mean porous, and that its outer layer has measure bounded by a power of its thickness. As a consequence, we show that a JohnNirenberg type inequality holds on an open subset Ω of X if, and often only if, Ω is an Hchain set. 0.
Harnack inequality and hyperbolicity for subelliptic pLaplacians with applications to Picard type theorems
, 2000
"... Contents 1 Introduction 2 2 Preliminaries 2 2.1 The gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 The doubling property . . . . . . . . . . . . . . . . . . . ..."
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Contents 1 Introduction 2 2 Preliminaries 2 2.1 The gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 The doubling property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 The Poincar'e inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.5 The pLaplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.6 The nonsmooth case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 pparabolicity and phyperbolicity 10 3.1 An inequality for supersolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Volume growth and pparabolicity . . . . . . . . . . . . . . . . .
The scaling limit of looperased random walk in three dimensions
"... ABSTRACT. We show that the scaling limit exists and is invariant to dilations and rotations. We give some tools that might be useful to show universality. ..."
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ABSTRACT. We show that the scaling limit exists and is invariant to dilations and rotations. We give some tools that might be useful to show universality.
Graphs between the elliptic and parabolic Harnack inequalities.
"... We present graphs that satisfy the uniform elliptic Harnack inequality, for harmonic functions, but not the stronger parabolic one, for solutions of the discrete heat equation. It is known from [9] that the parabolic Harnack inequality is equivalent to the conjunction of a volume regularity and a L ..."
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We present graphs that satisfy the uniform elliptic Harnack inequality, for harmonic functions, but not the stronger parabolic one, for solutions of the discrete heat equation. It is known from [9] that the parabolic Harnack inequality is equivalent to the conjunction of a volume regularity and a L 2 Poincaré inequality. The first example of graph satisfying the elliptic but not the parabolic Harnack inequality is due to M. Barlow and R. Bass, see [2]. It satisfies the volume regularity and not the Poincaré inequality. We construct another example that does not satisfy the volume regularity.
Geometric Analysis Aspects of Infinite Semiplanar Graphs with Nonnegative Curvature
, 2011
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Essential selfadjointness of Schrödinger operators on vector bundles over infinite graphs
, 2013
"... Given a Hermitian vector bundle over an infinite weighted graph, we define the Laplacian associated to a unitary connection on this bundle and study the essential selfadjointness of a perturbation of this Laplacian by an operatorvalued potential. Additionally, we give a sufficient condition for ..."
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Given a Hermitian vector bundle over an infinite weighted graph, we define the Laplacian associated to a unitary connection on this bundle and study the essential selfadjointness of a perturbation of this Laplacian by an operatorvalued potential. Additionally, we give a sufficient condition for the resulting Schrödinger operator to serve as the generator of a strongly continuous contraction semigroup in the corresponding `pspace.