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51
Sobolev inequalities in disguise
 Indiana Univ. Math. J
, 1995
"... We present a simple and direct proof of the equivalence of various functional inequalities such as Sobolev or Nash inequalities. This proof applies in the context of Riemannian or subelliptic geometry, as well as on graphs and to certain nonlocal Sobolev norms. It only uses elementary cutoff argu ..."
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Cited by 91 (14 self)
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We present a simple and direct proof of the equivalence of various functional inequalities such as Sobolev or Nash inequalities. This proof applies in the context of Riemannian or subelliptic geometry, as well as on graphs and to certain nonlocal Sobolev norms. It only uses elementary cutoff arguments. This method has interesting consequences concerning Trudinger type inequalities. 1. Introduction. On R n, the classical Sobolev inequality [27] indicates that, for every smooth enough function f with compact support,
On the relation between elliptic and parabolic Harnack inequalities
, 2001
"... We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in que ..."
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Cited by 67 (6 self)
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We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for on M , (i.e., for @ t + ) and elliptic Harnack inequality for @ 2 t + on R M . 1
Gaussian Upper Bounds For The Heat Kernel On Arbitrary Manifolds
, 1997
"... In this paper, we develop a universal way of obtaining Gaussian upper bounds of the heat kernel on Riemannian manifolds. By the word "Gaussian" we mean those estimates which contain a Gaussian exponential factor similar to one which enters the explicit formula for the heat kernel of the co ..."
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Cited by 63 (2 self)
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In this paper, we develop a universal way of obtaining Gaussian upper bounds of the heat kernel on Riemannian manifolds. By the word "Gaussian" we mean those estimates which contain a Gaussian exponential factor similar to one which enters the explicit formula for the heat kernel of the conventional Laplace operator in R...
SubGaussian estimates of heat kernels on infinite graphs
 Duke Math. J
, 2000
"... We prove that a two sided subGaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay. ..."
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Cited by 55 (10 self)
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We prove that a two sided subGaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay.
Higher Eigenvalues and Isoperimetric Inequalities on Riemannian manifolds and graphs
, 1999
"... ..."
The logrithmic Sobolev inequality along the Ricci flow, arXiv:0707.2424v4
"... 2. The Sobolev inequality 3. The logarithmic Sobolev inequality on a Riemannian manifold 4. The logarithmic Sobolev inequality along the Ricci flow 5. The Sobolev inequality along the Ricci flow ..."
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Cited by 35 (2 self)
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2. The Sobolev inequality 3. The logarithmic Sobolev inequality on a Riemannian manifold 4. The logarithmic Sobolev inequality along the Ricci flow 5. The Sobolev inequality along the Ricci flow
Manifolds and Graphs With Slow Heat Kernel Decay
 Invent. Math
, 1999
"... We give upper estimates on the long time behaviour of the heat kernel on a noncompact Riemannian manifold and infinite graphs, which only depend on a lower bound of the volume growth. We also show that these estimates are optimal. ..."
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Cited by 35 (4 self)
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We give upper estimates on the long time behaviour of the heat kernel on a noncompact Riemannian manifold and infinite graphs, which only depend on a lower bound of the volume growth. We also show that these estimates are optimal.
Ondiagonal lower bounds for heat kernels and Markov chains
 Duke Math. J
, 1997
"... Let M be a Riemannian manifold, and ∆ be the LaplaceBeltrami operator on M. It is known that there exists a unique minimal positive fundamental solution to the associated heat equation, which is referred to as the heat kernel and denoted by pt(x, y) (x, y ∈ M, t> 0). ..."
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Cited by 24 (2 self)
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Let M be a Riemannian manifold, and ∆ be the LaplaceBeltrami operator on M. It is known that there exists a unique minimal positive fundamental solution to the associated heat equation, which is referred to as the heat kernel and denoted by pt(x, y) (x, y ∈ M, t> 0).
The Art of Random Walks
 Lecture Notes in Mathematics 1885
, 2006
"... 1.1 Basic definitions and preliminaries................ 8 ..."
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Cited by 21 (4 self)
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1.1 Basic definitions and preliminaries................ 8
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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Cited by 20 (4 self)
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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 . . . . . . . . . . . . . . . . .