Results 1  10
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27
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 39 (4 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,
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 30 (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.
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 26 (3 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
Higher Eigenvalues and Isoperimetric Inequalities on Riemannian manifolds and graphs
"... this paper is to demonstrate in a rather general setup how isoperimetric inequalities and lower bounds of the eigenvalues of the Laplacian can be derived from existence of a distance function with controllable Laplacian. For x 2 # let us denote ae(x)=jxj =( P i x i ) . It is obvious that ..."
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Cited by 24 (2 self)
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this paper is to demonstrate in a rather general setup how isoperimetric inequalities and lower bounds of the eigenvalues of the Laplacian can be derived from existence of a distance function with controllable Laplacian. For x 2 # let us denote ae(x)=jxj =( P i x i ) . It is obvious that wehave the following two relations ) = 2n# (1.1) jraej = 1# x 6=0: (1.2) By integrating (1.1) over the ball B(r)ofradiusr centered at the origin, weobtain 2nVol(B(r)) = ) dVol(x)= @B(r) 2ae @ dA=2rA(@B(r)) where wehave used the fact that on the boundary @ = jraej = 1. Therefore, we have the following identity for the volume function V (r):=Vol(B(r)) V (r)= r (r): (1.3) Of course, the relation (1.3) of the volume and the boundary area of the Euclidean ball is well known from the elementary geometry.However, (1.1)(1.2) can also be used in a rather sophisticated waytoprove the following isoperimetric inequality between the volume and the boundary area of any bounded (assume for simplicity that the boundary is smooth) A(@ cVol : (1.4) The constant c obtained in this way, is not the sharp one. As is wellknown, the exact constant c in (1.4) is one for which both sides of (1.4) coincide for\Omega being a ball
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 24 (2 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.
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 conventional ..."
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Cited by 22 (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...
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 18 (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
The Art of Random Walks
 Lecture Notes in Mathematics 1885
, 2006
"... 1.1 Basic definitions and preliminaries................ 8 ..."
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Cited by 13 (4 self)
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1.1 Basic definitions and preliminaries................ 8
Riesz transform and L p cohomology for manifolds with Euclidean ends
 Duke Math. J
"... Abstract. Let M be a smooth Riemannian manifold which is the union of a compact part and a finite number of Euclidean ends, R n \ B(0, R) for some R> 0, each of which carries the standard metric. Our main result is that the Riesz transform on M is bounded from L p (M) → L p (M; T ∗ M) for 1 < p < n ..."
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Cited by 8 (2 self)
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Abstract. Let M be a smooth Riemannian manifold which is the union of a compact part and a finite number of Euclidean ends, R n \ B(0, R) for some R> 0, each of which carries the standard metric. Our main result is that the Riesz transform on M is bounded from L p (M) → L p (M; T ∗ M) for 1 < p < n and unbounded for p ≥ n if there is more than one end. It follows from known results that in such a case the Riesz transform on M is bounded for 1 < p ≤ 2 and unbounded for p> n; the result is new for 2 < p ≤ n. We also give some heat kernel estimates on such manifolds. We then consider the implications of boundedness of the Riesz transform in L p for some p> 2 for a more general class of manifolds. Assume that M is a ndimensional complete manifold satisfying the Nash inequality and with an O(r n) upper bound on the volume growth of geodesic balls. We show that boundedness of the Riesz transform on L p for some p> 2 implies a Hodgede Rham interpretation of the L p cohomology in degree 1, and that the map from L 2 to L p cohomology in this degree is injective. 1.
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 5 (1 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 . . . . . . . . . . . . . . . . .