Results 1  10
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27
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 32 (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 27 (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
L p spectral theory of higherorder elliptic differential operators
 Bull. London Math. Soc
, 1997
"... 2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518 ..."
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Cited by 13 (1 self)
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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
Maximal inequalities and Riesz transform estimates on L p spaces for Schrödinger operators with nonnegative potentials, Ann. Inst. Fourier, Grenoble 57 no 6
, 2007
"... The paper concerns the magnetic Schrödinger operator H(a, V) = ..."
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Cited by 9 (3 self)
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The paper concerns the magnetic Schrödinger operator H(a, V) =
Persistencebased segmentation of deformable shapes
 Computer Vision and Pattern Recognition Workshops (CVPRW), 2010 IEEE Computer Society Conference on
"... In this paper, we combine two ideas: persistencebased clustering and the Heat Kernel Signature (HKS) function to obtain a multiscale isometry invariant mesh segmentation algorithm. The key advantages of this approach is that it is tunable through a few intuitive parameters and is stable under near ..."
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Cited by 8 (1 self)
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In this paper, we combine two ideas: persistencebased clustering and the Heat Kernel Signature (HKS) function to obtain a multiscale isometry invariant mesh segmentation algorithm. The key advantages of this approach is that it is tunable through a few intuitive parameters and is stable under nearisometric deformations. Indeed the method comes with feedback on the stability of the number of segments in the form of a persistence diagram. There are also spatial guarantees on part of the segments. Finally, we present an extension to the method which first detects regions which are inherently unstable and segments them separately. Both approaches are reasonably scalable and come with strong guarantees. We show numerous examples and a comparison with the segmentation benchmark and the curvature function. 1.
Large time behavior of the heat kernel
, 2002
"... In this paper we study the large time behavior of the (minimal) heat kernel kM P (x, y, t) of a general time independent parabolic operator L = ut+P(x, ∂x) which is defined on a noncompact manifold M. More precisely, we prove that lim t→ ∞ eλ0t k M P (x, y, t) always exists. Here λ0 is the generaliz ..."
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Cited by 8 (2 self)
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In this paper we study the large time behavior of the (minimal) heat kernel kM P (x, y, t) of a general time independent parabolic operator L = ut+P(x, ∂x) which is defined on a noncompact manifold M. More precisely, we prove that lim t→ ∞ eλ0t k M P (x, y, t) always exists. Here λ0 is the generalized principal eigenvalue of the operator P in M.
HARDY SPACES ASSOCIATED TO NONNEGATIVE SELFADJOINT OPERATORS SATISFYING DAVIESGAFFNEY ESTIMATES
"... Abstract. Let X be a metric space with doubling measure, and L be a nonnegative, selfadjoint operator satisfying DaviesGaffney bounds on L 2 (X). In this article we develop a theory of Hardy and BMO spaces associated to L, including an atomic (or molecular) decomposition, square function characte ..."
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Cited by 7 (0 self)
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Abstract. Let X be a metric space with doubling measure, and L be a nonnegative, selfadjoint operator satisfying DaviesGaffney bounds on L 2 (X). In this article we develop a theory of Hardy and BMO spaces associated to L, including an atomic (or molecular) decomposition, square function characterization, duality of Hardy and BMO spaces. Further specializing to the case that L is a Schrödinger operator on R n with a nonnegative, locally integrable potential, we establish addition characterizations of such Hardy space in terms of maximal functions. Finally, (X) for p> 1, which may or may not coincide with the space L p (X), and show that they interpolate with H 1 L(X) spaces by the complex method. we define Hardy spaces H p L The authors gratefully acknowledge support from NSF as follows: S. Hofmann (DMS
An asymptotic dimension for metric spaces, and the 0th NovikovShubin invariant
"... A nonnegative number d∞, called asymptotic dimension, is associated with any metric space. Such number detects the asymptotic properties of the space (being zero on bounded metric spaces), fulfills the properties of a dimension, and is invariant under rough isometries. It is then shown that for a cl ..."
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Cited by 6 (5 self)
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A nonnegative number d∞, called asymptotic dimension, is associated with any metric space. Such number detects the asymptotic properties of the space (being zero on bounded metric spaces), fulfills the properties of a dimension, and is invariant under rough isometries. It is then shown that for a class of open manifolds with bounded geometry the asymptotic dimension coincides with the 0th NovikovShubin number α0 defined in a previous paper [D. Guido, T. Isola, J. Funct. Analysis, 176 (2000)]. Thus the dimensional interpretation of α0 given in the mentioned paper in the framework of noncommutative geometry is established on metrics grounds. Since the asymptotic dimension of a covering manifold coincides with the polynomial growth of its covering group, the stated equality generalises to open manifolds a result by Varopoulos. 0. Introduction.
Transition operators on cocompact Gspaces
 Rev. Mat. Iberoamericana
, 2006
"... We develop methods for studying transition operators on metric spaces that are invariant under a cocompact group which acts properly. A basic requirement is a decomposition of such operators with respect to the group orbits. We then introduce “reduced ” transition operators on the compact factor sp ..."
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Cited by 6 (6 self)
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We develop methods for studying transition operators on metric spaces that are invariant under a cocompact group which acts properly. A basic requirement is a decomposition of such operators with respect to the group orbits. We then introduce “reduced ” transition operators on the compact factor space whose norms and spectral radii are upper bounds for the Lpnorms and spectral radii of the original operator. If the group is amenable then the spectral radii of the original and reduced operators coincide, and under additional hypotheses, this is also sufficient for amenability. Further bounds involve the modular function of the group. In this framework, we prove among other things that the bottom of the spectrum of the Laplacian on a cocompact Riemannian manifold is 0 if and only if the group is amenable and unimodular. The same result holds for Euclidean simplicial complexes. On a geodesic, proper metric space with cocompact isometry group action, the averaging operator over balls with a fixed radius has norm equal to 1 if and only if the group is amenable and unimodular. The technique also allows explicit computation of spectral radii when the group is amenable. 1. Introduction: Four
Dirichlet Heat Kernel in the Exterior of a Compact Set
"... this paper do not directly use the Ricci curvature assumption. In fact, we will show that Theorems 1.1, 1.2 hold true for any manifold which is quasiisometric (even roughlyisometric, under any reasonable bounded local geometry assumption) to a manifold with nonnegative Riccicurvature. In particu ..."
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Cited by 5 (1 self)
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this paper do not directly use the Ricci curvature assumption. In fact, we will show that Theorems 1.1, 1.2 hold true for any manifold which is quasiisometric (even roughlyisometric, under any reasonable bounded local geometry assumption) to a manifold with nonnegative Riccicurvature. In particular, the bounds of Examples 1.1, 1.2, hold true if the Laplace operator is replaced by a uniformly elliptic operator in divergence form. Thus, the bounds stated above are reasonably stable. The adequate hypothesis for our purpose is expressed in terms of a parabolic Harnack inequality or, equivalently, in terms of certain Poincar'e inequality and volume growth (see below Section 2.2). The present work originated from our desire to understand the behavior of the heat kernel on manifolds with more than one ends. Indeed, together with good estimates of certain hitting probabilities obtained in [14], the result presented here is one of the main building blocks in the proof of the sharp estimates for the heat kernel on manifolds with ends that have been announced in [11] and are proved in [12]. The following result complements Theorems 1.1, 1.2 in this direction. Given a Riemannian manifold with k ends, let U be a relatively compact open set in M with smooth boundary such that M n U has exactly k unbounded connected components E 1 ; : : : ; E k . Let K i = @U " E i , and consider E i as a manifold with boundary ffi E i := K i . Denote by p i the heat kernel on E i and by p\Omega i the Dirichlet heat kernel on\Omega i = E i n K i (in other words, p i satisfies the Neumann condition on K i , whereas p\Omega i satisfies the Dirichlet condition on K i ). Let also V i (x; t) be the volume function on E i . For each end E i , fix a point o i 2 K i and define the functions H i , D...