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28
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
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 26 (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 26 (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.
Várilly, “Dixmier traces on noncompact isospectral deformations
 J. Funct. Anal
"... We extend the isospectral deformations of Connes, Landi and DuboisViolette to the case of Riemannian spin manifolds carrying a proper action of the noncompact abelian group R l. Under deformation by a torus action, a standard formula relates Dixmier traces of measurable operators to integrals of fu ..."
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Cited by 18 (8 self)
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We extend the isospectral deformations of Connes, Landi and DuboisViolette to the case of Riemannian spin manifolds carrying a proper action of the noncompact abelian group R l. Under deformation by a torus action, a standard formula relates Dixmier traces of measurable operators to integrals of functions on the manifold. We show that this relation persists for actions of R l, under mild restrictions on the geometry of the manifold which guarantee the Dixmier traceability of those operators.
The Art of Random Walks
 Lecture Notes in Mathematics 1885
, 2006
"... 1.1 Basic definitions and preliminaries................ 8 ..."
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Cited by 15 (5 self)
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1.1 Basic definitions and preliminaries................ 8
Heat kernels on metricmeasure spaces and an application to semilinear elliptic equations
 Trans. Amer. Math. Soc
, 2003
"... Abstract. We consider a metric measure space (M, d,µ) andaheat kernel pt(x, y) on M satisfying certain upper and lower estimates, which depend on two parameters α and β. We show that under additional mild assumptions, these parameters are determined by the intrinsic properties of the space (M, d, µ) ..."
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Cited by 13 (4 self)
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Abstract. We consider a metric measure space (M, d,µ) andaheat kernel pt(x, y) on M satisfying certain upper and lower estimates, which depend on two parameters α and β. We show that under additional mild assumptions, these parameters are determined by the intrinsic properties of the space (M, d, µ). Namely, α is the Hausdorff dimension of this space, whereas β, called the walk dimension, is determined via the properties of the family of Besov spaces W σ,2 on M. Moreover, the parameters α and β are related by the inequalities 2 ≤ β ≤ α +1. We prove also the embedding theorems for the space W β/2,2, and use them to obtain the existence results for weak solutions to semilinear elliptic equations on M of the form −Lu + f(x, u) =g(x), where L is the generator of the semigroup associated with pt. The framework in this paper is applicable for a large class of fractal domains, including the generalized Sierpiński carpet in Rn. 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.
Local monotonicity and mean value formulas for evolving Riemannian manifolds
"... We derive identities for general ‡ows of Riemannian metrics that may be regarded as local meanvalue, monotonicity, or Lyapunov formulae. These generalize previous work of the …rst author for mean curvature ‡ow and other nonlinear di¤usions. Our results apply in particular to Ricci ‡ow, where they ..."
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Cited by 7 (4 self)
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We derive identities for general ‡ows of Riemannian metrics that may be regarded as local meanvalue, monotonicity, or Lyapunov formulae. These generalize previous work of the …rst author for mean curvature ‡ow and other nonlinear di¤usions. Our results apply in particular to Ricci ‡ow, where they yield a local monotone quantity directly analogous to Perelman’s reduced volume ~ V and a local identity related to Perelman’s average energy F.
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