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10
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 43 (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.
L pSpectral theory of locally symmetric spaces with Qrank one
, 2008
"... We study the L pspectrum of the LaplaceBeltrami operator on certain complete locally symmetric spaces M = Γ\X with finite volume and arithmetic fundamental group Γ whose universal covering X is a symmetric space of noncompact type. We also show, how the obtained results for locally symmetric spac ..."
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We study the L pspectrum of the LaplaceBeltrami operator on certain complete locally symmetric spaces M = Γ\X with finite volume and arithmetic fundamental group Γ whose universal covering X is a symmetric space of noncompact type. We also show, how the obtained results for locally symmetric spaces can be generalized to manifolds with cusps of rank one. Keywords: Arithmetic lattices, heat semigroup on L pspaces, LaplaceBeltrami operator, locally symmetric space, L pspectrum, manifolds with cusps of rank one.
SQUARE FUNCTION AND HEAT FLOW ESTIMATES ON DOMAINS
, 812
"... Abstract. The first purpose of this note is to provide a proof of the usual square function estimate on L p (Ω). It turns out to follow directly from a generic Mikhlin multiplier theorem obtained by Alexopoulos, which mostly relies on Gaussian bounds on the heat kernel. We also provide a simple proo ..."
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Abstract. The first purpose of this note is to provide a proof of the usual square function estimate on L p (Ω). It turns out to follow directly from a generic Mikhlin multiplier theorem obtained by Alexopoulos, which mostly relies on Gaussian bounds on the heat kernel. We also provide a simple proof of a weaker version of the square function estimate, which is enough in most instances involving dispersive PDEs. Moreover, we obtain, by a relatively simple integration by parts, several useful L p (Ω; H) bounds for the derivatives of the heat flow with values in a given Hilbert space H. 1.
RIESZ TRANSFORMS ON CONNECTED SUMS
, 2006
"... Let (M, g) be a complete Riemannian manifold with infinite volume, we denote by ∆ = ∆ g its Laplace operator, it has an unique selfadjoint extension on L 2 (M, dvolg) which is also denoted by ∆. The Green formula and the spectral theorem show that for any ϕ ∈ C ∞ 0 (M): ..."
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Let (M, g) be a complete Riemannian manifold with infinite volume, we denote by ∆ = ∆ g its Laplace operator, it has an unique selfadjoint extension on L 2 (M, dvolg) which is also denoted by ∆. The Green formula and the spectral theorem show that for any ϕ ∈ C ∞ 0 (M):
SQUARE FUNCTION AND HEAT FLOW ESTIMATES ON DOMAINS O.IVANOVICI AND F.PLANCHON
"... Abstract. The first purpose of this note is to provide a proof of the usual square function estimate on L p (Ω). It turns out to follow directly from a generic Mikhlin multiplier theorem obtained by Alexopoulos, which mostly relies on Gaussian bounds on the heat kernel. We also provide a simple proo ..."
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Abstract. The first purpose of this note is to provide a proof of the usual square function estimate on L p (Ω). It turns out to follow directly from a generic Mikhlin multiplier theorem obtained by Alexopoulos, which mostly relies on Gaussian bounds on the heat kernel. We also provide a simple proof of a weaker version of the square function estimate, which is enough in most instances involving dispersive PDEs. Moreover, we obtain, by a relatively simple integration by parts, several useful L p (Ω; H) bounds for the derivatives of the heat flow with values in a given Hilbert space H. hal00347161, version 3 30 Apr 2009 1.
Asymptotic dimension and NovikovShubin invariants for Open Manifolds
, 1996
"... A trace on the C ∗algebra A of quasilocal operators on an open manifold is described, based on the results in [36]. It allows a description à la NovikovShubin [31] of the low frequency behavior of the LaplaceBeltrami operator. The 0th NovikovShubin invariant defined in terms of such a trace is ..."
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A trace on the C ∗algebra A of quasilocal operators on an open manifold is described, based on the results in [36]. It allows a description à la NovikovShubin [31] of the low frequency behavior of the LaplaceBeltrami operator. The 0th NovikovShubin invariant defined in terms of such a trace is proved to coincide with a metric invariant, which we call asymptotic dimension, thus giving a large scale “Weyl asymptotics ” relation. Moreover, in analogy with the ConnesWodzicki result [7, 8, 45], the asymptotic dimension d measures the singular traceability (at 0) of the LaplaceBeltrami operator, namely we may construct a (type II1) singular trace which is finite on the ∗bimodule over A generated by ∆ −d/2. 1 Asymptotic dimension and NovikovShubin invariants 2 0 Introduction. The inspiration of this paper came from the idea of Connes ’ [8] of defining the dimension of a noncommutative compact manifold in terms of the Weyl asymptotics,
Dedicated to E.M.Landis
"... This paper is a survey of some recent results on the heat kernel of a noncompact complete Riemannian manifold. The fast progress during the last 1015 years has turned this field into a welldeveloped theory which on the one hand has its own traditions and stimulations for further investigations an ..."
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This paper is a survey of some recent results on the heat kernel of a noncompact complete Riemannian manifold. The fast progress during the last 1015 years has turned this field into a welldeveloped theory which on the one hand has its own traditions and stimulations for further investigations and, on the other hand, is connected heavily with adjacent spheres of geometry, potential theory, partial differential equations, theory