Results 1  10
of
19
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
On Random Walks on Wreath Products
 Ann. Probab
, 2001
"... Wreath products are a type of semidirect products. They play an important role in group theory. This paper studies the basic behavior of simple random walks on such groups and shows that these walks have interesting, somewhat exotic behaviors. The crucial fact is that the probability of return to th ..."
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Cited by 23 (1 self)
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Wreath products are a type of semidirect products. They play an important role in group theory. This paper studies the basic behavior of simple random walks on such groups and shows that these walks have interesting, somewhat exotic behaviors. The crucial fact is that the probability of return to the starting point of certain walks on wreath products is closely related to some functionals of the local times of a walk taking place on a simpler factor group.
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
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.
SPECTRAL DISTRIBUTION AND L 2ISOPERIMETRIC PROFILE OF LAPLACE OPERATORS ON GROUPS
, 901
"... Abstract. We give a formula relating the L 2isoperimetric profile to the spectral distribution of the Laplace operator associated to a finitely generated group Γ or a Riemannian manifold with a cocompact, isometric Γaction. As a consequence, we can apply techniques from geometric group theory to e ..."
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Cited by 1 (1 self)
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Abstract. We give a formula relating the L 2isoperimetric profile to the spectral distribution of the Laplace operator associated to a finitely generated group Γ or a Riemannian manifold with a cocompact, isometric Γaction. As a consequence, we can apply techniques from geometric group theory to estimate the spectral distribution of the Laplace operator in terms of the growth and the Følner’s function of the group, generalizing previous estimates by Gromov and Shubin. This leads, in particular, to sharp estimates of the spectral distributions for several classes of solvable groups. Furthermore, we prove the asymptotic invariance of the spectral distribution under changes of measures with finite second moment. 1.
The heat kernel and its estimates
"... After a short survey of some of the reasons that make the heat kernel an important object of study, we review a number of basic heat kernel estimates. We then describe recent results concerning (a) the heat kernel on certain manifolds with ends, and (b) the heat kernel with Neumann or Dirichlet boun ..."
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After a short survey of some of the reasons that make the heat kernel an important object of study, we review a number of basic heat kernel estimates. We then describe recent results concerning (a) the heat kernel on certain manifolds with ends, and (b) the heat kernel with Neumann or Dirichlet boundary condition in Euclidean domains. This text is a revised version of the four lectures given by the author at the First MSJSI in Kyoto during the summer of 2008. The structure of the lectures has been mostly preserved although some material has been added, deleted, or shifted around. The goal is to present an
unknown title
, 2000
"... An asymptotic dimension for metric spaces, and the 0th NovikovShubin invariant ..."
Abstract
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An asymptotic dimension for metric spaces, and the 0th NovikovShubin invariant