Results 1  10
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16
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 66 (6 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
Harnack inequalities and subGaussian estimates for random walks
 Math. Annalen
, 2002
"... We show that a fiparabolic Harnack inequality for random walks on graphs is equivalent, on one hand, to so called fiGaussian estimates for the transition probability and, on the other hand, to the conjunction of the elliptic Harnack inequality, the doubling volume property, and the fact that the m ..."
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Cited by 60 (5 self)
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We show that a fiparabolic Harnack inequality for random walks on graphs is equivalent, on one hand, to so called fiGaussian estimates for the transition probability and, on the other hand, to the conjunction of the elliptic Harnack inequality, the doubling volume property, and the fact that the mean exit time in any ball of radius R is of the order R . The latter condition can be replaced by a certain estimate of a resistance of annuli.
The Art of Random Walks
 Lecture Notes in Mathematics 1885
, 2006
"... 1.1 Basic definitions and preliminaries................ 8 ..."
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Cited by 21 (4 self)
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1.1 Basic definitions and preliminaries................ 8
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 19 (2 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...
Riesz transform and Lp 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, Rn \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 Lp(M) → Lp(M;T ∗M) for 1 < p < n ..."
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Cited by 17 (5 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, Rn \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 Lp(M) → Lp(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 Lp 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(rn) upper bound on the volume growth of geodesic balls. We show that boundedness of the Riesz transform on Lp for some p> 2 implies a Hodgede Rham interpretation of the Lp cohomology in degree 1, and that the map from L2 to Lp cohomology in this degree is injective. 1.
Sobolev inequalities in familiar and unfamiliar settings
 In S. Sobolev Centenial Volumes, (V. Maz’ja, Ed
"... Abstract The classical Sobolev inequalities play a key role in analysis in Euclidean spaces and in the study of solutions of partial differential equations. In fact, they are extremely flexible tools and are useful in many different settings. This paper gives a glimpse of assortments of such applica ..."
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Abstract The classical Sobolev inequalities play a key role in analysis in Euclidean spaces and in the study of solutions of partial differential equations. In fact, they are extremely flexible tools and are useful in many different settings. This paper gives a glimpse of assortments of such applications in a variety of contexts. 1
Prime ends for domains in metric spaces
, 2012
"... Abstract. In this paper we propose a new definition of prime ends for domains in metric spaces under rather general assumptions. We compare our prime ends to those of Carathéodory and Näkki. Modulus ends and prime ends, defined by means of the pmodulus of curve families, are also discussed and r ..."
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Cited by 7 (4 self)
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Abstract. In this paper we propose a new definition of prime ends for domains in metric spaces under rather general assumptions. We compare our prime ends to those of Carathéodory and Näkki. Modulus ends and prime ends, defined by means of the pmodulus of curve families, are also discussed and related to the prime ends. We provide characterizations of singleton prime ends and relate them to the notion of accessibility of boundary points, and introduce a topology on the prime end boundary. We also study relations between the prime end boundary and the Mazurkiewicz boundary. Generalizing the notion of John domains, we introduce almost John domains, and we investigate prime ends in the settings of John domains, almost John domains and domains which are finitely connected at the boundary.
OBTAINING UPPER BOUNDS OF HEAT KERNELS FROM LOWER
"... Abstract. We show that a neardiagonal lower bound of the heat kernel of a Dirichlet form on a metric measure space with a regular measure implies an ondiagonal upper bound. If in addition the Dirichlet form is local and regular then we obtain a full offdiagonal upper bound of the heat kernel prov ..."
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Cited by 4 (0 self)
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Abstract. We show that a neardiagonal lower bound of the heat kernel of a Dirichlet form on a metric measure space with a regular measure implies an ondiagonal upper bound. If in addition the Dirichlet form is local and regular then we obtain a full offdiagonal upper bound of the heat kernel provided the Dirichlet heat kernel on any ball satisfies a neardiagonal lower estimate. This reveals a new phenomenon in the relationship between the lower and upper bounds of the heat kernel. Contents
The heat kernel and its estimates
, 2008
"... 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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Cited by 3 (0 self)
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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.
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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Cited by 2 (0 self)
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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):