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
Convergence of laplacian eigenmaps
 In NIPS
, 2006
"... Geometrically based methods for various tasks of machine learning have attracted considerable attention over the last few years. In this paper we show convergence of eigenvectors of the point cloud Laplacian to the eigenfunctions of the LaplaceBeltrami operator on the underlying manifold, thus esta ..."
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Cited by 26 (2 self)
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Geometrically based methods for various tasks of machine learning have attracted considerable attention over the last few years. In this paper we show convergence of eigenvectors of the point cloud Laplacian to the eigenfunctions of the LaplaceBeltrami operator on the underlying manifold, thus establishing the first convergence results for a spectral dimensionality reduction algorithm in the manifold setting. 1
Weighted norm inequalities, offdiagonal estimates and elliptic operators, Part II: Offdiagonal estimates on spaces of homogeneous type
, 2005
"... Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincar ..."
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Cited by 23 (6 self)
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Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and
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
Pseudolocality for Ricci flow and applications
"... Abstract. In [26], Perelman established a differential LiYauHamilton (LYH) type inequality for fundamental solutions of the conjugate heat equation corresponding to the Ricci flow on compact manifolds (also see [23]). As an application of the LYH inequality, Perelman proved a pseudolocality result ..."
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Cited by 8 (1 self)
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Abstract. In [26], Perelman established a differential LiYauHamilton (LYH) type inequality for fundamental solutions of the conjugate heat equation corresponding to the Ricci flow on compact manifolds (also see [23]). As an application of the LYH inequality, Perelman proved a pseudolocality result for the Ricci flow on compact manifolds. In this article we provide the details for the proofs of these results in the case of a complete noncompact Riemannian manifold. Using these results we prove that under certain conditions, a finite time singularity of the Ricci flow must form within a compact set. We also prove a long time existence result for the KählerRicci flow flow on complete nonnegatively curved Kähler manifolds.
Gradient estimates and Harnack inequalities on noncompact Riemannian manifolds, Stochastic Process
 Appl
"... Abstract. A new type of gradient estimate is established for diffusion semigroups on noncompact complete Riemannian manifolds. As applications, a global Harnack inequality with power and a heat kernel estimate are derived for diffusion semigroups on arbitrary complete Riemannian manifolds. 1. The m ..."
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Cited by 6 (3 self)
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Abstract. A new type of gradient estimate is established for diffusion semigroups on noncompact complete Riemannian manifolds. As applications, a global Harnack inequality with power and a heat kernel estimate are derived for diffusion semigroups on arbitrary complete Riemannian manifolds. 1. The main result Let M be a noncompact complete Riemannian manifold, and Pt be the Dirichlet diffusion semigroup generated by L = ∆ + ∇V for some C 2 function V. We intend to establish reasonable gradient estimates and Harnack type inequalities for Pt. In case that Ric − HessV is bounded below, a dimensionfree Harnack inequality was established in [15], which according to [17], is indeed equivalent to the corresponding curvature condition. See e.g. [2] for equivalent statements on heat kernel functional inequalities; see also [8, 3, 9] for a parabolic Harnack inequality using the dimensioncurvature condition by shifting time, which goes back to the classical local parabolic Harnack inequality of Moser [10]. Recently, some sharp gradient estimates have been derived in [13, 19] for the Dirichlet semigroup on relatively compact domains. More precisely, for V = 0 and a relatively compact open C2 domain D, the Dirichlet heat semigroup P D t satisfies (1.1) ∇P D t f(x) ≤ C(x, t)P D t f(x), x ∈ D, t> 0, for some locally bounded function C: D ×]0, ∞ [ →]0, ∞ [ and all f ∈ B + b, the space of bounded nonnegative measurable functions on M. Obviously, this implies the Harnack inequality (1.2) P D t f(x) ≤ ˜ C(x, y, t)P D t f(y), t> 0, x, y ∈ D, f ∈ B+ b, for some function ˜ C: M 2 ×]0, ∞ [ →]0, ∞[. The purpose of this paper is to establish inequalities analogous to (1.1) and (1.2) globally on the whole manifold M. On the other hand however, both (1.1) and (1.2) are in general wrong for Pt in place of P D t. A simple counterexample is already the standard heat semigroup on R d. Hence, we turn to search for the following slightly weaker version of gradient
Hitting probabilities for Brownian motion on Riemannian manifolds
"... this paper is to estimate the hitting probability function / K (t; x) := P x (9 s 2 [0; t] ; X s 2 K) where K ae M is a fixed compact set. In words, / K (t; x) is the probability that Brownian motion started at x hits K by time t. Our goal is to obtain precise estimates on / K for all t ? 0 and x ..."
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Cited by 4 (2 self)
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this paper is to estimate the hitting probability function / K (t; x) := P x (9 s 2 [0; t] ; X s 2 K) where K ae M is a fixed compact set. In words, / K (t; x) is the probability that Brownian motion started at x hits K by time t. Our goal is to obtain precise estimates on / K for all t ? 0 and x outside a neighborhood of K, hence avoiding the somewhat different question of the behavior of / K near the boundary of K. In the context of Riemannian manifolds, this natural question has been considered only in a handful of papers including [2], [4]. We were led to study / K in our attempt to develop sharp heat kernel estimates on manifolds with more than one end. Indeed, the proof of the heat kernel estimates announced in [20] depends in a crucial way on the results of the present paper (see [21]). In this context, it turns out to be important to estimate also the time derivative @ t / K (t; x) which is a positive function. We develop a general approach which allows to obtain estimates of / K in terms of the heat kernel p(t; x; y) or closely related objects such as the Dirichlet heat kernel p U (t; x; y) of some open set U . In the case when X t is transient, that is, M is nonparabolic, we show that the behavior of / K (t; x), away from K, is comparable to that of Z t 0 p(s; x; y)ds; where y is a reference point on @K. If (X t ) t?0 is recurrent, that is, M is parabolic, we obtain similar estimates through Z t 0 p U (s; x; y)ds where U is a certain region slightly larger than\Omega := M n K. We also show that @ t / K (t; x) is comparable to p\Omega (t; x; y) where y stays at a certain distance from @K. For precise statements, see Theorems 3.3, 3.5, 3.7 and Corollaries 3.9, 3.10. Using the known results concerning the heat kernel p(t; x; y) and the results of [23...
Strichartz estimates for the Schrödinger equation on polygonal domains
 J. Geom. Anal
"... Abstract. We prove Strichartz estimates with a loss of derivatives for the Schrödinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates on the polygon follow from those on Euclidean surfaces with conical si ..."
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Cited by 4 (1 self)
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Abstract. We prove Strichartz estimates with a loss of derivatives for the Schrödinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates on the polygon follow from those on Euclidean surfaces with conical singularities. We develop a LittlewoodPaley squarefunction estimate with respect to the spectrum of the Laplacian on these spaces. This allows us to reduce matters to proving estimates at each frequency scale. The problem can be localized in space provided the time intervals are sufficiently small. Strichartz estimates then follow from a recent result of the second author regarding the Schrödinger equation on the Euclidean cone. 1.
Strichartz estimates for the wave equation on flat cones
 IMRN
"... Abstract. We consider the solution operator for the wave equation on the flat Euclidean cone over the circle of radius ρ> 0, the manifold R+ × ( R / 2πρZ) equipped with the metric g(r, θ) = dr2 + r2 dθ2. Using explicit representations of the solution operator in regions related to flat wave propag ..."
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Cited by 2 (1 self)
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Abstract. We consider the solution operator for the wave equation on the flat Euclidean cone over the circle of radius ρ> 0, the manifold R+ × ( R / 2πρZ) equipped with the metric g(r, θ) = dr2 + r2 dθ2. Using explicit representations of the solution operator in regions related to flat wave propagation and diffraction by the cone point, we prove dispersive estimates and hence scale invariant Strichartz estimates for the wave equation on flat cones. We then show that this yields corresponding inequalities on wedge domains, polygons, and Euclidean surfaces with conic singularities. This in turn yields wellposedness results for the nonlinear wave equation on such manifolds. Morawetz estimates on the cone are also treated. 1.