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 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
Gaussian heat kernel upper bounds via the PhragmnLindelf theorem
 Proc. Lond. Math. Soc
"... Abstract. We prove that in presence of L 2 Gaussian estimates, socalled DaviesGaffney estimates, ondiagonal upper bounds imply precise offdiagonal Gaussian upper bounds for the kernels of analytic families of operators on metric measure spaces. Contents ..."
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Cited by 7 (0 self)
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Abstract. We prove that in presence of L 2 Gaussian estimates, socalled DaviesGaffney estimates, ondiagonal upper bounds imply precise offdiagonal Gaussian upper bounds for the kernels of analytic families of operators on metric measure spaces. Contents
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
Transition operators on cocompact Gspaces
 Rev. Mat. Iberoamericana
, 2006
"... We develop methods for studying transition operators on metric spaces that are invariant under a cocompact group which acts properly. A basic requirement is a decomposition of such operators with respect to the group orbits. We then introduce “reduced ” transition operators on the compact factor sp ..."
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Cited by 6 (6 self)
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We develop methods for studying transition operators on metric spaces that are invariant under a cocompact group which acts properly. A basic requirement is a decomposition of such operators with respect to the group orbits. We then introduce “reduced ” transition operators on the compact factor space whose norms and spectral radii are upper bounds for the Lpnorms and spectral radii of the original operator. If the group is amenable then the spectral radii of the original and reduced operators coincide, and under additional hypotheses, this is also sufficient for amenability. Further bounds involve the modular function of the group. In this framework, we prove among other things that the bottom of the spectrum of the Laplacian on a cocompact Riemannian manifold is 0 if and only if the group is amenable and unimodular. The same result holds for Euclidean simplicial complexes. On a geodesic, proper metric space with cocompact isometry group action, the averaging operator over balls with a fixed radius has norm equal to 1 if and only if the group is amenable and unimodular. The technique also allows explicit computation of spectral radii when the group is amenable. 1. Introduction: Four
On uniformly subelliptic operators and stochastic area
, 2006
"... We consider uniformly subelliptic operators on certain unimodular Lie groups of polynomial growth. It was shown by SaloffCoste and Stroock that classical results of De Giorgi, Nash, Moser, Aronson extend to this setting. It was then observed by Sturm that many proofs extend naturally to the setting ..."
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Cited by 5 (4 self)
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We consider uniformly subelliptic operators on certain unimodular Lie groups of polynomial growth. It was shown by SaloffCoste and Stroock that classical results of De Giorgi, Nash, Moser, Aronson extend to this setting. It was then observed by Sturm that many proofs extend naturally to the setting of locally compact Dirichlet spaces. We relate these results to what is known as rough path theory by showing that they provide a natural and powerful analytic machinery for construction and study of (random) geometric Hölder rough paths. (In particular, we obtain a simple construction of the LyonsStoica stochastic area for a diffusion process with uniformly elliptic generator in divergence form.) Our approach then enables us to establish a number of farreaching generalizations of classical theorems in diffusion theory including WongZakai approximations, FreidlinWentzell sample path large deviations and the StroockVaradhan support theorem. The latter was conjectured by T. Lyons in his recent St. Flour lecture. 1
Eigenfunction expansions for generators of Dirichlet forms
, 2002
"... Dedicated to the memory of Klaus Floret Abstract We present an eigenfunction expansion theorem for generators of strongly local, regular Dirichlet forms. Conditions are phrased in terms of the intrinsic metric. The result covers many cases of Hamiltonians which appear in Mathematical Physics and Geo ..."
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Cited by 5 (2 self)
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Dedicated to the memory of Klaus Floret Abstract We present an eigenfunction expansion theorem for generators of strongly local, regular Dirichlet forms. Conditions are phrased in terms of the intrinsic metric. The result covers many cases of Hamiltonians which appear in Mathematical Physics and Geometry.
Stability of the Essential Spectrum of SecondOrder Complex Elliptic Operators
 J. Reine Angew. Math
"... We prove compactness of the resolvent difference for second order divergence form operators whose matrix functions are close enough at infinity. Our theorems include certain subelliptic cases as well as cases with unbounded coefficients. AMS Subject Classification:47F05, secondary 35P05, 47A10 Intr ..."
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Cited by 4 (0 self)
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We prove compactness of the resolvent difference for second order divergence form operators whose matrix functions are close enough at infinity. Our theorems include certain subelliptic cases as well as cases with unbounded coefficients. AMS Subject Classification:47F05, secondary 35P05, 47A10 Introduction In this paper we consider the stability of the essential spectrum of second order divergence operators with complex coefficients. More precisely, we give L p \Gammaconditions on the difference of b and a under which the operators formally given by B = D bD and A = D aD have the same essential spectrum by proving the compactness of the resolvent difference. Here b; a 2 L 1 loc (R d ; C d+1;d+1 ) are matrixvalued functions and D aD is shorthand for \Gamma d X k;j=1 D j (a jk D k + a j;d+1 ) + a d+1;k D k + a d+1;d+1 which formally reduces to D aD by setting D = (D 1 ; :::; D d ; I) t : We think of A as being a comparison operator, whose spectrum is unders...
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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Cited by 3 (1 self)
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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
Small time heat kernel behavior on Riemannian complexes
"... SaloffCoste Abstract. We study how bounds on the local geometry of a Riemannian polyhedral complex yield uniform local Poincaré inequalities. These inequalities have a variety of applications, including bounds on the heat kernel and a uniform local Harnack inequality. We additionally consider the e ..."
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Cited by 2 (1 self)
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SaloffCoste Abstract. We study how bounds on the local geometry of a Riemannian polyhedral complex yield uniform local Poincaré inequalities. These inequalities have a variety of applications, including bounds on the heat kernel and a uniform local Harnack inequality. We additionally consider the example of a complex, X, which has a finitely generated group of isomorphisms, G, such that X/G = Y is a complex consisting of a finite number of polytopes. We show that when this group, G, haspolynomial volume growth, there is a uniform global Poincaré inequality on