Results 1  10
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20
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 26 (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
The AllegrettoPiepenbrink Theorem for Strongly Local Dirichlet Forms
 DOCUMENTA MATH.
, 2009
"... The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator. ..."
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Cited by 6 (5 self)
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The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator.
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
Divergence Form Operators on FractalLike Domains
, 2000
"... We consider elliptic operators L in divergence form on certain domains in R d with fractal volume growth. The domains we look at are preSierpinski carpets, which are derived from higher dimensional Sierpinski carpets. We prove a Harnack inequality for nonnegative Lharmonic functions on these domai ..."
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Cited by 4 (1 self)
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We consider elliptic operators L in divergence form on certain domains in R d with fractal volume growth. The domains we look at are preSierpinski carpets, which are derived from higher dimensional Sierpinski carpets. We prove a Harnack inequality for nonnegative Lharmonic functions on these domains and establish upper and lower bounds for the corresponding heat equation.
Holomorphic Functions and Subelliptic Heat Kernels over Lie groups. ∗
, 2008
"... A Hermitian form q on the dual space, g ∗ , of the Lie algebra, g, of a Lie group, G, determines a subLaplacian, ∆, on G. It will be shown that Hörmander’s condition for hypoellipticity of the subLaplacian holds if and only if the associated Hermitian form, induced by q on the dual of the universa ..."
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Cited by 4 (1 self)
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A Hermitian form q on the dual space, g ∗ , of the Lie algebra, g, of a Lie group, G, determines a subLaplacian, ∆, on G. It will be shown that Hörmander’s condition for hypoellipticity of the subLaplacian holds if and only if the associated Hermitian form, induced by q on the dual of the universal enveloping algebra, U ′ , is nondegenerate. The subelliptic heat semigroup, e t∆/4, is given by convolution by a C ∞ probability density ρt. When G is complex and u: G → C is a holomorphic function, the collection of derivatives of u at the identity in G gives rise to an element, û(e) ∈ ∗Key words and phrases. Subelliptic, heat kernel, complex groups, universal enveloping algebra, Taylor map.
Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms
, 2009
"... We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples. ..."
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Cited by 4 (4 self)
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We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples.
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
Differential Harnack estimates for timedependent heat equations with potential, Geom
 Funct. Anal
"... Abstract. In this paper, we prove a differential Harnack inequality for positive solutions of timedependent heat equations with potentials. We also prove a gradient estimate for the positive solution of the timedependent heat equation. 1. ..."
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Cited by 2 (1 self)
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Abstract. In this paper, we prove a differential Harnack inequality for positive solutions of timedependent heat equations with potentials. We also prove a gradient estimate for the positive solution of the timedependent heat equation. 1.