Results 1  10
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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 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
Sobolev Spaces, Laplacian, And Heat Kernel On Alexandrov Spaces
, 1998
"... . We prove the compactness of the imbedding of the Sobolev space W 1;2 0 (\Omega\Gamma into L 2(\Omega\Gamma for any relatively compact open subset\Omega of an Alexandrov space. As a corollary, the generator induced from the Dirichlet (energy) form has discrete spectrum. The generator can be appr ..."
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Cited by 17 (6 self)
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. We prove the compactness of the imbedding of the Sobolev space W 1;2 0 (\Omega\Gamma into L 2(\Omega\Gamma for any relatively compact open subset\Omega of an Alexandrov space. As a corollary, the generator induced from the Dirichlet (energy) form has discrete spectrum. The generator can be approximated by the Laplacian induced from the DCstructure on the Alexandrov space. We also prove the existence of the locally Holder continuous Dirichlet heat kernel. 1. Introduction Consider a family M of ndimensional closed Riemannian manifolds with a uniform lower bound of sectional curvature and a uniform upper bound of diameter for a fixed n 2 N . In order to investigate various properties of manifolds in M, it is very useful to study its closure M with respect to the GromovHausdorff distance dGH , which is compact by the Gromov compactness theorem [15]. Since the closure M consists of Alexandrov spaces introduced in [2], the study of Alexandrov spaces is nowadays an important topic i...
Sch’nol’s theorem for strongly local forms
, 2009
"... We prove a variant of Sch’nol’s theorem in a general setting: for generators of strongly local Dirichlet forms perturbed by measures. As an application, we discuss quantum graphs with δ or Kirchhoff boundary conditions. ..."
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Cited by 10 (6 self)
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We prove a variant of Sch’nol’s theorem in a general setting: for generators of strongly local Dirichlet forms perturbed by measures. As an application, we discuss quantum graphs with δ or Kirchhoff boundary conditions.
(Nonsymmetric) Dirichlet Operators On L¹: Existence, Uniqueness And Associated Markov Processes
"... Let L be a nonsymmetric operator of type Lu = P a ij @ i @ j u + P b i @ i u on an arbitrary subset U ae R d . We analyse L as an operator on L 1 (U; ¯) where ¯ is an invariant measure, i.e., a possibly infinite measure ¯ satisfying the equation L ¯ = 0 (in the weak sense). We explicitel ..."
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Cited by 9 (2 self)
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Let L be a nonsymmetric operator of type Lu = P a ij @ i @ j u + P b i @ i u on an arbitrary subset U ae R d . We analyse L as an operator on L 1 (U; ¯) where ¯ is an invariant measure, i.e., a possibly infinite measure ¯ satisfying the equation L ¯ = 0 (in the weak sense). We explicitely construct, under mild regularity assumptions, extensions of L generating subMarkovian C0 semigroups on L 1 (U; ¯) as well as associated diffusion processes. We give sufficient conditions on the coefficients so that there exists only one extension of L generating a C0 semigroup and apply the results to prove uniqueness of the invariant measure ¯. Our results imply in particular that if ' 2 H 1;2 loc (R d ; dx), ' 6= 0 dxa.e., the generalized Schrödinger operator (\Delta + 2' \Gamma1 r' \Delta r;C 1 0 (R d )) has exactly one extension generating a C0 semigroup if and only if the Friedrich's extension is conservative. We also give existence and uniqueness results for ...
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.
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.
A dual characterization of length spaces with application to Dirichlet metric spaces
, 2009
"... ..."
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.
Dirichlet forms and stochastic completeness of graphs and subgraphs, to appear
 J. Reine Angew. Math. (Crelle’s Journal
"... Abstract. We characterize stochastic completeness for regular Dirichlet forms on discrete sets. We then study how stochastic completeness of a subgraph is related to stochastic completeness of the whole graph. We show that any graph is a subgraph of a stochastically complete graph and that stochasti ..."
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Cited by 3 (1 self)
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Abstract. We characterize stochastic completeness for regular Dirichlet forms on discrete sets. We then study how stochastic completeness of a subgraph is related to stochastic completeness of the whole graph. We show that any graph is a subgraph of a stochastically complete graph and that stochastic incompleteness of a suitably modified subgraph implies stochastic incompleteness of the whole graph. Along our way we give a sufficient condition for essential selfadjointness of generators of Dirichlet forms on discrete sets and explicitely determine the generators on all ℓ p, 1 ≤ p < ∞, in this case.
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