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29
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 approx ..."
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Cited by 42 (7 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.
A dual characterization of length spaces with application to Dirichlet metric spaces
, 2009
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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 17 (9 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.
Functional inequalities for Markov semigroups
 PROBABILITY MEASURES ON GROUPS, MUMBAI: INDE
, 2004
"... In these notes, we describe some of the most interesting inequalities related to Markov semigroups, namely spectral gap inequalities, Logarithmic Sobolev inequalities and Sobolev inequalities. We show different aspects of their meanings and ..."
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Cited by 14 (3 self)
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In these notes, we describe some of the most interesting inequalities related to Markov semigroups, namely spectral gap inequalities, Logarithmic Sobolev inequalities and Sobolev inequalities. We show different aspects of their meanings and
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 10 (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
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 8 (6 self)
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The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator.
Dirichlet forms, Poincaré inequalities and the Sobolev spaces of Korevaar and Schoen, Potential Analysis 21
, 2004
"... We answer a question of Jost on the validity of Poincare ́ inequalities for metric spacevalued functions in a Dirichlet domain. We also investigate the relationship between Dirichlet functions and elements of the Sobolevtype space of functions introduced by Korevaar and Schoen. 1 ..."
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Cited by 8 (0 self)
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We answer a question of Jost on the validity of Poincare ́ inequalities for metric spacevalued functions in a Dirichlet domain. We also investigate the relationship between Dirichlet functions and elements of the Sobolevtype space of functions introduced by Korevaar and Schoen. 1
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 8 (5 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.