## Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms (2009)

Citations: | 4 - 4 self |

### BibTeX

@MISC{Lenz09generalizedeigenfunctions,

author = {Daniel Lenz and Peter Stollmann and Ivan Veselić},

title = { Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms},

year = {2009}

}

### OpenURL

### Abstract

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.

### Citations

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Citation Context ...scuss applications in Section 6. 1. Strongly local Dirichlet forms In this section we describe the set-up used throughout the paper. We refer to [35] as to the classical standard reference as well as =-=[21, 28, 36, 47]-=- for literature on Dirichlet forms. We treat real and complex function spaces at the same time and write K to denote either R or C. Throughout we will work with a locally compact, separable metric spa... |

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Citation Context ...scuss applications in Section 6. 1. Strongly local Dirichlet forms In this section we describe the set-up used throughout the paper. We refer to [35] as to the classical standard reference as well as =-=[21, 28, 36, 47]-=- for literature on Dirichlet forms. We treat real and complex function spaces at the same time and write K to denote either R or C. Throughout we will work with a locally compact, separable metric spa... |

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Citation Context ...scuss applications in Section 6. 1. Strongly local Dirichlet forms In this section we describe the set-up used throughout the paper. We refer to [35] as to the classical standard reference as well as =-=[21, 28, 36, 47]-=- for literature on Dirichlet forms. We treat real and complex function spaces at the same time and write K to denote either R or C. Throughout we will work with a locally compact, separable metric spa... |

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Citation Context ...ceeding subsection are strongly local. In the case of the Laplacian in Euclidean space, the measure Γ is given by (∇u|∇v)dx appearing above. We discuss properties of the energy measure next (see e.g. =-=[21, 35, 67]-=-). The energy measure inherits strong locality from E viz χUdΓ(η,u) = 0 holds for any open U ∈ X and any η,u ∈ D with η constant on U. This directly allows one to extend Γ to Dloc defined as {u ∈ L 2 ... |

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Citation Context ...n which the Harnack principle is satisfied:12 D. LENZ, P. STOLLMANN, AND I. VESELIĆ For ν ≡ 0 and λ = 0 a Harnack inequality holds, whenever E satisfies a Poincaré and a volume doubling property; cf =-=[20]-=- and the discussion there. The most general results for H0 = −∆ in terms of the measures ν that are allowed seem to be found in [37]. The uniformity of the estimates from [37] immediately gives that t... |

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Citation Context ...= H0 + V , where H0 is associated to a strictly local Dirichlet form and the function V is a suitable potential. In fact, we can even include measures as potentials. Here, we follow the approach from =-=[64, 65]-=-. Measure perturbations have been regarded by a number of authors in different contexts, see e.g. [11, 37, 67] and the references there. We denote by MR(U) the signed Radon measures on the open subset... |

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Citation Context ...ntial. In fact, we can even include measures as potentials. Here, we follow the approach from [64, 65]. Measure perturbations have been regarded by a number of authors in different contexts, see e.g. =-=[11, 37, 67]-=- and the references there. We denote by MR(U) the signed Radon measures on the open subset U of X and by MR,0(U) the subset of measures ν that do not charge sets of capacity 0, i.e., those measures wi... |

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