The Hölder-Poincaré duality for L q,p -cohomology

Gol’dshtein, Vladimir; Troyanov, Marc

doi:10.1007/s10455-011-9269-x

Cited by 5 publications

(9 citation statements)

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“…The so-obtained results are applied for establishing the Hölder-Poincaré duality for the reduced Orlicz cohomology of X, which extends the Hölder-Poincaré duality for L q,p -cohomology proved by Gol dshtein and Troyanov in [11].…”

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confidence: 70%

“…In studying the asymptotic invariants of infinite groups and manifolds with pinched negative curvature, Gromov and Pansu also considered L p -differential forms and l p -simplicial cochains (see [12,18,19]). Gol dstein and Troyanov obtained deep results about the L qp -cohomology of Riemannian manifolds for q = p in [9,10,11].…”

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confidence: 99%

“…In [11], Gol dshtein and Troyanov realized the kth L q,p -cohomology as the kth cohomology of some Banach complex. Here we apply this approach to L Φ I , Φ II -cohomology.…”

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confidence: 99%

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Orlicz Spaces of Differential Forms on Riemannian Manifolds: Duality and Cohomology

Kopylov¹

2017

Issues of Analysis

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Abstract. We consider Orlicz spaces of differential forms on a Riemannian manifold. A Riesz-type theorem about the functionals on Orlicz spaces of forms is proved and other duality theorems are obtained therefrom. We also extend the results on the Hölder-Poincaré duality for reduced L q,p -cohomology by Gol dshtein and Troyanov to L Φ I ,Φ II -cohomology, where Φ I and Φ II are N -functions of class ∆ 2 ∩ ∇ 2 . Introduction. This article is devoted to the study of the dual spaces of Orlicz spaces of differential forms on an oriented Riemannian manifold X.L p -theory of differential forms on Riemannian manifolds has been the subject of many papers and several books since the beginning of the 1980s. In 1976, Atiyah defined L 2 -cohomology for a Riemannian manifold and initiated various applications of L 2 -methods to the study of noncompact manifolds and quotient spaces of Riemannian manifolds by discrete groups of isometries. The L 2 -cohomology of such manifolds was studied by Gromov, Cheeger-Gromov and others (see, for example, [2,3,12]). In the 1980's, Goldshtein, Kuz minov, and Shvedov defined the L p -de Rham complex on a Riemannian manifold M for arbitrary p ∈ [1, ∞] and began to investigate its cohomology, which they called the L p -cohomology of M ; they obtained many results concerning the density of smooth forms in L p (see, for example, [5]); the nontriviality and the Hausdorff property of L pcohomology on important classes of manifolds (see, for instance, [7,8,17]),

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Orlicz Spaces of Differential Forms on Riemannian Manifolds: Duality and Cohomology

Kopylov¹

2017

Issues of Analysis

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“…In order to complete the proof of the first point we are left to show (15). To this aim it is enough to show that for each point (12) and (14). Using again [15] Th.…”

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confidence: 99%

Symplectic manifolds, L-cohomology and q-parabolicity

Bei

2019

Differential Geometry and its Applications

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Let (M, ω, J, g) be a non-compact almost Kähler manifold. In this paper we provide various criteria that assure that ω k induces a non trivial class in the reduced L p maximal/minimal cohomology of (M, g). Furthermore in the last part we explore some topological applications of our results. Theorem 0.3. Let (X, h) be a compact and irreducible Kähler space of complex dimension m. Assume that every point p ∈ sing(X) has a local base of open neighborhoods whose regular parts are connected. Then H 2k (X, R) = {0}for each k = 0, 1..., m. In particular if X is a compact and irreducible normal Kähler space thenfor each k = 0, ..., m.Acknowledgments. This work was performed within the framework of the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). It is a pleasure to thank Markus Banagl, Jacopo Gandini, Luca Migliorini and Jon Woolf for helpful conversations.

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“…for any complete manifold. Our proof can also be extended to the more general L q,p -cohomology, see [6].…”

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confidence: 90%