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

Abstract

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]),

De Rham's theorem for Orlicz cohomology

“…Orlicz cohomology has been studied recently (see [4,6,9,10,11,12]) as a natural generalization of L p -cohomology. These cohomology theories provide quasi-isometry invariants (and therefore have applications to classification problems, see for example [4,14,17]), and have connections with Sobolev and Poincaré type inequalities ( [6,8]) and harmonic functions ( [11,15]).…”