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Orlicz spaces of differential forms on Riemannian manifolds: duality and cohomology. (English) Zbl 1394.58002

The paper under review is devoted to the study of Orlicz spaces of differential forms on Riemannian manifolds. Let \(X\) be a Riemannian manifold of dimension \(n\) and \(\omega\) be a \(k\)-form on \(X\). For an \(N\)-function \(\Phi\), let \(\rho_\Phi(\omega)=\int_X\Phi(|\omega(x)|)d\mu_X\), where \(d\mu_X\) stands for the volume element of \(X\). Let \(L^\Phi(X,\Lambda^k)\) be the Orlicz space of all measurable \(k\)-forms \(\omega\) such that \(\rho_\Phi(\alpha\omega)<\infty\) for some \(\alpha>0\) and \(M^\Phi(X,\Lambda^k)\) be the corresponding Morse-Transue space of all measurable \(k\)-forms \(\omega\) such that \(\rho_\Phi(\alpha\omega)<\infty\) for all \(\alpha>0\). Let \(\|\cdot\|_\Phi\) and \(\|\cdot\|_{(\Phi)}\) denote the Luxemburg and the Orlicz norms, respectively.
It is shown that if \(\Phi\) and \(\Psi\) are complementary \(N\)-functions, then the correspondence \(\theta\mapsto F_\theta\), where \(F_\theta(\omega)=\int_X \omega\wedge\theta\), yields isometric isomorphisms between the Orlicz space \((L^\Psi(X,\Lambda^{n-k}),\|\cdot\|_{(\Psi)})\) and the dual of the Morse-Transue space \((M^\Phi(X,\Lambda^k),\|\cdot\|_\Phi)'\), as well as, between \((L^\Psi(X,\Lambda^{n-k}),\|\cdot\|_{\Psi})\) and \((M^\Phi(X,\Lambda^k),\|\cdot\|_{(\Phi)})'\). This result implies other duality theorems. The obtained duality results are applied to establish the Hölder-Poincaré duality for the reduced Orlicz cohomology of \(X\).

MSC:

58A12 de Rham theory in global analysis
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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