We give the atomic decomposition of the inhomogeneous Besov spaces defined on symmetric Riemannian spaces of noncompact type. As an application we get a theorem of Bernstein type for the Helgason-Fourier transform.
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The aim of the paper is twofold. First we give a survey of some recent results concerning the asymptotic behavior of the entropy and approximation numbers of compact Sobolev embeddings. Second we prove new estimates of approximation numbers of embeddings of weighted Besov spaces in the so called limiting case.
We study embeddings of spaces of Besov-Morrey type, $id_{Ω}: 𝓝^{s₁}_{p₁,u₁,q₁}(Ω) ↪ 𝓝^{s₂}_{p₂,u₂,q₂}(Ω)$, where $Ω ⊂ ℝ^{d}$ is a bounded domain, and obtain necessary and sufficient conditions for the continuity and compactness of $id_{Ω}$. This continues our earlier studies relating to the case of $ℝ^{d}$. Moreover, we also characterise embeddings into the scale of $L_{p}$ spaces or into the space of bounded continuous functions.
We investigate borderline traces of Besov and Triebel-Lizorkin spaces. The function spaces are defined on noncompact Riemannian manifolds with bounded geometry. We described spaces of traces on noncompact submanifolds that are also of bounded geometry.
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We continue our earlier investigations of radial subspaces of Besov and Lizorkin-Triebel spaces on $ℝ^d$. This time we study characterizations of these subspaces by differences.
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