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Homogeneous Besov spaces. (English) Zbl 1437.42034

Summary: This note is based on a series of lectures delivered at Kyoto University in 2015. This note surveys the homogeneous Besov space \(\dot{B}^s_{pq}\) on \(\mathbb{R}^n\) with \(1\le p\), \(q\le \infty\) and \(s\in\mathbb{R}\) in a rather self-contained manner. Among other results, we show that \(\mathcal{S}'_{\infty}\) and \(\mathcal{S}^\prime/\mathcal{P}\) are isomorphic, and we also discuss the realizations in \(\dot{B}^s_{pq}\). The fact that \(\mathcal{S}^\prime_{\infty}\) and \(\mathcal{S}^\prime/\mathcal{P}\) are isomorphic can be found in textbooks. The realization of \(\dot{B}^s_{pq}\) can be found in works by H. Bahouri et al. [Fourier analysis and nonlinear partial differential equations. Berlin: Heidelberg (2011; Zbl 1227.35004)] and by G. Bourdaud [Ark. Mat. 26, No. 1, 41–54 (1988; Zbl 0661.46026)] for example. Here, we prove these facts using fundamental results in functional analysis such as the Hahn-Banach extension theorem.

MSC:

42B35 Function spaces arising in harmonic analysis
46A04 Locally convex Fréchet spaces and (DF)-spaces
30H05 Spaces of bounded analytic functions of one complex variable
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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