Frazier, Michael; Jawerth, Björn Decomposition of Besov spaces. (English) Zbl 0551.46018 Indiana Univ. Math. J. 34, 777-799 (1985). Using a general decomposition result for distributions and a classical theorem of Plancherel-Pólya, we obtain two decompositions of elements of the Besov spaces \(\dot B_ p^{\alpha q}({\mathbb{R}}^ n)\) and \(B_ p^{\alpha q}({\mathbb{R}}^ n),\) \(-\infty <\alpha <+\infty\), \(0<p,q\leq +\infty\), into sums of basic functions. In the first decomposition, the basic functions are similar to the atoms in the atomic decomposition of the Hardy spaces. In the second, they are taken from a fixed family of rapidly decreasing functions whose Fourier transforms have compact support. We also give a decomposition of the second type for BMO. Simple consequences are that the trace of \(\dot B_ p^{q/p}\) or \(B_ p^{q/p}\) is \(L^ p\) if \(q\leq \min (1,p),\) \(0<p<+\infty\), and that the Besov spaces have the lower majorant property if \(0<p\leq 1\). Cited in 12 ReviewsCited in 303 Documents MSC: 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 42B30 \(H^p\)-spaces Keywords:decompositions; Besov spaces; atomic decomposition of the Hardy spaces; Fourier transforms; BMO; trace; lower majorant property PDFBibTeX XMLCite \textit{M. Frazier} and \textit{B. Jawerth}, Indiana Univ. Math. J. 34, 777--799 (1985; Zbl 0551.46018) Full Text: DOI