On the boundary value of Besov-Bergman spaces (Q1087792)

The author of this paper has previously, in 9 papers, introduced and studied some function spaces, denoted by \(B^ p\). These spaces are: \[ B^ p=\{f| f: [-\pi,\pi)\to {\mathbb{R}},\quad f(t)=\sum c_ nb_ n(t),\quad \sum | c_ n| <\infty \}, \] where \((c_ n)\) are sequences of real numbers and \((b_ n)\) are special atoms, that is functions \(b:[-\pi,\pi)\to {\mathbb{R}}\), defined by \(b(t)=\pi\), or for each interval \(I\subseteq [-\pi,\pi)\), \(b(t)=-| I|^{-1/p}\chi_ R(t)+| I|^{-1/p}\chi_ L(t)\), \(\chi_ E\) is the characteristic function of E. These space \(B^ p\) are endowed with a norm: \(\| f\|_{B^ p}=\inf \sum | c_ n|\), where the infimum is taken over all possible representations of f. In this paper, the author proves an embedding theorem of the space \(B^ p\) in a Besov-Bergman space and that \(B^ p\) is dense in this space. With these results he proves that \(f\in B^ p\), for \(1<p<\infty\), if and only if f belongs to a Besov-Bergman space. Well, this characterization is for a better understanding of the Besov- Bergman spaces, but it is for a better understanding of the \(B^ p\) spaces, too.