Wavelets and geometric structure for function spaces (Q1887428)

The dual spaces \(\dot{D}^{s,q}_ p\) of the homogeneous Triebel--Lizorkin spaces \(\dot{F}^{s,q}_ p\), \(s\in\mathbb R\), \(0<p\leq1\leq q\leq\infty\), are investigated. In this case, the structure of the dual spaces \(\dot{D}^{s,q}_ p\) is very different from that of Besov spaces or that of Triebel--Lizorkin spaces and cannot be analysed easily in the Littlewood--Paley analysis. The author characterizes the spaces \(\dot{D}^{s,q}_ p\), where \(s\in\mathbb R\), \(p=1\) and \(1\leq q\leq\infty\), in tent spaces with wavelets. Moreover, some inclusion relations between the spaces \(\dot{D}^{s,q}_ p\) and some relations between the spaces \(\dot{B}^{0,q}_1\), \(\dot{B}^{0,q}_1\) and \(L^1\) are studied.