Wavelets and Geometric Structure for Function Spaces

With LittlewoodPaley analysis, Peetre and Triebel classified, systematically, almost all the usual function spaces into two classes of spaces: Besov spaces \ifmmode\expandafter\else\expandafter\.\fiBs,qp(sR,0>p,q) and TriebelLizorkin spaces \ifmmode\expandafter\else\expandafter\.\fiFs,qp(sR,0>p>,0>q); but the structure of dual spaces \ifmmode\expandafter\else\expandafter\.\fiDs,qp of \ifmmode\expandafter\else\expandafter\.\fiFs,qp(sR,0>p1q) is very different from that of Besov spaces or that of TriebelLizorkin spaces, and their structure cannot be analysed easily in the LittlewoodPaley analysis. Our main goal is to characterize \ifmmode\expandafter\else\expandafter\.\fiDs,qp(sR,0>p=1q) in tent spaces with wavelets. By the way, some applications are given: (i) TriebelLizorkin spaces for p = defined by LittlewoodPaley analysis cannot serve as the dual spaces of TriebelLizorkin spaces for p = 1; (ii) Some inclusion relations among these above spaces and some relations among \ifmmode\expandafter\else\expandafter\.\fiB0,q1,\ifmmode\expandafter\else\expandafter\.\fiF0,q1 and L 1 are studied.