New 2-microlocal Besov and Triebel-Lizorkin spaces via the Littlewood-Paley decomposition

Abstract: In this paper we introduce and investigate new 2-microlocal Besov and Triebel-Lizorkin space via the Littlewood-Paly decomposition. We establish characterizations of these function spaces by the $phi$-transform, the atomic and molecular decomposition and the wavelet decomposition. As applications we prove boundedness of Caldeon-Zygmund operators and the pseudo-differential operators on the function spaces. Moreover, we give characterizations via oscillations and differences.