New $2$-microlocal Besov and Triebel–Lizorkin spaces via the Litllewood–Paley decomposition

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