Characterizations of \(B^u_\omega \) type Morrey–Triebel–Lizorkin spaces with variable smoothness and integrability

In this paper, we first obtain Fourier multiplier theorem, the approximation characterization and embedding for \(B^u_\omega \) $B_{\omega }^u$ type Morrey–Triebel–Lizorkin spaces with variable smoothness and integrability. Then, we characterize these spaces via Peetre’s maximal functions, the Lusin area function, and the Littlewood–Paley \(g^*_\lambda \) $g_{\lambda }^{\ast }$ -function. Finally, we obtain the boundedness of the pseudo-differential operators on these spaces.