Weighted Triebel-Lizorkin spaces with variable smoothness and integrability

In this paper, we introduce the weighted Triebel-Lizorkin spaces of variable integral, smooth and summation exponents with variable Muckenhoupt weights. To make these spaces definite, we provide the weighted vector-valued convolution inequality and the Fourier multiplier theorem on these spaces. We then obtain a characterization of these spaces via approximation by analytic functions. Furthermore, we obtain embeddings, the lifting property, and duality of these spaces, respectively. Finally, we study the real interpolation in these spaces.