Herz-type Triebel–Lizorkin Spaces, I

Let s ∈ ℝ, 0 < β ≤ ∞, 0 < q, p < ∞ and –n/q < α. In this paper the authors introduce the Herz-type Triebel–Lizorkin spaces, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ K^{{\alpha ,p}}_{q} F^{s}_{\beta } {\left( {\mathbb{R}^{n} } \right)} $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \dot{K}^{{\alpha ,p}}_{q} F^{s}_{\beta } {\left( {\mathbb{R}^{n} } \right)}, $$\end{document}which are the generalizations of the well-known Herz-type spaces and the inhomogeneous Triebel–Lizorkin spaces. Some properties on these Herz-type Triebel–Lizorkin spaces are also given.