Variable Exponent Herz Spaces

We introduce a new type of variable exponent function spaces  Ḣp(·),q(·),α(·)(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^n}$$\end{document}) and Hp(·),q(·),α(·)(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^n}$$\end{document}) of Herz type, homogeneous and non-homogeneous versions, where all the three parameters are variable, and give comparison of continual and discrete approaches to their definition. Under the only assumption that the exponents p, q and α are subject to the log-decay condition at infinity, we prove that sublinear operators, satisfying the size condition known for singular integrals and bounded in Lp(·)(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^n}$$\end{document}), are also bounded in the nonhomogeneous version of the introduced spaces, which includes the case maximal and Calderón-Zygmund singular operators.