Fractional Type Integral Operators on Variable Hardy Spaces

Given certain n × n invertible matrices A1, . . . , Am and 0 ≦ α < n, we obtain the Hp(.)(Rn)→Lq(.)(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H^{p(.)}(\mathbb{R}^n) \to L^{q(.)}(\mathbb{R}^n)}$$\end{document} boundedness of the integral operator with kernel k(x,y)=|x-A1y|-α1...|x-Amy|-αm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${k(x, y) = |x - A_1y|^{-\alpha_1} . . . |x - A_my|^{-\alpha_m}}$$\end{document}, where α1 +  . . . + αm = n − α and p(.), q(.) are exponent functions satisfying log-Hölder continuity conditions locally and at infinity related by 1q(.)=1p(.)-αn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\frac{1}{q(.)} = \frac{1}{p(.)} - \frac{\alpha}{n}}$$\end{document}. We also obtain the Hp(.)(Rn)→Hq(.)(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H^{p(.)}(\mathbb{R}^n) \to H^{q(.)}(\mathbb{R}^n)}$$\end{document} boundedness of the Riesz potential operator.