Boundedness of Riesz-Type Potential Operators on Variable Exponent Herz–Morrey Spaces

We show the boundedness of the Riesz-type potential operator of variable order β(x) from the variable exponent Herz – Morrey spaces MK̇p1,q1⋅α⋅,λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ M{\dot{K}}_{p_1,{q}_1\left(\cdot \right)}^{\upalpha \left(\cdot \right),\lambda } $$\end{document} (ℝn) into the weighted space MK̇p2,q2⋅α⋅,λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ M{\dot{K}}_{p_2,{q}_2\left(\cdot \right)}^{\upalpha \left(\cdot \right),\lambda } $$\end{document} (ℝn,ω) where 𝛼(x) 𝜖 L∞(ℝn) is log-Hölder continuous both at the origin and at infinity, ω = (1 + |x|)−γ(x) with some γ(x) > 0, and1/q1(x) − 1/q2(x) = β(x)/n when q1(x) is not necessarily constant at infinity. It is assumed that the exponent q1(x) satisfies the logarithmic continuity condition both locally and at infinity and, moreover, 1 < (q1)∞ ≤ q1(x) ≤ (q1) +  <  ∞ ,  x∈ ℝn.