We show the boundedness of the Riesz-type potential operator of variable order β(x) from the variable exponent Herz – Morrey spaces \( M{\dot{K}}_{p_1,{q}_1\left(\cdot \right)}^{\upalpha \left(\cdot \right),\lambda } \) (ℝn) into the weighted space \( M{\dot{K}}_{p_2,{q}_2\left(\cdot \right)}^{\upalpha \left(\cdot \right),\lambda } \) (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.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 9, pp. 1187–1197, September, 2017.
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Wu, JL. Boundedness of Riesz-Type Potential Operators on Variable Exponent Herz–Morrey Spaces. Ukr Math J 69, 1379–1392 (2018). https://doi.org/10.1007/s11253-018-1438-7
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DOI: https://doi.org/10.1007/s11253-018-1438-7