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Starting from the known positive two-sided comparison for the kernel of $H_a^{-s/2}$, we determine the complete strong non-endpoint mapping range for two power weights. If $\\sigma=(d-2-\\sqrt{(d-2)^2+4a})/2$ and $0<s<d-2\\sigma$, then [\n||x|^{-\\beta}H_a^{-s/2}f|{L^q} \\lesssim ||x|^\\alpha f|{L^p} ]\nholds for $1<p,q<\\infty$ precisely under the exponent ordering, scaling, sum, and origin conditions stated in the main theorem. 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Starting from the known positive two-sided comparison for the kernel of $H_a^{-s/2}$, we determine the complete strong non-endpoint mapping range for two power weights. If $\\sigma=(d-2-\\sqrt{(d-2)^2+4a})/2$ and $0<s<d-2\\sigma$, then [\n||x|^{-\\beta}H_a^{-s/2}f|{L^q} \\lesssim ||x|^\\alpha f|{L^p} ]\nholds for $1<p,q<\\infty$ precisely under the exponent ordering, scaling, sum, and origin conditions stated in the main theorem. 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