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This paper concerns the validity of the optimal Moser inequality\n  \\[ \\left(\\int_M |u|^r\\; dv_g \\right)^{\\frac{\\tau}{p}} \\leq \\left( A(p,n)^{\\frac{\\tau}{p}} \\left(\\int_M |\\nabla_g u|^p\\; dv_g\\right)^{\\frac{\\tau}{p}} + B_{opt} \\left(\\int_M |u|^p\\; dv_g\\right)^{\\frac{\\tau}{p}} \\right) \\left( \\int_M |u|^p\\; dv_g \\right)^{\\frac{\\tau}{n}} \\; . \\]\n  This kind of inequality was already studied in the last years in the particular cases $1 < p < n$. 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