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We propose and prove a new fractional Leibniz rule for $D^s=(-\\Delta)^{s/2}$ and similar operators, generalizing the Kenig-Ponce-Vega estimate \\cite{KPV93} to all $s>0$. We also prove a family of generalized and refined Kato-Ponce type inequalities which include many commutator estimates as special cases. 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This solves a question raised in Kato-Ponce \\cite{KP88}. We propose and prove a new fractional Leibniz rule for $D^s=(-\\Delta)^{s/2}$ and similar operators, generalizing the Kenig-Ponce-Vega estimate \\cite{KPV93} to all $s>0$. We also prove a family of generalized and refined Kato-Ponce type inequalities which include many commutator estimates as special cases. 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