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We re-examine the combined semi-classical and mean-field limit in the$N$-body fermionic Schr\"odinger equation with pure state initial data usingthe Husimi measure framework. The Husimi measure equation involves threeresidue types: kinetic, semiclassical, and mean-field. The main result of thispaper is to provide better estimates for the kinetic and mean-field residuethan those in \cite{Chen2021JSP}. Especially, the estimate for the mean-fieldresidue is shown to be smaller than the semiclassical residue by a mixed-normestimate of the two-particle reduced density matrix factorization. Our analysisalso updates the oscillation estimate parts in the residual term estimatesappeared in \cite{Chen2021JSP}.

A mixed-norm estimate of the two-particle reduced density matrix of many-body Schr\"odinger dynamics for deriving the Vlasov equation