We remark that in the first theorem, γ 2 can be arbitrarily large, but s 1 > −1, while in the second theorem γ 2 < 2n/p, but for sufficiently large γ 1 and sufficiently small γ 2 , s 1 > γ 2 − n/p − 1 can be less than −1. Thus the non-standard product estimate allows us to obtain existence results for initial **data** with lower regularity, but requires γ 2 to be small and requires the use of Besov spaces.