  In this paper we present an improvement of [Math. Ann. 345 (2009), 213--243],
where the authors proved a result concerning continuous dependence for backward
parabolic operators whose coefficients are Log-Lipschitz in $t$ and $C^2$ in
$x$. The $C^2$ regularity with respect to $x$ had to be assumed for technical
reasons. Here we remove this assumption, replacing it with
Lipschitz-continuity. The main tools in the proof are Littlewood-Paley theory
and Bony's paraproduct as well as a result of Coifman and Meyer [Ast\'erisque
57, 1978, Th. 35].
