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In this paper we present an improvement of [Math. Ann. 345 (2009), 213--243],where the authors proved a result concerning continuous dependence for backwardparabolic 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 technicalreasons. Here we remove this assumption, replacing it withLipschitz-continuity. The main tools in the proof are Littlewood-Paley theoryand Bony's paraproduct as well as a result of Coifman and Meyer [Ast\'erisque57, 1978, Th. 35].

Conditional stability for backward parabolic equations with $\rm{Log}\rm{Lip}_t \times \rm{Lip}_x$-coefficients