Conditional stability for backward parabolic equations with LogLipt×Lipx-coefficients

In this work we present an improvement of Del Santo and Prizzi (2009), where the authors proved a result concerning continuous dependence for backward-parabolic operators whose coefficients are Log-Lipschitz in tt and C2C2 in xx. In that paper, the C2C2 regularity with respect to xx 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.