Conditional stability for backward parabolic equations with LogLip_txLip_x-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 C^2 in x. In that paper, 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.