A well-posedness result for hyperbolic operators with Zygmund coefficients

Abstract

In this paper we prove an energy estimate with no loss of derivatives for a strictly hyperbolic operator with Zygmund continuous second order coefficients both in time and in space. In particular, this estimate implies the well-posedness for the related Cauchy problem. On the one hand, this result is quite surprising, because it allows to consider coefficients which are not Lipschitz continuous in time. On the other hand, it holds true only in the very special case of initial data in $H^{1/2}\times H^{-1/2}$. Paradifferential calculus with parameters is the main ingredient to the proof.