Gaussian estimates for fundamental solutions to certain parabolic systems (Q1768147)

\textit{P. Auscher} [J. Lond. Math. Soc., II. Ser. 54, No. 2, 284--296 (1996; Zbl 0863.35020)] proved Gaussian upper bound estimates for the fundamental solutions to parabolic equations with complex coefficients in the case when the coefficients are time-independent and a small perturbation of real coefficients. In the present paper the authors prove the equivalence between the local boundedness property of solutions to a parabolic system and a Gaussian upper bound for its fundamental matrix. As a consequence, the Auscher result is extended to the time dependent case.