Note on quantitative homogenization results for parabolic systems in [Formula: see text]

In [Formula: see text] , we consider a semigroup [Formula: see text] , [Formula: see text] , generated by a matrix elliptic second-order differential operator [Formula: see text] . Coefficients of [Formula: see text] are periodic, depend on [Formula: see text] , and oscillate rapidly as [Formula: see text] . Approximations for [Formula: see text] were obtained by Suslina (Funktsional Analiz i ego Prilozhen 38(4):86–90, 2004) and Suslina (Math Model Nat Phenom 5(4):390–447, 2010) via the spectral method and by Zhikov and Pastukhova (Russ J Math Phys 13(2):224–237, 2006) via the shift method. In the present note, we give another short proof based on the contour integral representation for the semigroup and approximations for the resolvent with two-parametric error estimates obtained by Suslina (2015).