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The main goal of the paper is to establish time semidiscrete and space-timefully discrete maximal parabolic regularity for the lowest order timediscontinuous Galerkin solution of linear parabolic equations withtime-dependent coefficients. Such estimates have many applications. As one ofthe applications we establish best approximations type results with respect tothe $L^p(0,T;L^2(\Omega))$ norm for $1\le p\le \infty$.

Discrete maximal parabolic regularity for Galerkin finite element methods for non-autonomous parabolic problem s