We show that if « is a solution to £]" ,=1 a¡¡(x, t)D¡jU(x, t) -Dtu(x, t) = (¡>(x) on a cylinder ÎÎ7-= Q. x (0, T), where ii is a bounded open set in R" , T > 0 , and u vanishes continuously on the parabolic boundary of Clj . Then the maximum of u on the cylinder is bounded by a constant C depending on the ellipticity of the coefficient matrix (a¡j(x, t)), the diameter of Q , and the dimension n times the L" norm of in Cl.