  We suggest a modification of the estimate for weighted Sobolev norms of
solutions of parabolic equations such that the matrix of the higher order
coefficients is included into the weight for the gradient. More precisely, we
found the upper limit estimate that can be achieved by variations of the zero
order coefficient. As an example of applications, an asymptotic estimate was
obtained for the gradient at initial time. The constant in the estimates is the
same for all possible choices of the dimension, domain, time horizon, and the
coefficients of the parabolic equation. As an another example of application,
existence and regularity results are obtained for parabolic equations with time
delay for the gradient.
