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We deal with solutions of the Cauchy problem to linear both homogeneous andnonhomogeneous parabolic second order equations with real constant coefficientsin the layer ${\mathbb R}^{n+1}_T={\mathbb R}^n\times (0, T)$, where $n\geq 1$and $T<\infty$. The homogeneous equation is considered with initial data in$L^p({\mathbb R}^n)$, $1\leq p \leq \infty $. For the nonhomogeneous equationwe suppose that initial function is equal to zero and the function in theright-hand side belongs to $f\in L^p({\mathbb R}^{n+1}_T)\cap C^\alpha \big(\bar{{\mathbb R}^{n+1}_T} \big ) $ , $p>n+2$ and $\alpha \in (0, 1)$. Explicitformulas for the sharp coefficients in pointwise estimates for the length ofthe gradient to solutions to these problems are obtained.

Sharp pointwise estimates for the gradients of solutions to linear parabolic second order equation in the layer