Let $G=\mathbb D\times\mathbb C$, where $\mathbb D$ is the open unit disk on the complex plane $\mathbb C$. In $G$ we consider the analytic solutions $u(t,z)$ $(t\in \mathbb D$, $z\in\mathbb C$) of the heat equation $2u_t=u_{zz}$ with initial data $f(z)=u(0,z)$ belonging to the Fock space $F$, i.e., to the space of entire functions square summable with the weight $e^{-|z|^2}$. Conditions on a nonnegative measure $\mu$ on $G$ are described under which for all $f\in F$ we have $$ \|u,L^2(G,\mu )\|\le C\|f,L^2(\mathbb C,e^{-|z|^2})\|. $$