Sharp upper bounds on hitting probabilities for the solution to the stochastic heat equation on the line

For Gaussian random fields with values in $\mathbb{R}^d$, sharp upper and lower bounds on the probability of hitting a fixed set have been available for many years. These apply in particular to the solutions of systems of linear SPDEs. For non-Gaussian random fields, the available bounds are less sharp. For nonlinear systems of stochastic heat equations, a sharp lower bound was obtained in a previous paper by two of the authors. Here, we obtain the corresponding sharp upper bound. The proof requires a bound on the joint probability density function of a two-dimensional random vector whose components are the solution to the {\em nonlinear} stochastic heat equation and the supremum over a small rectangle of the solution to the {\em linear} stochastic heat equation, in terms of the size of the rectangle. This bound makes use of a formula that expresses the density of a {\em locally nondegenerate} random vector as an iterated Skorohod integral. The main effort is to estimate, using Malliavin calculus, each of the terms that arise from this formula.