the so-called Kannai transform expressing solutions of the heat equation in terms of solutions of the wave equation, and the work [@Kannai92] proposes a study of singular problems based on a representation formula adding one dimension to the problem.

In the context of control theory in which we focus here, the so-called Fourier Bros Iagnoniltzer transform, linking solutions of the wave equation to a suitable elliptic operator, provides an efficient tool to prove quantified unique continuation properties, see e.g. [@Robbiano91; @Robbiano95; @Phung09].

Similarly, using a suitable transformation linking the wave and heat equations, Miller in [@Miller06a; @Miller06b] derived estimates on the cost of controllability of the heat equations in small time, later improved in [@TucsnakTenenbaumTransAMS]. Similar estimates, related to the characterization of the reachable set for heat equations, can be found in [@ErvZuazuaARMA]. The main difference between the transforms in [@Miller06a; @Miller06b] and [@ErvZuazuaARMA] is that, whereas the articles [@Miller06a; @Miller06b] express solutions of the heat equations in terms of solutions of the wave equation, [@ErvZuazuaARMA] is based on a transform expressing solutions of the wave equation in terms of the solutions of the heat equation. Note that the robustness of the transformation of [@ErvZuazuaARMA] is illustrated by the fact 