Let $f$ be a function satisfying the diffusion equation, the three domination conditions, and the boundary condition $\lim_{t\to 0^+} f(t,x) = g_\sigma(x)$. Justify that, for any real $t > 0$ and any real $\xi$, the function $x \mapsto f(t, x) \exp(-2\mathrm{i}\pi \xi x)$ is integrable on $\mathbb{R}$.