Consider the heat-Schrodinger evolution $e^{(1+i)t\Delta}$, $t\geq 0$. For simplicity, suppose to work in dimension three. Due to the Strichartz estimates for the Schrodinger equation, and the smoothing effect of the heat flow, I expect that the following estimate holds true:
\begin{equation}\label{tre}
(1)\qquad \left\|\int_0^te^{(1+i)(t-s)\Delta}F(s)ds\right\|_{L^2((0,t),W^{1,6}(\mathbb{R}^3))}\lesssim \|F\|_{L^2((0,t),L^2(\mathbb{R}^3))}
\end{equation}
Nevertheless, I have not been able to prove it.

**Is estimate (1) actually true? In case, is it already known in the literature?**

Thank you for your suggestions.