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_{\mathbb{R^+}}e^{(1+i)(t-s)\Delta}F(s)ds\right\|_{L^2(\mathbb{R^+},W^{1,6}(\mathbb{R}^3))}\lesssim \|F\|_{L^2(\mathbb{R^+},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.