Estimates on evolution operator

Question

Let's consider the following evolution operator in $\mathbb{R}^3$ $$S(t)=e^{(i+\delta)t\Delta }$$ How to get the following estimate $$\Vert S(t)f\Vert_2\leq C_\varepsilon t^{-\frac{1}{4}}\Vert f\Vert_{\frac{3}{2}}$$? ($\Vert f\Vert_p$ is the standard Lebesgue norm and $C_\delta$ is a constant depending on $\delta$)

Answer 1

Since $e^{it\Delta}$ is unitary, it suffices to consider $e^{\delta t\Delta}f=K_{\delta t}*f$, where $$ K_s(x)= \frac{1}{(4\pi s)^{d/2}} e^{-|x|^2/(4s)} $$ is the heat kernel (I'll do it for general dimension $d$, which isn't any harder). By Young's inequality, $\|K*f\|_2\le \|K\|_{6/5}\|f\|_{3/2}$, and $$ \|K_{\delta t}\|_{6/5} = C_0(\delta t)^{-d/2} \left(\int e^{-3|x|^2/(10\delta t)} dx \right)^{5/6} = C (\delta t)^{-d/12} , $$ as desired. The last step is by the substitution $y=x/(\delta t)^{1/2}$.