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 $\delta$)



Estimates on evolution operator
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 $y=x/(\delta t)^{1/2}$.


