1
$\begingroup$

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 $\delta$)

$\endgroup$
0

1 Answer 1

2
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.