The question is originally related to Hardy's uncertainty principle, convexity and Schrodinger evolutions. In this work the authors deduce a convex property of Schrodinger equation by doing it first with the heat equation of complex time.

The first step to go is to get an energy estimate of the heat equation of complex time with the time-depending Guassian weight, simply saying which is:

$$\|e^{\frac{A\alpha|.|^2}{A+4\alpha t(A^2+B^2)}}u(t)\|_{L^2} \le \|e^{\alpha|.|^2}u(0)\|_{L^2}$$

where $u(t)$ satifies $\partial_t u = (A+iB)\Delta_x u$ in $L^2$, $A, \alpha>0$, $B$ is real, and $\|e^{\alpha|.|^2}u(0)\|_{L^2} < \infty$.

This weighted energy estimate confirms the validity of later calculations, and **it's done by the energy method**. Note that the decay constant ${A\alpha}/(A+4\alpha t(A^2+B^2))$ is what I guess optimal here, though this character wasn't used in that work.

What I'm thinking about is a version of this estimate for higher order heat equation of complex time, like if $u(t)$ satifies $\partial_t u = -(A+iB)(-\Delta_x)^m u$ in $L^2$, where $m$ is a positive integer, $A, p, \alpha>0$, $B$ is real, and $\|e^{\alpha|.|^p}u(0)\|_{L^2} < \infty$.

Q1: What $p$ will permit an estimate like $\|e^{\beta(t)|.|^p}u(t)\|_{L^2} \le \|e^{\alpha|.|^p}u(0)\|_{L^2}$ for this equation? Actually I can do it when $p = \frac{2m}{2m-1}$ which is the decay index of this equation's kernel, and also **I get the same result by using the heat kernel when $m=1$**.

What I'm more intrested in is the next one

Q2: What's the **optimal** time-depending relation $\beta(t)$ for a given $\alpha > 0$ if $p = \frac{2m}{2m-1}$?