Evolution equation generated by Fourier multiplier I am on the hunt for techniques regarding a field which I am not familiar with. 
More precisely, I am considering equation of the form
$$ i \partial_{t} u(t,x) + p(D)u(t,x) = 0, \ \ u_{|_{t=0}}=u_0(x), \ \ x \in \mathbb{R}^{d} $$
where $p(D)$ is the Fourier multiplier with symbol $p(\xi)$. 
In particular, I am interested in studying the possible $L^{p} \rightarrow L^{p}$ estimates for the solution operator.
I would be grateful if anyone could provide with a sketch as to where we currently stand for this problem, many thanks in advance!
 A: Formally you get
$
u(t)=e^{it p(D)} u_0
$
and this means that 
$$
u(t,x)=\int e^{i2π x\cdot \xi}e^{it p(\xi)}\widehat{u_0}(\xi) d\xi. 
$$
Obviously, you have to require something to give a meaning to that integral, even in a weak sense. If you assume first that $p$ is real-valued on $\mathbb R^n$ and $u_0$ belongs to the Schwartz space, then the above integral is absolutely converging and extends to a bounded operator on $L^2(\mathbb R^n)$ since it corresponds to a bounded Fourier multiplier. The same is true ne varietur if
$$
\Im p(\mathbb R^n)\subset \mathbb R_+.
$$
Let us stick to the case $p(\mathbb R^n)\subset \mathbb R$, definitely not parabolic as pointed out in a comment. The kernel of the above operator is 
$$
k(t,x,y)=\int e^{i2π (x-y)\cdot \xi}e^{it p(\xi)} d\xi, \quad\text{its symbol is } a(t,\xi)=e^{it p(\xi)}.
$$
The classical conditions of the Hörmander-Mikhlin $L^p$ boundedness Theorem are not met by this symbol, whose first derivative could be large. 
On the other hand if you assume some positive ellipticity for the imaginary part, e.g. 
$p(\xi)=q(\xi) +i\vert \xi\vert$, $q$ real-valued with $\vert{q'(\xi)}\vert\le C$,
you find
$$
a(t,\xi)= e^{it q(\xi)} e^{-t \vert \xi\vert}, \quad 
\partial_\xi a(t,\xi)= e^{it q(\xi)} e^{-t \vert \xi\vert}\bigl(it q'(\xi)-t\frac{\xi}{\vert \xi\vert}\bigr)\text{ is bounded},
$$
and 
$$
\vert\partial_\xi a(t,\xi)\vert\le e^{-t \vert \xi\vert}\frac{1}{\vert \xi\vert}\bigl(Ct\vert \xi \vert+ t\vert \xi\vert\bigr)\le (C+1)\vert \xi\vert^{-1},
$$
and you are in good shape to obtain $L^p$ boundedness for $p\in (1,+\infty)$.
