I want to prove that the wave operator $P=\partial_t - \Delta_x$ isn't hypoelliptic in $\Bbb R \times\Bbb R^n$. My attempt is to use the fact that an operator with constant coefficients is hypoelliptic iff (given $E$ fundamental solution of $P$) $\text{sing supp}(E)=\{0\}$
If we call the family of distributions $E_t(\varphi)=\left\{ \begin{array}{ll} T_t(\varphi)=\frac{t}{4\pi} \int_{S^{n-1}}\varphi(tw)d\sigma(w) & \mbox{if } t \geq 0 \\ 0 & \mbox{if } t < 0 \end{array} \right.$
(for $\varphi$ test function in $\Bbb R^3$, where $d\sigma(w)$ is the induced measure in $\Bbb R^3$ by the unit sphere and $w$ angular variable)
Then a fundamental solution for $P$ is $E=\int\langle E_t,\psi(t,\cdot)\rangle\,dt$ with $\psi$ test function in $\Bbb R^{1+3}$. Now I'm not sure how to prove that the singular support of $E$ is not $\{0\}$. My attempt is:
If $t>0$, $T_t$ is a compact support distribution with support in $\{x \in\Bbb R^3 : |x|=|t|\}\Rightarrow\operatorname{supp}E=\{(t,x) : t \geq 0 ,|x|=t\}$.
Intuitively I think the distribution is discontinue in all its support, but (if it is true) I don't know how to actually prove it. Any idea?
I think you omitted a $t$ subscript, wave operator should be $P=\partial_{tt}- \Delta_x$
or you omitted a $2$ superscript, should be $P=\partial_t^2- \Delta_x$
If $P$ is hypoelliptic, then singsupp(u) ⊆ singsupp(Pu) for all u ∈ D′(R×Rn).
Let $H$ denote the Heaviside step function,
wave operator $P$ is not hypoelliptic since $u(x, t)=H\left(x-x_0-\left(t-t_0\right)\right)$ is a solution of $P{u}=0$, so$$\mathrm{singsupp}(Pu)=\emptyset$$but$$\mathrm{singsupp}({u})=\left\{({x}, {t}): x-x_0={t}-{t}_0\right\}$$is not contained in $\emptyset$.
I find the above in https://www.mat.univie.ac.at/~stein/lehre/SoSem09/distrvo.pdf