Hörmander’s propagation of singularities in two variables

Question

I am trying to apply the propagation of singularities theorem to a distribution $u \in D’(M \times M)$ that verifies $Pu = f$, with $P$ a linear differential operator and $f \in D’(M \times M)$, as usual. But I do no see how to do it properly.

My first idea is to apply the theorem to $P_xu_y(x)=f_y(x)$ for $x \in M$ and for $y$ in some small enough compact set (are there results that could help this kind of proof ?)

And my second idea : $(P_x \otimes 1_y)u(x,y) = f(x,y) $.

At the end I want to know how to pass from the wavefront set of $u_y$ to the one of $u$. Is there something as the pullback of a wavefront set to do so ?

Any ideas, suggestions or references would be nice, thanks.

Edit (after Deane Yang and Bazin’s comments) : Thank you very much for your answers, here are more details about the context.

$M = \mathbb{R} \times \Sigma$ is a Lorentzian manifold equipped with a lorentzian metric $g$, that is globally hyperbolic (i.e. there exists a (Cauchy) hypersurface $\Sigma$). We have $g = -dt^2 +h$, $t$ being the variable in $\mathbb{R} $ and $h$ the induced Riemannian metric on $\Sigma$.

$P$ is a second order linear differential operator acting on sections of $S \rightarrow M$, complex vector bundle over $M$, that is normally hyperbolic, meaning that its principal symbol is given by the metric : $\sigma_P(\xi)= g(\xi^{\#},\xi ^{\#})\cdot id_S $. Moreover, $P$ has product structure over $M$, i.e. $P$ takes the form $$ P = \frac{\partial^2}{\partial t^2} + \Delta $$ with $\Delta$ a Laplace-type operator on $\Sigma$, independent of $t$.

Answer 1

The Propagation-of-Singularities Theorem is telling you that for a real-principal type operator $P$, and a given equation $Pu=f$, $$ WF(u)\backslash WF(f)\quad \text{is invariant by the flow of $H_p$}, $$ where $H_p$ is the Hamiltonian vector field of the principal symbol $p$ of $P$.

Now it seems that your question is what is the wave-front set of $u(x,y)$ knowing the wave-front-set of $x\mapsto u(x,y)=u_y(x)$ and $y\mapsto u(x,y)=v_x(y)$ when it makes sense, say when $u$ is a continuous function. For this I can only refer you to Hörmander's Theorem 8.2.9 (page 267) in the first volume of the ALPDO and I am afraid that nothing more could be said in full generality.

Answer 2

I am a bit of confused by what you want to ask. It seems that your operator is 'principally scalar' and the original propagation of singularity (in particular, the positive commutator argument) should go through directly?

And for your very first version of the question, I think there is no general result on this type of propagation... since the 'Hamilton flow' is not even defined in this case? (When the principal symbol is a general matrix)