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I gave the entry wave front set more of an Idea-section, and I added pointer to Hörmander’s book.
I have added to the Idea-section at wave front set pointer to the propagation of singularities theorem (so far just a pointer to references).
Added the statement that taking derivatives does not increase the wave front set of a distribution: here
I keep being unsatisfied with the expositional aspect of the entry wave front set. The perspective offered there goes back to Tim van Beek’s original rev 1, which emphasizes the Paley-Wiener-Schwartz theorem-characterization of the Fourier-Laplace transform. While there is nothing wrong with this technically speaking, I feel like the complex analysis invoked this way, subsuming the Laplace tansform, makes the concept of the wave front set come across unnecessarily involved and indirect.
It seems to me that a good idea of what’s going on is rather provided by the “purely real” characterization that Hörmander uses in the relevant section 8.1 of his book, which I have stated now as a stand-alone proposition here, right after the full-blown Paley-Wiener-Schwartz theorem.
I have written out what I imagine as the more-easily-readable-definition now at compactly supported distribution (here) in the course of adding several other things to this entry, too. But unless somebody objects, and maybe after some further polishing, I feel like this material could replace what we currently have at the beginning of wave front set.
Hm, does anyone have an elementary way to compute the wave front set of the causal propagator $\Delta$ on Minkowski spacetime, using the knowldge that its Fourier transform is simply $\delta\left( k_\mu k^\mu +m^2 \right) sgn(k_0)$ ?
I tried this:
For $b$ any bump function we need to analyze the decay of the Fourier transform of
$x \mapsto b(x- a) \Delta(x)$for all $a$. This Fourier transform is the convolution of
$k \mapsto \hat b(k) e^{i k a}$with $\delta\left( k_\mu k^\mu +m^2 \right) sgn(k_0)$, so effectively it is the mass shell smeared by the Fourier transform $\hat b$ of $b$.
Now if $\hat b$ had compact support, and using that the mass shell asymptotes to the light cone, it would be clear that along a direction $k$ not on the light cone, this function would decay, in fact it would eventually vanish in that direction, while for $k$ along the direction of the light cone it would not decay, but asymptote to a constant.
In reality $\hat b$ does not have compact support, if $b$ does. All we know (?) is that it decays polynomially. First I thought that the previous kind of argument would still work, now concluding that for $k$ in a direction not along the lightcone the function will vanish polynomially.
But it cannot be that simple, because we know that the wave front set is supported over the singular locus, hence has to depend on the coordinate $a$ where the bump function is centered, while nothing in this naive argument depends on $a$.
So this naive argument is too naive. Is there a way to promote this to an actual proof? Or is there an alternative explicit computation?
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