Literature on the product of two distributions satisfying the Hörmander condition

I am currently studying some basic questions concerning the product $uv\in \mathscr{D}'(\mathbb R^n)$ of two Schwartz distributions $u,v\in \mathscr{D}'(\mathbb R^n)$ satisfying the Hörmander wavefront set condition $(\mathrm{WF}(u)+\mathrm{WF}(v))\cap \mathbb R^n\times\{0\}=\emptyset$. I noticed that unfortunately most of the literature on products of distributions does not deal with this situation but rather with the (problematic) issue of defining some generalized notion of multiplication for all distributions (and even more general objects).

Who knows a reference where “Hörmander products” of distributions, as above, are studied in detail for their own sake, i.e., not for the sake of discussing the (limitations of) possible generalizations of the multiplication?

The corresponding section in Hörmander’s book The Analysis of linear partial differential operators I is extremely brief. I hope that there are books, lecture notes, or articles which are more comprehensive and give an overview on some basic non-trivial known facts. For example, a good start would be to know a reference in which it is shown that if $u,v\in \mathrm{L}^1_\mathrm{loc}\cap \mathrm{L}^2_\mathrm{loc}$ and $u,v$ satisfy the wavefront set condition, then $uv\in \mathscr{D}'(\mathbb R^n)$ is the ordinary product of $u$ and $v$ as functions. (I do know how one sees this, I just wanted to mention an example).

Edit: In view of the first answer I got, I again want to stress that I am not interested in the problem of extending the distribution product beyond the Hörmander condition. I am entirely fine with the Hörmander product and I am not asking why the Hörmander condition is required or why the definition makes sense (I do know the wavefront set calculus). What interests me are the properties of the product, which can be quite delicate. At the moment I feel that I am reproving/trying to reprove some things myself which could be found in the literature. Even very basic things seem far from obvious to me: For example, suppose that $u$ is a continuous nowhere-vanishing function. Then, as continuous functions, one has $1=u\frac{1}{u}$ and this also holds as distributions provided $u$ and $\frac{1}{u}$ satisfy the Hörmander condition. But in general, the wavefront sets of $u$ and $\frac{1}{u}$ can be pretty bad. It would be cool to have a reference where such issues are discussed.


Not very clear what you are asking. I don’t think Hörmander wrote his theorem in order to show one cannot multiply distributions. I think, on the contrary, he came up with a reasonable condition which if satisfied tells you that you can multiply two distributions. Note that one can always multiply distributions $S(x)$ and $T(x)$, but the caveat/joke is we get a distribution $S(x)T(y)$, namely, the tensor product.
The difficulty is restricting this distribution to the diagonal $x=y$. If I remember well, this is how Hörmander proves his theorem in his book, as a particular case of a more general restriction theorem.

For a pedagogical reference on this kind of result, see
“A smooth introduction to the wavefront set” by Brouder, Dang, and Hélein.

Note that there are other constructions of products of distributions, for instance using paraproducts and a condition on the (possibly negative) Hölder/Besov exponents of the factors. This is done in the book “Fourier Analysis and Nonlinear Partial Differential Equations” by Bahouri, Chemin and Danchin.

Finally, if I may mention some of my work, in a different direction of defining products of distributions for almost all of them in a probabilistic sense, see my CMP article “A Second-Quantized Kolmogorov-Chentsov Theorem via the Operator Product Expansion”.

Source : Link , Question Author : B K , Answer Author : Abdelmalek Abdesselam

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