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.


1 Answer 1


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".

  • $\begingroup$ Thank you for pointing out that my question was perhaps not clear enough. I made an edit. I know the paper by Brouder, Dang, and Hélein which is very nice and has interesting examples but which does not go into details concerning the properties of the Hörmander product (apart from the product rule). $\endgroup$
    – B K
    Feb 25, 2021 at 0:22
  • $\begingroup$ Sorry my answer was not quite to your satisfaction, but I will leave it since it could be useful to someone else. You may be able to answer the questions you have, e.g., the one about $uv$, by yourself as follows. The definition of Hormander's product is not whatever is produced by Hormander's Theorem black box. The definition (which predates Hormander) is: let $u_n$ be a sequence in $\mathscr{E}$ (smooth functions) which converges to $u$ in the $\mathscr{D}'$ topology, and likewise for $v_n$ converging to $v$, then $uv:=\lim_{n\rightarrow\infty} u_n v_n$ in $\mathscr{D}'$... $\endgroup$ Feb 25, 2021 at 15:56
  • $\begingroup$ ...of course $u_n v_n$ is the ordinary product. Let $P\subset \mathscr{D}'\times\mathscr{D}'$ be the set of pairs $(u,v)$ where for all approximating sequences the limit exists. Then by interlacing and the Hausdorff property of $\mathscr{D}'$, the limit is uniquely defined. What Hormander does is just to tell you there is subset $H$ of pairs which satisfy the wavefront condition and $H\subset P$. $\endgroup$ Feb 25, 2021 at 16:15
  • $\begingroup$ For proving properties satisfied by the product, the definition $\lim u_n v_n$ may be all you need. $\endgroup$ Feb 25, 2021 at 16:18
  • $\begingroup$ Dear Abdelmalek, what you have just explained in the comments is a great example of the kind of interesting content which I hope can be found in the literature that my question is asking for. Do you have references where the statements/explanations you gave can be found? I am totally fine with references predating Hörmander. $\endgroup$
    – B K
    Feb 25, 2021 at 18:05

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