In short, my question is: are there conditions that one can impose on two tempered distributions $u$ and $v$ that will guarantee that the paraproduct remainder $R(u,v)$ is well-defined and is "weaker" than the conditions most commonly imposed to ensure such? One example of a "weaker" condition would be Hörmander's criterion, which gives a criterion for ensuring when the product of two distributions is well-defined. For all of the details, see below.
Let $\mathscr{S}(\mathbb{R}^d)$ denote the space of Schwartz functions on $\mathbb{R}^d$ and let $\mathscr{S}^\prime(\mathbb{R}^d)$ denote the space of tempered distributions on $\mathbb{R}^d$. Then, it is a well-known result in analysis, going back to the work of Bony in the early 80's, that we can, at least formally, decompose the product of $u,v\in\mathscr{S}^\prime(\mathbb{R}^d)$ as $$uv = T_u v + T_v u + R(u,v),$$ where $T_uv$ is the paraproduct $$T_uv = \sum_{p\geq 2} S_{p-2}u\Delta_p v.$$ Of course, the frequency gap of two in the definition above could be otherwise (i.e., replace $p-2$ by $p-N$ for $N\gg1$) and, in some situations, one might need to tailor the frequency gap to the particulars of the problem at hand. The remainder $R(u,v)$ above is given by $$R(u,v) = \sum_{|p-q| \leq 2}\Delta_pu\Delta_qv.$$
I will not clutter up the post by going over basic Littlewood-Paley theory, constructing the operators $S_p$ and $\Delta_p$ and so on. But, I will say a few words about these operators for the sake of clarity so that anyone reading (and hopefully answering) will be on the same page if they are not familiar with any notation I am using. The operators $S_p$ and $\Delta_p$, $p \in \mathbb{N}_0$, are the frequency filters and dyadic blocks of Littlewood-Paley theory. That is, given a tempered distribution $u$, the support of $\mathcal{F}(S_pu)$ is contained in the dyadic ball $\{ |\xi| \leq 2^p \}$ and the support of $\mathcal{F}(\Delta_pu)$ is contained in the dyadic shell $\{ 2^{p-1} \leq |\xi| \leq 2^{p+1} \}$. Here, $\mathcal{F}$ denotes the Fourier transform. This gives rise to a dyadic partition of unity: $$\mathrm{id} = S_0 + \sum_{p \geq 0} \Delta_p.$$
We know that the sum defining the paraproduct $T_uv$ converges, in the topology of $\mathscr{S}^\prime(\mathbb{R}^d)$, and thus that the paraproduct is well-defined, for any two arbitrary tempered distributions $u$ and $v$. On the other hand, the remainder $R(u,v)$ is not well-defined for arbitrary tempered distributions $u$ and $v$. Of course, this makes perfect sense as we cannot multiply arbitrary tempered distributions, which goes back to the Schwartz impossibility theorem. For example, $\delta_0 \in \mathscr{S}^\prime(\mathbb{R}^d)$, where $\delta_0$ denotes the delta-distribution, but we cannot make sense of $\delta_0^2$ (at least in standard distribution theory).
There are a number of criteria which one can impose upon $u,v \in \mathscr{S}^\prime(\mathbb{R}^d)$ in order to ensure that $R(u,v)$ is well-defined. In fact, a number of these go back, more or less, to Bony's work. For example, if $r,s\in\mathbb{R}$ are such that $r+s>0$, $u \in H^r(\mathbb{R}^d)$ and $v \in H^s(\mathbb{R}^d)$, then $R(u,v)$ is well-defined by the sum given above. There is a fairly similar result for Besov spaces $B_{p,r}^s(\mathbb{R}^d)$, however one can have the sum of the regularity indices $s_1 + s_2 = 0$ for some values of $p_1,p_2,r_1,r_2$. It is also quite common to work with paraproducts on Hölder spaces $C^\alpha(\mathbb{R}^d)$ and there are, again, some similar results in this case.
These sorts of quantitative criteria make perfect sense in terms of how paraproducts and paradifferential calculus is actually applied. Indeed, the purpose (broadly speaking) of paradifferential analysis is to localize singularities of solutions to nonlinear partial differential equations as precisely as possible. Necessarily, these singularities will all be with respect to some threshold of regularity and one then considers terms more regular than this threshold to be error terms. This is a rather common principle in microlocal analysis in general: smooth = error. So, to return, for example, to the Sobolev space example noted above, if $u \in H^r(\mathbb{R}^d)$ and $v \in H^s(\mathbb{R}^d)$ with $r+s>0$, then $R(u,v) \in H^{r+s-\frac{d}{2}}(\mathbb{R}^d)$. That is to say, $R(u,v)$ is (substantially) more regular than $u$ and $v$, which could allow us to consider such a term as an error term. So, I understand that it makes perfect sense, and is even quite natural, to have these sorts of quantitative criteria imposed upon $u$ and $v$ to ensure that $R(u,v)$ is well-defined. Indeed, they are rather key to doing the analysis for which paraproducts, and paradifferential operators more generally, were intended.
My question is then not so much motivated out of a desire to solve a particular problem, but out of simply wanting to know. I would like to understand if it is possible to ensure that the remainder $R(u,v)$ is well-defined by imposing weaker conditions upon $u,v \in \mathscr{S}^\prime(\mathbb{R}^d)$. For example, we know by Hörmander's criterion that we can define the product $uv$ of $u,v\in\mathscr{S}^\prime(\mathbb{R}^d)$ as long as there is no $(x_0,\xi_0) \in \mathbb{R}^d \times \mathbb{R}^d$ such that $(x_0,\xi_0) \in \mathrm{WF}(u)$ and $(x_0,-\xi_0) \in \mathrm{WF}(v)$. Would imposing Hörmander's criterion upon $u$ and $v$ guarantee that $R(u,v)$ is well-defined?
I would be very interested in either an answer or references to the literature. The example above of Hörmander's criterion was just that, an example. I did not intend to say that I was only interested in that criterion, but just wanted to give an example of the kind of criterion that I had in mind. If you made it this far, I already owe you my thanks! I look forward to reading what you all have to say!
