Bounding derivatives of the Wigner function using phase-space tails

Question

Suppose I have a Wigner function that falls off faster than any polynomial for all directions in phase space. That is, for all $a,b>0$, $$\lim_{|x|\to\infty} |x^a p^b W(x,p)| =0=\lim_{|p|\to\infty} |x^a p^b W(x,p)|,$$ which, when combined with the boundedness of its magnitude, is known to be equivalent to $|W|_{(a,b),(0,0)} < \infty$ where $$|W|_{(a,b),(c,d)}\equiv \sup_{x,p} |x^a p^b \partial_x^c \partial_p^d W(x,p)|$$ using some standard tricks in Fourier analysis. Does this means $W$ is a Schwartz function, i.e., that $|W|_{(a,b),(c,d)} < \infty$ for all $a,b,c,d$?

This of course is not true for a generic function on phase space, but I am pretty sure it holds for Wigner functions corresponding to the density matrix (a positive operator). Intuitively, the Wiger function's high-frequency wiggles in the $x$ direction correspond to coherence over long distances in $p$, but if the support of (most of) the Wigner function's mass is basically bounded in $p$, these $x$ wiggles must be suppressed above some frequency cut-off. And vice versa.

I thought I remember this being demonstrated in Folland's Harmonic Analysis in Phase Space, but I can't dig it up even after a lot of searching.

Answer 1

Felipe Hernández and I ended up showing that the answer is yes for both pure and mixed states: "Rapidly Decaying Wigner Functions are Schwartz Functions" [arXiv:2103.14183].

We were able to laboriously find an explicit bound (Theorem 3.9 in v1 of the arXiv paper): $$|W|_{(a,b),(c,d)} \le (2\pi)^{5n} 2^{4(|a|+|b|+|c|+|d|+n)} \|W_\chi\|_{(a,b),(c,d)} \|W_\chi\|_{(2a+2d+12,2b+2c+12),(0,0)}$$ $$\qquad\times \|W\|_{(2a+2d+8,2b+2c+8),(0,0)}.$$ Here, $W_\chi$ is the Wigner function for an arbitrary Schwartz function (wavepacket) $\chi$ (which can be taken to be, e.g., Gaussian), $n$ is the dimension, $a=(a_1,\ldots,a_n)$ is a multi-index, $|a|=\sum_{i=1}^n a_i$ is the multi-index norm, $|W|_{(a,b),(c,d)} = \sup_{x,p} |x^a p^b \partial_x^c \partial_p^d W(x,p)|$ is the Schwartz semi-norm (with, e.g., $x^a = x_1^{a_1}\cdots x_n^{a_n}$ and $\partial_p^d =\partial_{p_1}^{d_1}\cdots \partial_{p_n}^{d_n}$), and $$\|W\|_{(a,b),(c,d)} = \sum_{a'\le a}\sum_{b'\le b}\sum_{c'\le c}\sum_{d'\le d} |W|_{(a',b'),(c',d')}$$ the Schwartz norm (with $a\le a'$ meaning $a_i \le a_i^\prime$ for all $i=1,\ldots,n$).

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(EDIT: After posting the preprint, we were alerted to an earlier theorem by K. Gröchenig that superficially seemed to prove this in the pure case, but a more careful reading revealed that Gröchenig's theorem relied on important additional assumptions, unlike our results. See Theorem 11.2.5 and its Corollary 11.2.6, on pages 228-229 of K. Gröchenig, "Foundations of Time-frequency Analysis" (2001).)