Stein universality lemma for second-Wiener-chaos targets

Let $F_\infty$ be a random variable in the second Wiener chaos, with associated differential operator $\mathcal{A}_\infty$ of order $d$, and let $\mathcal{H}$ be an appropriate class of test functions. For $h\in\mathcal{H}$, consider the Stein equation

$$\mathcal{A}_\infty f(x)=h(x)-\mathbb{E}[h(F_\infty)].$$

Stein universality lemma. For every $h\in\mathcal{H}$, this equation admits a bounded solution $f_h$ that is $d$ times differentiable, with

$$\lVert f_h^{(r)}\rVert_\infty<+\infty\qquad\text{for }r=1,\ldots,d,$$

and these bounds are independent of $h$. The lemma is intended to provide the non-Gaussian counterpart of Stein's characterization and would complete the Stein part of the approach in this setting; the appropriate class $\mathcal{H}$ and the asserted uniform bounds remain to be established.