Following up to the question raised here, I am searching for a reference (or a simple argument) to establish (in the whole space) the following (suggested) equivalence :

Given $N_1$ and $N_2$ two (homogeneous spaces semi-) norms with scaling exponents $t$ ans $s$ (as in the answer given by @fedja in the cited post), a convolution operator $f\mapsto f\star \varphi_\delta$ is exact on polynomials of degree equal to or less than $r=t-s$ if and only if it satisfies $N_1(f-f\star \varphi_\delta) \lesssim \delta^r > N_2(f)$.

For instance, if $\varphi$ as the first 3 moments vanishing, do we have $\|f-f\star\varphi_\delta\|_2 \lesssim \delta^2 \|D^2 f\|_2$ on $H^2(\mathbb{R}^d)$ ? And why, if these moments are not vanishing, such a result is hopeless ?

I am quite sure that this result (maybe a bit modified in its statement) should be doable via Fourier analysis, however I would very much appreciate a "direct" proof of such an inequality, in the spirit of the ordre $1$ case which is based on $\|f-\tau_\delta f\|_2 \leq \|\nabla f\|_2$, that can be proven by Taylor formula.