Uniform decomposition of a test function into rescaled test functions $S^\delta_{s,x}\eta$

A uniform anisotropic partition of unity at scale $\delta$.

Fix a scaling $s=(s_1,\dots,s_d)$ with $s_i\ge 1$, an order $r\in\mathbb N$, and a test function $\varphi\in C^\infty_0(\mathbb R^d)$. Then there are a compact set $K\subseteq\mathbb R^d$ and a constant $M$, depending only on $\varphi$, $s$ and $r$, with the following property. For every $\delta\in(0,1]$ there exist finitely many points $x_1,\dots,x_n\in K$, reals $c_1,\dots,c_n$ and functions $\eta_1,\dots,\eta_n\in\mathcal B^r_{s,0}$ such that

Here $\mathcal B^r_{s,0}$ is Hairer's class of test functions supported in the unit ball of the scaled quasi-norm $\|\cdot\|_s$ whose derivatives up to order $r$ are bounded by $1$, and $(S^\delta_{s,x}\eta)(y)=\delta^{-|s|}\eta\big((y_i-x_i)/\delta^{s_i}\big)_i$.

The decomposition is obtained from a smooth partition of unity subordinate to the anisotropic boxes $\prod_i [x_i-\delta^{s_i},x_i+\delta^{s_i}]$ centred at the points of the lattice $\prod_i \delta^{s_i}\mathbb Z$ that meet the support of $\varphi$. There are $O(\delta^{-|s|})$ such boxes, while each piece $\varphi\psi_i$, rescaled to the unit box, has $C^r$ norm $O(1)$ because every derivative taken in the rescaled variable carries a factor $\delta^{s_j}\le 1$; hence each coefficient is $O(\delta^{|s|})$ and the total mass $\sum_i |c_i|$ stays bounded as $\delta\to 0$.

The statement is the exact quantitative input needed for the uniqueness clause of Hairer's Theorem 3.10: a distribution that is $O(\delta^{\gamma})$ with $\gamma>0$ on all rescaled test functions $S^\delta_{s,x}\eta$, uniformly over compact sets, annihilates every test function. Because a distribution here is a plain linear functional on $C^\infty_0$, with no continuity assumed, the decomposition must be an exact finite linear combination rather than an approximation.