The nonexistence of a sufficient decomposition in a generalized function algebra

Let $d$ be a positive integer, let ${\rm Gen}_{\mathscr{S}}(\mathbb{R}^d)$ denote the generalized function algebra used in the paper, and let $\widetilde{\mathbb{R}}^d_{ss}$ denote its strongly singular generalized points. For $u\in{\rm Gen}_{\mathscr{S}}(\mathbb{R}^d)$, consider vanishing against every Schwartz test function and decompositions $u=v+w$ with $v,w\in{\rm Gen}_{\mathscr{S}}(\mathbb{R}^d)$, $v(\tilde x)=0$, and $\widehat w(\tilde x)=0$ for every $\tilde x\in\widetilde{\mathbb{R}}^d_{ss}$.

Nonexistence conjecture. There exists $u\in{\rm Gen}_{\mathscr{S}}(\mathbb{R}^d)$ such that

$$\int_{\mathbb{R}^d}u\phi=0,\qquad\forall\phi\in\mathscr{S}(\mathbb{R}^d),$$

yet no decomposition $u=v+w$ has $v,w\in{\rm Gen}_{\mathscr{S}}(\mathbb{R}^d)$ and

$$v(\tilde x)=\widehat w(\tilde x)=0,\qquad\forall\tilde x\in\widetilde{\mathbb{R}}^d_{ss}.$$

The preceding theorem proves that such a decomposition would be sufficient for vanishing against all Schwartz test functions. The conjecture asserts that this sufficient condition is not necessary, so the pointwise characterization cannot be completed by this decomposition alone.