Manuscript XVIII-A@2 · XVIII-A-v2.tex · proposition · 0f4336ec

Proposition: Smooth compact approximants can acquire singular wavefront

Microlocal operations and their boundaries › Weak limits do not preserve admission · lines 791–808 · proof included or linked

XVIII-A@2#prop:weaklimit

There are smooth compact $u_{τ}$ with $WF (u_{τ}) = \emptyset$ for every $τ > 0$ such that $u_{τ} \to δ_{0}$ in $D^{'} (R^{d})$ .

Proof. Let $ρ \in C_{c}^{\infty} (R^{d})$ with integral one and set $u_{τ} (x) = τ^{- d} ρ (x / τ)$ . For every compact smooth test,

⟨ u_{τ}, ϕ ⟩ = \int ρ (y) ϕ (τ y) d y ⟶ ϕ (0) .

The limit has every nonzero covector in its wavefront at zero.

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