is the distributional dual to $-i{\mathrm{d}}/{\mathrm{d}}t$.

Microlocal analysis applied to quantum field theory {#microlocal review}
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The Sobolev wave-front set {#microlocal review 1}
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Since the publication of Radzikowski’s equivalence theorem [@radzikowski] microlocal analysis, in particular Hörmander’s concept of wave-front set, has been successfully applied to quantum field theory[^5]. The proof of the most general QWEIs rely on microlocal analysis at various stages in their argument, e.g. [@dawson; @fewster; @fewster; @fewster; @pfenning]. A refined version of the usual (smooth) wave-front set, an exposition of which may be found in chapter VIII of [@hormander], has already been employed in the proof of an absolute quantum energy inequality for the Klein-Gordon field [@fewster; @smith]. We shall briefly review the necessary details of this refinement, known as the Sobolev wave-front set.

For $s \in {\mathbb{R}}$, the *Sobolev space* $H^s({\mathbb{R}}^n)$ is the set of all tempered distributions $u$ such that $\widehat{u}$ is a measurable function and $${\parallel}u {\parallel}_{H^s({\mathbb{R}}^n)}^2=\int_{{\mathbb{R}}^n} {\mathrm{d}}^n\xi \, (1+|\xi|^2)^s |\widehat{u}(\xi)|^2 < \infty \, .$$ It is clear that ${\mathcal{S}}({\mathbb{R}}^n)\subset H^s({\mathbb{R}}^n)$ for each $s\in{\mathbb{R}}$. Moreover, one may show that ${\mathcal{S}}({\mathbb{R}}^n)$ is dense in $H^s({\mathbb{R}}^n)$, see chapter 1 §3 of [@taylor] for a brief argument.

Sobolev space theory is usually introduced into the study 