Differentiating the Dirac Delta distribution

Question

More generally, let $\psi (D)$ denote a pseudodifferential operator on $\mathbb{R}^n$ given by the function $\psi \in S^m_{\rho, \eta}$, the usual symbol class. My question is: can we interpret $\psi (D) \delta_0(x)$ in any meaningful way? I think the answer should lie in $L^2(\mathbb{R}^n)$, but I am not totally sure how to see this. If it does, is it just in $L^2$, or may be have even better properties, like smoothness and decay?

Answer 1

How about using the duality?

For T smooth enough, you have

$$\langle \psi(D)T, \varphi \rangle = \int_{\mathbb{R}} [\psi(D)T](x) \varphi(x) dx $$

$$= \int_{\mathbb{R}} \varphi(x) \int_{\mathbb{R}} e^{i x \xi} \psi(\xi) \hat{T}(\xi) \, d\xi dx$$

$$= \int_{\mathbb{R}} \frac{1}{2 \pi} \psi(\xi) \hat{T}(\xi) \int_{\mathbb{R}} e^{i x \xi}\varphi(x) \, dx d\xi$$

$$= \int_{\mathbb{R}} \psi(\xi) \hat{T}(\xi) \mathcal{F}^{-1}[\varphi](\xi) d\xi$$

$$= \int_{\mathbb{R}} \psi(\xi) \mathcal{F}^{-1}[\varphi](\xi) \int_{\mathbb{R}} T(x) e^{-ix\xi}d\xi$$

$$= \int_{\mathbb{R}} T(x) \mathcal{F}[\psi\mathcal{F}^{-1}[\varphi]] dx$$

$$= \langle T, \mathcal{F}[\psi\mathcal{F}^{-1}[\varphi]] \rangle$$

Hence, by duality, what you seek should be

$$\langle\psi(D)\delta_0, \varphi\rangle = \mathcal{F}[\psi\mathcal{F}^{-1}[\varphi]](0) $$