(I asked this question on math.stackexchange (see here) but didn't receive any reaction, hence I try it here. If it does not fit within here, just let me know in the comments.)
In the definition of the wave front set of a distribution $T\in\mathcal{D}^{\prime}(\mathbb{R}^{d})$, one needs the concept of "regular directed points": a pair $(p,\xi)\in\mathbb{R}^{d}\times\mathbb{R}^{d}\backslash\{0\}$ is called "regular directed", if there exists a test function $f\in\mathcal{D}(\mathbb{R}^{d})$ satisfying $f(p)=1$ as well as a closed conical neighbourhood $V\subset\mathbb{R}^{d}$ of $\xi$, such that $\mathcal{F}(fT)$ is "fast decreasing", which means that for every $N\in\mathbb{Z}$ there is a constant $C_{N}>0$ such that $$\vert\mathcal{F}(fT)(x)\vert\leq\frac{C_{N}}{(1+\vert x\vert)^{N}}$$ for all $x\in V$, where $\mathcal{F}$ denotes the Fourier transform. There are two small (and related) things I don't understand in this definition. I was not able to find an explanation in any book or on the internet. So, maybe they are obvious, but I am new to distribution theory:
- Why do we know that the Fourier transform is defined for $fT$? Isn't the Fourier transform only defined for "tempered distributions"? As far as I know, it is then usually defined by duality, i.e. $\langle \mathcal{F}(T),f\rangle:=\langle T,\mathcal{F}(f)\rangle$ for all $f\in\mathcal{S}(\mathbb{R}^{d})$ for some tempered distribution $T\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$.
- What does $\vert\mathcal{F}(fT)(x)\vert$ mean? The product $fT$ is a distribution, so it should take functions as arguments right? Is it defined by duality? I mean, the product $fT$ for some test function $f$ is defined by $\langle fT,\varphi\rangle:=\langle T,f\varphi\rangle$ for all $\varphi\in\mathcal{D}^{\prime}(\mathbb{R}^{d})$, at least as far as I know. So, what does $\mathcal{F}(fT)(x)$ even mean? (Of course, if $T$ is regular and can be identified with a function, then it is clear how this should be interpreted. However, not every distribution is regular. I mean, the definition of above is exactly meant to distinuish regular from irregular distributions.)
