I have the following, physically motivated problem:
Consider the convolution $F(x):=\int f(y)\Delta_0(x-y)d^4y$ where $f\in\mathcal{C}_0^\infty(\mathbb{R}^4)$, i.e. smooth and compactly supported, while $\Delta_0\in\mathcal{S}'(\mathbb{R}^4)$ is the massless two-point function (Wightman function). It can be written as
- $\Delta_0(x)=\int\delta(k^2)\theta(k^0)e^{-ikx}d^4k$
- $\Delta_0(x)=\lim_{\epsilon\rightarrow 0}\frac{1}{x^2-i\epsilon x^0}$
Theorem 4.1.1 in Hörmander's "The Analysis of partial differential operators I" asserts that $F\in\mathcal{C}^\infty(\mathbb{R}^4)$. However, I would like to make some statements on the decay properties of $F$ (1. suggest that it decays like $1/r^2$ in spacelike directions).
More generally, how can I determine the decay properties of a function $f$ through its Fourier transform $\hat{f}$?
If $f$ was integrable I would try to use the connection $x_i f(x)\leftrightarrow \frac{\partial}{\partial k_i}\hat{f}(k)$, but how can I proceed if $f$ is smooth but not integrable, e.g. a smooth (tempered) distribution?
Thank you very much for your help :)
My idea for a solution:
Let $x=(x_0,\vec{x})\in\mathbb{R}^4$ and $\operatorname{supp} f\subset O$. Then
$\int f(y)\Delta_0(x-y)d^4x=\int_O f(y)\Delta_0(x-y)d^4x$.
I am interested in the behavior for fixed $x_0$ and $|\vec{x}|\rightarrow\infty$.
For $|\vec{x}|$ large enough, we leave $\operatorname{singsupp}\Delta_0=\{x\in\mathbb{R}^4:x^2=x_0^2-|\vec{x}|^2=0\}$ and we can treat $\Delta_0$ as a smooth function:
$\left|\int_O f(y)\Delta(x-y)d^4x\right|\leq\int_O\left|f(y)\Delta(x-y)\right|d^4x\leq\sup_{y\in O}\frac{1}{|\vec{x}-\vec{y}|^2-(x_0-y_0)^2}\int_O |f(y)|d^4y$
The asymptotic behavior is governed by the supremum term.