Let $r > 0$ and $\mathbf{w}\in\mathbb{R}^n$, and denote by $rB_n$ the ball of radius $r$ in $n$ dimensions. Consider the functions: $f_1(\mathbf{x}) = \mathbf{1}(\langle\mathbf{w},\mathbf{x}\rangle > 0)$ and $f_2(\mathbf{x})$ is the Fourier transform of the indicator of a ball with radius $r$ (This has an explicit formula in Fourier transform of the indicator of the unit ball).
I want to calculate, or at least lower bound by some constant, the following integral:
$\int_{\mathbf{x} \in \mathbb{R}^n \setminus 2rB_n}|\widehat{f_1f_2}(\mathbf{x})|^2d\mathbf{x}$
I am trying to calculate this integral using the convolution theorem, and without using the explicit formula for $f_2$. I basically need some closed form to: $|\widehat{f_1}*\widehat{f_2}|^2$.
First, since the integral is over a symmetric domain, we can assume w.l.o.g that $\mathbf{w} = \mathbf{e_1}$. Now, we have that $\widehat{f_2}(\mathbf{x}) = \mathbf{1}(\|{x}\| \leq r)$, and the Fourier transform of the Heaviside function is also known (Fourier Transform of Heaviside Step Function), but is given via a tempered distribution. The problem I am having is how to calculate this convolution. The thing is that $f_1f_2\in L_2 $ so it should have a Fourier transform in the standard sense, but calculating it requires doing a convolution with a tempered distribution which I'm not sure how to do.