Let $f:\mathcal{D}\to\mathcal{D}$ be a function whose domain and co-domain are $\mathcal{D}$. Let $\hat{f}$ be its Hilbert transform, which is defined as
$\hat{f}(t)=\mathcal{H}(f(t))=\frac{1}{\pi} \mathop{p.v.}\int_{-\infty}^{\infty} \frac{f(\tau)}{t-\tau}\ d\tau.$
Now I can see that the domain of $\hat{f}(t)$ is $\mathcal{D}$. What is its co-domain? Is it $\mathcal{D}$ too or can it be different?
NOTE: When I say $\mathcal{D}$, I mean either $\mathbb{R}$ or $\mathbb{C}$