The Hilbert transform $Hf(x) = \lim_{\epsilon\to 0} \frac1\pi\int_{|s|>\epsilon} \frac{f(x-s)}s \ ds$ is bounded on $L^p(\mathbb R)$ for every $1<p<\infty$, but not at the endpoints.
Now, in higher dimensions $d>1$, every polygon can be written as the intersection of a finite number of half-planes.
Is there an extension of $H$ to higher dimensions?
Can we use that higher-dimensional extension to prove that some operators like Fourier projections to polygons must also be bounded on $L^p$ for all $1<p<\infty$?