Directional Hilbert transform ($H_{v}g(x)={\pi }^{-1}\mathrm{pv}{\int }_{\mathbb{R}}g(x-sv)ds/s$)

For a unit vector $v$, the directional Hilbert transform integrates along lines parallel to $v$. Its Fourier multiplier is $-i\mathrm{sgn}(\xi \cdot v)$ under the usual transform convention. Applied to a rectangle whose long axis is $v$, it reduces to the one-dimensional logarithmic tail of an interval indicator function, separately on each transverse coordinate.