Half-plane Fourier projection ($\widehat{P_{v}g}(\xi )=1_{\{\xi \cdot v<0\}}\hat{g}(\xi )$)

The half-plane Fourier projection is $(I-i{H}_v)/2$, where $H_v$ is the directional Hilbert transform. A disk of radius $R$, modulated to put a boundary point at the origin, has symbol $1_{\{|\xi +Rv|\le R\}}$, which converges almost everywhere to this half-plane symbol. The Plancherel theorem makes this an $L^2$ multiplier limit.