Remarks on space curve $L^2$ Fourier restriction

We re-examine and consider variants of the Fourier restriction theorem for space curves. In particular, we introduce a Stein–Tomas-type argument which recovers the sharp $L^2$ restriction theorem from classical oscillatory integral estimates, in the spirit of Arkhipov–Chubarikov–Karatsuba. In addition, we consider certain square function estimates which can be thought of as partial progress towards a space curve variant of the orthonormal Fourier restriction theorem of Frank–Sabin. These estimates are obtained by reducing the problem to a Kakeya-type maximal inequality, which we analyse using a formulation of the Bourgain–Guth broad/narrow method.