Space curve Fourier restriction for orthonormal functions

We prove a variant of the $L^2$ Fourier restriction inequality for non-degenerate curves in the setting of orthonormal systems. To achieve this, we develop a Hilbert space approach to studying the $L^2$ theory for Fourier restriction for curves. Our arguments also rely on novel $L^p$-improving bounds for associated convolution operators, which take averages over neighbourhoods of cones generated by the curve. The $L^p$-improving bound are established using a variant of the high-low method, in particular using induction-on-scale.