Mizohata–Takeuchi inequalities for orthonormal systems

Abstract We establish some weighted inequalities for Fourier extension operators in the setting of orthonormal systems. In the process, we develop a direct approach to such inequalities based on generalized Wigner distributions, complementing the Schatten space approach that is prevalent in the wider context of estimates for such orthonormal systems. Our results are set within a broader family of tentatively suggested () inequalities of Mizohata–Takeuchi type. For an even integer, we see that such weighted inequalities may be recast as questions of copositivity of tensor forms, and for we provide some evidence that they may hold in reverse provided the orthonormal sequence is complete.