Mizohata-Takeuchi inequalities for orthonormal systems

TL;DR

This paper develops new weighted $L^2$ inequalities for Fourier extension operators in orthonormal systems using a direct approach with generalized Wigner distributions, expanding the Mizohata--Takeuchi inequality framework.

Contribution

It introduces a novel direct method for establishing inequalities in orthonormal systems, complementing existing Schatten duality techniques, and explores their validity across different $L^p$ regimes.

Findings

Weighted inequalities are recast as co-positivity of tensor forms for even $p$.

Evidence suggests inequalities may hold in reverse for $p o 0$ with complete orthonormal sequences.

Results extend the Mizohata--Takeuchi inequality family to broader $L^p$ settings.

Abstract

We establish some weighted $L^2$ inequalities for Fourier extension operators in the setting of orthonormal systems. In the process we develop a direct approach to such inequalities based on generalised Wigner distributions, complementing the Schatten duality 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 ($L^p$) inequalities of Mizohata--Takeuchi type. For $p$ an even integer we see that such weighted inequalities may be recast as questions of co-positivity of tensor forms, and for $p\leq 1$ we provide some evidence that they may hold in reverse provided the orthonormal sequence is complete.