Determine whether Gaussians are global extremizers in the remaining exponent range

Determine whether Gaussians are global extremizers for the mixed-norm Strichartz inequalities for the Schrödinger equation in the admissible exponent and dimension ranges not ruled out by the paper’s counterexamples, including the ranges in which only local maximality has been established.

Background

The paper studies whether the Gaussian family maximizes the sharp mixed-norm Strichartz inequality for Schrödinger evolution. It proves that Gaussians cease to be local maximizers above dimension-dependent thresholds in dimensions one through four, while establishing stable local maximality in several complementary ranges.

The authors do not determine whether Gaussian functions are global extremizers throughout the remaining ranges. Thus, the unresolved issue is stronger than local stability: it concerns the global sharp inequality and whether any non-Gaussian function can attain a larger Strichartz norm.