Optimal regularity bounds in the intermediate summability range

Determine the optimal Sobolev regularity threshold for orthonormal maximal estimates for the free Schrödinger propagator, and consequently for the generalized Schrödinger propagators, the Hermite operator, and the special Hermite operator, when the dimension satisfies d ≥ 2 and the summability exponent satisfies 1 < β < 2.

Background

The paper establishes sufficient regularity conditions s > s(d, β) and necessary conditions s ≥ σ(d, β) for orthonormal maximal estimates of the free Schrödinger evolution. The two bounds coincide for β ≥ 2, yielding matching results up to the endpoint, but they do not coincide throughout the intermediate range 1 < β < 2 when d ≥ 2.

The equivalence principle proved in the paper transfers any improvement of the free sufficient bound to generalized Schrödinger propagators of the form S^ζ_a, and hence to Schrödinger operators with quadratic potentials, including the Hermite and special Hermite operators. Resolving the gap would therefore determine the optimal regularity for all of these related flows in the intermediate summability regime.