Improve the Strichartz derivative loss to one-sixth

Establish Strichartz estimates for the semiclassical Schrödinger equation on the Friedlander model domain in the regime under consideration with only a one-sixth derivative loss, rather than the one-quarter loss obtained from the dispersive estimates and the standard $TT^*$ argument.

Background

The paper proves dispersive estimates for strictly convex domains with a loss of one-quarter in the decay exponent relative to the boundaryless case, and the resulting Strichartz estimates inherit a corresponding loss. The authors explain that pointwise sharpness of the dispersive estimate does not by itself rule out improved Strichartz estimates, because the spatial region where the worst loss occurs is narrow and may yield better bounds after integration.

For the Friedlander model domain, earlier one-dimensional work obtained improved estimates in part of the relevant parameter regime. The authors therefore conjecture that the multidimensional Strichartz loss can be reduced from one-quarter to one-sixth, although they note that proving this would require refined exponential-sum estimates that are currently unavailable.

References

We conjecture that Strichartz estimates hold with only $1/6$ loss in this regime, (possibly by following the approach in cite[the proof of Thm.4 ]{doi25}).

— Dispersion for the Schrödinger equation inside strictly convex domains: the general case  (2610.06305 - Ivanovici et al., 5 Oct 2026) in Introduction, paragraph beginning “We now briefly discuss linear Strichartz estimates and their optimality”

While it is well known that a loss of at least $\frac 16$ derivatives in Strichartz estimates must hold for SE on the Friedlander model domain (obtained by taking the initial data to be a gallery mode; see Section~2), it is not clear whether this result should be sharp.

— Dispersion for the Schrödinger equation inside strictly convex domains: the general case  (2610.06305 - Ivanovici et al., 5 Oct 2026) in Introduction, paragraph beginning “We now briefly discuss linear Strichartz estimates and their optimality”