Orthonormal Strichartz estimates over fractal time

Abstract

We study Strichartz estimates for orthonormal families of initial data when time is integrated against an $\alpha$-dimensional Frostman measure $\mu$, $0<\alpha \le 1$, with coefficients measured in ${\ell }^{\beta }$. For the fractional Schrödinger, wave and Klein--Gordon equations, we obtain orthogonality gains in the admissible region and necessary conditions on $\beta$. For the fractional Schrödinger equation the exponents are sharp up to an endpoint; the optimality of the threshold $\beta =q/2$ requires $\mu$ to be Ahlfors regular. For the Schrödinger equation in dimensions $d\ge 2$, we prove a Lorentz endpoint estimate and, if $d>2\alpha$, the critical exponent $\beta =q/2$ at every pair $2\le q,r<\infty$ with \[\frac d2(\frac12-\frac1r)\ge\fracαq, \quad r>\frac{dq}{d-α};\] on $r=\infty$ we obtain a restricted weak-type estimate. Since the constants depend on $\mu$ only through its Frostman constant, Lebesgue measure yields global-in-time critical estimates in this range when $d\ge 3$. The proofs combine Frostman fractional integration, a spectral near--far argument at the endpoint, and asymmetric Hilbert--Schmidt bounds with orthogonal frequency-block summation. We also obtain further estimates in the low-dimensional cases $d=1$ (fractional Schrödinger) and $d=2$ (wave and Klein--Gordon) with $\alpha >1/2$, and formulate the consequences for densities of freely evolving states.

arXiv comment61 pages