Orthonormal Strichartz estimates over fractal time

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 ${\mathrm{\ell }}^{\beta }$. 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 $\beta =q\mathrm{/}2$ 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 $\beta =q\mathrm{/}2$ at every pair $2\le q,r<\mathrm{\infty }$ with $$\frac{d}{2}(\frac{1}{2}-\frac{1}{r})\ge \frac{\alpha }{q},r>\frac{dq}{d-\alpha };$$ on $r=\mathrm{\infty }$ 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 $\alpha >1\mathrm{/}2$, and formulate the consequences for densities of freely evolving states.