  We establish new Strichartz estimates for orthonormal families of initial
data in the case of the wave, Klein-Gordon and fractional Schr\"odinger
equations. Our estimates extend those of Frank-Sabin in the case of the wave
and Klein-Gordon equations, and generalize work of Frank-Lewin-Lieb-Seiringer
and Frank-Sabin for the Schr\"{o}dinger equation. Due to a certain technical
barrier, except for the classical Schr\"odinger equation, the Strichartz
estimates for orthonormal families of initial data have not previously been
established up to the sharp summability exponents in the full range of
admissible pairs. We obtain the optimal estimates in various notable cases and
improve the previous results. The main novelty of this paper is the use of
estimates for weighted oscillatory integrals which we combine with an approach
due to Frank and Sabin. This strategy also leads us to proving new estimates
for weighted oscillatory integrals with optimal decay exponents which we
believe to be of wider independent interest. Applications to the theory of
infinite systems of Hartree type, weighted velocity averaging lemmas for
kinetic transport equations, and refined Strichartz estimates for data in Besov
spaces are also provided.
