We consider the explicit asymptotic profile of massless Dirac fields on a Schwarzschild background. First, we prove for spin $s=\pm \frac{1}{2}$ components field uniform bound positive definite energy and an integrated local decay estimate from symmetric hyperbolic wave system. Based these estimates, further show that have globally pointwise $fv^{-3/2-s}\tau^{-5/2+s}$ as both upper lower outsid...

We study periodic Lyapunov matrix equations for a general discrete-time linear system Bpxp−Apxp−1=fp, where the coefficients Bp and Ap can be singular. The block of inverse operator are referred to as Green matrices. derive new decay estimates matrices in terms spectral norms special solutions equations. is based on Schur decomposition