Local and global estimates for hyperbolic equations in Besov–Lipschitz and Triebel–Lizorkin spaces

In this paper we establish optimal local and global Besov-Lipschitz Triebel-Lizorkin estimates for the solutions to linear hyperbolic partial differential equations. These are based on Fourier integral operators that span all possible scales (and in particular both Banach quasi-Banach scales) of spaces $B^s_{p,q}(\R^n)$, certain $F^s_{p,q}(\R^n)$