Weighted-$L^\infty$ and pointwise space-time decay estimates for wave equations with potentials and initial data of low regularity

We prove weighted-$L^\infty$ and pointwise space-time decay estimates for weak solutions of a class of wave equations with time-independent potentials and subject to initial data, both of low regularity, satisfying given decay bounds at infinity. The rate of their decay depends on the asymptotic behaviour of the potential and of the data. The technique is robust enough to treat also more regular solutions and provides decay estimates for arbitrary derivatives, provided the potential and the data have sufficient regularity, but it is restricted to potentials of bounded strength (such that $-\Delta-|V|$ has no negative eigenvalues).