A Remark on Regularity of Solutions to Wave Equations

Abstract

We prove a generalization of the Strichartz estimate for the inhomogeneous wave equation $\square u(t,x) =f(t,x)$ in the space-time $\boldsymbol{R}^{1+n}$. We estimate the solution in vector-valued homogeneous Besov spaces $\dot{B}^\theta_{q,2}(\boldsymbol{R}; \dot{B}^\sigma_{r,2}(\boldsymbol{R}^n))$. Such an estimate shows the time differentiability of the solution of fractional order.