Endpoint maximal estimates for the half-wave propagator

We present a proof of the strong maximal estimate for the Euclidean half-wave propagator at the critical pair $s_d=d/(d+1)$ and $q_d=2(d+1)/(d-1)$, in every dimension $d\ge2$, i.e. \begin{equation*} \biggl\|\sup_{0<t<1}|e^{it\sqrt{-Δ}}f(x)|\biggl\|_{L^{q_d}({\mathbb{R}}^d)} \le C_d\|f\|_{ H^{s_d}(