Endpoint estimates for maximal operators associated to the wave equation

We consider the $H^s$–$L^q$ maximal estimates associated to the wave operator \begin{equation*} e^{ it\sqrt{-Δ}}f(x) = \frac{1}{(2π)^d}\int_{$\mathbb{R}$^d} e^{i(x \cdot ξ, + t|ξ|)} \widehat{f}(ξ,) dξ. \end{equation*} Rogers–Villarroya proved $H^s$–$L^q$ estimates for the maximal operator $f\mapsto$ ${\sup }_t\mathrm{\mid }{e}^{it\sqrt{-\mathrm{\Delta }}}f\mathrm{\mid }$ up to the critical Sobolev exponents $s_c(q,d)$. However, the endpoint case estimates for the critical exponent $s=s_c(q,d)$ have remained open so far. We obtain the endpoint $H^{s_c(q,d)}$–$L^q$ bounds on the maximal operator $f\mapsto {\sup }_t\mathrm{\mid }{e}^{it\sqrt{-\mathrm{\Delta }}}f\mathrm{\mid }$. We also prove that several different forms of the maximal estimates considered by Rogers–Villarroya are basically equivalent to each other.