In this note, we consider a general maximal wave operator defined by \(\begin{aligned} W_{a,t}f(x)=\int _{\mathbb {R}^{n}}e^{i(x\cdot \xi +t(x)|\xi |)}a(x,\xi )\widehat{f}(\xi )d\xi , \end{aligned}\) where the amplitude \(a\in L^{\infty }S^{m}_{\rho }\) and \(t\in L^{\infty }\) . We prove that this operator is bounded on \(L^{2}\) provided \(\begin{aligned} m<\frac{(n-1)\rho -n}{2}. \end{aligned}\) As a direct application, we obtain the well-known result that the maximal wave operator \(W^*\) is bounded from the Sobolev space \(H^s=W^{s,2}\) to \(L^2\) if \(s>\frac{1}{2}\) . This result is known to be sharp for \(s>\frac{1}{2}.\) .
