A maximal inequality associated to Schrödinger type equation (Q2372692)

Let \(\Omega\) be a \(C^2\)-function and denote by \(\Omega(D)\) the pseudodifferential operator with symbol \(\Omega(|\xi|)\). The authors investigate the ``Schrödinger type'' equation \(\frac{\partial u}{\partial t}=i\Omega(D)u\) in \(\mathbb{R}^{n+1}\), \(u(x,0)=f(x)\). In particular they consider the maximal function \(\sup_{t\in\mathbb{R}}|n(x,t)|=\sup_{t\in \mathbb{R}}|e^{it\Omega(0)}f(x)|\) and prove certain boundedness results in (weighted) \(L^2\)-spaces depending on growth condition for \(\Omega\) and regularity condition for \(f\).