A maximal inequality associated to Schr\{o}dinger type equation.
2005
In this note, we consider a maximal operator $\sup_{t \in
\mathbb{R}}|u(x,t)| = \sup_{t \in \mathbb{R}}|e^{it\Omega(D)}f(x)|$,
where $u$ is the solution to the initial value problem $u_t =
i\Omega(D)u$, $u(0) = f$ for a $C^2$ function $\Omega$ with some
growth rate at infinity. We prove that the operator $\sup_{t \in
\mathbb{R}}|u(x,t)|$ has a mapping property from a fractional
Sobolev space $H^\frac14$ with additional angular regularity to
$L_{loc}^2$.
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