Endpoint maximal and smoothing estimates for Schroedinger equations

For $\alpha >1$ we consider the initial value problem for the dispersive equation $i{\mathrm{\partial }}_{t}u+(-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}u=0$. We prove an endpoint $L^p$ inequality for the maximal function ${\sup }_{t\in [0,1]}\mathrm{\mid }u(\cdot ,t)\mathrm{\mid }$ with initial values in $L^p$-Sobolev spaces, for $p\in (2+4\mathrm{/}(d+1),\mathrm{\infty })$. This strengthens the fixed time estimates due to Fefferman and Stein, and Miyachi. As an essential tool we establish sharp $L^p$ space-time estimates (local in time) for the same range of $p$.