Endpoint maximal and smoothing estimates for Schroedinger equations

Details Endpoint maximal and smoothing estimates for Schroedinger equations Endpoint maximal and smoothing estimates for Schroedinger equations

2008-10-26

arxiv.org/abs/0810.4651v2

Journal f\"ur die reine und angewandte Mathematik, 640 (2010), 47-66

Mathematics

Classical Analysis and ODEs

Scientific paper

For $\alpha >1$ we consider the initial value problem for the dispersive equation $i\partial_t u +(-\Delta)^{\alpha/2} u= 0$. We prove an endpoint $L^p$ inequality for the maximal function $\sup_{t\in[0,1]}|u(\cdot,t)|$ with initial values in $L^p$-Sobolev spaces, for $p\in(2+4/(d+1),\infty)$. 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 $p$.

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