Endpoint maximal and smoothing estimates for Schrödinger equations

• Published in: Journal für die reine und angewandte Mathematik Vol. 2010; no. 640; pp. 47 - 66
• Main Authors: Rogers, Keith M., Seeger, Andreas
• Format: Journal Article
• Language: English
• Published: Walter de Gruyter GmbH & Co. KG 01.03.2010
• ISSN: 0075-4102; 1435-5345
• DOI: 10.1515/crelle.2010.018

For α > 1 we consider the initial value problem for the dispersive equation i∂tu + (–Δ) α/2 u = 0. We prove an endpoint Lp inequality for the maximal function with initial values in Lp -Sobolev spaces, for p ∈ (2 + 4/(d + 1), ∞). This strengthens the fixed time estimates due to Fefferman and Stein, and Miyachi. As an essential tool we establish sharp Lp space-time estimates (local in time) for the same range of p.