Boundedness of Maximal Operators of Schrödinger Type with Complex Time

Details Boundedness of Maximal Operators of Schrödinger Type with Complex Time Boundedness of Maximal Operators of Schrödinger Type with Complex Time

2011-06-16

arxiv.org/abs/1106.3288v2

Mathematics

Analysis of PDEs

12 pages. Minor corrections and clarifications. To appear in the Revista Matem\'atica Iberoamericana

Scientific paper

Results of P. Sj\"olin and F. Soria on the Schr\"odinger maximal operator with complex-valued time are improved by determining up to the endpoint the sharp $s \geq 0$ for which boundedness from the Sobolev space $H^s(\mathbb{R})$ into $L^2(\mathbb{R})$ occurs. Bounds are established for not only the Schr\"odinger maximal operator, but further for a general class of maximal operators corresponding to solution operators for certain dispersive PDEs. As a consequence of additional bounds on these maximal operators from $H^s(\mathbb{R})$ into $L^2([-1, 1])$, sharp results on the pointwise almost everywhere convergence of the solutions of these PDEs to their initial data are determined.

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