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Boundedness of Maximal Operators of Schr\"odinger Type with Complex Time

Abstract

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 s0s \geq 0 for which boundedness from the Sobolev space Hs(R)H^s(\mathbb{R}) into L2(R)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 Hs(R)H^s(\mathbb{R}) into L2([1,1])L^2([-1, 1]), sharp results on the pointwise almost everywhere convergence of the solutions of these PDEs to their initial data are determined.Comment: 12 pages. One further minor correction. To appear in the Revista Matem\'atica Iberoamerican

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