Revista Matemática Iberoamericana


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Volume 29, Issue 2, 2013, pp. 531–546
DOI: 10.4171/RMI/729

Published online: 2013-04-22

Boundedness of maximal operators of Schrödinger type with complex time

Andrew D. Bailey[1]

(1) Tessella, Abingdon, Oxfordshire, Great Britain

Results of P. Sjölin and F. Soria on the Schrödinger 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 Schrödinger 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.

Keywords: Schrödinger maximal operator, complex time, dispersive equation, Sobolev space

Bailey Andrew: Boundedness of maximal operators of Schrödinger type with complex time. Rev. Mat. Iberoamericana 29 (2013), 531-546. doi: 10.4171/RMI/729