Boundedness of Maximal Operators of Schrödinger Type with Complex Time

Abstract

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 $s\ge 0$ 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.

12 pages. One further minor correction. To appear in the Revista Matemática Iberoamericana

Cited by in corpus (1)

• On the dimension of divergence sets of Schrödinger equation with complex time