A Survey on the Study of Generalized Schrödinger Operators along Curves

Abstract

In this survey, we review the historical development for the Carleson problem about the a.e. pointwise convergence in five aspects: the a.e. convergence for generalized Schrödinger operators along vertical lines; a.e. convergence for Schrödinger operators along arbitrary single curves; a.e. convergence for Schrödinger operators along a family of restricted curves; upper bounds of p for the Lp-Schrödinger maximal estimates; and a.e. convergence rate for generalized Schrödinger operators along curves. Finally, we list some open problems which need to be addressed.

Keywords

• Schrödinger operator
• pointwise convergence
• maximal estimate
• convergence rate
• tangential curves