We introduce the notion of Schrödinger integral operators, which are the oscillatory integrals that appear naturally in the study of Schrödinger equations, and prove sharp local and global regularity results for these (including propagators for the quantum mechanical harmonic oscillator). Furthermore we introduce general classes of oscillatory integral operators with inhomogeneous phase functions, whose local and global regularity are also established in classical function spaces (both in the Banach and quasi-Banach scales). The results are then applied to obtain optimal (local in time) estimates for the solution to the Cauchy problem for variable-coefficient Schrödinger equations as well as other evolutionary partial differential equations.
|Number of pages||56|
|Journal||Journal of the European Mathematical Society|
|Publication status||Submitted - 2020|
- Schrödinger integral operators
- Oscillatory integral operators