# H\"older regularity for the time fractional Schr\"odinger equation

## TL;DR

This paper studies the H"older regularity of solutions to the time fractional Schr"odinger equation of order between 1 and 2, providing new insights into its space and time regularity properties using Fourier analysis.

## Contribution

It establishes H"older regularity results for solutions to the fractional Schr"odinger equation, extending regularity theory to fractional orders and connecting it with classical Schr"odinger and wave equations.

## Key findings

- Proved space regularity using singular Fourier multipliers
- Established time regularity via asymptotic behavior of kernels
- Demonstrated pointwise convergence to initial data in H"older spaces

## Abstract

In this paper, we investigate that the H\"older regularity of solutions to the time fractional Schr\"odinger equation of order $1<\alpha<2$, which interpolates between the Schr\"odinger and wave equations. This is inspired by Hirata and Miao's work which studied the fractional diffusion-wave equation. First, we give the asymptotic behavior for the oscillatory distributional kernels and their Bessel potentials by using Fourier analytic techniques. Then, the space regularity is derived by employing some results on singular Fourier multipliers. Using the asymptotic behavior for the above kernels, we prove the time regularity. Finally, we use mismatch estimates to prove the pointwise convergence to the initial data in H\"older spaces. In addition, we also prove H\"older regularity result for the Schr\"odinger equation.