On the Hölder regularity for the fractional Schrödinger equation and its improvement for radial data

We consider the linear, time-independent fractional Schrödinger equation (−∆)ψ + V ψ = f on Ω ⊂ R . We are interested in the local Hölder exponents of distributional solutions ψ, assuming local L integrability of the functions V and f . By standard arguments, we obtain the formula 2s−N/p for the local Hölder exponent of ψ where we take some extra care regarding endpoint cases. For our main result, we assume that V and f (but not necessarily ψ) are radial functions, a situation which is commonplace in applications. We find that the regularity theory “becomes one-dimensional” in the sense that the Hölder exponent improves from 2s − N/p to 2s − 1/p away from the origin. Similar results hold for ∇ψ as well.