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

Abstract

We consider the linear, time-independent fractional Schrödinger equation (-Δ)^s ψ + Vψ = f on Ω ⊂ R^N. We are interested in the local Hölder exponents of distributional solutions ψ, assuming local L^p 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.

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