Paper detail

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

We consider the linear, time-independent fractional Schrödinger equation $$ (-Δ)^s ψ+Vψ=f. $$ 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 $\nablaψ$ as well.