C^{2,\alpha}$ regularity of flat free boundaries for the thin one-phase problem

We prove $C^{2,\alpha }$ regularity of sufficiently flat free boundaries, for the thin one-phase problem in which the free boundary occurs on a lower dimensional subspace. This problem appears also as a model of a one-phase free boundary problem in the context of the fractional Laplacian $(-\mathrm{\Delta }{)}^{1\mathrm{/}2}$.