Universal Hölder estimates for stable solutions to semilinear equations of fractional order

We establish universal interior Hölder and fractional energy estimates for stable solutions of $(-\mathrm{\Delta }{)}^{s}u=f(u)$, $0<s<1$, with $f\in C_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathbb{R})$ nonnegative and convex. The bounds depend only on a weighted $L^1$ norm of the solution and hold under optimal conditions on the dimension and the fractional order. In particular, the half-Laplacian estimate holds through dimension eight. The estimates extend to stable finite-energy Dirichlet $L^1$-weak solutions without an a priori local boundedness assumption, and an explicit logarithmic solution proves optimality in this class. Applications include global bounds in convex domains, bounded extremal solutions, and a Liouville theorem.