Nonlocal Evans-Krylov theorem

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The classical Evans-Krylov theorem [1] [2] says that convex or concave fully nonlinear elliptic equations have $C^{2,\alpha}$ (therefore classical) solutions. This type of equations can be written as a Hamilton-Jacobi-Bellman equation. $\sup_\beta a_{ij}^\beta \partial_{ij} u = f$ for a family of uniformly elliptic coefficients $a_{ij}^\alpha$.

A purely integro-differential version of this theorem[3] says that solutions of an integro-differential Bellman equation of the form $\sup_\beta \int_{\R^n} (u(x+y) - u(x)) K_\beta (y) \mathrm d y = 0 \qquad \text{in } B_1$ are $C^{s+\alpha}(B_{1/2})$ (which implies that they are classical) if the kernels satisfy the following assumptions \begin{align*} \frac{(2-s)\lambda}{|y|^{n+s}} \leq K(y) &\leq \frac{(2-s)\Lambda}{|y|^{n+s}} && \text{uniform ellipticity of order $s$} \\ D^2 K(y) &\leq \frac{(2-s)\Lambda}{|y|^{n+s+2}} && \text{Decay of the tails of $K$ in $C^2$} \\ K(y) &= K(-y) && \text{symmetry} \end{align*}

The $C^{s+\alpha}$ estimate does not blow up as $s \to 2$. Thus, the result is a true generalization of Evans-Krylov theorem.

Note that the result is relevant only if $s>1$, otherwise it is a weaker result compared to the $C^{1,\alpha}$ estimates.

The hypothesis above are most probably not optimal. Most likely a similar estimate would hold for kernels with $C^\alpha$ dependence respect $x$. Unlike the $C^{1,\alpha}$ estimates, no variation of this result is known.

References

1.
2. Krylov, N. V. (1982), "Boundedly inhomogeneous elliptic and parabolic equations", Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 46 (3): 487–523, ISSN 0373-2436
3. Caffarelli, Luis; Silvestre, Luis (to appear), "The Evans-Krylov theorem for non local fully non linear equations", Annals of Mathematics, ISSN 0003-486X