Hölder estimates

Holder continuity of the solutions can sometimes be proved only from ellipticity assumptions on the equation, without depending on smoothness of the coefficients. This allows great flexibility in terms of applications of the result. The corresponding result for elliptic equations of second order is the Krylov-Safonov theorem in the non-divergence form, or the De Giorgi-Nash-Moser theorem in the divergence form.

Typically a Holder estimate says that a solution to an integro-differential equation of the form $\int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \, \chi_{B_1}(y)) K(x,y) \, dy = f(x)$ in $B_1$, is $C^\alpha$ in $B_{1/2}$ for some $\alpha>0$ (small).

Since integro-differential equations allow for a great flexibility of equations, there are several variations on the result: different assumptions of the kernels, mixed local terms, evolution equations, etc.

It is very important to allow for a very rough dependence of $K$ with respect to $x$, since the result then applies to the linearization of fully nonlinear equations without any extra a priori estimate. On the other hand, the linearization of a fully nonlinear equation (for example the isaacs equation) would inherit the initial assumptions regarding for the kernels with respect to $y$. Therefore, smoothness (or even structural) assumptions for the kernels with respect to $y$ can be made keeping such result useful.

The holder estimates are closely related to the Harnack inequality.

Some estimates blow up as the order of the equation converges to two, and others pass to the limit. In the latter case the estimates are a true generalization of either De Giorgi-Nash-Moser or Krylov-Safonov]

The linear equation with rough coefficients is equivalent to the function $u$ satisfying two inequalities for extremal operators. This setting stresses the nonlinear character of the estimates.

Estimates which blowing up as the order goes to two

Non variational case

The first Holder estimates were obtained using probabilistic techniques^{[citation needed]}. The first purely analytic proof was given in^{[citation needed]}. The assumptions are that for each $x$ the kernel $K(x,.)$ belongs to a family satisfying certain set of assumptions. No regularity of any kind is assumed for $K$ with respect to $x$. The assumption for the family of kernels are

1. Scaling: If $K(y)$ belongs to the family, then so does $C_{\lambda,K} K(\lambda y)$ for any $\lambda<1$ and some $C_{\lambda,K}$ which could depend on $K$.
2. Nondegeneracy: $\frac{\int_{\R^n} \min(y^2,y^\alpha) K(y) \, dy} {\sup_{B_1} K} \leq C_1$ for some $C_1$ and $\alpha>0$ independent of $K$.

The right hand side $f$ is assumed to belong to $L^\infty$.

A particular cases in which this result applies is the uniformly elliptic case. $$\frac{\lambda}{|y|^{n+s(x)}} \leq K(x,y) \leq \frac{\Lambda}{|y|^{n+s(x)}}.$$ where $s$ is bounded below and above: $0 < s_0 \leq s(x) \leq s_1 < 2$, but no continuity of $s$ respect to $x$ is required. The kernel $K$ is assumed to be symmetric with respect to $y$: $K(x,y)=K(x,-y)$. However this assumption can be overcome in the following two situations.

• For $s<1$, the symmetry assumption can be removed if the equation does not contain the drift correction term: $\int_{\R^n} (u(x+y) - u(x)) K(x,y) \, dy = f(x)$ in $B_1$.
• For $s>1$, the symmetry assumption can be removed if the drift correction term is global: $\int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x)) K(x,y) \, dy = f(x)$ in $B_1$.