Hölder estimates

Hölder 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.

The holder estimates are closely related to the Harnack inequality.

There are integro-differential versions of both De Giorgi-Nash-Moser theorem and Krylov-Safonov theorem. The former uses variational techniques and is stated in terms of Dirichlet forms. The latter is based on comparison principles.

A Hölder estimate says that a solution to an integro-differential equation $L_x u(x) = f(x)$ in $B_1$, is $C^\alpha$ in $B_{1/2}$ for some $\alpha>0$ (small). It is very important to allow for a very rough dependence of $L_x$ 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.

In the non variational setting the integro-differential operators $L_x$ are assumed to belong to some family, but no continuity is assumed for its dependence with respect to $x$. Typically, $L_x u(x)$ has the form $$ L_x u(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \, \chi_{B_1}(y)) K(x,y) \, dy$$ Since integro-differential equations allow for a great flexibility of equations, there are several variations on the result: different assumptions on the kernels, mixed local terms, evolution equations, etc. The linear equation with rough coefficients is equivalent to the function $u$ satisfying two inequalities for the extremal operators corresponding to the family of operators $L$, which stresses the nonlinear character of the estimates.

As with other estimates in this field too, some Hölder estimates blow up as the order of the equation converges to two, and others pass to the limit. The blow-up is a matter of the techniques used in the proof. Only estimates which are robust are a true generalization of either the De Giorgi-Nash-Moser or Krylov-Safonov theorem.

Estimates which blow up as the order goes to two

Non variational case

The Hölder estimates were first obtained using probabilistic techniques , and then using purely analytic methods . 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 operators are

1. Scaling: If $L$ belongs to the family, then so does its scaled version

$L_r u(x) = C_{r,L} L [u(x/r)] (x)$ for any $r<1$ and some $C_{r,L}<1$ which could depend on $L$, but $C_{r,L} \to 0$ as $r \to 0$ uniformly in $L$.

1. Nondegeneracy: If $K$ is the kernel associated to $L$,

$\frac{\int_{\R^n} \min(y^2,y^\alpha) K(y) \, dy} {\inf_{B_1} K} \leq C_1$ for some $C_1$ and $\alpha>0$ independent of $K$.

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

A particular case 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 $B_1$.

The reason for the symmetry assumption, or the modification of the drift correction term, is that in the original formulation the term $y \cdot \nabla u(x) \, \chi_{B_1}(y)$ is not scale invariant.

Variational case

A typical example of a symmetric nonlocal Dirichlet form is a bilinear form $$E(u,v)$$ satisfying $$ E(u,v) = \iint_{\R^n \times \R^n} (v(y)-u(x))(v(y)-v(x)) K(x,y) \, dx \, dy $$ on the closure of all $L^2$-functions with respect to $$J(u)=E(u,u)$$. Note that $K$ can be assumed to be symmetric because the skew-symmetric part of $K$ would be ignored by the bilinear form. Nonsymmetric nonlocal Dirichlet looks differently.

Minimizers of the corresponding quadratic forms satisfy the nonlocal Euler equation $$ \int_{\R^n} (u(x+y) - 2 u(x) + u(x-y) ) K(x,y) \, dy = 0$$.

It is known that the gradient flow of a Dirichlet form (parabolic version of the result) becomes instantaneously Hölder continuous . The method of the proof builds an integro-differential version of the parabolic De Giorgi technique that was developed for the study of critical [[surface quasi-geostrophic equation]].

At some point in the original proof of De Giorgi, it is used that the characteristic functions of a set of positive measure do not belong to $H^1$. Moreover, a quantitative estimate is required about the measure of intermediate level sets for $H^1$ functions. In the integro-differential context, the required statement to carry out the proof would be the same with the $H^{s/2}$ norm. This required statement is not true for $s$ small, and would even require a non trivial proof for $s$ close to $2$. The difficulty is bypassed though an argument that takes advantage of the nonlocal character of the equation, and hence the estimate blows up as the order approaches two.

Estimates which pass to the second order limit

Non variational case

An integro-differential generalization of Krylov-Safonov theorem is available . The assumption on the kernels are

1. Symmetry: $K(x,y) = K(x,-y)$.
2. Uniform ellipticity: $\frac{(2-s)\lambda}{|y|^{n+s}} \leq K(x,y) \leq

\frac{(2-s) \Lambda}{|y|^{n+s}}$ for some fixed value $s \in (0,2)$.

The right hand side $f$ is assumed to be in $L^\infty$. The constants in the Hölder estimate do not blow up as $s \to 2$.

Variational case

In the stationary case, it is known that minimizers of Dirichlet forms are Hölder continuous by adapting Moser's proof of De Giorgi-Nash-Moser to the nonlocal setting .