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| Hölder continuity of the solutions can sometimes be proved only from ellipticity
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| assumptions on the equation, without depending on smoothness of the
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| coefficients. This allows great flexibility in terms of applications of the
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| result. The corresponding result for elliptic equations of second order is the
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| [[Krylov-Safonov]] theorem in the non-divergence form, or the [[De Giorgi-Nash-Moser theorem]] in the divergence form.
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| The Hölder estimates are closely related to the [[Harnack inequality]]. In most cases, one can deduce the Hölder estimates from the Harnack inequality. However, there are simple example of integro-differential equations for which the Hölder estimates hold and the Harnack inequality does not <ref name="rang2013h" /> <ref name="bogdan2005harnack" />. | | The Boltzmann equation is an evolution equation used to describe the configuration of particles in a gas, but only statistically. Specifically, if the probability that a particle in the gas lies in some region $A$ of phase space $\mathbb{R}^d\times \mathbb{R}^d$ at time $t$ is given by |
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| There are integro-differential versions of both [[De Giorgi-Nash-Moser theorem]]
| | \begin{equation*} |
| and [[Krylov-Safonov theorem]]. The former uses variational techniques and is
| | \int_A f(x,v,t)dxdy |
| stated in terms of Dirichlet forms. The latter is based on comparison
| | \end{equation*} |
| principles.
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| A Hölder estimate says that a solution to an integro-differential equation with rough coefficients
| | then $f(x,v,t)$ solves the non-local 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 when an estimate allows for a very rough dependence of
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| $L_x$ 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
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| hand, the linearization of a [[fully nonlinear integro-differential equation]] (for example the [[Isaacs equation]] or the [[Bellman equation]]) would inherit the initial assumptions regarding for the kernels with
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| respect to $y$. Therefore, smoothness (or even structural) assumptions for the
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| kernels with respect to $y$ can be made keeping such result applicable.
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| In the non variational setting the integro-differential operators $L_x$ are
| | \begin{equation*} |
| assumed to belong to some family, but no continuity is assumed for its
| | \partial_t f + v \cdot \nabla_x f = Q(f,f) |
| dependence with respect to $x$. Typically, $L_x u(x)$ has the form
| | \end{equation*} |
| $$ L_x u(x) = a_{ij}(x) \partial_{ij} u + b(x) \cdot \nabla u + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \, \chi_{B_1}(y))
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| K(x,y) \, dy$$
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| Within the context of nonlocal equations, we would be interested on a regularization effect caused by the integral term and not the second order part of the equation. Because o that, the coefficients $a_{ij}(x)$ are usually zero.
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|
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| Since [[linear integro-differential operators]] allow for a great flexibility of
| | where $Q(f,f)$ is the Boltzmann collision operator, given by |
| equations, there are several variations on the result: different assumptions on
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| the kernels, mixed local terms, evolution equations, etc. The linear equation
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| with rough coefficients is equivalent to the function $u$ satisfying two
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| inequalities for the [[extremal operators]] corresponding to the family of
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| operators $L$, which stresses the nonlinear character of the estimates.
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| As with other estimates in this field too, some Hölder estimates blow up as the
| | \begin{equation*} |
| order of the equation converges to two, and others pass to the limit. The
| | Q(f,f)(v) = \int_{\mathbb{R}^d}\int_{\mathbb{S}^{d-1}} B(v-v_*,e) (f(v')f(v'_*)-f(v)f(v_*) d\sigma(e) dv_* |
| blow-up is a matter of the techniques used in the proof. Only estimates which
| | \end{equation*} |
| are robust are a true generalization of either the [[De Giorgi-Nash-Moser theorem]] or
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| [[Krylov-Safonov theorem]].
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|
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| == The general statement ==
| | here $d\sigma$ denotes the Hausdorff measure on $\mathbb{S}^{d-1}$, and given $v,v_* \in \mathbb{R}^d$ and $e \in \mathbb{S}^{d-1}$ we write |
|
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| === Elliptic form ===
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| The general form of the Hölder estimates for an elliptic problem say that if we have an equation which holds in a domain, and the solution is globally bounded, then the solution is Hölder continuous in the interior of the domain. Typically this is stated in the following form: if $u : \R^d \to \R$ solves
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| \[
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| L(u,x) = 0 \ \ \text{in } B_1,
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| \]
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| and $u \in L^\infty(\R^d)$, then for some small $\alpha > 0$,
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| \[ \|u\|_{C^\alpha(B_{1/2})} \leq C \|u\|_{L^\infty(\R^d)}.\]
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|
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| There is no lack of generality in assuming that $L$ is a '''linear''' integro-differential operator, provided that there is no regularity assumption on its $x$ dependence.
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|
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| For non variational problems, in order to adapt the situation to the [[viscosity solution]] framework, the equation may be replaced by two inequalities.
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| \begin{align*} | | \begin{align*} |
| M^+u \geq 0 \ \ \text{in } B_1, \\
| | v' & = v-(v-v_*,e)e\\ |
| M^-u \leq 0 \ \ \text{in } B_1.
| | v'_* & = v_*+(v-v_*,e)e |
| \end{align*} | | \end{align*} |
| where $M^+$ and $M^-$ are [[extremal operators]] with respect to some class.
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|
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| === Parabolic form ===
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| The general form of the Hölder estimates for a parabolic problem is also an interior regularity statement for solutions of a parabolic equation. Typically this is stated in the following form: if $u : \R^d \times (-1,0] \to \R$ solves
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| \[
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| u_t - L(u,x) = 0 \ \ \text{in } (-1,0] \times B_1,
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| \]
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| and $u \in L^\infty(\R^d)$, then for some small $\alpha > 0$,
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| \[ \|u\|_{C^\alpha((-1/2,0] \times B_{1/2})} \leq C \|u\|_{L^\infty((-1,0] \times \R^d)}.\]
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|
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|
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| == Estimates which blow up as the order goes to two ==
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|
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| === Non variational case ===
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|
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| The Hölder estimates were first obtained using probabilistic techniques <ref
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| name="BL"/> <ref name="BK"/> , and then using purely analytic methods <ref
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| name="S"/>. The assumptions are that for each $x$ the kernel $K(x,.)$ belongs to
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| a family satisfying certain set of assumptions. No regularity of any kind is
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| assumed for $K$ with respect to $x$. The assumption for the family of operators
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| are
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| # '''Scaling''': If $L$ belongs to the family, then so does its scaled version
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| $L_r u(x) = C_{r,L} L [u(x/r)] (x)$ for any $r<1$ and some $C_{r,L}<1$ which
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| could depend on $L$, but $C_{r,L} \to 0$ as $r \to 0$ uniformly in $L$.
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| # '''Nondegeneracy''': If $K$ is the kernel associated to $L$,
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| $\frac{\int_{\R^n} \min(y^2,y^\alpha) K(y) \, dy} {\inf_{B_1} K} \leq C_1$ for
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| some $C_1$ and $\alpha>0$ independent of $K$.
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|
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| The right hand side $f$ is assumed to belong to $L^\infty$.
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|
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| A particular case in which this result applies is the uniformly elliptic case.
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| $$\frac{\lambda}{|y|^{n+s(x)}} \leq K(x,y) \leq \frac{\Lambda}{|y|^{n+s(x)}}.$$
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| where $s$ is bounded below and above: $0 < s_0 \leq s(x) \leq s_1 < 2$, but no
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| continuity of $s$ respect to $x$ is required.
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| The kernel $K$ is assumed to be symmetric with respect to $y$: $K(x,y)=K(x,-y)$.
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| However this assumption can be overcome in the following two situations.
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| * For $s<1$, the symmetry assumption can be removed if the equation does not
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| contain the drift correction term: $\int_{\R^n} (u(x+y) - u(x)) K(x,y) \, dy =
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| f(x)$ in $B_1$.
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| * For $s>1$, the symmetry assumption can be removed if the drift correction term
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| is global: $\int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x)) K(x,y) \, dy =
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| f(x)$ in $B_1$.
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|
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| The reason for the symmetry assumption, or the modification of the drift
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| correction term, is that in the original formulation the term $y \cdot \nabla
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| u(x) \, \chi_{B_1}(y)$ is not scale invariant.
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|
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| === Variational case ===
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|
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| A typical example of a symmetric nonlocal [[Dirichlet form]] is a bilinear form
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| $E(u,v)$ satisfying
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| $$ E(u,v) = \iint_{\R^n \times \R^n} (v(y)-u(x))(v(y)-v(x)) K(x,y) \, dx
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| \, dy $$
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| on the closure of all $L^2$-functions with respect to $J(u)=E(u,u)$. Note
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| that $K$ can be assumed to be symmetric because the skew-symmetric part
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| of $K$ would be ignored by the bilinear form.
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|
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| Minimizers of the corresponding quadratic forms satisfy the nonlocal Euler
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| equation
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| $$ \lim_{\varepsilon \to 0} \int_{|x-y|>\varepsilon} (u(y) - u(x) ) K(x,y) \, dy = 0,$$
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| which should be understood in the sense of distributions.
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|
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| It is known that the gradient flow of a Dirichlet form (parabolic version of the
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| result) becomes instantaneously Hölder continuous <ref name="CCV"/>. The method
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| of the proof builds an integro-differential version of the parabolic De Giorgi
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| technique that was developed for the study of critical [[surface
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| quasi-geostrophic equation]].
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|
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| At some point in the original proof of De Giorgi, it is used that the
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| characteristic functions of a set of positive measure do not belong to $H^1$.
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| Moreover, a quantitative estimate is required about the measure of
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| ''intermediate'' level sets for $H^1$ functions. In the integro-differential
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| context, the required statement to carry out the proof would be the same with
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| the $H^{s/2}$ norm. This required statement is not true for $s$ small, and would
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| even require a non trivial proof for $s$ close to $2$. The difficulty is
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| bypassed though an argument that takes advantage of the nonlocal character of
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| the equation, and hence the estimate blows up as the order approaches two.
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|
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| == Estimates which pass to the second order limit ==
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|
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| === Non variational case ===
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|
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| An integro-differential generalization of [[Krylov-Safonov]] theorem is
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| available both in the elliptic <ref name="CS"/> and parabolic <ref name="lara2011regularity"/> setting. The assumption on the kernels are
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| # '''Symmetry''': $K(x,y) = K(x,-y)$.
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| # '''Uniform ellipticity''': $\frac{(2-s)\lambda}{|y|^{n+s}} \leq K(x,y) \leq
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| \frac{(2-s) \Lambda}{|y|^{n+s}}$ for some fixed value $s \in (0,2)$.
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|
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| The right hand side $f$ is assumed to be in $L^\infty$. The constants in the
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| Hölder estimate do not blow up as $s \to 2$.
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|
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| === Variational case ===
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|
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| In the stationary case, it is known that minimizers of Dirichlet forms are
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| Hölder continuous by adapting Moser's proof of [[De Giorgi-Nash-Moser theorem]] to the
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| nonlocal setting <ref name="K"/>.
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|
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| == Other variants ==
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|
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| * There are Holder estimates for equations in divergence form that are non local in time <ref name="zacher2013" />
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| * If we allow for continuous dependence on the coefficients with respect to $x$, there are Hölder estimates for a very general class of integral equations <ref name="barles2011" />.
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|
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|
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| == References ==
| | and $B$, which is known as the Boltzmann collision kernel, measures the strength of collisions in different directions. |
| {{reflist|refs=
| |
| <ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Hölder
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| estimates for solutions of integro-differential equations like the fractional
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| Laplace | url=http://dx.doi.org/10.1512/iumj.2006.55.2706 |
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| doi=10.1512/iumj.2006.55.2706 | year=2006 | journal=Indiana University
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| Mathematics Journal | issn=0022-2518 | volume=55 | issue=3 |
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| pages=1155–1174}}</ref>
| |
| <ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre |
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| first2=Luis | title=Regularity theory for fully nonlinear integro-differential
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| equations | url=http://dx.doi.org/10.1002/cpa.20274 | doi=10.1002/cpa.20274 |
| |
| year=2009 | journal=[[Communications on Pure and Applied Mathematics]] |
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| issn=0010-3640 | volume=62 | issue=5 | pages=597–638}}</ref>
| |
| <ref name="CCV">{{Citation | last1=Caffarelli | first1=Luis | last2=Chan |
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| first2=Chi Hin | last3=Vasseur | first3=Alexis | title= |
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| doi=10.1090/S0894-0347-2011-00698-X | year=2011 | journal=[[Journal of the
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| American Mathematical Society]] | issn=0894-0347 | issue=24 |
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| pages=849–869}}</ref>
| |
| <ref name="K">{{Citation | last1=Kassmann | first1=Moritz | title=A priori
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| estimates for integro-differential operators with measurable kernels |
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| url=http://dx.doi.org/10.1007/s00526-008-0173-6 | doi=10.1007/s00526-008-0173-6
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| | year=2009 | journal=Calculus of Variations and Partial Differential Equations
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| | issn=0944-2669 | volume=34 | issue=1 | pages=1–21}}</ref>
| |
| <ref name="BK">{{Citation | last1=Bass | first1=Richard F. | last2=Kassmann |
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| first2=Moritz | title=Hölder continuity of harmonic functions with respect to
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| operators of variable order | url=http://dx.doi.org/10.1080/03605300500257677 |
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| doi=10.1080/03605300500257677 | year=2005 | journal=Communications in Partial
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| Differential Equations | issn=0360-5302 | volume=30 | issue=7 |
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| pages=1249–1259}}</ref>
| |
| <ref name="BL">{{Citation | last1=Bass | first1=Richard F. | last2=Levin |
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| first2=David A. | title=Harnack inequalities for jump processes |
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| url=http://dx.doi.org/10.1023/A:1016378210944 | doi=10.1023/A:1016378210944 |
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| year=2002 | journal=Potential Analysis. An International Journal Devoted to the
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| Interactions between Potential Theory, Probability Theory, Geometry and
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| Functional Analysis | issn=0926-2601 | volume=17 | issue=4 |
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| pages=375–388}}</ref>
| |
| <ref name="lara2011regularity">{{Citation | last1=Lara | first1= Héctor Chang | last2=Dávila | first2= Gonzalo | title=Regularity for solutions of non local parabolic equations | journal=Calculus of Variations and Partial Differential Equations | year=2011 | pages=1--34}}</ref>
| |
| <ref name="zacher2013">{{Citation | last1=Zacher | first1= Rico | title=A De Giorgi--Nash type theorem for time fractional diffusion equations | url=http://dx.doi.org/10.1007/s00208-012-0834-9 | journal=Math. Ann. | issn=0025-5831 | year=2013 | volume=356 | pages=99--146 | doi=10.1007/s00208-012-0834-9}}</ref>
| |
| <ref name="barles2011">{{Citation | last1=Barles | first1= Guy | last2=Chasseigne | first2= Emmanuel | last3=Imbert | first3= Cyril | title=H\"older continuity of solutions of second-order non-linear elliptic integro-differential equations | url=http://dx.doi.org/10.4171/JEMS/242 | journal=J. Eur. Math. Soc. (JEMS) | issn=1435-9855 | year=2011 | volume=13 | pages=1--26 | doi=10.4171/JEMS/242}}</ref>
| |
| <ref name="rang2013h">{{Citation | last1=Rang | first1= Marcus | last2=Kassmann | first2= Moritz | last3=Schwab | first3= Russell W | title=H$\backslash$" older Regularity For Integro-Differential Equations With Nonlinear Directional Dependence | journal=arXiv preprint arXiv:1306.0082}}</ref>
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| <ref name="bogdan2005harnack">{{Citation | last1=Bogdan | first1= Krzysztof | last2=Sztonyk | first2= Pawe\l | title=Harnack’s inequality for stable Lévy processes | journal=Potential Analysis | year=2005 | volume=22 | pages=133--150}}</ref>
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| <ref name="schwab2014regularity">{{Citation | last1=Schwab | first1= Russell W | last2=Silvestre | first2= Luis | title=Regularity for parabolic integro-differential equations with very irregular kernels | journal=arXiv preprint arXiv:1412.3790}}</ref>
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| <ref name="kassmann2013intrinsic">{{Citation | last1=Kassmann | first1= Moritz | last2=Mimica | first2= Ante | title=Intrinsic scaling properties for nonlocal operators | journal=arXiv preprint arXiv:1310.5371}}</ref>
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| }}
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