Extension technique and Harnack inequality: Difference between pages

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The Dirichlet-to-Neumann operator for the upper half-plane maps the boundary value $U(x, 0)$ of a harmonic function $U(x, y)$ in the upper half-space $\R^{n+1}_+ = \R^n \times [0, \infty)$ to its outer normal derivative $-\partial_y U(x, 0)$. This operator coincides with the square root of the Laplace operator, $(-\Delta)^{1/2}$. Extension technique is a similar identification of non-local operators (most notably the [[fractional Laplacian]] $(-\Delta)^s$) as Dirichlet-to-Neumann operators for (possibly degenerate) elliptic equations. This construction is frequently used to turn nonlocal problems involving the fractional Laplacian into local problems in one more space dimension.
$$
\newcommand{\dd}{\mathrm{d}}
\newcommand{\R}{\mathbb{R}}
$$


==Fractional Laplacian==
The Harnack inequality refers to a control of the maximum of a nonnegative solution of an equation by its minimum. Unlike the local case (either [[De Giorgi-Nash-Moser theorem]] or [[Krylov-Safonov theorem]]), for nonlocal equations one needs to assume that the function is nonnegative in the full space.


The extension problem for the [[fractional Laplacian]] $(-\Delta)^s$, $s \in (0, 1)$ takes the following form.<ref name="CS"/> Let
The Harnack inequality is tightly related to [[Holder estimates]] for solutions to elliptic/parabolic equations. For a large class of problems both statements are equivalent. But there are simple cases (stable processes with the spectral measure consisting of atoms) where the Harnack inequality fails but [[Hölder estimates]] still hold true.
$$U:\mathbb{R}^n \times \mathbb{R}_+ \longrightarrow \mathbb{R}$$
be a function satisfying
\begin{equation}
\label{eqn:Main}
\nabla \cdot (y^{1-2s} \nabla U(x,y)) = 0
\end{equation}
on the upper half-space, lying inside the appropriately weighted Sobolev space $\dot{H}(1-2s,\mathbb{R}^{n+1}_+)$. Then if we let $u(x) = U(x,0)$, we have
\begin{equation}
\label{eqn:Neumann}
(-\Delta)^s u(x) = -C_{n,s} \lim_{y\rightarrow 0} y^{1-2s} \partial_y U(x,y).
\end{equation}
The energy associated with the operator in \eqref{eqn:Main} is
\begin{equation}
\label{eqn:Energy}
\int y^{1-2s} |\nabla U|^2 dx dy
\end{equation}


The weight $y^{1-2s}$, for $0<s<1$, lies inside the Muckenhoupt $A_2$ class of weights. It is known that degenerate 2nd order elliptic PDEs with these weights satisfy many of the usual properties of uniformly elliptic PDEs, such as the maximum principle, the [[De Giorgi-Nash-Moser]] regularity theory, the [[boundary Harnack inequality]], and the Wiener criterion for regularity of a boundary point.<ref name="FKS"/><ref name="FKJ1"/><ref name="FKJ2"/>
The result can hold either in the parabolic or elliptic setting. The parabolic Harnack inequality trivially implies the elliptic one. The reverse implication is not automatic, and the proof in the parabolic case may have some extra difficulties compared to the elliptic case.


The translation invariance of the operator in the $x$-directions can be applied to obtain higher regularity results and Liouville type properties.<ref name="CSS"/>
== Elliptic case ==


The above extension technique is closely related to the concept of trace of a diffusion on a hyperplane.<ref name="MO"/><ref name="D"/>
In the elliptic setting, the Harnack inequality refers to the following type of result: if a function $u: \R^n \to \R$ satisfies an elliptic equation $ L_x u (x) = f(x)$ in the unit ball $B_1$ and is nonnegative in the full space $\R^n$, then
\[ \sup_{B_{1/2}} u \leq C \left( \inf_{B_{1/2}} u + \|f\| \right). \]


==Fractional powers of more general operators==
The norm $\|f\|$ may depend on the type of equation.
In a similar way, extension problem for fractional powers $L^s$ of a general self-adjoint nonnegative linear differential operators $L$ in a domain $\Omega \subset \mathbb{R}^n$ (or more generally, a Hilbert space) can be constructed. In this case, the extension into a "cylinder" $\Omega \times [0, \infty)$ is considered. Let $U$ be a solution of
$$\begin{cases}
\partial_y (y^{1-2s} \partial_y U(x, y)) - L_x U(x, y) = 0,&\hbox{in}~\Omega\times(0,\infty),\\
U(x,0)=u(x),&\hbox{on}~\Omega,
\end{cases}$$
with boundary conditions along $\partial \Omega \times [0,\infty)$ equal to the boundary conditions for $L$. Then
\begin{equation}
L^su(x) = C_{s} \lim_{y\rightarrow 0} y^{1-2s} U_y(x,y).
\end{equation}
Here $C_s$ is a constant depending only on $s$. Moreover, the solution can be explicitly written as
$$U(x,y)=\frac{y^{2s}}{4^s\Gamma(s)}\int_0^\infty e^{-tL}u(x)e^{-y^2/(4t)}\,\frac{dt}{t^{1+s}}.$$
For details see
If $L$ has a purely discrete spectrum on $\Omega$, the operator $(-L)^s$ has the same eigenfunctions as $L$, and its eigenvalues are $\{\lambda_i^s\}$, where the $\{\lambda_i\}$ are the eigenvalues of $(-L)$.


For example, $L$ can be the Dirichlet Laplacian in $\Omega$.<ref name="CT"/> Note that $(-L)^s$ is not the same as the fractional Laplacian, except when $\Omega = \mathbb{R}^n$.
== Parabolic case ==


==More general non-local operators==
In the parabolic setting, the Harnack inequality refers to the following type of result: if a function $u: [-1,0] \times \R^n \to \R$ satisfies a parabolic equation $ u_t - L_x u (x) = f(x)$ in the unit cylinder $(-1,0) \times B_1$ and is nonnegative in the full space $[-1,0] \times \R^n$, then
Let $L$ be as above, and consider the Dirichlet-to-Neumann operator $A$ related to the elliptic equation
\[ \sup_{[-1/2,-1/4] \times B_{1/2}} u \leq \left(\inf_{[-1/4,0] \times B_{1/2}} u + \|f\| \right). \]
\[ \partial_y (w(y) \partial_y U(x, y)) + L_x U(x, y) = 0 \]
in the upper half-plane. Then $A = f(-L)$ for some [[operator monotone function]] $f$. Conversely, for any operator monotone $f$, there is an appropriate extension problem for $f(-L)$. (This identification requires some conditions on $w(y)$ which ensure the extension problem is well-posed.)


The relation between $w$ and $f$ is equivalent to the Krein correspondence, and can be described as follows.<ref name="KW"/><ref name="SSV"/> For $\lambda \ge 0$, let $g_\lambda$ be the nonincreasing positive solution of the ODE
The norm $\|f\|$ may depend on the type of equation.
\[ \partial_y (w(y) \partial_y g_\lambda(y)) = \lambda g_\lambda(y) \]
for $y \ge 0$, satisfying $g_\lambda(0) = 1$. Furthermore, let $h$ be the nondecreasing solution of
\[ \partial_y (w(y) \partial_y h(y)) = 0 \]
satisfying $h(0) = 0$ and $h(1) = 1$. Then
\[ f(\lambda) = \lim_{y \to 0^+} \frac{1 - g_\lambda(y)}{h(y)} . \]
One can prove that $f$ defined above is operator monotone, and conversely, for any operator monotone $f$ one can find $w$ for which the above identity holds. Noteworthy, there are relatively few explicit pairs of corresponding $w$ and $f$.


Suppose now that $U(x, y)$ is a sufficiently regular solution of the extension problem
== Concrete examples ==
\[ \partial_y (w(y) \partial_y U(x, y)) + L_x U(x, y) = 0 . \]
For simplicity, suppose that the spectrum of $L$ is discrete. Let $\varphi$ be an eigenfunction of $-L$ with eigenvalue $\lambda$, and denote $U_\varphi(y) = \langle U(\cdot, y), \varphi \rangle$. Then $U_\varphi$ is a solution of the ODE
\[ \partial_y (w(y) \partial_y U_\varphi(y)) + \lambda U_\varphi(y) = 0 . \]
Hence, $U_\varphi(y) = U_\varphi(0) g_\lambda(y)$, and
\[ -\lim_{y \to 0^+} \frac{U_\varphi(y) - U_\varphi(0)}{h(y)} = f(\lambda) U_\varphi(0) . \]
It follows that
\[ -\lim_{y \to 0^+} \frac{U(\cdot, y) - U_\varphi(\cdot, 0)}{h(y)} = f(-L) U(\cdot, 0) , \]
or equivalently
\[ -\lim_{y \to 0^+} \frac{\partial_y U(\cdot, y)}{h'(y)} = f(-L) U(\cdot, 0) . \]
This proves that the Dirichlet-to-Neumann operator is indeed equal to $f(-L)$. The proof in the continuous spectrum case is similar.<ref name="K"/><ref name="KSV"/>


[[Operator monotone function]]s, often called complete Bernstein functions, form a subclass of [[Bernstein function]]s. Hence, existence of the extension problem is closely related to the concept of [[subordination]]. Operators of the form $f(-\Delta)$ for an operator monotone $f$ admit an explicit [[Operator monotone function#Operator monotone functions of the Laplacian|description]]. This gives a fairly explicit condition for the existence of the extension problem for a given translation-invariant non-local operator.
The Harnack inequality as above is known to hold in the following situations.


==Relationship with Scattering operators==
* '''Generalizad elliptic and parabolic [[Krylov-Safonov]]'''. If $L_x u(x)$ is a symmetric integro-differential operator of the form
There is an identification between the fractional Laplacian defined by the extension and the fractional Paneitz operator from Scattering Theory when the order of the operator is less than 1.<ref name="CG"/>
\[ L_x u(x) = \int_{\R^n} (u(x+y)-u(x)) K(x,y) \dd y \]
<!--This is a stub, to be expanded on later.-->
with $K$ symmetric ($K(x,y)=K(x,-y)$) and uniformly elliptic of order $s$: $(2-s)\lambda |y|^{-n-s} \leq K(x,y) \leq (2-s) \Lambda |y|^{-n-s}$.


==References==
In this case both the elliptic and parabolic Harnack inequality is known to hold with a constant $C$ which '''does not blow up as $s\to 2$''', and $||f||$ refers to $||f||_{L^\infty(B_1)}$ <ref name="CS"/><ref name="lara2011regularity"/>. It is a generalization of [[Krylov-Safonov]] theorem.
 
* '''Elliptic equations with variable order (but strictly less than 2)'''. If $L_x u(x)$ is an integro-differential operator of the form
\[ L_x u(x) = \int_{\R^n} (u(x+y)-u(x)- y \cdot \nabla u(x) \chi_{B_1}(y)) K(x,y) \dd y \]
with uniformly elliptic of variable order: $\lambda |y|^{-n-s_1} \leq K(x,y) \leq \Lambda |y|^{-n-s_2}$ and $0<s_1 < s_2 < 2$ and $s_2 - s_1 < 1$, then
the elliptic Harnack inequality holds if $f \equiv 0$<ref name="BK"/>. The constants in this result blow up as $s_2 \to 2$, so it does not generalize [[Krylov-Safonov]] theorem. The proof uses probability and was based on a previous result with fixed order <ref name="BL"/>.
 
It is conceivable that a purely analytic proof could be done using the method of the corresponding [[Holder estimate]] <ref name="S"/>, but such proof has never been done.
 
* '''Gradient flows of symmetric Dirichlet forms with variable order'''. If $u_t - L_x u(x)=0$ is the gradient flow of a [[Dirichlet form]]:
\[ \iint_{\R^n \times \R^n} (u(y)-u(x))^2 K(x,y)\, \dd x \dd y. \]
for kernels $K$ such that $K(x,y)=K(y,x)$ and $\lambda |x-y|^{-n-s_1} \leq K(x,y) \leq \Lambda |x-y|^{-n-s_2}$ for some $0<s_1<s_2<2$ and $|x-y|$ sufficiently small. Then the parabolic Harnack inequality holds if $f \equiv 0$ for some constant $C$ which a priori '''blows up as $s_2 \to 2$''' <ref name="BBCK"/>.
 
It is conceivable that a Harnack inequality for Dirichlet forms can be proved for an equation with fixed order, with constants that do not blow up as the order goes to two, using the ideas from the Holder estimates<ref name="K"/>.
 
== References ==
{{reflist|refs=
{{reflist|refs=
<ref name="CG">{{Citation | last1=González | first1=Maria del Mar | last2=Chang | first2=Sun-Yung Alice | title=Fractional Laplacian in conformal geometry | url=http://dx.doi.org/10.1016/j.aim.2010.07.016 | doi=10.1016/j.aim.2010.07.016 | year=2011 | journal=Advances in Mathematics | issn=0001-8708 | volume=226 | issue=2 | pages=1410–1432}}</ref>
<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Hölder estimates for solutions of integro-differential equations like the fractional Laplace | url=http://dx.doi.org/10.1512/iumj.2006.55.2706 | doi=10.1512/iumj.2006.55.2706 | year=2006 | journal=Indiana University Mathematics Journal | issn=0022-2518 | volume=55 | issue=3 | pages=1155–1174}}</ref>
<ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=An extension problem related to the fractional Laplacian | url=http://dx.doi.org.ezproxy.lib.utexas.edu/10.1080/03605300600987306 | doi=10.1080/03605300600987306 | year=2007 | journal=Communications in Partial Differential Equations | issn=0360-5302 | volume=32 | issue=7 | pages=1245–1260}}</ref>
<ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=Regularity theory for fully nonlinear integro-differential equations | url=http://dx.doi.org/10.1002/cpa.20274 | doi=10.1002/cpa.20274 | year=2009 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=62 | issue=5 | pages=597–638}}</ref>
<ref name="CSS">{{Citation | last1=Caffarelli | first1=Luis | last2=Salsa | first2=Sandro | last3=Silvestre | first3=Luis | title=Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian | url=http://dx.doi.org/10.1007/s00222-007-0086-6 | doi=10.1007/s00222-007-0086-6 | year=2008 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=171 | issue=2 | pages=425–461}}</ref>
<ref name="BK">{{Citation | last1=Bass | first1=Richard F. | last2=Kassmann | first2=Moritz | title=Harnack inequalities for non-local operators of variable order | url=http://dx.doi.org/10.1090/S0002-9947-04-03549-4 | doi=10.1090/S0002-9947-04-03549-4 | year=2005 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=357 | issue=2 | pages=837–850}}</ref>
<ref name="CT">{{Citation | last1=Tan | first1=Jinggang | last2=Cabré | first2=Xavier | title=Positive solutions of nonlinear problems involving the square root of the Laplacian | url=http://dx.doi.org/10.1016/j.aim.2010.01.025 | doi=10.1016/j.aim.2010.01.025 | year=2010 | journal=Advances in Mathematics | issn=0001-8708 | volume=224 | issue=5 | pages=2052–2093}}</ref>
<ref name="BL">{{Citation | last1=Bass | first1=Richard F. | last2=Levin | first2=David A. | title=Harnack inequalities for jump processes | url=http://dx.doi.org/10.1023/A:1016378210944 | doi=10.1023/A:1016378210944 | year=2002 | journal=Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis | issn=0926-2601 | volume=17 | issue=4 | pages=375–388}}</ref>
<ref name="FKS">{{Citation | last1=Fabes | first1=Eugene B. | last2=Kenig | first2=Carlos E. | last3=Serapioni | first3=Raul P. | title=The local regularity of solutions of degenerate elliptic equations | url=http://dx.doi.org/10.1080/03605308208820218 | doi=10.1080/03605308208820218 | year=1982 | journal=Communications in Partial Differential Equations | issn=0360-5302 | volume=7 | issue=1 | pages=77–116}}</ref>
<ref name="BBCK">{{Citation | last1=Barlow | first1=Martin T. | last2=Bass | first2=Richard F. | last3=Chen | first3=Zhen-Qing | last4=Kassmann | first4=Moritz | title=Non-local Dirichlet forms and symmetric jump processes | url=http://dx.doi.org/10.1090/S0002-9947-08-04544-3 | doi=10.1090/S0002-9947-08-04544-3 | year=2009 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=361 | issue=4 | pages=1963–1999}}</ref>
<ref name="FKJ1">{{Citation | last1=Fabes | first1=Eugene B. | last2=Kenig | first2=Carlos E. | last3=Jerison | first3=David | title=Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) | publisher=Wadsworth | series=Wadsworth Math. Ser. | year=1983 | chapter=Boundary behavior of solutions to degenerate elliptic equations | pages=577–589}}</ref>
<ref name="K">{{Citation | last1=Kassmann | first1=Moritz | title=A priori estimates for integro-differential operators with measurable kernels | url=http://dx.doi.org/10.1007/s00526-008-0173-6 | doi=10.1007/s00526-008-0173-6 | year=2009 | journal=Calculus of Variations and Partial Differential Equations | issn=0944-2669 | volume=34 | issue=1 | pages=1–21}}</ref>
<ref name="FKJ2">{{Citation | last1=Fabes | first1=Eugene B. | last2=Jerison | first2=David | last3=Kenig | first3=Carlos E. | title=The Wiener test for degenerate elliptic equations | url=http://www.numdam.org/item?id=AIF_1982__32_3_151_0 | year=1982 | journal=[[Annales de l'Institut Fourier|Université de Grenoble. Annales de l'Institut Fourier]] | issn=0373-0956 | volume=32 | issue=3 | pages=151–182}}</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="D">{{Citation | last1=DeBlassie | first1=R. D. | title=The first exit time of a two-dimensional symmetric stable process from a wedge | url=http://dx.doi.org/10.1214/aop/1176990735 | doi=10.1214/aop/1176990735 | year=1990 | journal=Ann. Probab. | issn=0091-1798 | volume=18 | pages=1034–1070}}</ref>
<ref name="MO">{{Citation | last1=Molchanov | first1=S. A. | last2=Ostrovski | first2=E. | title=Symmetric stable processes as traces of degenerate diffusion processes | url=http://dx.doi.org/10.1137/1114012 | doi=10.1137/1114012 | year=1969 | journal=Theor Probab. Appl. | volume=14 | pages=128–131}}</ref>
<ref name="KW">{{Citation | last1=Kotani | first1=S. | last2=Watanabe | first1=S. | title=Krein’s spectral theory of strings and
generalized diffusion processes | url=http://dx.doi.org/10.1007/BFb0093046 | doi=10.1007/BFb009304 | year=1982 | pages=235–259 | booktitle=Functional Analysis in Markov Processes | series=Lecture Notes in Mathematics | volume=923 | editor1-last=Fukushima | editor1-first=M. | publisher=Springer, Berlin / Heidelberg | isbn=978-3-540-11484-0}}</ref>
<ref name="SSV">{{Citation | last1=Schilling | first1=R. | last2=Song | first2=R. | last3=Vondraček | first3=Z. | title=Bernstein functions. Theory and Applications | year=2010 | publisher=de Gruyter, Berlin | series=Studies in Mathematics | volume=37 | url=http://dx.doi.org/10.1515/9783110215311 | doi=10.1515/9783110215311}}</ref>
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}}
}}
{{stub}}

Revision as of 17:05, 8 April 2015

$$ \newcommand{\dd}{\mathrm{d}} \newcommand{\R}{\mathbb{R}} $$

The Harnack inequality refers to a control of the maximum of a nonnegative solution of an equation by its minimum. Unlike the local case (either De Giorgi-Nash-Moser theorem or Krylov-Safonov theorem), for nonlocal equations one needs to assume that the function is nonnegative in the full space.

The Harnack inequality is tightly related to Holder estimates for solutions to elliptic/parabolic equations. For a large class of problems both statements are equivalent. But there are simple cases (stable processes with the spectral measure consisting of atoms) where the Harnack inequality fails but Hölder estimates still hold true.

The result can hold either in the parabolic or elliptic setting. The parabolic Harnack inequality trivially implies the elliptic one. The reverse implication is not automatic, and the proof in the parabolic case may have some extra difficulties compared to the elliptic case.

Elliptic case

In the elliptic setting, the Harnack inequality refers to the following type of result: if a function $u: \R^n \to \R$ satisfies an elliptic equation $ L_x u (x) = f(x)$ in the unit ball $B_1$ and is nonnegative in the full space $\R^n$, then \[ \sup_{B_{1/2}} u \leq C \left( \inf_{B_{1/2}} u + \|f\| \right). \]

The norm $\|f\|$ may depend on the type of equation.

Parabolic case

In the parabolic setting, the Harnack inequality refers to the following type of result: if a function $u: [-1,0] \times \R^n \to \R$ satisfies a parabolic equation $ u_t - L_x u (x) = f(x)$ in the unit cylinder $(-1,0) \times B_1$ and is nonnegative in the full space $[-1,0] \times \R^n$, then \[ \sup_{[-1/2,-1/4] \times B_{1/2}} u \leq \left(\inf_{[-1/4,0] \times B_{1/2}} u + \|f\| \right). \]

The norm $\|f\|$ may depend on the type of equation.

Concrete examples

The Harnack inequality as above is known to hold in the following situations.

  • Generalizad elliptic and parabolic Krylov-Safonov. If $L_x u(x)$ is a symmetric integro-differential operator of the form

\[ L_x u(x) = \int_{\R^n} (u(x+y)-u(x)) K(x,y) \dd y \] with $K$ symmetric ($K(x,y)=K(x,-y)$) and uniformly elliptic of order $s$: $(2-s)\lambda |y|^{-n-s} \leq K(x,y) \leq (2-s) \Lambda |y|^{-n-s}$.

In this case both the elliptic and parabolic Harnack inequality is known to hold with a constant $C$ which does not blow up as $s\to 2$, and $||f||$ refers to $||f||_{L^\infty(B_1)}$ [1][2]. It is a generalization of Krylov-Safonov theorem.

  • Elliptic equations with variable order (but strictly less than 2). If $L_x u(x)$ is an integro-differential operator of the form

\[ L_x u(x) = \int_{\R^n} (u(x+y)-u(x)- y \cdot \nabla u(x) \chi_{B_1}(y)) K(x,y) \dd y \] with uniformly elliptic of variable order: $\lambda |y|^{-n-s_1} \leq K(x,y) \leq \Lambda |y|^{-n-s_2}$ and $0<s_1 < s_2 < 2$ and $s_2 - s_1 < 1$, then the elliptic Harnack inequality holds if $f \equiv 0$[3]. The constants in this result blow up as $s_2 \to 2$, so it does not generalize Krylov-Safonov theorem. The proof uses probability and was based on a previous result with fixed order [4].

It is conceivable that a purely analytic proof could be done using the method of the corresponding Holder estimate [5], but such proof has never been done.

  • Gradient flows of symmetric Dirichlet forms with variable order. If $u_t - L_x u(x)=0$ is the gradient flow of a Dirichlet form:

\[ \iint_{\R^n \times \R^n} (u(y)-u(x))^2 K(x,y)\, \dd x \dd y. \] for kernels $K$ such that $K(x,y)=K(y,x)$ and $\lambda |x-y|^{-n-s_1} \leq K(x,y) \leq \Lambda |x-y|^{-n-s_2}$ for some $0<s_1<s_2<2$ and $|x-y|$ sufficiently small. Then the parabolic Harnack inequality holds if $f \equiv 0$ for some constant $C$ which a priori blows up as $s_2 \to 2$ [6].

It is conceivable that a Harnack inequality for Dirichlet forms can be proved for an equation with fixed order, with constants that do not blow up as the order goes to two, using the ideas from the Holder estimates[7].

References

  1. Caffarelli, Luis; Silvestre, Luis (2009), "Regularity theory for fully nonlinear integro-differential equations", Communications on Pure and Applied Mathematics 62 (5): 597–638, doi:10.1002/cpa.20274, ISSN 0010-3640, http://dx.doi.org/10.1002/cpa.20274 
  2. Lara, Héctor Chang; Dávila, Gonzalo (2011), "Regularity for solutions of non local parabolic equations", Calculus of Variations and Partial Differential Equations: 1--34 
  3. Bass, Richard F.; Kassmann, Moritz (2005), "Harnack inequalities for non-local operators of variable order", Transactions of the American Mathematical Society 357 (2): 837–850, doi:10.1090/S0002-9947-04-03549-4, ISSN 0002-9947, http://dx.doi.org/10.1090/S0002-9947-04-03549-4 
  4. Bass, Richard F.; Levin, David A. (2002), "Harnack inequalities for jump processes", Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis 17 (4): 375–388, doi:10.1023/A:1016378210944, ISSN 0926-2601, http://dx.doi.org/10.1023/A:1016378210944 
  5. Silvestre, Luis (2006), "Hölder estimates for solutions of integro-differential equations like the fractional Laplace", Indiana University Mathematics Journal 55 (3): 1155–1174, doi:10.1512/iumj.2006.55.2706, ISSN 0022-2518, http://dx.doi.org/10.1512/iumj.2006.55.2706 
  6. Barlow, Martin T.; Bass, Richard F.; Chen, Zhen-Qing; Kassmann, Moritz (2009), "Non-local Dirichlet forms and symmetric jump processes", Transactions of the American Mathematical Society 361 (4): 1963–1999, doi:10.1090/S0002-9947-08-04544-3, ISSN 0002-9947, http://dx.doi.org/10.1090/S0002-9947-08-04544-3 
  7. Kassmann, Moritz (2009), "A priori estimates for integro-differential operators with measurable kernels", Calculus of Variations and Partial Differential Equations 34 (1): 1–21, doi:10.1007/s00526-008-0173-6, ISSN 0944-2669, http://dx.doi.org/10.1007/s00526-008-0173-6