Hölder estimates: Difference between revisions

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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.
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]].
The Hölder 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.
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 Holder 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.
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 $L_x$ will be 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
In the non variational setting the integro-differential operators $L_x$ are
$$ L_x u(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \, \chi_{B_1}(y)) K(x,y) \, dy$$
assumed to belong to some family, but no continuity is assumed for its
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.
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.


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]]]
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 ==
== Estimates which blow up as the order goes to two ==
Line 17: Line 44:
=== Non variational case ===
=== Non variational case ===


The Holder estimates were first obtained using probabilistic techniques <ref name="BL"/> <ref name="BK"/> , and then using purely analytic methods <ref name="S"/>. 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
The Hölder estimates were first obtained using probabilistic techniques <ref
# '''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 $L$.
name="BL"/> <ref name="BK"/> , and then using purely analytic methods <ref
# '''Nondegeneracy''': If $K$ is the kernel associated to $L$, $\frac{\int_{\R^n} \min(y^2,y^\alpha) K(y) \, dy} {\sup_{B_1} K} \leq C_1$ for some $C_1$ and $\alpha>0$ independent of $K$.
name="S"/>. 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
# '''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 $L$.
# '''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 $L^\infty$.
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.
A particular case in which this result applies is the uniformly elliptic case.
$$\frac{\lambda}{|y|^{n+s(x)}} \leq K(x,y) \leq \frac{\Lambda}{|y|^{n+s(x)}}.$$
$$\frac{\lambda}{|y|^{n+s(x)}} \leq K(x,y) \leq \frac{\Lambda}{|y|^{n+s(x)}}.$$
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.
where $s$ is bounded below and above: $0 < s_0 \leq s(x) \leq s_1 < 2$, but no
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.
continuity of $s$ respect to $x$ is required.
* 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 $B_1$.
The kernel $K$ is assumed to be symmetric with respect to $y$: $K(x,y)=K(x,-y)$.
* 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 $B_1$.
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.
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 ===
=== Variational case ===


A [[Dirichlet forms]] is a quadratic functional of the form
A typical example of a symmetric nonlocal [[Dirichlet form]] is a bilinear form
$$ J(u) = \iint_{\R^n \times \R^n} |u(x)-u(y)|^2 K(x,y) \, dx \, dy $$.
$E(u,v)$ 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.  


Minimizers of Dirichlet forms are a nonlocal version of minimizers of integral functionals as in [[De Giorgi-Nash-Moser]] theorem.
Minimizers of the corresponding quadratic forms satisfy the nonlocal Euler
equation
$$ \int_{\R^n} (u(x+y) - 2 u(x) + u(x-y) ) K(x,y) \, dy = 0$$.


The symmetry assumption $K(x,y)=K(y,x)$ is natural since the skew-symmetric part of $K$ would be ignored by the quadratic functional.
It is known that the gradient flow of a Dirichlet form (parabolic version of the
result) becomes instantaneously Hölder continuous <ref name="CCV"/>. 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]].  


It is known that the gradient flow of a Dirichlet form (parabolic version of the result) becomes instantaneously Holder continuous <ref name="CCV"/>. 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 $H^1$.
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, similarly as in {{Citation needed}}.
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 ==
== Estimates which pass to the second order limit ==
Line 49: Line 110:
=== Non variational case ===
=== Non variational case ===


An integro-differential generalization of [[Krylov-Safonov]] theorem is available <ref name="CS"/>. The assumption on the kernels are
An integro-differential generalization of [[Krylov-Safonov]] theorem is
available <ref name="CS"/>. The assumption on the kernels are
# '''Symmetry''': $K(x,y) = K(x,-y)$.
# '''Symmetry''': $K(x,y) = K(x,-y)$.
# '''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 $s \in (0,2)$.
# '''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 Holder estimate do not blow up as $s \to 2$.
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 ===
=== Variational case ===


In the stationary case, it is known that minimizers of Dirichlet forms are Holder continuous by adapting Moser's proof of [[De Giorgi-Nash-Moser]] to the nonlocal setting <ref name="K"/>.
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 <ref name="K"/>.


== References ==
== References ==
{{reflist|refs=
{{reflist|refs=
<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="S">{{Citation | last1=Silvestre | first1=Luis | title=Hölder
<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>
estimates for solutions of integro-differential equations like the fractional
<ref name="CCV">{{Citation | last1=Caffarelli | first1=Luis | last2=Chan | first2=Chi Hin | last3=Vasseur | first3=Alexis | title= | doi=10.1090/S0894-0347-2011-00698-X | year=2011 | journal=[[Journal of the American Mathematical Society]] | issn=0894-0347 | issue=24 | pages=849–869}}</ref>
Laplace | url=http://dx.doi.org/10.1512/iumj.2006.55.2706 |
<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>
doi=10.1512/iumj.2006.55.2706 | year=2006 | journal=Indiana University
<ref name="BK">{{Citation | last1=Bass | first1=Richard F. | last2=Kassmann | first2=Moritz | title=Hölder continuity of harmonic functions with respect to operators of variable order | url=http://dx.doi.org/10.1080/03605300500257677 | doi=10.1080/03605300500257677 | year=2005 | journal=Communications in Partial Differential Equations | issn=0360-5302 | volume=30 | issue=7 | pages=1249–1259}}</ref>
Mathematics Journal | issn=0022-2518 | volume=55 | issue=3 |
<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>
pages=1155–1174}}</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="CCV">{{Citation | last1=Caffarelli | first1=Luis | last2=Chan |
first2=Chi Hin | last3=Vasseur | first3=Alexis | title= |
doi=10.1090/S0894-0347-2011-00698-X | year=2011 | journal=[[Journal of the
American Mathematical Society]] | issn=0894-0347 | issue=24 |
pages=849–869}}</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="BK">{{Citation | last1=Bass | first1=Richard F. | last2=Kassmann |
first2=Moritz | title=Hölder continuity of harmonic functions with respect to
operators of variable order | url=http://dx.doi.org/10.1080/03605300500257677 |
doi=10.1080/03605300500257677 | year=2005 | journal=Communications in Partial
Differential Equations | issn=0360-5302 | volume=30 | issue=7 |
pages=1249–1259}}</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>
}}
}}

Revision as of 18:37, 16 February 2012

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 Hölder 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 [1] [2] , and then using purely analytic methods [3]. 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.

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 [4]. 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 [5]. 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 [6].

References

  1. 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 
  2. Bass, Richard F.; Kassmann, Moritz (2005), [http://dx.doi.org/10.1080/03605300500257677 "Hölder continuity of harmonic functions with respect to operators of variable order"], Communications in Partial Differential Equations 30 (7): 1249–1259, doi:10.1080/03605300500257677, ISSN 0360-5302, http://dx.doi.org/10.1080/03605300500257677 
  3. Silvestre, Luis (2006), [http://dx.doi.org/10.1512/iumj.2006.55.2706 "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 
  4. Caffarelli, Luis; Chan, Chi Hin; Vasseur, Alexis (2011), [[Journal of the American Mathematical Society]] (24): 849–869, doi:10.1090/S0894-0347-2011-00698-X, ISSN 0894-0347 
  5. Caffarelli, Luis; Silvestre, Luis (2009), [http://dx.doi.org/10.1002/cpa.20274 "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 
  6. Kassmann, Moritz (2009), [http://dx.doi.org/10.1007/s00526-008-0173-6 "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