Hölder estimates and Conformally invariant operators: Difference between pages

From nonlocal pde
(Difference between pages)
Jump to navigation Jump to search
imported>Luis
 
imported>Tianling
No edit summary
 
Line 1: Line 1:
Hölder continuity of the solutions can sometimes be proved only from ellipticity
On a general compact Riemannian manifold $M$ with metric $g$, a metrically defined operator $A$ is said to be conformally invariant if under the conformal change in the metric $g_w=e^{2w}g$, the pair of the corresponding operators $A_w$ and $A$ are related by
assumptions on the equation, without depending on smoothness of the
\[
coefficients. This allows great flexibility in terms of applications of the
A_w(\varphi)=e^{-bw} A(e^{aw}\varphi)\quad\mbox{for all }\varphi \in C^{\infty}(M),
result. The corresponding result for elliptic equations of second order is the
\]
[[Krylov-Safonov]] theorem in the non-divergence form, or the [[De Giorgi-Nash-Moser theorem]] in the divergence form.
where $a, b$ are constant.


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" />.
Examples of conformally invariant operators include:


There are integro-differential versions of both [[De Giorgi-Nash-Moser theorem]]
* The conformal Laplacian:
and [[Krylov-Safonov theorem]]. The former uses variational techniques and is
\[
stated in terms of Dirichlet forms. The latter is based on comparison
L_g=-\Delta_g + \frac{n-2}{4(n-1)}R_g,
principles.
\]  
where $n$ is the dimension of the manifold, $-\Delta_g$ is the Laplace–Beltrami operator of $g$, and $R_g$ is the scalar curvature of $g$. This is a second order differential operator. One can check that in this case, $a=\frac{n-2}{2}$ and $b=\frac{n+2}{2}$.


A Hölder estimate says that a solution to an integro-differential equation with rough coefficients
* The Paneitz operator <ref name="paneitz1983quartic"/> <ref name="paneitz2008quartic"/>:
$L_x u(x) = f(x)$ in $B_1$, is $C^\alpha$ in $B_{1/2}$ for some $\alpha>0$
\[
(small). It is very important when an estimate allows for a very rough dependence of
P=(-\Delta_g)^2-\mbox{div}_g (a_n R_g g+b_n Ric_g)d+\frac{n-4}{2}Q,
$L_x$ with respect to $x$, since the result then applies to the linearization of
\]
(fully) nonlinear equations without any extra a priori estimate. On the other
where $\mbox{div}_g$ is the divergence operator, $d$ is the differential operator, $Ric_g$ is the Ricci tensor,
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
respect to $y$. Therefore, smoothness (or even structural) assumptions for the
kernels with respect to $y$ can be made keeping such result applicable.
 
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
$$ L_x u(x) = a_{ij}(x) \partial_{ij} u + b(x) \cdot \nabla u + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \, \chi_{B_1}(y))
K(x,y) \, dy$$
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.
 
Since [[linear integro-differential operators]] 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 theorem]] or
[[Krylov-Safonov theorem]].
 
== The general statement ==
 
=== Elliptic form ===
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
\[
\[
L(u,x) = 0 \ \ \text{in } B_1,
Q=c_n|Ric_g|^2+d_nR_g^2-\frac{1}{2(n-2)}\Delta_gR
\]
\]
and $u \in L^\infty(\R^d)$, then for some small $\alpha > 0$,
and
\[ \|u\|_{C^\alpha(B_{1/2})} \leq C \|u\|_{L^\infty(\R^d)}.\]
 
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.
 
For non variational problems, in order to adapt the situation to the [[viscosity solution]] framework, the equation may be replaced by two inequalities.
\begin{align*}
M^+u \geq 0 \ \ \text{in } B_1, \\
M^-u \leq 0 \ \ \text{in } B_1.
\end{align*}
where $M^+$ and $M^-$ are [[extremal operators]] with respect to some class.
 
=== Parabolic form ===
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
\[
\[
u_t - L(u,x) = 0 \ \ \text{in } (-1,0] \times B_1,
a_n=\frac{(n-2)^2+4}{2(n-1)(n-2)}, b_n=-\frac{4}{n-2}, c_n=-\frac{2}{(n-2)^2}, d_n=\frac{n^3-4n^2+16n-16}{8(n-1)^2(n-2)^2}.
\]
\]
and $u \in L^\infty(\R^d)$, then for some small $\alpha > 0$,
This is a fourth order operator with leading term $(-\Delta_g)^2$.
\[ \|u\|_{C^\alpha((-1/2,0] \times B_{1/2})} \leq C \|u\|_{L^\infty((-1,0] \times \R^d)}.\]


== List of results ==
* GJMS operators <ref name="GJMS"/>: this is a family of conformally invariant differential operators with leading term $(-\Delta_g)^k$ for all integers $k$ if $n$ is odd, and for $k\in \{1,2,\cdots,\frac{n}{2}\}$ if $n$ is even. A nonexistence result can be found in <ref name="gover2004conformally"/> for $k>\frac n2$ and $n\ge 4$ even. An explicit formula and a recursive formula each for GJMS operators and Q-curvatures have been found by Juhl <ref name="Juhl1"/><ref name="Juhl2"/> (see also Fefferman-Graham<ref name="FG13"/> ). The formula are more explicit when they are on the standard spheres.


There are several Hölder estimates for elliptic and parabolic integro-differential equations which have been obtained. Here we list some of these results with a brief description of their main assumptions.
*Scattering operators <ref name="graham2003scattering"/>, or the conformally invariant fractional powers of the Laplacian <ref name="chang2011fractional"/>: This is a family of conformally invariant pseudo-differential operators $P_\sigma$ defined on the conformal infinity of asymptotically hyperbolic manifolds with leading term $(-\Delta_g)^\sigma$ for all real numbers $\sigma\in (0,\frac n2)$ except at most finite values. The authors <ref name="chang2011fractional"/> reconciled the way of defining $P_\sigma$ in <ref name="graham2003scattering"/> and the localization method of Caffarelli-Silvestre  <ref name="CSextension"/> for the fractional Laplacian $(-\Delta)^\sigma$ in the Euclidean space $\mathbb{R}^n$.


=== Variational equations ===
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
Special cases:
equation
$$ \lim_{\varepsilon \to 0} \int_{|x-y|>\varepsilon} (u(y) - u(x) ) K(x,y) \, dy = 0,$$
which should be understood in the sense of distributions.


It is known that the gradient flow of a Dirichlet form (parabolic version of the
* On the Euclidean space $\mathbb{R}^n$: the operators mentioned above are just the fractional Laplacians.
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]].  


At some point in the original proof of De Giorgi, it is used that the
* On the standard sphere $(\mathbb{S}^n, g_{\mathbb{S}^n})$ (which is the conformal infinity of the standar Poincare disk): they are the following intertwining operator <ref name="branson1987group"/> of explicit formula:
characteristic functions of a set of positive measure do not belong to $H^1$.
\[
Moreover, a quantitative estimate is required about the measure of
P_\sigma=\frac{\Gamma(B+\frac{1}{2}+\sigma)}{\Gamma(B+\frac{1}{2}-\sigma)},\quad B=\sqrt{-\Delta_{g_{\mathbb{S}^n}}+\left(\frac{n-1}{2}\right)^2},
''intermediate'' level sets for $H^1$ functions. In the integro-differential
\]
context, the required statement to carry out the proof would be the same with
where $\Gamma$ is the Gamma function and $\Delta_{g_{\mathbb{S}^n}}$ is the Laplace-Beltrami operator on $(\mathbb{S}^n, g_{\mathbb{S}^n})$. Moreover, the operator $P_{\sigma}$  
the $H^{s/2}$ norm. This required statement is not true for $s$ small, and would
* is the pull back of $(-\Delta)^{\sigma}$ under stereographic projections,  
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.


{{stub}}
* has the eigenfunctions of spherical harmonics, and


=== Non variational equations ===
* is the inverse of a spherical Riesz potential.
The results below are nonlocal versions of [Krylov-Safonov theorem].  No regularity needs to be
assumed for $K$ with respect to $x$.


* The first result was obtained by Bass and Levin using probabilistic techniques.<ref
name="BL"/> It applies to elliptic integro-differential equations with symmetric and uniformly elliptic kernels (bounded pointwise). The constants obtained in the estimates are not uniform as the order of the equation goes to two. An extension of this result was obtained by Song and Vondracev <ref name="song2004" />.


The estimate says the following. Assume a bounded function $u: \R^n \to \R$ solves
== References ==
$$ \int_{\R^n} (u(x+y) - u(x)) K(x,y) \mathrm{d}y = 0 \qquad \text{in } B_1, $$
{{reflist|refs=
where $K$ satisfies
\begin{equation} \label{pointwisebound}
\frac{\lambda}{|y|^{n+s}} \leq K(x,y) \leq \frac{\Lambda}{|y|^{n+s}} \qquad \text{for all } x,y \in B_1,
\end{equation}
where $s \in (0,2)$ and $\Lambda \geq \lambda > 0$ are given parameters. Then
\[ \|u\|_{C^\alpha(B_{1/2})} \leq C \|u\|_{L^\infty(\R^n)}.\]
 
* A result by Bass and Kassmann also uses probabilistic techniques.<ref
name="BL"/> It applies to elliptic integro-differential equations with a rather general set of assumptions in the kernels. The main novelty is that the order of the equation may vary (continuously) from point to point. The constants obtained in the estimates are not uniform as the order of the equation goes to two.
* The first purely analytic proof was obtained by Luis Silvestre <ref
name="S"/>. The assumptions on the kernel are similar to those of Bass and Kassmann except that the order of the equation may change abruptly from point to point. The constants obtained in the estimates are not uniform as the order of the equation goes to two.
* The first estimate which remains uniform as the order of the equation goes to two was obtained by Caffarelli and Silvestre <ref name="CS"/>. The equations here are elliptic, with symmetric and uniformly elliptic kernels (pointwise bounded as in \ref{pointwisebound}).
* An estimate for parabolic equations of order one with a bounded drift was obtained by Silvestre <ref name="silvestre2011differentiability"/>. The equation here is parabolic, with symmetric and uniformly elliptic kernels (pointwise bounded as in \ref{pointwisebound}). The order of the equation is set to be one, but the proof also gives the estimate for any order greater than one, or also less than one if there is no drift. The constants in the estimates blow up as the order of the equation converges to two.
* Davila and Chang-Lara studied equations with nonsymmetric kernels, elliptic and parabolic. Their first result is for elliptic equations so that the odd part of the kernels is of lower order compared to their symmetric part <ref name="lara2012regularity" />. A later work provides estimates for parabolic equations which stay uniform as the order of the equation goes to two.<ref name="lara2014regularity" />. In a newer paper they improved both of their previous results by providing estimates for parabolic equations, with nonsymmetric kernels <ref name="chang2014h" />. For these three results,the kernels are required to be uniformly elliptic (pointwise bounded as in \ref{pointwisebound}).
* The first relaxation of the pointwise bound \eqref{pointwisebound} on the kernels was given by Bjorland, Caffarelli and Figalli <ref name="bjorland2012" /> for elliptic equations. They consider symmetric kernels which are bounded above everywhere, but are bounded below only in a cone of directions. A similar result is obtained by Kassmann, Rang and Schwab <ref name="rang2013h"/>.
* A generalization of all previous results was obtained by Schwab and Silvestre <ref name="schwab2014regularity" />. They consider parabolic equations. The estimates stay uniform as the order of the equation goes to two. The kernels are assumed to be bounded above only in average and bounded below only in sets of positive density. There is no symmetry assumption. It corresponds to the class of operators described [[Linear_integro-differential_operator#More singular/irregular kernels|here]].
* A non scale invariant family of kernels was studied by Kassmann and Mimica <ref name="kassmann2013intrinsic"/>. They study elliptic equations with symmetric kernels that satisfy pointwise bounds. However, the assumption is different from \eqref{pointwisebound} in the sense that $K(x,y)$ is required to be comparable to a function of $|y|$ which is not necessarily power-like.
 
== Estimates which blow up as the order goes to two ==
 
=== Non variational case ===
 
The Hölder 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 needs to be
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 $L^\infty$.
 
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
<ref name="branson1987group">{{Citation | last1=Branson | first1= Thomas P | title=Group representations arising from Lorentz conformal geometry | journal=Journal of functional analysis | year=1987 | volume=74 | pages=199--291}}</ref>
$E(u,v)$ 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
<ref name="CSextension">{{Citation | last1=Caffarelli | first1= Luis | last2=Silvestre | first2= Luis | title=An extension problem related to the fractional Laplacian | journal=Communications in Partial Differential Equations | year=2007 | volume=32 | pages=1245--1260}}</ref>
equation
$$ \lim_{\varepsilon \to 0} \int_{|x-y|>\varepsilon} (u(y) - u(x) ) K(x,y) \, dy = 0,$$
which should be understood in the sense of distributions.


It is known that the gradient flow of a Dirichlet form (parabolic version of the
<ref name="chang2011fractional">{{Citation | last1=Chang | first1= Sun-Yung Alice | last2=González | first2= Maria del Mar | title=Fractional Laplacian in conformal geometry | journal=Advances in Mathematics | year=2011 | volume=226 | pages=1410--1432}}</ref>
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]].


At some point in the original proof of De Giorgi, it is used that the
<ref name="FG13">{{Citation | last1=Fefferman | first1= Charles | last2=Graham | first2= C | title=Juhl’s formulae for GJMS operators and 𝑄-curvatures | journal=Journal of the American Mathematical Society|year=2013 | volume=26 | pages=1191--1207}}</ref>
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 ==
<ref name="gover2004conformally">{{Citation | last1=Gover | first1= A | last2=Hirachi | first2= Kengo | title=Conformally invariant powers of the Laplacian—a complete nonexistence theorem | journal=Journal of the American Mathematical Society |year=2004 |volume=17 | pages=389--405}}</ref>


=== Non variational case ===
<ref name="GJMS">{{Citation | last1=Graham | first1= C Robin | last2=Jenne | first2= Ralph | last3=Mason | first3= Lionel J | last4=Sparling | first4= George AJ | title=Conformally invariant powers of the Laplacian, I: Existence | journal=Journal of the London Mathematical Society | year=1992 | volume=2 | pages=557--565}}</ref>


An integro-differential generalization of [[Krylov-Safonov]] theorem is
<ref name="graham2003scattering">{{Citation | last1=Graham | first1= C Robin | last2=Zworski | first2= Maciej | title=Scattering matrix in conformal geometry | journal=Inventiones mathematicae | year=2003 | volume=152 | pages=89--118}}</ref>
available both in the elliptic <ref name="CS"/> and parabolic <ref name="lara2011regularity"/> setting. The assumption on the kernels are
# '''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 $s \in (0,2)$.


The right hand side $f$ is assumed to be in $L^\infty$. The constants in the
<ref name="Juhl1">{{Citation | last1=Juhl | first1= Andreas | title=On the recursive structure of Branson’s Q-curvature | journal=arXiv preprint arXiv:1004.1784}}</ref>
Hölder estimate do not blow up as $s \to 2$.


=== Variational case ===
<ref name="Juhl2">{{Citation | last1=Juhl | first1= Andreas | title=Explicit formulas for GJMS-operators and Q-curvatures | journal=Geometric and Functional Analysis | year=2013|volume=23 | pages=1278--1370}}</ref>


In the stationary case, it is known that minimizers of Dirichlet forms are
<ref name="paneitz1983quartic">{{Citation | last1=Paneitz | first1= S | title=A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds |year=1983 | journal=preprint}}</ref>
Hölder continuous by adapting Moser's proof of [[De Giorgi-Nash-Moser theorem]] to the
nonlocal setting <ref name="K"/>.


== Other variants ==
<ref name="paneitz2008quartic">{{Citation | last1=Paneitz | first1= S | title=A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary) | url=http://dx.doi.org/10.3842/SIGMA.2008.036 | doi:10.3842/SIGMA.2008.036 | year=2008 | journal=SIGMA Symmetry Integrability Geom. Methods Appl. | issue=4 | Paper=036}}</ref>


* There are Holder estimates for equations in divergence form that are non local in time <ref name="zacher2013" />
* 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" />.
== References ==
{{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="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>
<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>
<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>
<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>
<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>
<ref name="song2004">{{Citation | last1=Song | first1= Renming | last2=Vondra\vcek | first2= Zoran | title=Harnack inequality for some classes of Markov processes | url=http://dx.doi.org/10.1007/s00209-003-0594-z | journal=Math. Z. | issn=0025-5874 | year=2004 | volume=246 | pages=177--202 | doi=10.1007/s00209-003-0594-z}}</ref>
<ref name="silvestre2011differentiability">{{Citation | last1=Silvestre | first1= Luis | title=On the differentiability of the solution to the Hamilton--Jacobi equation with critical fractional diffusion | journal=Advances in mathematics | year=2011 | volume=226 | pages=2020--2039}}</ref>
<ref name="lara2012regularity">{{Citation | last1=Lara | first1= Héctor Chang | last2=Dávila | first2= Gonzalo | title=Regularity for solutions of nonlocal, nonsymmetric equations | year=2012 | volume=29 | pages=833--859}}</ref>
<ref name="chang2014h">{{Citation | last1=Chang-Lara | first1= Hector | last2=Davila | first2= Gonzalo | title=H$\backslash$" older estimates for non-local parabolic equations with critical drift | journal=arXiv preprint arXiv:1408.0676}}</ref>
<ref name="lara2014regularity">{{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=2014 | volume=49 | pages=139--172}}</ref>
<ref name="bjorland2012">{{Citation | last1=Bjorland | first1= C. | last2=Caffarelli | first2= L. | last3=Figalli | first3= A. | title=Non-local gradient dependent operators | url=http://dx.doi.org/10.1016/j.aim.2012.03.032 | journal=Adv. Math. | issn=0001-8708 | year=2012 | volume=230 | pages=1859--1894 | doi=10.1016/j.aim.2012.03.032}}</ref>
}}
}}

Revision as of 22:05, 23 September 2013

On a general compact Riemannian manifold $M$ with metric $g$, a metrically defined operator $A$ is said to be conformally invariant if under the conformal change in the metric $g_w=e^{2w}g$, the pair of the corresponding operators $A_w$ and $A$ are related by \[ A_w(\varphi)=e^{-bw} A(e^{aw}\varphi)\quad\mbox{for all }\varphi \in C^{\infty}(M), \] where $a, b$ are constant.

Examples of conformally invariant operators include:

  • The conformal Laplacian:

\[ L_g=-\Delta_g + \frac{n-2}{4(n-1)}R_g, \] where $n$ is the dimension of the manifold, $-\Delta_g$ is the Laplace–Beltrami operator of $g$, and $R_g$ is the scalar curvature of $g$. This is a second order differential operator. One can check that in this case, $a=\frac{n-2}{2}$ and $b=\frac{n+2}{2}$.

\[ P=(-\Delta_g)^2-\mbox{div}_g (a_n R_g g+b_n Ric_g)d+\frac{n-4}{2}Q, \] where $\mbox{div}_g$ is the divergence operator, $d$ is the differential operator, $Ric_g$ is the Ricci tensor, \[ Q=c_n|Ric_g|^2+d_nR_g^2-\frac{1}{2(n-2)}\Delta_gR \] and \[ a_n=\frac{(n-2)^2+4}{2(n-1)(n-2)}, b_n=-\frac{4}{n-2}, c_n=-\frac{2}{(n-2)^2}, d_n=\frac{n^3-4n^2+16n-16}{8(n-1)^2(n-2)^2}. \] This is a fourth order operator with leading term $(-\Delta_g)^2$.

  • GJMS operators [3]: this is a family of conformally invariant differential operators with leading term $(-\Delta_g)^k$ for all integers $k$ if $n$ is odd, and for $k\in \{1,2,\cdots,\frac{n}{2}\}$ if $n$ is even. A nonexistence result can be found in [4] for $k>\frac n2$ and $n\ge 4$ even. An explicit formula and a recursive formula each for GJMS operators and Q-curvatures have been found by Juhl [5][6] (see also Fefferman-Graham[7] ). The formula are more explicit when they are on the standard spheres.
  • Scattering operators [8], or the conformally invariant fractional powers of the Laplacian [9]: This is a family of conformally invariant pseudo-differential operators $P_\sigma$ defined on the conformal infinity of asymptotically hyperbolic manifolds with leading term $(-\Delta_g)^\sigma$ for all real numbers $\sigma\in (0,\frac n2)$ except at most finite values. The authors [9] reconciled the way of defining $P_\sigma$ in [8] and the localization method of Caffarelli-Silvestre [10] for the fractional Laplacian $(-\Delta)^\sigma$ in the Euclidean space $\mathbb{R}^n$.


Special cases:

  • On the Euclidean space $\mathbb{R}^n$: the operators mentioned above are just the fractional Laplacians.
  • On the standard sphere $(\mathbb{S}^n, g_{\mathbb{S}^n})$ (which is the conformal infinity of the standar Poincare disk): they are the following intertwining operator [11] of explicit formula:

\[ P_\sigma=\frac{\Gamma(B+\frac{1}{2}+\sigma)}{\Gamma(B+\frac{1}{2}-\sigma)},\quad B=\sqrt{-\Delta_{g_{\mathbb{S}^n}}+\left(\frac{n-1}{2}\right)^2}, \] where $\Gamma$ is the Gamma function and $\Delta_{g_{\mathbb{S}^n}}$ is the Laplace-Beltrami operator on $(\mathbb{S}^n, g_{\mathbb{S}^n})$. Moreover, the operator $P_{\sigma}$

  • is the pull back of $(-\Delta)^{\sigma}$ under stereographic projections,
  • has the eigenfunctions of spherical harmonics, and
  • is the inverse of a spherical Riesz potential.


References

  1. Paneitz, S (1983), "A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds", preprint 
  2. Paneitz, S (2008), "A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary)", SIGMA Symmetry Integrability Geom. Methods Appl. (4), http://dx.doi.org/10.3842/SIGMA.2008.036 
  3. Graham, C Robin; Jenne, Ralph; Mason, Lionel J; Sparling, George AJ (1992), "Conformally invariant powers of the Laplacian, I: Existence", Journal of the London Mathematical Society 2: 557--565 
  4. Gover, A; Hirachi, Kengo (2004), "Conformally invariant powers of the Laplacian—a complete nonexistence theorem", Journal of the American Mathematical Society 17: 389--405 
  5. Juhl, Andreas, "On the recursive structure of Branson’s Q-curvature", arXiv preprint arXiv:1004.1784 
  6. Juhl, Andreas (2013), "Explicit formulas for GJMS-operators and Q-curvatures", Geometric and Functional Analysis 23: 1278--1370 
  7. Fefferman, Charles; Graham, C (2013), "Juhl’s formulae for GJMS operators and 𝑄-curvatures", Journal of the American Mathematical Society 26: 1191--1207 
  8. 8.0 8.1 Graham, C Robin; Zworski, Maciej (2003), "Scattering matrix in conformal geometry", Inventiones mathematicae 152: 89--118 
  9. 9.0 9.1 Chang, Sun-Yung Alice; González, Maria del Mar (2011), "Fractional Laplacian in conformal geometry", Advances in Mathematics 226: 1410--1432 
  10. Caffarelli, Luis; Silvestre, Luis (2007), "An extension problem related to the fractional Laplacian", Communications in Partial Differential Equations 32: 1245--1260 
  11. Branson, Thomas P (1987), "Group representations arising from Lorentz conformal geometry", Journal of functional analysis 74: 199--291