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= Problems for integro-differential equations with rough coefficients or nonlinear equations =
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
== Hölder estimates for singular integro-differential equations ==
\[
A_w(\varphi)=e^{-bw} A(e^{aw}\varphi)\quad\mbox{for all }\varphi \in C^{\infty}(M),
\]
where $a, b$ are constant.


Consider an integro-differential equation of the form
Examples of conformally invariant operators include:
\[ \int_{\R^d} \left(u(x+y) - u(x) \right) \mathrm{d} \mu_x(y) = 0 \qquad \text{for all } x \in B_1.\]
(An extra gradient correction term may be necessary if the measure $\mu_x$ is too singular at the origin)


[[Hölder estimates]] are known to hold under certain 'ellipticity' assumptions for the measures $\mu_x(y)$. In many cases, we consider the absolutely continuous version $\mathrm{d} \mu_x(y) = K(x,y) \mathrm{d}y$ and write the assumptions in terms of the kernel $K$. One would expect that the estimates should hold every time the measures $\mu_x$ satisfy.
* The conformal Laplacian:
\[ \int_{B_{2R} \setminus B_R} (x \cdot e)^2 \mathrm{d} \mu_x(y) \approx R^{2-\alpha}, \]
\[
for all radius $R>0$ and $x \in B_1$, for some given constant $\alpha \in (0,2)$. This is the sharp assumption for stable operators that are independent of $x$ <ref name="ros2014regularity" />.
L_g=-\Delta_g + \frac{n-2}{4(n-1)}R_g,
\]
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 $b=\frac{n+2}{2}$.


[[Hölder estimates]] are not known to hold under such generality. For the current methods, singular measures $\mu_x$ (without an absolutely continuous part) are out of reach. A new idea is needed in order to solve this problem.
* The Paneitz operator <ref name="paneitz1983quartic"/> <ref name="paneitz2008quartic"/>:
\[
P=(-\Delta_g)^2-\mbox{div}_g (a_n R_g g+b_n Ric_g)d+\frac{n-4}{2}Q,
\]
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 $(-\Delta_g)^2$.


Note that a key part of this problem is that the measures $\mu_x$ should not have any regularity assumption respect to $x$.
* 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.  


== An integral ABP estimate ==
*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 $\mathbb{R}^n$.


The nonlocal version of the [[Alexadroff-Bakelman-Pucci estimate]] holds either for a right hand side in $L^\infty$ <ref name="CS"/> (in which the integral right hand side is approximated by a discrete sum) or under very restrictive assumptions on the kernels <ref name="GS"/>. Would the following result be true?


Assume $u_n \leq 0$ outside $B_1$ and for all $x \in B_1$,
Special cases:
\[ \int_{\R^n} (u(x+y)-u(x)) K(x,y) \mathrm d y \geq \chi_{A_n}(x). \]
Where $\chi_{A_n}$ stands for the characteristic function of the sets $A_n$. Assume that the kernels $K$ satisfy symmetry and a uniform ellipticity condition
\begin{align*}
K(x,y) &= K(x,-y) \\
\lambda |y|^{-n-s} \leq K(x,y) &\leq \Lambda |y|^{-n-s} \qquad \text{for some } 0<\lambda<\Lambda \text{ and } s \in (0,2).
\end{align*}
If $|A_n|\to 0$ as $n \to +\infty$, is it true that $\sup u_n^+ \to 0$ as well?


This type of estimate is currently known only under strong structural hypothesis on the kernels $K$.<ref name="GS"/>
* On the Euclidean space $\mathbb{R}^n$: the operators mentioned above are just the fractional Laplacians.


== Holder estimates for parabolic equations with variable order ==
* 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:
\[
P_\sigma=\frac{\Gamma(B+\frac{1}{2}+\sigma)}{\Gamma(B+\frac{1}{2}-\sigma)},\quad B=\sqrt{-\Delta_{g_{\mathbb{S}^n}}+\left(\frac{n-1}{2}\right)^2},
\]
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,


[[Holder estimates]] are known for elliptic and parabolic integro-differential equations with rough kernels. For elliptic equations, these estimates are available even when the order of the equation changes from point to point <ref name="BK"/> <ref name="S" />. Such estimate is not available for parabolic equations and it is not clear whether it holds.
* has the eigenfunctions of spherical harmonics, and  


More precisely, we would like to study a parabolic equation of the form
* is the inverse of a spherical Riesz potential.
\[ u_t(t,x) = \int_{\R^n} (u(t,x+y) - u(t,x)) K(t,x,y) dy.\]
Here $K$ is symmetric (i.e. $K(t,x,y) = K(t,x,-y)$) and satisfies the bounds
\[ \frac \lambda {|y|^{n+s(t,x)}} \leq K(t,x,y) \leq \frac \Lambda {|y|^{n+s(t,x)}}.\]
The order of the equation $s(t,x) \in (0,1)$ changes from point to point and it should stay strictly away from zero. It would also make sense to study other families of [[linear integro-differential operators]]. Does a parabolic [[Holder estimate]] hold in this case?


== A [[comparison principle]] for $x$-dependent nonlocal equations which are '''not''' in the Levy-Ito form ==
Consider two continuous functions $u$ and $v$ such that
\begin{align*}
u(x) &\leq v(x) \qquad \text{for all $x$ outside some set } \Omega,\\
F(x,\{I_\alpha u(x)\}) &\geq F(x,\{I_\alpha v(x)\})\qquad \text{for all $x \in \Omega$}.
\end{align*}
Is it true that $u \leq v$ in $\Omega$ as well?


It is natural to expect this result to hold if $F$ is continuous respect to $x$ and the [[linear integro-differential operators]] $I_\alpha$ satisfy some nondegeneracy condition and continuity respect to $x$, e.g.
== References ==
\begin{align*}
{{reflist|refs=
I[u] = \int (u(x+z) - u(x) - Du(x)\cdot z 1_{B}(z))\mu_x(dz)
\end{align*}
where $(\mu_x)_x$ is a family of L\'evy measures, H\"older continous with respect to $x$?
 
Currently the comparison principle is only known if the kernels are continuous when written in the Levy-Ito form.<ref name="BI"/>
 
== Holder estimates for drift-diffusion equations (sharp assumptions for $b$ in the case $s>1/2$) ==
 
Consider a [[drift-diffusion equation]] of the form
\[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0.\]
 
The solution $u$ is known to become Holder continuous under a variety of assumptions on the vector field $b$. If we assume that $\mathrm{div}\, b = 0$, we may expect that the required assumptions are slightly more flexible. Indeed, if $s=1/2$, the solution $u$ becomes Holder for positive time if $b \in L^\infty(BMO)$ <ref name="CV"/>. On the other hand, if $s=1$, the solution $u$ becomes Holder continuous for positive time if $b \in L^\infty(BMO^{-1})$ (if $b$ is the sum of derivatives of $BMO$ functions) <ref name="FV"/> <ref name="SSSZ"/>. A natural conjecture would be that the same result applies for $s \in (1/2,1)$ if $b \in L^\infty(BMO^{2s-1})$ (meaning that $(-\Delta)^{1-2s} b \in L^\infty(BMO)$).
 
The case $s < 1/2$ is completely understood and the assumption $\mathrm{div}\, b =0$ is not even necessary. For $s \in (1/2,1)$, only some perturbative results seem to be known under stronger assumptions. It is conceivable that the approach of Caffarelli and Vasseur <ref name="CV"/> can be worked out assuming that $b \in L^\infty(L^p)$ for a critical power $p$.
 
= Open problems for equations related to fluids =  
 
== Well posedness of the supercritical [[surface quasi-geostrophic equation]] and related problems ==
Let $\theta_0 : \R^2 \to \R$ be a smooth function either with compact support or periodic. Let $s \in (0,1/2)$. Is there a global classical solution $\theta :\R^2 \to \R$ for the SQG equation?
\begin{align*}
\theta(x,0) &= \theta_0(x) \\
\theta_t + u \cdot \nabla \theta &= 0 \qquad \text{in } \R^2 \times (0,+\infty)
\end{align*}
where $u = R^\perp \theta$ and $R$ stands for the Riesz transform.
 
This is a very difficult open problem. It is believed that a solution would be a major step towards the understanding of Navier-Stokes equation. In the supercritical regime $s\in (0,1/2)$, the effect if the drift term is larger than the diffusion in small scales. Therefore, it seems unlikely that a proof of well posedness could be achieved with the methods currently known and listed in this wiki.


== Well posedness of the Hilbert flow problem ==
<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>


The Hilbert flow problem is a simple 1D toy model for fluid equations in higher dimensions. It was originally suggested in a paper by Cordoba, Cordoba and Fontelos.<ref name="cordobacordoba2005" /> The equation is in terms of a scalar function $\theta(t,x)$. Here $x \in \R$ is a one dimensional variable.
<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>
\[ \theta_t + \mathrm H\theta \, \theta_x = 0.\]
There are several independent proofs that this equation develops singularities in finite time.<ref name="cordobacordoba2005" />
<ref name="CCF2"/> <ref name="HDong"/> <ref name="K"/> <ref name="SV" /> The equation still develops singularities in finite time if we add fractional diffusion
\[ \theta_t + \mathrm H\theta \, \theta_x + (-\Delta)^s \theta = 0,\]
provided that $s < 1/4$.<ref name="HDong"/> <ref name="SV"/> <ref name="K"/> <ref name="li2011one" /> The equation is known to be well posed for $s \geq 1/2$. In the range $s \in [1/4,1/2)$, it is not known whether singularities may occur in finite time.


Silvestre and Vicol conjectured that the solution $\theta$ satisfies an a priori estimate in $C^{1/2}$ for positive time, both in the viscous and inviscid model.<ref name="SV" /> If this conjecture turns out to be true, the equation above will be well posed when $s > 1/4$.
<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>


= Open problems related to minimal surfaces and free boundaries =
<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>


== Regularity of [[nonlocal minimal surfaces]] ==
<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>


A nonlocal minimal surface that is sufficiently flat is known to be smooth <ref name="CRS"/>. The possibility of singularities in the general case reduces to the analysis of a possible existence of nonlocal minimal cones. The problem can be stated as follows.
<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>


For any $s \in (0,1)$, and any natural number $n$, is there any set $A \in \R^n$, other than a half space, such that
<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>
# $A$ is a cone: $\lambda A = A$ for any $\lambda > 0$.
# If $B$ is any set in $\R^n$ which coincides with $A$ outside of a compact set $C$, then the following inequality holds
\[ \int_C \int_{C} \frac{|\chi_A(x) - \chi_A(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y + 2 \int_C \int_{\R^n \setminus C} \frac{|\chi_A(x) - \chi_A(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y \leq \int_C \int_{C} \frac{|\chi_B(x) - \chi_B(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y + 2\int_C \int_{\R^n \setminus C} \frac{|\chi_B(x) - \chi_B(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y. \]


When $s$ is sufficiently close to one, such set does not exist if $n < 8$.
<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>


== Optimal regularity for the [[obstacle problem]] for a general integro-differential operator ==
<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>


Let $u$ be the solution to the [[obstacle problem for the fractional laplacian]],
<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>
\begin{align*}
u &\geq \varphi \qquad \text{in } \R^n, \\
(-\Delta)^{s/2} u &\geq 0 \qquad \text{in } \R^n, \\
(-\Delta)^{s/2} u &= 0 \qquad \text{in } \{u>\varphi\}, \\
\end{align*}
where $\varphi$ is a smooth compactly supported function. It is known that $u \in C^{1,s/2}$ (where $s$ coincides with the order of the fractional Laplacian). This regularity is optimal in the sense that one can construct solutions that are not in $C^{1,s/2+\varepsilon}$ for any $\varepsilon>0$. One can consider the same problem replacing the fractional Laplacian by any other nonlocal operator. In fact, this problem corresponds to the [[optimal stopping problem]] in stochastic control, with applications to mathematical finance. The fractional Laplacian is just the particular case when the [[Levy  process]] involved is $\alpha$-stable and radially symmetric. The optimal regularity for the general problem is currently an open problem. Even in the linear case with constant coefficients this is nontrivial. If $u$ is a solution of
\begin{align*}
u &\geq \varphi \qquad \text{in } \R^n, \\
L u &\leq 0 \qquad \text{in } \R^n, \\
L u &= 0 \qquad \text{in } \{u>\varphi\}, \\
\end{align*}
where $L$ is a [[linear integro-differential operator]], then what is the optimal regularity we can obtain for $u$?
 
The optimal regularity would naturally depend on some assumptions on the linear operator $L$. If $L$ is a purely integro-differential with a kernel $K$ satisfying the usual ellipticity conditions
\begin{align*}
K(y) &= K(-y) \\
\frac{\lambda(2-s)}{ |y|^{n+s}} \leq K(y) &\leq \frac{\Lambda(2-s)}{ |y|^{n+s}} \qquad \text{for some } 0<\lambda<\Lambda \text{ and } s \in (0,2),
\end{align*}
it is natural to expect the solution $u$ to be $C^s$, but this regularity is not optimal. Is the optimal regularity going to be $C^{1,s/2}$ as in the fractional Laplacian case? Most probably some extra assumption on the kernel will be needed.
 
A solution to this problem would be very interesting if it provides an optimal regularity result for a natural family of kernels. If the assumption is something hard to check (like for example that there exists an extension problem whose Dirichlet to Neumann map is $L$), then the result may not be that interesting.
 
UPDATE: This problem has been recently solved by Caffarelli, Ros-Oton, and Serra <ref name="CRS16" />.
 
== Complete understanding of free boundary points in the [[fractional obstacle problem]] ==
 
Some free boundary points of the [[fractional obstacle problem]] are classified as regular and the free boundary is known to be smooth around them <ref name="CSS"/>. Other points on the free boundary are classified as singular, and for $s=\frac12$ they are shown to be contained in a lower dimensional differentiable surface, and therefore to be rare <ref name="GP"/>. However, there may be other points on the free boundary that do not fall under those two categories. Two questions need to be answered.\
# Can there be any point on the free boundary that is neither regular nor singular? It is easy to produce examples in the [[thin obstacle problem]], using the [[extension technique]]. However, it is not clear if such examples can be made in the original formulation of the [[fractional obstacle problem]] because of the decay at infinity requirement.
# In case that a point of a third category exist, is the free boundary smooth around these points in the ''third category''?
 
== References ==
{{reflist|refs=
<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="CV">{{Citation | last1=Caffarelli | first1=Luis A. | last2=Vasseur | first2=Alexis | title=Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation | url=http://dx.doi.org/10.4007/annals.2010.171.1903 | doi=10.4007/annals.2010.171.1903 | year=2010 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=171 | issue=3 | pages=1903–1930}}</ref>
<ref name="SSSZ">{{Citation | last1=Seregin | first1=G. | last2=Silvestre | first2=Luis | last3=Sverak | first3=V. | last4=Zlatos | first4=A. | title=On divergence-free drifts | year=2010 | journal=Arxiv preprint arXiv:1010.6025}}</ref>
<ref name="FV">{{Citation | last1=Friedlander | first1=S. | last2=Vicol | first2=V. | title=Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics | year=2011 | journal=Annales de l'Institut Henri Poincare (C) Non Linear Analysis}}</ref>
<ref name="CRS">{{Citation | last1=Caffarelli | first1=Luis A. | last2=Roquejoffre | first2=Jean Michel |last3= Savin | first3= Ovidiu | title= Nonlocal Minimal Surfaces | url=http://onlinelibrary.wiley.com/doi/10.1002/cpa.20331/abstract | doi=10.1002/cpa.20331 | year=2010 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0003-486X | volume=63 | issue=9 | pages=1111–1144}}</ref>


<ref name="CRS16">{{Citation | last1=Caffarelli | first1=Luis A. | last2=Ros-Oton | first2=Xavier |last3= Serra | first3= Joaquim | title= Obstacle problems for integro-differential operators: Regularity of solutions and free boundaries | year=2016 | journal=[[preprint arXiv (2016)]]}}</ref>
<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>


<ref name="GS">{{Citation | last1=Guillen | first1=N. | last2=Schwab | first2=R. | title=Aleksandrov-Bakelman-Pucci Type Estimates For Integro-Differential Equations | year=2010 | journal=Arxiv preprint arXiv:1101.0279}}</ref>
<ref name="CSS">{{Citation | last1=Caffarelli | first1=Luis A. | 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="GP">{{Citation | last1=Petrosyan | first1=A. | last2=Garofalo | first2=N. | title=Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=2009 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=177 | issue=2 | pages=415–461}}</ref>
<ref name="GS">{{Citation | last1=Guillen | first1=N. | last2=Schwab | first2=R. | title=Aleksandrov-bakelman-pucci type estimates for integro-differential equations | year=2010 | journal=Arxiv preprint arXiv:1101.0279}}</ref>
<ref name="BI">{{Citation | last1=Barles | first1=Guy | last2=Imbert | first2=Cyril | title=Second-order elliptic integro-differential equations: viscosity solutions' theory revisited | url=http://dx.doi.org/10.1016/j.anihpc.2007.02.007 | doi=10.1016/j.anihpc.2007.02.007 | year=2008 | journal=Annales de l'Institut Henri Poincaré. Analyse Non Linéaire | issn=0294-1449 | volume=25 | issue=3 | pages=567–585}}</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="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="SV">{{Citation | last1=Silvestre | first1= Luis | last2=Vicol | first2= Vlad | title=On a transport equation with nonlocal drift | journal=arXiv preprint arXiv:1408.1056}}</ref>
<ref name="li2011one">{{Citation | last1=Li | first1= Dong | last2=Rodrigo | first2= José L | title=On a one-dimensional nonlocal flux with fractional dissipation | journal=SIAM Journal on Mathematical Analysis | year=2011 | volume=43 | pages=507--526}}</ref>
<ref name="cordobacordoba2005">{{Citation | last1=Córdoba | first1= Antonio | last2=Córdoba | first2= Diego | last3=Fontelos | first3= Marco A. | title=Formation of singularities for a transport equation with nonlocal velocity | url=http://dx.doi.org/10.4007/annals.2005.162.1377 | journal=Ann. of Math. (2) | issn=0003-486X | year=2005 | volume=162 | pages=1377--1389 | doi=10.4007/annals.2005.162.1377}}</ref>
<ref name="ros2014regularity">{{Citation | last1=Ros-Oton | first1= Xavier | last2=Serra | first2= Joaquim | title=Regularity theory for general stable operators | journal=arXiv preprint arXiv:1412.3892}}</ref>
<ref name="HDong">{{Citation | last1=Dong | first1= Hongjie | title=Well-posedness for a transport equation with nonlocal velocity | url=http://dx.doi.org/10.1016/j.jfa.2008.08.005 | journal=J. Funct. Anal. | issn=0022-1236 | year=2008 | volume=255 | pages=3070--3097 | doi=10.1016/j.jfa.2008.08.005}}</ref>
<ref name="K">{{Citation | last1=Kiselev | first1= A. | title=Regularity and blow up for active scalars | url=http://dx.doi.org/10.1051/mmnp/20105410 | journal=Math. Model. Nat. Phenom. | issn=0973-5348 | year=2010 | volume=5 | pages=225--255 | doi=10.1051/mmnp/20105410}}</ref>
<ref name="CCF2">{{Citation | last1=Córdoba | first1= Antonio | last2=Córdoba | first2= Diego | last3=Fontelos | first3= Marco A. | title=Integral inequalities for the Hilbert transform applied to a nonlocal transport equation | url=http://dx.doi.org/10.1016/j.matpur.2006.08.002 | journal=J. Math. Pures Appl. (9) | issn=0021-7824 | year=2006 | volume=86 | pages=529--540 | doi=10.1016/j.matpur.2006.08.002}}</ref>
}}
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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