Fractional Laplacian and Conformally invariant operators: Difference between pages

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The fractional Laplacian $(-\Delta)^s$ is a classical operator which can be defined in several equivalent ways.
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.


It is the most typical elliptic operator of order $2s$.
Examples of conformally invariant operators include:


== Definitions ==
* 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 $b=\frac{n+2}{2}$.


=== As a pseudo-differential operator ===
* The Paneitz operator <ref name="paneitz1983quartic"/> <ref name="paneitz2008quartic"/>:
The fractional Laplacian is the pseudo-differential operator with symbol $|\xi|^{2s}$. In other words, the following formula holds
\[
\[ \widehat{(-\Delta)^s f}(\xi) = |\xi|^{2s} \hat f(\xi).\]
P=(-\Delta_g)^2-\mbox{div}_g (a_n R_g g+b_n Ric_g)d+\frac{n-4}{2}Q,
for any function (or tempered distribution) for which the right hand side makes sense.
\]
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 $(-\Delta_g)^2$.


This formula is the simplest to understand and it is useful for problems in the whole space. On the other hand, it is hard to obtain local estimates from it.
* 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.  


=== From functional calculus ===
*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 $\mathbb{R}^n$.


Since the operator $-\Delta$ is a self-adjoint positive definite operator in a dense subset $D$ of $L^2(\R^n)$, one can define $F(-\Delta)$ for any continuous function $F:\R^+ \to \R$. In particular, this serves as a more or less abstract definition of $(-\Delta)^s$.


This definition is not as useful for practical applications, since it does not provide any explicit formula.
Special cases:


=== As a singular integral ===
* On the Euclidean space $\mathbb{R}^n$: the operators mentioned above are just the fractional Laplacians.
If $f$ is regular enough and $s \in (0,1)$, $(-\Delta)^s f(x)$ can be computed by the formula
\[ (-\Delta)^s f(x) = c_{n,s} \int_{\R^n} \frac{f(x) - f(y)} {|x-y|^{n+2s}} \mathrm d y .\]


Where $c_{n,s}$ is a constant depending on dimension and $s$.
* 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,


This formula is the most useful to study local properties of equations involving the fractional Laplacian and regularity for critical semilinear problems.
* has the eigenfunctions of spherical harmonics, and  


=== As a generator of a [[Levy process]] ===
* is the inverse of a spherical Riesz potential.
The operator can be defined as the generator of $\alpha$-stable Levy processes. In other words, if $X_t$ is an $\alpha$-stable process starting at zero and $f$ is a smooth function, then
\[ (-\Delta)^s f(x) = \lim_{h \to 0^+} \frac 1 {h} \mathbb E [f(x) - f(x+X_h)]. \]


This definition is important for applications to probability.
 
== References ==
{{reflist|refs=
 
<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>
 
<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>
 
<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>
 
<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>
 
<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>
 
<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>
 
<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>
 
<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>
 
<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>
 
<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>
 
<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>
 
}}

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