---
title: Conformal Logarithmic Laplacian
url: https://www.emergentmind.com/topics/conformal-logarithmic-laplacian
type: topic
---

# Conformal Logarithmic Laplacian

The **conformal logarithmic Laplacian** most commonly denotes the order-zero derivative of the conformal fractional Laplacian on the round sphere \((\mathbb S^N,g)\). In the formulation developed on \(\mathbb S^N\), it is the operator
\[
\mathscr P_g^{\log}u(z):=\left.\frac{d}{ds}\right|_{s=0}[\mathscr P_g^s u](z),
\]
where \(\mathscr P_g^s\) is the conformal fractional Laplacian. It is a nonlocal singular integral operator with an explicit kernel \(|z-\zeta|^{-N}\), a precise conformal covariance law containing a logarithmic correction term, a complete spectral description in spherical harmonics, and a stereographic correspondence with the Euclidean logarithmic Laplacian \(L_\Delta\) modified by a conformal weight term [2507.21779].

## 1. Definition and basic construction

On the round sphere, the conformal fractional Laplacian is written for \(s\in(0,1)\) as
\[
\mathscr P_g^s u(z)=c_{N,s}\,\mathrm{P.V.}\int_{\mathbb S^N}\frac{u(z)-u(\zeta)}{|z-\zeta|^{N+2s}}\,dV_g(\zeta)+A_{N,s}u(z),
\]
with
\[
A_{N,s}:=\frac{\Gamma\!\left(\frac N2+s\right)}{\Gamma\!\left(\frac N2-s\right)}, \qquad
c_{N,s}:=4^s\pi^{-N/2}\frac{\Gamma\!\left(\frac N2+s\right)}{\Gamma(2-s)}\,s(1-s).
\]
Since \(\mathscr P_g^0=I\), the logarithmic operator is obtained by differentiating at \(s=0\):
\[
\mathscr P_g^{\log}u=\left.\frac{d}{ds}\right|_{s=0}\mathscr P_g^su
=\lim_{s\to0^+}\frac{\mathscr P_g^su-u}{s}.
\]
For \(u\in C^\beta(\mathbb S^N)\), \(\beta>0\), the resulting operator has the explicit form
\[
\mathscr P_g^{\log}u(z)=c_N\int_{\mathbb S^N}\frac{u(z)-u(\zeta)}{|z-\zeta|^N}\,dV_g(\zeta)+A_Nu(z),
\]
where
\[
c_N:=\pi^{-N/2}\Gamma\!\left(\frac N2\right), \qquad
A_N:=2\psi\!\left(\frac N2\right).
\]
The convergence
\[
\frac{\mathscr P_g^su-u}{s}\to \mathscr P_g^{\log}u
\quad\text{in }L^p(\mathbb S^N), \qquad 1\le p\le\infty,
\]
is part of the basic construction, so the derivative is not merely formal but an actual \(L^p\)-limit of the conformal fractional family [2507.21779].

The operator is “logarithmic” in two senses already visible at the level of definition. First, it is the derivative in the order parameter \(s\) of a power-type family. Second, the differentiated kernel lands exactly at the borderline singularity \(|z-\zeta|^{-N}\), which is the conformal-order-zero analogue of the Euclidean symbol \(2\log|\xi|\). Constants are eigenfunctions rather than kernel elements:
\[
\mathscr P_g^{\log}(1)=A_N.
\]
Accordingly, the first eigenvalue is negative for \(N=1,2\) and positive for \(N\ge 3\) [2507.21779].

## 2. Spectral description on the sphere

The spectral theory of \(\mathscr P_g^{\log}\) is completely aligned with the spherical harmonic decomposition. If \(Y_{i,j}\) is a spherical harmonic of degree \(i\), then
\[
-\Delta_gY_{i,j}=b_iY_{i,j}, \qquad b_i=i(i+N-1).
\]
For the fractional family, the eigenvalue multiplier is
\[
\varphi_{N,s}(\lambda)
=\frac{\Gamma\!\left(\frac12+s+\sqrt{\lambda+\left(\frac{N-1}{2}\right)^2}\right)}
{\Gamma\!\left(\frac12-s+\sqrt{\lambda+\left(\frac{N-1}{2}\right)^2}\right)}.
\]
Differentiating at \(s=0\) gives the logarithmic multiplier
\[
\varphi_N(\lambda)
=2\psi\!\left(\sqrt{\frac14(N-1)^2+\lambda}+\frac12\right),
\]
so that
\[
\mathscr P_g^{\log}\Phi=\varphi_N(\lambda)\Phi
\]
whenever \(-\Delta_g\Phi=\lambda\Phi\). On the degree-\(i\) spherical harmonics this becomes
\[
\varphi_N(b_i)=2\psi\!\left(i+\frac N2\right).
\]
The multiplier is strictly increasing, since
\[
\varphi_N'(\lambda)
=
\frac{2\,\psi^{(1)}\!\left(\sqrt{\frac14(N-1)^2+\lambda}+\frac12\right)}
{\sqrt{4\lambda+(N-1)^2}}
>0.
\]
Hence the eigenspaces of \(\mathscr P_g^{\log}\) are exactly the eigenspaces of \(-\Delta_g\), and the operator is spectrally diagonal in the spherical harmonic basis [2507.21779].

A later extension places \(\mathscr P_g^{\log}\) inside a one-parameter differentiated family,
\[
\mathscr P_g^{s+\ln}u(z):=\left.\frac{d}{dt}\mathscr P_g^t u(z)\right|_{t=s}, \qquad s\in(0,1),
\]
called the **conformal fractional--logarithmic Laplacian**. Its spherical-harmonic multiplier is
\[
\varphi_N^{s+\ln}(\lambda)
=
\varphi_{N,s}(\lambda)\Bigl[
\psi\!\Bigl(\tfrac12+s+\sqrt{\lambda+\tfrac14(N-1)^2}\Bigr)
+\psi\!\Bigl(\tfrac12-s+\sqrt{\lambda+\tfrac14(N-1)^2}\Bigr)
\Bigr],
\]
and \(\mathscr P_g^{s+\ln}u\to \mathscr P_g^{\ln}u\) uniformly and in \(L^p(\mathbb S^N)\) as \(s\to0^+\). This places the order-zero operator as the endpoint of a broader derivative-in-order conformal family [2603.21146].

## 3. Conformal covariance and stereographic correspondence

The defining structural property of \(\mathscr P_g^{\log}\) is its inhomogeneous conformal covariance. If \(\eta\in C^\infty(\mathbb S^N)\) is positive, then
\[
\mathscr P_{\eta g}^{\log}(\varphi)
=
\eta^{-N/4}\,\mathscr P_g^{\log}(\eta^{N/4}\varphi)-\varphi\ln\eta.
\]
If the conformal metric is written as
\[
\widetilde g=u^{4/N}g,
\]
then the same law becomes
\[
\mathscr P_{\widetilde g}^{\log}(\varphi)
=
u^{-1}\mathscr P_g^{\log}(u\varphi)-\frac{4}{N}\varphi\ln u.
\]
Unlike the conformal fractional Laplacians, whose covariance is purely multiplicative, the logarithmic derivative produces the additive correction term \(-\varphi\ln\eta\). That term is the distinctive conformal signature of the logarithmic operator [2507.21779].

The associated logarithmic \(Q\)-curvature is defined by
\[
Q_{\widetilde g}^{\log}:=\mathscr P_{\widetilde g}^{\log}(1).
\]
For the round metric,
\[
Q_g^{\log}=A_N=2\psi\!\left(\frac N2\right).
\]
Under \(\widetilde g=u^{4/N}g\), one obtains
\[
Q_{\widetilde g}^{\log}
=
u^{-1}\mathscr P_g^{\log}(u)-\frac{4}{N}\ln u,
\]
equivalently
\[
\mathscr P_g^{\log}(u)=\frac{4}{N}u\ln u+uQ_{\widetilde g}^{\log}.
\]
This is the conformal curvature identity behind the logarithmic Yamabe problem [2507.21779].

The sphere operator is linked to the Euclidean logarithmic Laplacian through stereographic projection. With
\[
\sigma(z)=\frac{z'}{1+z_{N+1}}, \qquad
\sigma^{-1}(x)=\left(\frac{2x}{1+|x|^2},\frac{1-|x|^2}{1+|x|^2}\right),
\]
and conformal factor
\[
\phi(x)=\frac{2}{1+|x|^2},
\]
the pullback
\[
\iota(u)(x):=\phi(x)^{N/2}u(\sigma^{-1}(x))
\]
satisfies the exact intertwining identity
\[
\iota(\mathscr P_g^{\log}u)(x)=L_\Delta(\iota u)(x)-2(\iota u)(x)\ln\phi(x).
\]
Thus \(\mathscr P_g^{\log}\) does not correspond directly to \(L_\Delta\), but to \(L_\Delta-2\ln\phi\) after conformal conjugation [2507.21779].

The Euclidean operator \(L_\Delta\) itself is the derivative at \(s=0\) of the fractional Laplacian:
\[
L_\Delta=\left.\partial_s\right|_{s=0}(-\Delta)^s,
\qquad
\widehat{L_\Delta u}(\zeta)=2\log|\zeta|\,\hat u(\zeta),
\]
and on \(\mathbb R^N\) it has the exact singular integral representation
\[
L_\Delta u(x)=c_N\int_{\mathbb R^N}\frac{u(x)1_{B_1(x)}(y)-u(y)}{|x-y|^N}\,dy+\rho_Nu(x),
\]
with
\[
c_N=\pi^{-N/2}\Gamma\!\left(\frac N2\right), \qquad
\rho_N=2\log 2+\psi\!\left(\frac N2\right)-\gamma.
\]
That operator is not presented as conformally covariant; the conformal modification enters through the stereographic factor in the sphere formula [1710.03416].

## 4. Yamabe-type problems and variational framework

The principal nonlinear equation attached to the conformal logarithmic Laplacian is the logarithmic Yamabe, or constant logarithmic \(Q\)-curvature, equation
\[
\mathscr P_g^{\log}u=\frac{4}{N}u\ln|u|+\mu u
\qquad\text{on }\mathbb S^N.
\]
Its weak formulation is set in a Hilbert space
\[
\mathbb H(\mathbb S^N)
=
\left\{u\in L^2(\mathbb S^N):\|u\|_{\mathbb H(\mathbb S^N)}<\infty\right\},
\]
with norm
\[
\|u\|_{\mathbb H(\mathbb S^N)}
=
\left(
\frac{c_N}{2}\iint_{\mathbb S^N\times\mathbb S^N}
\frac{(u(z)-u(\zeta))^2}{|z-\zeta|^N}\,dV_g(z)\,dV_g(\zeta)
+\kappa\int_{\mathbb S^N}u^2\,dV_g
\right)^{1/2},
\]
where \(\kappa>|2\psi(N/4)|\). The corresponding bilinear form can also be written as
\[
\int_{\mathbb S^N}u_1\,\mathscr P_g^{\log}u_2\,dV_g
+(\kappa-A_N)\int_{\mathbb S^N}u_1u_2\,dV_g.
\]
The density statement
\[
\overline{C_c^\infty(\mathbb S^N\setminus\{-e_{N+1}\})}^{\|\cdot\|_{\mathbb H(\mathbb S^N)}}
=
\mathbb H(\mathbb S^N)
\]
is part of this framework [2507.21779].

On \(\mathbb R^N\), the corresponding weak equation is
\[
L_\Delta v=\frac{4}{N}v\ln|v|+\mu v
\qquad\text{in }\mathbb R^N,
\]
and the natural energy space is
\[
D^{\log}(\mathbb R^N)
=
\left\{
v\in L^2(\mathbb R^N):
E(v,v)+\int_{\mathbb R^N}v(x)^2\ln(e+|x|^2)\,dx<\infty
\right\},
\]
with
\[
E(v_1,v_2)
=
\frac{c_N}{2}\int_{\mathbb R^N}\int_{B_1(x)}
\frac{(v_1(x)-v_1(y))(v_2(x)-v_2(y))}{|x-y|^N}\,dy\,dx.
\]
The full bilinear form of \(L_\Delta\) is
\[
E_L(v_1,v_2)
=
E(v_1,v_2)
-c_N\iint_{|x-y|\ge1}\frac{v_1(x)v_2(y)}{|x-y|^N}\,dx\,dy
+\rho_N\int_{\mathbb R^N}v_1v_2\,dx.
\]
Although \(E_L\) is not positive definite, \(D^{\log}(\mathbb R^N)\) is a Hilbert space, \(C_c^\infty(\mathbb R^N)\) is dense in it, and the embedding
\[
D^{\log}(\mathbb R^N)\hookrightarrow L^2(\mathbb R^N)
\]
is compact [2507.21779].

The stereographic correspondence is exact at the weak level:
\[
u \text{ is a weak solution on }\mathbb S^N
\iff
v=\iota(u) \text{ is a weak solution on }\mathbb R^N.
\]
This transfers the classification theory. If \(v\) is a nonnegative nontrivial weak solution of
\[
L_\Delta v=\frac{4}{N}v\ln|v|+\mu v
\quad\text{in }\mathbb R^N,
\]
then
\[
v(x)=e^{\frac N4(A_N-\mu)}
\left(\frac{2t}{t^2+|x-a|^2}\right)^{N/2}
\qquad\text{for some }t>0,\ a\in\mathbb R^N.
\]
The sphere-side equation at \(\mu=A_N\) coincides with the equation used by Frank–König–Tang in their classification theorem, so the conformal logarithmic Laplacian furnishes the geometric bridge between the spherical and Euclidean logarithmic Yamabe problems [2507.21779].

## 5. Fractional--logarithmic extension and sharp inequalities

The conformal logarithmic Laplacian is the endpoint \(s=0\) of the broader family
\[
\mathscr P_g^{s+\ln}u(z):=\left.\frac{d}{dt}\mathscr P_g^t u(z)\right|_{t=s},
\qquad s\in(0,1).
\]
On \(\mathbb S^N\), this operator has the kernel formula
\[
\mathscr{P}^{s+\ln}_g u(z)
=
c_{N,s}\,\mathrm{p.v.}\!\int_{\mathbb{S}^N}
\frac{u(z)-u(\zeta)}{|z-\zeta|^{N+2s}}
\bigl(-2\ln|z-\zeta|+b_{N,s}\bigr)\,dV_g(\zeta)
+
A'_{N,s}\,u(z),
\]
where
\[
b_{N,s}
=
\ln 4 + \psi\!\Big(\tfrac N2+s\Big) + \psi(2-s) + \frac{1}{s} - \frac{1}{1-s},
\]
and
\[
A'_{N,s}
=
A_{N,s}\Bigl(
\psi\!\Bigl(\tfrac N2+s\Bigr)+\psi\!\Bigl(\tfrac N2-s\Bigr)
\Bigr).
\]
Its conformal covariance law is obtained by differentiating the covariance of \(\mathscr P_g^s\); the resulting formula preserves the same conformal weight as \(\mathscr P_g^s\) but introduces additional \(\ln\eta\)-correction terms. The endpoint limit
\[
\mathscr P_g^{s+\ln}u\to \mathscr P_g^{\ln}u
\quad\text{as }s\to0^+
\]
holds uniformly and in \(L^p(\mathbb S^N)\) [2603.21146].

This family supports a fractional--logarithmic Yamabe equation on \(\mathbb S^N\), proved equivalent under stereographic projection to the corresponding Euclidean equation. The explicit constant spherical solutions \(u_C\equiv C\) correspond to Euclidean bubbles
\[
v_{s,C}(x)=C\left(\frac{2}{1+|x|^2}\right)^{\frac{N-2s}{2}}.
\]
At the level of inequalities, the framework recovers the sharp logarithmic Sobolev inequality, shows that a naive fractional--logarithmic analogue fails, and yields new sharp fractional--logarithmic inequalities based on \(\langle u,(-\Delta)^{s+\ln}u\rangle\). A plausible implication is that the operator \(\mathscr P_g^{\log}\) is best viewed not as an isolated endpoint object but as the first member of a differentiated conformal family whose analytic behavior is already visible for \(s\in(0,1)\) [2603.21146].

## 6. Related constructions, antecedents, and distinctions

The term should be distinguished from several neighboring operators. It is not the same as the ordinary conformal Laplacian or Yamabe operator
\[
Y_g=-\Delta_g+\frac{n-2}{4(n-1)}R_g,
\]
whose spectral behavior is different. For \(Y_g\), \(0\) is not an eigenvalue for generic smooth metrics on a compact manifold, but the number of negative eigenvalues can become arbitrarily large along geometrically degenerating sequences. Those results are directly relevant to attempts to define \(\log Y_g\), zeta-determinants, or related spectral regularizations, but they do not by themselves define the sphere operator \(\mathscr P_g^{\log}\). This suggests two distinct “logarithmic” programs in conformal geometry: spectral logarithms of the Yamabe operator, and derivatives in order of conformal fractional operators [1511.08524].

It is also distinct from the general-manifold spectral logarithmic Laplacian
\[
\log(-\Delta):=\int_0^\infty \log\lambda\,dE(\lambda),
\]
developed on complete Riemannian manifolds by spectral calculus and heat semigroups. In that setting the basic identity is the Bochner formula
\[
\log(-\Delta)f=\int_0^\infty \frac{e^{-t}f-e^{t\Delta}f}{t}\,dt,
\]
and on manifolds with Ricci lower bounds the operator admits a pointwise representation through the kernels
\[
K_1(x,y)=\int_0^1 \frac{p_t(x,y)}{t}\,dt,
\qquad
K_2(x,y)=\int_1^\infty \frac{p_t(x,y)}{t}\,dt.
\]
That theory is metric and spectral rather than conformally covariant, and the paper explicitly compares spectral and heat-kernel definitions through a discrepancy governed by the mass loss function and stochastic completeness [2506.19311].

Earlier conformal-geometry work addressed logarithmic phenomena at the level of Green functions rather than a standalone operator. For conformal powers \(P_{k,g}\), including GJMS operators and fractional conformal powers, the Green kernel has an expansion with a logarithmic singularity, and the coefficient of that logarithmic term is given by
\[
2\,\Gamma(k)^{-1}(4\pi)^{-n/2}\,\widetilde a_{n-2k}(\Delta_g;x).
\]
In low orders this yields explicit formulas involving \(|W|^2\) and higher Weyl invariants, together with characterizations of local conformal flatness and of the round sphere. That background shows that “logarithmic” behavior was already intrinsic to conformal powers before the derivative-at-zero operator \(\mathscr P_g^{\log}\) was isolated explicitly [1306.3104].

In this sense, the conformal logarithmic Laplacian is best understood as the operator-level realization of a broader conformal pattern: differentiation in the order parameter of a conformally covariant family, explicit borderline kernels of type \(|z-\zeta|^{-N}\), and inhomogeneous covariance laws with additive logarithmic terms. The sphere construction makes that pattern fully explicit, while the Euclidean, Yamabe, Green-kernel, and general-manifold theories delineate the neighboring meanings of “logarithmic Laplacian” in current geometric analysis [2507.21779].

Source: https://www.emergentmind.com/topics/conformal-logarithmic-laplacian