Conformal Logarithmic Laplacian

• Conformal logarithmic Laplacian is defined as the derivative at zero of the conformal fractional Laplacian on the sphere, introducing a logarithmic correction term.
• It features an explicit singular kernel |z-ζ|⁻ᴺ, complete spectral decomposition via spherical harmonics, and an inhomogeneous conformal covariance law.
• It underpins variational frameworks and Yamabe-type problems, linking spherical and Euclidean equations through stereographic projection.

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 (Fernández et al., 29 Jul 2025).

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 $\mathbb S^N$0: $\mathbb S^N$1 For $\mathbb S^N$2, $\mathbb S^N$3, the resulting operator has the explicit form

$\mathbb S^N$4

where

$\mathbb S^N$5

The convergence

$\mathbb S^N$6

is part of the basic construction, so the derivative is not merely formal but an actual $\mathbb S^N$7-limit of the conformal fractional family (Fernández et al., 29 Jul 2025).

The operator is “logarithmic” in two senses already visible at the level of definition. First, it is the derivative in the order parameter $\mathbb S^N$8 of a power-type family. Second, the differentiated kernel lands exactly at the borderline singularity $\mathbb S^N$9, which is the conformal-order-zero analogue of the Euclidean symbol $$\mathscr P_g^{\log}u(z):=\left.\frac{d}{ds}\right|_{s=0}[\mathscr P_g^s u](z),$$0. Constants are eigenfunctions rather than kernel elements: $$\mathscr P_g^{\log}u(z):=\left.\frac{d}{ds}\right|_{s=0}[\mathscr P_g^s u](z),$$1 Accordingly, the first eigenvalue is negative for $$\mathscr P_g^{\log}u(z):=\left.\frac{d}{ds}\right|_{s=0}[\mathscr P_g^s u](z),$$2 and positive for $$\mathscr P_g^{\log}u(z):=\left.\frac{d}{ds}\right|_{s=0}[\mathscr P_g^s u](z),$$3 (Fernández et al., 29 Jul 2025).

2. Spectral description on the sphere

The spectral theory of $$\mathscr P_g^{\log}u(z):=\left.\frac{d}{ds}\right|_{s=0}[\mathscr P_g^s u](z),$$4 is completely aligned with the spherical harmonic decomposition. If $$\mathscr P_g^{\log}u(z):=\left.\frac{d}{ds}\right|_{s=0}[\mathscr P_g^s u](z),$$5 is a spherical harmonic of degree $$\mathscr P_g^{\log}u(z):=\left.\frac{d}{ds}\right|_{s=0}[\mathscr P_g^s u](z),$$6, then

$$\mathscr P_g^{\log}u(z):=\left.\frac{d}{ds}\right|_{s=0}[\mathscr P_g^s u](z),$$7

For the fractional family, the eigenvalue multiplier is

$$\mathscr P_g^{\log}u(z):=\left.\frac{d}{ds}\right|_{s=0}[\mathscr P_g^s u](z),$$8

Differentiating at $$\mathscr P_g^{\log}u(z):=\left.\frac{d}{ds}\right|_{s=0}[\mathscr P_g^s u](z),$$9 gives the logarithmic multiplier

$\mathscr P_g^s$0

so that

$\mathscr P_g^s$1

whenever $\mathscr P_g^s$2. On the degree-$\mathscr P_g^s$3 spherical harmonics this becomes

$\mathscr P_g^s$4

The multiplier is strictly increasing, since

$\mathscr P_g^s$5

Hence the eigenspaces of $\mathscr P_g^s$6 are exactly the eigenspaces of $\mathscr P_g^s$7, and the operator is spectrally diagonal in the spherical harmonic basis (Fernández et al., 29 Jul 2025).

A later extension places $\mathscr P_g^s$8 inside a one-parameter differentiated family,

$\mathscr P_g^s$9

called the conformal fractional--logarithmic Laplacian. Its spherical-harmonic multiplier is

$|z-\zeta|^{-N}$0

and $|z-\zeta|^{-N}$1 uniformly and in $|z-\zeta|^{-N}$2 as $|z-\zeta|^{-N}$3. This places the order-zero operator as the endpoint of a broader derivative-in-order conformal family (Chen et al., 22 Mar 2026).

3. Conformal covariance and stereographic correspondence

The defining structural property of $|z-\zeta|^{-N}$4 is its inhomogeneous conformal covariance. If $|z-\zeta|^{-N}$5 is positive, then

$|z-\zeta|^{-N}$6

If the conformal metric is written as

$|z-\zeta|^{-N}$7

then the same law becomes

$|z-\zeta|^{-N}$8

Unlike the conformal fractional Laplacians, whose covariance is purely multiplicative, the logarithmic derivative produces the additive correction term $|z-\zeta|^{-N}$9. That term is the distinctive conformal signature of the logarithmic operator (Fernández et al., 29 Jul 2025).

The associated logarithmic $L_\Delta$0-curvature is defined by

$L_\Delta$1

For the round metric,

$L_\Delta$2

Under $L_\Delta$3, one obtains

$L_\Delta$4

equivalently

$L_\Delta$5

This is the conformal curvature identity behind the logarithmic Yamabe problem (Fernández et al., 29 Jul 2025).

The sphere operator is linked to the Euclidean logarithmic Laplacian through stereographic projection. With

$L_\Delta$6

and conformal factor

$L_\Delta$7

the pullback

$L_\Delta$8

satisfies the exact intertwining identity

$L_\Delta$9

Thus $s\in(0,1)$0 does not correspond directly to $s\in(0,1)$1, but to $s\in(0,1)$2 after conformal conjugation (Fernández et al., 29 Jul 2025).

The Euclidean operator $s\in(0,1)$3 itself is the derivative at $s\in(0,1)$4 of the fractional Laplacian: $s\in(0,1)$5 and on $s\in(0,1)$6 it has the exact singular integral representation

$s\in(0,1)$7

with

$s\in(0,1)$8

That operator is not presented as conformally covariant; the conformal modification enters through the stereographic factor in the sphere formula (Chen et al., 2017).

4. Yamabe-type problems and variational framework

The principal nonlinear equation attached to the conformal logarithmic Laplacian is the logarithmic Yamabe, or constant logarithmic $s\in(0,1)$9-curvature, equation

$$\mathscr P_g^s u(z)=c_{N,s}\,\mathrm{P.V.}\int_{\mathbb S^N}\frac{u(z)-u(\zeta)}{|z-\zeta|^{N+2s}}\,dV_g(\zeta)+A_{N,s}u(z),$$0

Its weak formulation is set in a Hilbert space

$$\mathscr P_g^s u(z)=c_{N,s}\,\mathrm{P.V.}\int_{\mathbb S^N}\frac{u(z)-u(\zeta)}{|z-\zeta|^{N+2s}}\,dV_g(\zeta)+A_{N,s}u(z),$$1

with norm

$$\mathscr P_g^s u(z)=c_{N,s}\,\mathrm{P.V.}\int_{\mathbb S^N}\frac{u(z)-u(\zeta)}{|z-\zeta|^{N+2s}}\,dV_g(\zeta)+A_{N,s}u(z),$$2

where $$\mathscr P_g^s u(z)=c_{N,s}\,\mathrm{P.V.}\int_{\mathbb S^N}\frac{u(z)-u(\zeta)}{|z-\zeta|^{N+2s}}\,dV_g(\zeta)+A_{N,s}u(z),$$3. The corresponding bilinear form can also be written as

$$\mathscr P_g^s u(z)=c_{N,s}\,\mathrm{P.V.}\int_{\mathbb S^N}\frac{u(z)-u(\zeta)}{|z-\zeta|^{N+2s}}\,dV_g(\zeta)+A_{N,s}u(z),$$4

The density statement

$$\mathscr P_g^s u(z)=c_{N,s}\,\mathrm{P.V.}\int_{\mathbb S^N}\frac{u(z)-u(\zeta)}{|z-\zeta|^{N+2s}}\,dV_g(\zeta)+A_{N,s}u(z),$$5

is part of this framework (Fernández et al., 29 Jul 2025).

On $$\mathscr P_g^s u(z)=c_{N,s}\,\mathrm{P.V.}\int_{\mathbb S^N}\frac{u(z)-u(\zeta)}{|z-\zeta|^{N+2s}}\,dV_g(\zeta)+A_{N,s}u(z),$$6, the corresponding weak equation is

$$\mathscr P_g^s u(z)=c_{N,s}\,\mathrm{P.V.}\int_{\mathbb S^N}\frac{u(z)-u(\zeta)}{|z-\zeta|^{N+2s}}\,dV_g(\zeta)+A_{N,s}u(z),$$7

and the natural energy space is

$$\mathscr P_g^s u(z)=c_{N,s}\,\mathrm{P.V.}\int_{\mathbb S^N}\frac{u(z)-u(\zeta)}{|z-\zeta|^{N+2s}}\,dV_g(\zeta)+A_{N,s}u(z),$$8

with

$$\mathscr P_g^s u(z)=c_{N,s}\,\mathrm{P.V.}\int_{\mathbb S^N}\frac{u(z)-u(\zeta)}{|z-\zeta|^{N+2s}}\,dV_g(\zeta)+A_{N,s}u(z),$$9

The full bilinear form of $$A_{N,s}:=\frac{\Gamma\!\left(\frac N2+s\right)}{\Gamma\!\left(\frac N2-s\right)}, \qquad c_{N,s}:=4^s\pi^{-N/2}\frac{\Gamma\!\left(\frac N2+s\right)}{\Gamma(2-s)}\,s(1-s).$$0 is

$$A_{N,s}:=\frac{\Gamma\!\left(\frac N2+s\right)}{\Gamma\!\left(\frac N2-s\right)}, \qquad c_{N,s}:=4^s\pi^{-N/2}\frac{\Gamma\!\left(\frac N2+s\right)}{\Gamma(2-s)}\,s(1-s).$$1

Although $$A_{N,s}:=\frac{\Gamma\!\left(\frac N2+s\right)}{\Gamma\!\left(\frac N2-s\right)}, \qquad c_{N,s}:=4^s\pi^{-N/2}\frac{\Gamma\!\left(\frac N2+s\right)}{\Gamma(2-s)}\,s(1-s).$$2 is not positive definite, $$A_{N,s}:=\frac{\Gamma\!\left(\frac N2+s\right)}{\Gamma\!\left(\frac N2-s\right)}, \qquad c_{N,s}:=4^s\pi^{-N/2}\frac{\Gamma\!\left(\frac N2+s\right)}{\Gamma(2-s)}\,s(1-s).$$3 is a Hilbert space, $$A_{N,s}:=\frac{\Gamma\!\left(\frac N2+s\right)}{\Gamma\!\left(\frac N2-s\right)}, \qquad c_{N,s}:=4^s\pi^{-N/2}\frac{\Gamma\!\left(\frac N2+s\right)}{\Gamma(2-s)}\,s(1-s).$$4 is dense in it, and the embedding

$$A_{N,s}:=\frac{\Gamma\!\left(\frac N2+s\right)}{\Gamma\!\left(\frac N2-s\right)}, \qquad c_{N,s}:=4^s\pi^{-N/2}\frac{\Gamma\!\left(\frac N2+s\right)}{\Gamma(2-s)}\,s(1-s).$$5

is compact (Fernández et al., 29 Jul 2025).

The stereographic correspondence is exact at the weak level: $$A_{N,s}:=\frac{\Gamma\!\left(\frac N2+s\right)}{\Gamma\!\left(\frac N2-s\right)}, \qquad c_{N,s}:=4^s\pi^{-N/2}\frac{\Gamma\!\left(\frac N2+s\right)}{\Gamma(2-s)}\,s(1-s).$$6 This transfers the classification theory. If $$A_{N,s}:=\frac{\Gamma\!\left(\frac N2+s\right)}{\Gamma\!\left(\frac N2-s\right)}, \qquad c_{N,s}:=4^s\pi^{-N/2}\frac{\Gamma\!\left(\frac N2+s\right)}{\Gamma(2-s)}\,s(1-s).$$7 is a nonnegative nontrivial weak solution of

$$A_{N,s}:=\frac{\Gamma\!\left(\frac N2+s\right)}{\Gamma\!\left(\frac N2-s\right)}, \qquad c_{N,s}:=4^s\pi^{-N/2}\frac{\Gamma\!\left(\frac N2+s\right)}{\Gamma(2-s)}\,s(1-s).$$8

then

$$A_{N,s}:=\frac{\Gamma\!\left(\frac N2+s\right)}{\Gamma\!\left(\frac N2-s\right)}, \qquad c_{N,s}:=4^s\pi^{-N/2}\frac{\Gamma\!\left(\frac N2+s\right)}{\Gamma(2-s)}\,s(1-s).$$9

The sphere-side equation at $\mathscr P_g^0=I$0 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 (Fernández et al., 29 Jul 2025).

5. Fractional--logarithmic extension and sharp inequalities

The conformal logarithmic Laplacian is the endpoint $\mathscr P_g^0=I$1 of the broader family

$\mathscr P_g^0=I$2

On $\mathscr P_g^0=I$3, this operator has the kernel formula

$\mathscr P_g^0=I$4

where

$\mathscr P_g^0=I$5

and

$\mathscr P_g^0=I$6

Its conformal covariance law is obtained by differentiating the covariance of $\mathscr P_g^0=I$7; the resulting formula preserves the same conformal weight as $\mathscr P_g^0=I$8 but introduces additional $\mathscr P_g^0=I$9-correction terms. The endpoint limit

$\mathbb S^N$00

holds uniformly and in $\mathbb S^N$01 (Chen et al., 22 Mar 2026).

This family supports a fractional--logarithmic Yamabe equation on $\mathbb S^N$02, proved equivalent under stereographic projection to the corresponding Euclidean equation. The explicit constant spherical solutions $\mathbb S^N$03 correspond to Euclidean bubbles

$\mathbb S^N$04

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 $\mathbb S^N$05. A plausible implication is that the operator $\mathbb S^N$06 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 $\mathbb S^N$07 (Chen et al., 22 Mar 2026).

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

$\mathbb S^N$08

whose spectral behavior is different. For $\mathbb S^N$09, $\mathbb S^N$10 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 $\mathbb S^N$11, zeta-determinants, or related spectral regularizations, but they do not by themselves define the sphere operator $\mathbb S^N$12. This suggests two distinct “logarithmic” programs in conformal geometry: spectral logarithms of the Yamabe operator, and derivatives in order of conformal fractional operators (Gover et al., 2015).

It is also distinct from the general-manifold spectral logarithmic Laplacian

$\mathbb S^N$13

developed on complete Riemannian manifolds by spectral calculus and heat semigroups. In that setting the basic identity is the Bochner formula

$\mathbb S^N$14

and on manifolds with Ricci lower bounds the operator admits a pointwise representation through the kernels

$\mathbb S^N$15

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 (Chen, 24 Jun 2025).

Earlier conformal-geometry work addressed logarithmic phenomena at the level of Green functions rather than a standalone operator. For conformal powers $\mathbb S^N$16, 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

$\mathbb S^N$17

In low orders this yields explicit formulas involving $\mathbb S^N$18 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 $\mathbb S^N$19 was isolated explicitly (Ponge, 2013).

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 $\mathbb S^N$20, 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 (Fernández et al., 29 Jul 2025).