Logarithmic Kernel Φ in Analysis

• Logarithmic kernel Φ is a fundamental solution defined via spectral and integral representations that encodes nonlocal, logarithmic interactions and weak singularities.
• Its asymptotic behavior, characterized by scale-invariant tails and weak singularities, is essential in analyzing Laplacians, pseudodifferential operators, and related PDEs.
• Applications span operator theory, potential theory, and complex geometry, linking energy, capacity, and dispersion relations in both continuous and discrete settings.

The logarithmic kernel $Φ$ is a central object across diverse domains, encapsulating logarithmic interactions in potential theory, integral representations of nonlocal and pseudodifferential operators, quantum and statistical mechanics, and complex geometry. Its archetypal role arises as a fundamental solution or as the integral kernel for logarithmic (pseudo)differential operators, reflecting intrinsic structures of logarithmic growth, singularity, and energy. $Φ$ appears in the analysis of Laplacians with logarithmic symbols, the Kramers-Kronig transform, projective capacity and energy in pluripotential theory, and modifications of the Bergman kernel. A unified feature across contexts is that $Φ$ encodes nonlocal, weakly singular, and often scale-invariant interactions.

1. Integral and Symbolic Constructions

The general construction of a logarithmic kernel $Φ$ follows from functional calculus applied to an infinitesimal generator, typically the Laplacian $-Δ$, on a suitable space (Riemannian manifold, graph, $\mathbb{R}^n$). Through the spectral theorem, one defines

$$\log(-Δ)f = \int_0^\infty \left( e^{-t}f - e^{tΔ}f \right)\frac{dt}{t}$$

with convergence in $L^2$ in the appropriate domain $H^{\log}$ (Chen, 24 Jun 2025, Chen et al., 8 Jul 2025). For pseudodifferential and fractional Schrödinger operators, e.g., $(I + (-Δ)^s)^{\log}$, the Fourier symbol is $\log(1+|\xi|^{2s})$; the associated kernel is given by inverse Fourier transform:

$$Φ_s(z) = (2\pi)^{-N} \int_{\mathbb{R}^N} e^{i \xi \cdot z} \log(1 + |\xi|^{2s}) \, d\xi$$

or equivalently by Bochner subordination and stable heat semigroups (Feulefack, 2023).

For the "log-Laplacian" with symbol $2\log|\zeta|$, the time-dependent “heat” kernel is

$$P_{\text{ln}}(t, x) = (2\pi)^{-n/2} \int_{\mathbb{R}^n} e^{ix \cdot \zeta} |\zeta|^{-2t} d\zeta = P_0(t) |x|^{2t-n},$$

with $P_0(t)$ involving Gamma functions (Chen et al., 2023).

2. Asymptotic Properties and Singularities

Logarithmic kernels are characterized by weakly singular, non-integrable behavior at the diagonal and scale-invariant tails. In Euclidean space, the fundamental logarithmic kernel is

$Φ_{\mathbb{R}^n}(x, y) = c_n |x - y|^{-n}$

with $c_n = \pi^{-n/2}\Gamma(n/2) = 2/|\mathbb{S}^{n-1}|$, singular of order $|x-y|^{-n}$ (Chen, 24 Jun 2025). For general fractional symbols,

$$Φ_s(z) \sim κ_{N, s}|z|^{-N} \ \text{as } |z| \to 0; \quad Φ_s(z) \sim C_{N,s}|z|^{-(N+2s)} \ \text{as } |z| \to \infty,$$

with constants as specified in (Feulefack, 2023). For the log-Laplacian on $\mathbb{R}^N$,

$$Φ_{\text{ln}}(x) \sim c_0 (\ln|x|)^2 + O(|x|^3 (\ln|x|)^3) \ \text{as} \ |x| \to 0; \quad Φ_{\text{ln}}(x) = O(|x|^{2-n}) \ \text{as} \ |x| \to \infty,$$

with $c_0 = 2^{1-n}\omega_n$ (Chen et al., 2023).

On weighted graphs, the log kernel $W_{\log}(x, y)$ and its tails are derived from the heat kernel, with sharp two-sided bounds: for the lattice $\mathbb{Z}^d$, $W(x, y) \approx d(x,y)^{-d}$ for large $d(x,y)$, and $$W_{\log}(x, y) \lesssim e^{d(x,y)}/d(x,y)^{d(x,y)+1}$$ at short range (Chen et al., 8 Jul 2025).

3. Applications in Operator Theory and PDEs

The logarithmic kernel governs a variety of nonlocal, pseudodifferential, and integral operators:

• For the logarithmic Laplacian on Riemannian manifolds, the action is nonlocal:

$$\log(-Δ) f(x) = \int_M [f(x) - f(y)] Φ(x, y)\, dV(y) - \int_M K_2(x, y) f(y)\, dV(y) + \text{const} \cdot f(x)$$

with $Φ(x, y) = \int_0^1 p_t(x, y) dt/t$ (Chen, 24 Jun 2025).

• In Euclidean space, $(I + (-Δ)^s)^{\log}$ acts via

$$(I + (-Δ)^s)^{\log} u(x) = \text{P.V.} \int_{\mathbb{R}^N} [u(x) - u(y)] Φ_s(x-y) dy$$

where $Φ_s$ is as above (Feulefack, 2023).

• The Kramers–Kronig transform for the reflection phase is written with the logarithmic kernel as

$$Φ(\omega) = \frac{1}{π} \int_0^\infty \ln\left| \frac{ω' + ω}{ω' - ω} \right| \frac{d \ln R(ω')}{dω'} dω' - π$$

(André et al., 2011).

• On graphs, the Bochner-type formula delivers a discrete convolution with the log kernel $W_{\log}(x, y)$ in the pointwise representation for $\log(-Δ)$ (Chen et al., 8 Jul 2025).

4. Potential Theory, Energy, and Capacity

In complex geometry and pluripotential theory, $Φ$ encodes energy and capacity in both Euclidean and projective spaces:

• On complex projective space $\mathbb{P}^n$ with the Fubini–Study form, the projective logarithmic kernel is

$$Φ([z], [w]) = \log \sin \left( \frac{d_{\mathrm{FS}}([z], [w])}{\sqrt{2}} \right)$$

and equivalently in terms of homogeneous coordinates as

$$Φ([z], [w]) = \log \frac{ \sum_{0 \leq i < j \leq n} | z_i w_j - z_j w_i | }{ \|z\| \|w\| }$$

(Asserda et al., 2019).

• The logarithmic energy of a measure $\mu$ is

$$I(\mu) = -\iint_{P^n \times P^n} Φ([z], [w])\, d\mu([z]) d\mu([w])$$

and capacity is defined as $K(E) = \exp(- y(E))$, where $y(E)$ is the infimum of the energy over probability measures supported on $E$ (Asserda et al., 2019).

• There is a transfinite diameter interpretation: $D(E) = \lim_{s \to \infty} D_s(E) = K(E)$ with $D_s(E)$ the projective diameter at order $s$.

5. Specializations: Manifolds, Graphs, and Geometry

The structure of $Φ$ adapts to ambient geometry:

• On general Riemannian manifolds, $Φ(x, y)$ is defined through integrals of the corresponding heat kernel; asymptotically, $Φ(x, y) \approx d(x, y)^{-n}$ near the diagonal (Chen, 24 Jun 2025).
• In real hyperbolic space $H^n$, the kernel $K_1(r)$ (for $r = d(x, y)$) satisfies $K_1(r) \approx r^{-n}$ for $r \leq 1$, and decays super-exponentially for $r \gg 1$ (Chen, 24 Jun 2025).
• On graphs, kernel definitions involve heat kernels $p(t, x, y)$, with explicit asymptotics on the integer lattice and concrete Fourier multipliers:

$$\widehat{ \log(-Δ) u } (\xi) = \ln(\Phi(\xi)) \widehat{u}(\xi), \quad \Phi(\xi) = \sum_{j=1}^d (2 - 2 \cos \xi_j)$$

(Chen et al., 8 Jul 2025).

6. Connections to Complex Analysis and Random Geometry

In geometric analysis and random complex geometry, logarithmic modifications of reproducing kernels are central:

• For a compact Kähler manifold $X$ and positive line bundle $L$, the logarithmic Bergman kernel relative to a subvariety $V$ is

$Φ_{p, V}(z) = \frac{1}{p} \log \rho_{p, V}(z)$

where $\rho_{p, V}(z)$ is the on-diagonal logarithmic Bergman kernel for sections vanishing on $V$ (Sun, 2019).

• The asymptotics of $Φ_{p, V}(z)$ show three regimes (far, core, neck) dictated by the distance to $V$, with explicit expressions for each. This potential controls the conditional density of zeros for Gaussian random holomorphic fields vanishing along $V$, with

$$\mathbb{E}\left[ Z_s | s|_V = 0 \right] = \frac{i\, p}{2\pi} \partial \bar{\partial} Φ_{p, V}(z) + p\, \omega$$

(Sun, 2019).

7. Further Applications and Theoretical Significance

The logarithmic kernel provides a universal structure for constructing nonlocal operators of “small order” and for quantifying weakly singular interactions. In particular:

• In the theory of fractional Laplacians, the logarithmic kernel demarcates the “borderline” between local and nonlocal behavior; its weak singularity places its associated energy spaces $H^{\log, s}$ strictly between Sobolev spaces $H^m$ and spaces of continuous functions (Feulefack, 2023).
• In potential theory and capacity, $Φ$ serves as a bridge connecting capacity, energy, extremal problems, and pluripolarity. The projective logarithmic kernel establishes the equivalence of capacity and transfinite diameter and provides Evans-type theorems for polar sets (Asserda et al., 2019).
• The logarithmic Kramers-Kronig kernel provides robust dispersion relations in optical spectroscopy for retrieving the reflection phase from magnitude data under causality and analyticity constraints (André et al., 2011).

The logarithmic kernel $Φ$ thus encapsulates a class of kernels of critical regularity and universality in diverse analytic, geometric, and probabilistic frameworks. Its detailed structure, singularity, spectral properties, and geometric adaptability underscore its fundamental role across analysis, geometry, and mathematical physics.