On m-order logarithmic Laplacians and related propeties

Abstract: In this article, we study $m$-order logarithmic Laplacian $\mathcal{L}m$, which is a singular integro-differential operator with symbol $\big(2\ln |\cdot|\big)^m$ by the Fourier transform. With help of these logarithmic Laplacians, we build the $n$-th order Taylor expansion for fractional Laplacian with respect to the order and the Riesz operators: for $u \in C^\infty_c(\mathbb{R}^N)$ and $x \in \mathbb{R}^N$, $$ (-Δ)^s u (x) = u(x) + \sum^{n}{m=1} \frac{s^m}{m!}\mathcal{L}_mu(x) + o(s^n) \quad {\rm as}\ \, s\to 0^+ $$ and $$ \big(Φs\ast u\big)(x) = u(x) + \sum^{n}{m=1}(-1)^m\frac{s^m}{m!}\mathcal{L}_mu(x) + o(s^n) \quad {\rm as}\ \, s\to 0^+, $$ where $ (-Δ)^s$ is the $s$-fractional Laplacian, $Φs\ast u$ is $s$-order of Riesz operator with the form $Φ_s(x)=κ{N,s}|x|^{2s-N}$ in $\mathbb{R}^N\setminus{0}$. Moreover, we analyze qualitative properties of these operators based on the order $m$, such as basic regularity and the Dirichlet eigenvalues.

• The paper introduces m-order logarithmic Laplacians by deriving a Taylor expansion of the fractional Laplacian as s → 0, generalizing the classical logarithmic case.
• It rigorously establishes operator regularity, variational formulations, and a complete spectral theory with Dirichlet eigenvalues and L²-orthonormal eigenfunctions.
• The study proves a Faber-Krahn inequality for these operators, highlighting optimal geometric conditions that minimize the first Dirichlet eigenvalue.

m-Order Logarithmic Laplacians: Definition, Properties, and Spectral Analysis

Introduction

The study of nonlocal operators, particularly the fractional Laplacian $(-\Delta)^s$, has informed a broad spectrum of analysis in PDEs and functional analysis. The present work constructs and rigorously analyzes a family of higher-order logarithmic Laplacians $\mathcal{L}_m$, whose Fourier symbol is $(2\log|\xi|)^m$, generalizing the well-studied logarithmic Laplacian (the case $m=1$). The framework interconnects these operators with Taylor expansions for both Riesz convolutions and fractional Laplacians as the order parameter $s$ approaches zero. The work addresses formal expansions, regularity, spectral theory, and geometric inequalities.

Construction of m-Order Logarithmic Laplacians

Central to the article is the introduction of the $m$-th order logarithmic Laplacian $\mathcal{L}_m$, defined via the $m$-th derivative with respect to order $s$ of the Riesz and fractional Laplacians, in the limit $s\to0$. The Fourier symbol $|2\log|\xi||^m$ is realized for each $m\in\mathbb{N}$ through careful analysis of the kernel expansions. For a smooth, compactly supported function $u$, the following expansion holds: $$(-\Delta)^s u(x) = u(x) + \sum_{m=1}^n \frac{s^m}{m!} \mathcal{L}_m u(x) + o(s^n) \quad \text{as } s\downarrow0,$$ and a corresponding series for the Riesz convolution operator.

$\mathcal{L}_m$ is explicitly written as an integro-differential operator with logarithmic-type kernel singularities, and its principal part $K_m$ is dominant for small arguments, highlighting its weak differential character. The analysis identifies and resolves subtle aspects of kernel singularity, convergence, and operator regularity.

Analytical Properties and Functional Setting

The study demonstrates that $\mathcal{L}_m$ is well-defined for $u$ in the appropriate function spaces, notably for those satisfying uniform $m$-order Dini continuity. The work systematically develops the space $H_{m,0}(\Omega)$—a Hilbert space tailored for the quadratic form induced by $\mathcal{L}_m$, which plays a central role in the variational formulation of eigenvalue problems.

Key regularity results are established: if $u$ is uniformly $n$-order Dini continuous for $n > m$, the mapping $x\mapsto\mathcal{L}_m u(x)$ is continuous, linking the differentiability class of $u$ to the spatial regularity of $\mathcal{L}_m u$.

Spectral Theory and Dirichlet Eigenvalues

The Dirichlet eigenvalue problem for $\mathcal{L}_m$ on a bounded Lipschitz domain $\Omega$ is formulated variationally via its associated closed, symmetric, and semi-bounded quadratic form. The existence and variational characterization of an increasing sequence of real eigenvalues, with corresponding $L^2$-orthonormal eigenfunctions, are proven. For even $m$, the lowest eigenvalue is nonnegative.

The paper provides detailed analysis of the spectral structure:

• The principal eigenvalue is simple and, under a sign condition on the kernel in small balls, has an eigenfunction that is positive almost everywhere.
• The expansion for fractional Laplacians' eigenvalues as $s\to0$ is rigorously constructed using the new operators $\mathcal{L}_m$.
• The eigenfunctions form a complete basis in $L^2(\Omega)$.

Faber-Krahn Inequality and Geometric Optimization

A key functional analytic result is a Faber-Krahn inequality: among all Lipschitz domains of given volume, the Euclidean ball uniquely minimizes the first Dirichlet eigenvalue of $\mathcal{L}_m$. The proof leverages rearrangement inequalities and relies on the monotonicity properties of the kernel when restricted to sufficiently small domains.

This result is analogous to classical and fractional Laplacian cases but identifies limitations—such as failure of a general result for larger domains or for the regional fractional Laplacian—and makes progress for $m\ge2$ where previous proofs for the single logarithmic Laplacian do not directly extend.

Theoretical and Practical Implications

This framework provides a robust analytical tool for studying asymptotics of nonlocal operators as the order parameter changes, informing the understanding of spectral stability, kernel singularities, and regularity of eigenfunctions in the nonlocal regime. The decomposition of the fractional Laplacian into Taylor-like expansions using logarithmic Laplacians offers new avenues for fine analysis of nonlocal PDEs, spectral geometry, and limit theorems.

On the theoretical side, the work clarifies the mapping $s\mapsto(-\Delta)^s u(x)$ as a smooth function in the order, rigorously justifying and generalizing first and higher-order expansions for nonlocal operators. Practically, these results will inform analytical and numerical treatments of nonlocal problems in mathematical physics, potential theory, and probability, especially in regimes where $s\to0$ or in the perturbative analysis of nonlocal models.

Potential Directions for Future Research

Several avenues arise:

• Extension of Faber-Krahn type inequalities to more general domains and operators, or identifying potential counterexamples.
• Deepened exploration of the regularity and sign structure of higher eigenfunctions for $\mathcal{L}_m$.
• Application of these logarithmic Laplacians in stochastic processes (e.g., stable and geometric-stable Lévy flights), geometric flows, or nonlocal curvature problems.
• Exploration of explicit kernel representations and the singular integrals appearing for $m>2$.

Conclusion

The construction and analysis of the $m$-order logarithmic Laplacian $\mathcal{L}_m$ establishes a precise tool for understanding the small-order regime of nonlocal operators such as the fractional Laplacian and Riesz potentials. This work elucidates the analytic structure, spectral theory, and geometric properties of these operators, and positions them as central objects in nonlocal analysis and its diverse applications.