The Fractional-Logarithmic Laplacian:Fundamental Properties and Eigenvalues

TL;DR

This paper introduces the fractional-logarithmic Laplacian, explores its properties, develops associated functional frameworks, and analyzes eigenvalues and spectral behavior, bridging fractional and logarithmic Laplacians.

Contribution

It is the first to define and analyze the fractional-logarithmic Laplacian, including its integral, Fourier, spectral, and extension formulations, and studies its spectral properties and eigenvalue asymptotics.

Findings

Established pointwise integral representation of the operator.

Developed functional spaces with compact embeddings at critical exponents.

Derived Weyl-type asymptotic law for eigenvalues, showing combined fractional and logarithmic growth.

Abstract

In this paper, we introduce, for the first time, the fractional--logarithmic Laplacian \( (-\Delta)^{s+\log} \), defined as the derivative of the fractional Laplacian \( (-\Delta)^t \) at \( t=s \). It is a singular integral operator with Fourier symbol \( |\xi|^{2s}(2\ln|\xi|) \), and we prove the pointwise integral representation \[ (-\Delta)^{s+\log}u(x) = c_{n,s}\,\mathrm{PV}\!\int_{\mathbb{R}^n} \frac{u(x)-u(y)}{|x-y|^{n+2s}}\bigl(-2\ln|x-y|\bigr)\,dy + b_{n,s}(-\Delta)^s u(x), \] where \( c_{n,s} \) is the normalization constant of the fractional Laplacian and \( b_{n,s}:=\frac{d}{ds}c_{n,s}.\) We also establish several equivalent formulations of \( (-\Delta)^{s+\log} \), including the singular-integral representation, the Fourier-multiplier representation, the spectral-calculus definition, and an extension characterization. We develop the associated functional framework on both \( \mathbb{R}^n \) and bounded Lipschitz domains, introducing the natural energy spaces and proving embedding results. In particular, we obtain a compact embedding at the critical exponent \( 2_s^*=\frac{2n}{n-2s},\) a phenomenon that differs from the classical Sobolev and fractional Sobolev settings. We further study the Poisson problem, proving existence and \( L^\infty \)-regularity results. We then investigate the Dirichlet eigenvalue problem and establish qualitative spectral properties. Finally, we derive a Weyl-type asymptotic law for the eigenvalue counting function and for the \( k \)-th Dirichlet eigenvalue, showing that the high-frequency behavior combines the fractional Weyl scaling with a logarithmic growth factor, thereby interpolating between the fractional Laplacian and the logarithmic Laplacian.