The Fractional-Logarithmic Laplacian: Fundamental Properties and Eigenvalues

In this paper, we introduce the fractional--logarithmic Laplacian \( (-Δ)^{s+\log} \), defined as the derivative of the fractional Laplacian \( (-Δ)^t \) at \(t=s\). It is a singular integral operator with Fourier symbol \( |ξ|^{2s}\log|ξ|^2 \). We derive its pointwise integral representation \[ (-Δ