The Fractional-Logarithmic Laplacian:Fundamental Properties and Eigenvalues

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.