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.