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The fractional-logarithmic Laplacian: fundamental properties and eigenvalues
Daniel Hauer
Brandenburgische Technische Universität Cottbus-Senftenberg, Germany
In this talk, I introduce the fractional--logarithmic Laplacian \( (-\Delta)^{s+\Log} \) as the derivative of the fractional Laplacian \begin{displaymath} (-\Delta)^{s+\Log}:=\big.\frac{d}{dt}(-\Delta)^t u(x)\big|_{t=s} \end{displaymath} and show that, alternatively, it can be realized as the singular integral operator \begin{displaymath} (-\Delta)^{s+\Log}= c_{n,s}\,\mathcal \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), \end{displaymath} where \(c_{n,s}\) is the normalization constant of the fractional Laplacian, and $b_{n,s}:=\frac{d}{ds}c_{n,s}$. We also present other equivalent formulations of the fractional--logarithmic Laplacian \( (-\Delta)^{s+\Log}\), including its definition via spectral calculus or as a singular pseudo-differential operator, and provide a characterization through an extension problem. We develop the associated functional analytical framework for studying the Poisson problem governed by the fractional--logarithmic operator \( (-\Delta)^{s+\Log} \) on \(\R^n\) and on bounded Lipschitz domains. We introduce the natural corresponding energy spaces and establish Sobolev and Poincaré embeddings; in particular, we show that at the critical exponent $2_s^*=\frac{2n}{n-2s}$, the Sobolev embedding remains compact, a phenomenon that differs from the classical and fractional Sobolev settings. We present first global \(L^\infty\)-regularity results of weak solutions of the Poisson problem and we investigate the associated Dirichlet eigenvalue problem. If time permits, then we show Weyl-type asymptotics for the eigenvalue counting function and for the $k$-th Dirichlet eigenvalue, showing that the high-frequency behaviour combines the fractional Weyl scaling with the logarithmic growth factor, thus interpolating between the fractional Laplacian and the logarithmic Laplacian. These results have been obtained in joint work with Huyuan Chen (Shanghai Institute for Mathematics and Interdisciplinary Sciences and Fudan University, China) and Rui Chen (PhD student at BTU Cottbus-Senftenberg and Fudan University) and recently been accepted for publication in the journal Mathematische Annalen
