I will discuss various regularity aspects of Kolmogorov type equations from kinetic theory, in particular kinetic maximal $L_p$-regularity in the strong setting as well as boundedness, Harnack- and Hölder estimates for problems with rough coefficients. I will further explain the new method of critical kinetic trajectories, which allows to prove several sharp functional analytic estimates such as the kinetic Sobolev inequality with optimal exponent, without relying on the fundamental solution. This is joint work with Lukas Niebel and partly with Helge Dietert and Clément Mouhot.

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

The Caffarelli-Kohn-Nirenberg (CKN) inequality is a first-order scale-invariant generalization of the Sobolev inequality. Differently from the latter, as observed by Felli-Schneider (2003), minimizers under a radial symmetry constraint need not be unconstrained local minimizers for the CKN inequality. This induces a remarkable transition from symmetry to symmetry-breaking of optimizers, which has been completely understood by Dolbeault-Esteban-Loss (2016). In this talk, I will present some new results about a fractional-order variant of the CKN inequality, for which to date the above questions remain wide open. More precisely, we show non-degeneracy of minimizers for a certain parameter range and, as a consequence, obtain a quantitative stability inequality. Moreover, we exhibit a new region for which every minimizer must be radially symmetric. Along the way, we rely on careful regularity considerations for distributional solutions to fractional-order equations. This is joint work with Nicola De Nitti (Bari) and Federico Glaudo (Princeton).