The fractional logarithmic Schrödinger operator: properties and functional spaces (Q6606865)

The paper is devoted to the study of the fractional (relativistic) logarithmic Schrödinger operator \((I+(-\Delta)^s)^{\log}\), having the logarithmic Fourier symbol \(\log(1+|\xi|^{2s})\), \(s>0\). Therefore, this work can be seen as a natural continuation of the previous work by the same author [J. Math. Anal. Appl. 517, No. 2, Article ID 126656, 33 p. (2023; Zbl 1498.60183)], where the logarithmic Schrödinger operator corresponding to the particular case \(s=1\) was studied.\N\NIn the present work, the author deduces that for compactly supported Dini continuous functions, \(u : \mathbb{R}^N \to \mathbb{R}\), the operator \({(I+(-\Delta)^s)^{\log}}\) can be identified as a singular integral operator\N\[\N(I+(-\Delta)^s)^{\log} u(x) =F^{-1}\big(\log(I+|\xi|^{2s}) F(u)\big)(x)\]\N\[= p.v.~\int_{\mathbb{R}^N}\big(u(x)-u(y)\big)K_s(x-y) \ dy,\N\]\Nfor a certain kernel function \(K_s\) for which its asymptotics is differently identified in the two cases \(s=1\) and \(s\in (0, 1)\).\N\NThe Green function for \({(I+(-\Delta)^s)^{\log}}\) is expressed as\N\[\NG(x) = \int_0^{\infty} \frac{1}{\Gamma(t)}\int_{0}^{\infty}p_s(x,\tau)\tau^{t-1}e^{-\tau}\ d\tau\ dt,\N\]\Nand it is shown that it satisfies the asymptotics\N\begin{align*}\NG(x)\sim \begin{cases} \frac{1}{|x|^{N}(\log\frac{1}{|x|})^2}&\qquad\text{as}\quad |x|\to 0\\\N\\\N\frac{C_{N,s}}{|x|^{N+2s}(\log\frac{1}{|x|})^2} &\qquad\text{ as }\quad |x|\to \infty. \end{cases}\N\end{align*}\NIn addition, some functional analytic framework for Dirichlet problems involving the operator \({(I+(-\Delta)^s)^{\log}}\) is also considered, as well as existence results and some properties of solutions for a Poisson problem involving this operator in open bounded sets (of \(\mathbb{R}^N\)).