标题： 分数对数薛定谔算子：性质和函数空间 显示英文标题

标题： The fractional logarithmic Schrödinger operator: properties and functional spaces

摘要： 在本文中，我们研究分数对数薛定谔算子$(I+(-\Delta)^s)^{\log}$以及用于变分研究的相应能量空间。 分数（相对论）对数薛定谔算子是具有对数傅里叶符号的伪微分算子， $\log(1+|\xi|^{2s})$，$s>0$。 我们首先建立与该算子对应的积分表示，并提供相关核的渐近性质。 我们引入了允许从偏微分方程观点研究该算子的泛函分析理论，以及在$\mathbb{ R}^N.$的开集中的相关狄利克雷问题。 我们还建立了某些变分不等式，提供了基本解以及对应格林函数在零点和无穷处的渐近性质。 显示英文摘要

摘要： In this note, we deal with the fractional Logarithmic Schr\"{o}dinger operator $(I+(-\Delta)^s)^{\log}$ and the corresponding energy spaces for variational study. The fractional (relativistic) Logarithmic Schr\"{o}dinger operator is the pseudo-differential operator with logarithmic Fourier symbol, $\log(1+|\xi|^{2s})$, $s>0$. We first establish the integral representation corresponding to the operator and provide an asymptotics property of the related kernel. We introduce the functional analytic theory allowing to study the operator from a PDE point of view and the associated Dirichlet problems in an open set of $\mathbb{ R}^N.$ We also establish some variational inequalities, provide the fundamental solution and the asymptotics of the corresponding Green function at zero and at infinity.