The fractional logarithmic Schrödinger operator: properties and functional spaces

In this note, we deal with the fractional Logarithmic Schrödinger operator $(I+(-\mathrm{\Delta }{)}^s{)}^{\log }$ and the corresponding energy spaces for variational study. The fractional (relativistic) Logarithmic Schrödinger operator is the pseudo-differential operator with logarithmic Fourier symbol, $\log (1+\mathrm{\mid }\xi {\mathrm{\mid }}^{2s})$, $s>0$. 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 ${\mathbb{R}}^N\mathrm{.}$ We also establish some variational inequalities, provide the fundamental solution and the asymptotics of the corresponding Green function at zero and at infinity.