The fractional logarithmic Schrödinger operator: properties and functional spaces

In this note, we deal with the fractional Logarithmic Schrödinger operator (I+(-Δ)^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+|ξ|^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 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.