The fractional Poisson problem and the logarithmic Laplacian

I will discuss some recent results on the family of fractional Poisson problems $(-\Delta)^s u =f$ in $\Omega$, $u=0$ on $\Omega^c$ of order $2s$ and its connection to the logarithmic Laplacian operator. This connection allows, in particular, to characterize the $s$-dependence of solutions to this family. Special attention will be given to the case $f\equiv 1$, i.e., the fractional torsion problem. This is joint work with Sven Jarohs and Alberto Saldana.