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A new look at the fractional Poisson problem via the Logarithmic Laplacian

We analyze the $s$-dependence of solutions $u_s$ to the family of fractional Poisson problems $(-Δ)^s u =f$ in $Ω$, $u \equiv 0$ on $\mathbb{R}^N\setminus Ω$ in an open bounded set $Ω\subset \mathbb{R}^N$, $s \in (0,1)$. In the case where $Ω$ is of class $C^2$ and $f \in C^α(\barΩ)$ for some $α>0$, we show that the map $(0,1) \to L^\infty(Ω)$, $s\mapsto u_s$ is of class $C^1$, and we characterize the derivative $\partial_s u_s$ in terms of the logarithmic Laplacian of $f$. As a corollary, we derive pointwise monotonicity properties of the solution map $s \mapsto u_s$ under suitable assumptions on $f$ and $Ω$. Moreover, we derive explicit bounds for the corresponding Green operator on arbitrary bounded domains which are new even for the case $s=1$, i.e., for the local Dirichlet problem $-Δu = f$ in $Ω$, $u \equiv 0$ on $\partial Ω$.