On a fractional boundary version of Talenti's inequality in the unit ball

Inspired by recent work of Ferone and Volzone arXiv:2007.13195, we derive sufficient conditions for the validity and non-validity of a boundary version of Talenti’s comparison principle in the context of Dirichlet-Poisson problems for the fractional Laplacian $(-\mathrm{\Delta }{)}^s$ in the unit ball $\mathrm{\Omega }=B_1(0)\subset {\mathbb{R}}^N$. In particular, our results imply a universial failure of the classical pointwise Talenti inequality in the fractional radial context which sheds new light on the one-dimensional counterexamples given in arXiv:2007.13195.