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

Inspired by recent work of Ferone and Volzone (2021), 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 (−𝛥) 𝑠 in the unit ball 𝛺 = 𝐵1 (0) ⊂ R𝑁 , 𝑠 ∈ (0, 1). In particular, our results imply a universal failure of the classical pointwise Talenti inequality in the fractional radial context. In contrast, a boundary Talenti type inequality holds for radial functions in the higher order case 𝑠 > 1.