On comparison of fractional Laplacians. (May 2022) - Nazarov, Alexander I.
Abstract: For s > − 1, s ∉ N 0, we compare two natural types of fractional Laplacians ( − Δ ) s, namely, the restricted Dirichlet and the spectral Neumann ones. We show that the quadratic form of their difference taken on the space H ˜ s ( Ω ) is positive or negative depending on whether the integer part of s is even or odd. For s ∈ ( 0, 1 ) and convex domains we prove also that the difference of these operators is positivity preserving on H ˜ s ( Ω ) . This paper complements (Musina and Nazarov, 2014; 2016) where similar statements were proved for the spectral Dirichlet and the restricted Dirichlet fractional Laplacians.