Fractional Sobolev spaces from a complex analytic viewpoint

Abstract

For any non-zero function u in the Schwartz space S(Rn), we prove that s↦[u]s,22 can be extended to C as a transcendental meromorphic function, which establishes a connection between the Bourgain-Brezis-Mironescu's formula, Maz'ya-Shaponshikova's formula and the residues of the transcendental meromorphic function of s↦[u]s,22 at s=0 and s=1 separately. Moreover, we study the function properties of [u]s,22 and obtain the convergence rate version of the Bourgain-Brezis-Mironescu's formula and Maz'ya-Shaponshikova's formula. And we also obtain the following sharp interpolation inequality. For any u∈W1,2(Rn) and s∈(0,1), we have[u]s,22≤πn2+122s−1Γ(n2+s)Γ(1+s)sin⁡πs‖u‖22(1−s)‖∇u‖22s.