Fractional Sobolev spaces from a complex analytic viewpoint

Abstract For any non-zero function u in the Schwartz space S ( R n ) , we prove that s ↦ [ u ] s , 2 2 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 , 2 2 at s = 0 and s = 1 separately. Moreover, we study the function properties of [ u ] s , 2 2 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 ∈ W 1 , 2 ( R n ) and s ∈ ( 0 , 1 ) , we have [ u ] s , 2 2 ≤ π n 2 + 1 2 2 s − 1 Γ ( n 2 + s ) Γ ( 1 + s ) sin ⁡ π s ‖ u ‖ 2 2 ( 1 − s ) ‖ ∇ u ‖ 2 2 s .