Characterization of Sobolev spaces on the sphere

We introduce a new characterization of the Sobolev spaces H-proportional to on the unit sphere Sd-1, where the smoothness index a is any positive real number and d >= 2. This characterization is in terms of a square function and it avoids the notion of differentiability. The result is fully based on zonal Fourier multipliers, special functions and the subtle use of classical analysis tools. (C) 2020 Elsevier Inc. All rights reserved.