Characterization of Sobolev spaces on the sphere

We prove a characterization of the Sobolev spaces $H^\alpha$ on the unit sphere $\mathbb{S}^{d-1}$, where the smoothness index $\alpha$ is any positive real number and $d\geq 2$. This characterization does not use differentiation and it is given in terms of $([\alpha/2]+1)$-multidimensional square functions $S_\alpha$. For $[\alpha/2]=0,$ a function $f\in L^2(\mathbb{S}^{d-1})$ belongs to $H^\alpha(\mathbb{S}^{d-1})$ if and only if $S_\alpha (f)\in L^2(\mathbb{S}^{d-1})$. If $n=[\alpha/2]>0$, the membership of $f$ is equivalent to the existence of $g_1,\cdots,g_n$ in $L^2(\mathbb{S}^{d-1})$ such that $S_\alpha(f,g_1,\ldots,g_n)\in L^2(\mathbb{S}^{d-1})$ and in this case, $g_j=T_j((-\Delta_S)^j f)$, where $T_j$ is a zonal Fourier multiplier in the sphere and $\Delta_S$ is the Laplace-Beltrami operator. The square functions $S_\alpha $ are based on averaging operators over euclidean balls (caps) in the sphere that may be viewed as zonal multipliers. The results in the paper are in the spirit of the characterization of fractional Sobolev spaces given in $\mathbb{R}^d$ proved in \cite{AMV}. The development of the theory is fully based on zonal Fourier multipliers and special functions.