  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.
