Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

21
The categories of this paper are: adap-org
31
The categories of this paper are: ao-sci
16
The categories of this paper are: stat
12
The categories of this paper are: nucl-th
27
The categories of this paper are: comp-gas
9
The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th
0
The categories of this paper are: hep-ph
25
The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat
26
The categories of this paper are: bayes-an
30
The categories of this paper are: q-alg
7
The categories of this paper are: hep-th
6
The categories of this paper are: astro-ph
5
The categories of this paper are: gr-qc
10
The categories of this paper are: q-bio