  Solution of some boundary value problems and initial problems in unique ball
leads to the convergence and sumability problems of Fourier series of given
function by eigenfunctions of Laplace operator on a sphere - spherical
harmonics. Such a series are called as Fourier-Laplace series on sphere. There
are a number of works devoted investigation of these expansions in different
topologies and for the functions from the various functional spaces. In the
present work we consider only localization problems in both usual and
generalized (almost everywhere localization) senses in the classes of summable
functions. We use Chezaro means of the partial sums for study of summability
problems. In order to prove the main theorems we obtaine estimations for so
called maximal operator estimating it by Hardy-Littlwood's maximal function.
Significance of this function is that it majors of many important operators of
mathematical physics. For instance, Hardy-Littlewood's maximal function majors
Poisson's integral in the space.
