The n-dimensional sphere, E, can be seen as the quotient between the group of rotations of R n+1 and the subgroup of all the rotations that fix one point. Using representation theory, one can see that any operator on Lp (Sigma n) that commutes with the action of the group of rotations (called multiplier) may be associated with a sequence of complex numbers. We prove that, if a certain discrete derivative of a given sequence represents a bounded multiplier on LP (E 1), then the given sequence represents a bounded multiplier on Lp (Sigma n). As a corollary of this, we obtain the multidimensional version of the Marcinkiewicz theorem on multipliers. An associated problem related to expansions in ultraspherical polynomials is also studied.
@article{urn:eudml:doc:41518,
title = {Some multiplier theorems on the sphere.},
journal = {Collectanea Mathematica},
volume = {51},
year = {2000},
pages = {157-203},
zbl = {0968.42007},
mrnumber = {MR1776831},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41518}
}
Gandulfo, R. O.; Gigante, G. Some multiplier theorems on the sphere.. Collectanea Mathematica, Tome 51 (2000) pp. 157-203. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41518/