Resumen de Some multiplier theorems on the sphere

R. O. Gandulfo, Giacomo Gigante

• The $n$-dimensional sphere,$\Sigma_n$, can be seen as the quotient between the group of rotations of $\mathbb{R}^{n+1}$ and the subgroup of all the rotations that fix one point. Using representation theory, one can see that any operator on $L^p(\Sigma_n)$ 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 $L^p(\Sigma_1)$, then the given sequence represents a bounded multiplier on $L^p(\Sigma_n)$. 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