Some multiplier theorems on the sphere

• Autores: R. O. Gandulfo, Giacomo Gigante
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 51, Fasc. 2, 2000, págs. 157-204
• Idioma: inglés
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• Resumen

  • 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