Spherical multipliers

V. Z. Meshkov

Matematičeskie zametki, Tome 23 (1978) no. 1, pp. 105-112.

Résumé

It is proven in the paper that if function $f(x)\in L^p(R^n)$, where $1/p>1/2+1/(n+1)$, then the restriction of the Fourier transform $\widehat{f}(\xi)$ to the unit sphere $S^{n-1}$ lies in $L^2(S^{n-1})$. As was shown by Fefferman [1], it follows from this that, when $\alpha>(n-1)/(2(n+1))$, the Riesz–Bochner multiplieragr acts in $L^p(R^n)$, if $(n-1-2\alpha)/(2n)1/p(n+1+2\alpha)/(2n)$.