Stein's higher Riesz transforms are translation invariant operators on L2(Rn) built from multipliers whose restrictions to the unit sphere are eigenfunctions of the Laplace-Beltrami operators. In this article, generalizing Stein's higher Riesz transforms, we construct a family of translation invariant operators by using discrete series representations for hyperboloids associated to the indenite quadratic form of signature (p,q). We prove that these operators extend to Lr-bounded operators for 1 < r < if the parameter of discrete series representations is generic.
[ RIMS preprint(pdf) | RIMS preprint(ps.gz) | SpringerLink | related papers ]
The original publication is available at www.springerlink.com.
© Toshiyuki Kobayashi