Indefinite higher Riesz transforms

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 indefinite quadratic form of signature (p, q). We prove that these operators extend to L-bounded operators for 1 < r < ∞ if the parameter of discrete series representations is generic. Partially supported by Grant-in-Aid for Scientific Research 18340037, Japan Society for the Promotion of Science Partially supported by Japan Society for the Promotion of Science 2000 Mathematics Subject Classification. Primary 22E46, Secondary 42B15