Indefinite higher Riesz transforms

Stein’s higher Riesz transforms are translation invariant operators on L ²(R ⁿ ) 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 ^r -bounded operators for 1<r<∞ if the parameter of the discrete series representations is generic.