Indefinite higher Riesz transforms

Abstract

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^r-bounded operators for 1< r<∞ if the parameter of the discrete series representations is generic.