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.

Given a domain Ω⊂ℂ^{$n$}, the Lempert function is a functional on the space $\text{Hol}(\mathbb{D},\Omega)$ of analytic disks with values in Ω, depending on a set of poles in Ω. We generalize its definition to the case where poles have multiplicities given by local indicators (in the sense of Rashkovskii) to obtain a function which still dominates the corresponding Green function, behaves relatively well under limits, and is monotonic with respect to the local indicators. In particular, this is an improvement over the previous generalization used by the same authors to find an example of a set of poles in the bidisk so that the (usual) Green and Lempert functions differ.

In this paper we introduce Coleff–Herrera residue currents defined by systems of c-holomorphic functions and prove a Lelong–Poincaré and a Cauchy-type formula as well as the transformation law for these currents.

Let$X$be a complex manifold and let$f$:$X$→ℂ^{$p$}be a holomorphic mapping defining a complete intersection. We prove that the iterated Mellin transform of the residue integral associated with$f$has an analytic continuation to a neighborhood of the origin in ℂ^{$p$}.

We consider several modified versions of the Positivstellensatz for global analytic functions that involve infinite sums of squares and/or positive semidefinite analytic functions. We obtain a general local-global criterion which localizes the obstruction to have such a global result. This criterion allows us to get completely satisfactory results for analytic curves, normal analytic surfaces and real coherent analytic sets whose connected components are all compact.