Real principal value integrals
type: Item
instance of: scholarly article
title: Real principal value integrals
zbMATH Open document ID: 0983.32007
author: Philippe Jacobs
published in: Monatshefte für Mathematik
publication date: 2001-03-14T00:00:00Z
full work available at URL: https://lirias.kuleuven.be/handle/123456789/345553
review text: The aim of this paper is to define the principal value \(PV \int_X\rho|\omega|^{1/d}\) where \(X\) is a real-analytic manifold of dimension \(n\), \(\rho\) is a \(C^\infty\)-function with compact support from \(X\) to \(\mathbb{R}\) and \(\omega\) is a real-meromorphic differential \(m\)-form of degree \(d\) on \(X\). Using a suitable resolution of singularities, a \(C^\infty\)-partition of unity and suitable charts, one has to consider the case when \(X\) is an open subset of \(\mathbb{R}^m\) and \(\omega=\xi x^\beta (dx)^d\) where \(\xi\) is a nowhere vanishing real-analytic function and \(x^\beta= x^{\beta_1}_1 \cdots x_m^{\beta_m}\) with \(\beta_k\in \mathbb{Z}\). The integral \(\int_X\rho |x|^s |\omega |^{1/d}\) is then a well-defined holomorphic function for \({\mathfrak R}(s)\) sufficiently large which has a meromorphic continuation to \(\mathbb{C}^m\). If \(\omega^{1/d}\) has no integral poles, that is \(\beta_i/d \notin \mathbb{Z}\backslash \mathbb{N}\), this continuation is holomorphic at \(s=0\) and its value at \(s=0\) is by definition \(PV\int_X\rho |\omega |^{1/d}\). The author checks very carefully that the definition is independent of the different choices (resolution, partition of unity, charts). He then gives some examples of computation of principal values. When \(X\) is compact one can take \(\rho=1\) and define \(PV\int_X|\omega|^{1/d}\). Unfortunately this can not be generalized to non compact manifolds via a compactification of \(X\) since, as shown in this paper, the result can depend on the compactification. These principal values have been introduced by Langlands in view of the Langlands program. In the introduction the author explains some connections with oscillating integrals and distributions \(|f|^\lambda\) and gives a cohomological condition which is sufficient for the vanishing of principal value integrals.
Mathematics Subject Classification ID: 32C05
Mathematics Subject Classification ID: 32S45
zbMATH DE Number: 1533082
zbMATH Keywords: real-meromorphic differential \(m\)-form
zbMATH Keywords: principal value
zbMATH Keywords: real-analytic manifold
zbMATH Keywords: resolution of singularities
zbMATH Keywords: oscillating integrals
zbMATH Keywords: distributions
reviewed by: P-A Mazet
MaRDI profile type: Publication
OpenAlex ID: W2044621875
DOI: 10.1007/S006050070027
description: scientific article; zbMATH DE number 1533082