Real principal value integrals
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.
real-meromorphic differential \(m\)-form
scientific article; zbMATH DE number 1533082