On principal value and standard extension of distributions

Authors

  • Daniel Barlet

DOI:

https://doi.org/10.7146/math.scand.a-134458

Abstract

For a holomorphic function $f$ on a complex manifold $\mathscr {M}$ we explain in this article that the distribution associated to $\lvert  f\rvert^{2\alpha } (\textrm{Log} \lvert f\rvert^2)^q f^{-N}$ by taking the corresponding limit on the sets $\{ \lvert f\rvert \geq \varepsilon \}$ when $\varepsilon $ goes to $0$, coincides for $\Re (\alpha ) $ non negative and $q, N \in \mathbb {N}$, with the value at $\lambda = \alpha $ of the meromorphic extension of the distribution $\lvert f\rvert^{2\lambda } (\textrm{Log} \lvert f\rvert^2)^qf^{-N}$. This implies that any distribution in the $\mathcal {D}_{\mathscr {M}}$-module generated by such a distribution has the standard extension property. This implies a non $\mathcal {O}_\mathscr {M}$ torsion result for the $\mathcal {D}_{\mathscr {M}}$-module generated by such a distribution. As an application of this result we determine generators for the conjugate modules of the regular holonomic $\mathcal {D}$-modules introduced and studied in [Barlet, D., On symmetric partial differential operators, Math. Z. 302 (2022), no. 3, 1627–1655] and [Barlet, D., On partial differential operators which annihilate the roots of the universal equation of degree $k$, arXiv:2101.01895] associated to the roots of universal equation of degree $k$, $z^k + \sum _{h=1}^k (-1)^h\sigma _hz^{k-h} = 0$.

References

Barlet, D., Développement asymptotique des fonctions obtenues par intégration sur les fibres, Invent. Math. 68 (1982), no. 1, 129–174. https://doi.org/10.1007/BF01394271

Barlet, D., Fonctions de type trace, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 2, 43–76. http://www.numdam.org/item?id=AIF_1983__33_2_43_0

Barlet, D., On symmetric partial differential operators, Math. Z. 302 (2022), no. 3, 1627–1655. https://doi.org/10.1007/s00209-022-03116-2

Barlet, D. On partial differential operators which annihilate the roots of the universal equation of degree k, arXiv:2101.01895

Barlet, D., and Kashiwara, M., Le réseau $L^2$ d'un système holonome régulier, Invent. Math. 86 (1986), no. 1, 35–62. https://doi.org/10.1007/BF01391494

Barlet, D., and Magnússon, J., Cycles analytiques complexes I: théorèmes de préparation des cycles, Cours Spécialisés 22, Société Mathématique de France, Paris, 2014.

Barlet, D., and Maire, H.-M., Asymptotique des intégrales-fibres, Ann. Inst. Fourier (Grenoble) 43 (1993), no. 5, 1267–1299. http://www.numdam.org/item?id=AIF_1993__43_5_1267_0

Bernstein, I. N., Analytic continuation of generalized functions with respect to a parameter, Funkcional. Anal. i Priložen. 6 (1972), no. 4, 26–40.

Björk, J. E., Ring of differential operators, North-Holland Publishing Co., Amsterdam-New York, 1979.

Björk, J. E., Analytical $D$-modules and applications, Mathematics and its Applications, 247. Kluwer Academic Publishers Group, Dordrecht, 1993. https://doi.org/10.1007/978-94-017-0717-6

Björk, J. E., Residus and $D$-modules, The Legacy of Niels Henrik Abel, 605–651, Laudal, O. A. and Peine, R., eds. Springer-Verlag Berlin, 2004.

Herrera, M., and Lieberman, D., Residues and principal values on complex spaces, Math. Ann. 194 (1971), 259–294. https://doi.org/10.1007/BF01350129

Hörmander, L., The analysis of partial differential operators, I, Springer-Verlag, Berlin, 2003. https://doi.org/10.1007/978-3-642-61497-2

Kashiwara, M. $b$-function and holonomic systems, rationality of roots of $b$-functions, Invent. Math. 38 (1976/77), no. 1, 33–53. https://doi.org/10.1007/BF01390168

Kashiwara, M., Regular holonomic $D$-modules and distributions on a complex manifolds, Complex analytic singularities, 199–206, Adv. Stud. Pure Math., 8, North-Holland, Amsterdam, 1987. https://doi.org/10.2969/aspm/00810199

Published

2023-06-05

How to Cite

Barlet, D. (2023). On principal value and standard extension of distributions. MATHEMATICA SCANDINAVICA, 129(2). https://doi.org/10.7146/math.scand.a-134458

Issue

Section

Articles