Stratified Whitney jets and tempered ultradistributions on the subanalytic site

N. Honda; G. Morando

Bulletin de la Société Mathématique de France (2011)

  • Volume: 139, Issue: 3, page 389-435
  • ISSN: 0037-9484

Abstract

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In this paper we introduce the sheaf of stratified Whitney jets of Gevrey order on the subanalytic site relative to a real analytic manifold X . Then, we define stratified ultradistributions of Beurling and Roumieu type on X . In the end, by means of stratified ultradistributions, we define tempered-stratified ultradistributions and we prove two results. First, if X is a real surface, the tempered-stratified ultradistributions define a sheaf on the subanalytic site relative to X . Second, the tempered-stratified ultradistributions on the complementary of a 1 -regular closed subset of X coincide with the sections of the presheaf of tempered ultradistributions.

How to cite

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Honda, N., and Morando, G.. "Stratified Whitney jets and tempered ultradistributions on the subanalytic site." Bulletin de la Société Mathématique de France 139.3 (2011): 389-435. <http://eudml.org/doc/272690>.

@article{Honda2011,
abstract = {In this paper we introduce the sheaf of stratified Whitney jets of Gevrey order on the subanalytic site relative to a real analytic manifold $X$. Then, we define stratified ultradistributions of Beurling and Roumieu type on $X$. In the end, by means of stratified ultradistributions, we define tempered-stratified ultradistributions and we prove two results. First, if $X$ is a real surface, the tempered-stratified ultradistributions define a sheaf on the subanalytic site relative to $X$. Second, the tempered-stratified ultradistributions on the complementary of a $1$-regular closed subset of $X$ coincide with the sections of the presheaf of tempered ultradistributions.},
author = {Honda, N., Morando, G.},
journal = {Bulletin de la Société Mathématique de France},
keywords = {sheaves on subanalytic sites; tempered ultradistributions; Whitney jets},
language = {eng},
number = {3},
pages = {389-435},
publisher = {Société mathématique de France},
title = {Stratified Whitney jets and tempered ultradistributions on the subanalytic site},
url = {http://eudml.org/doc/272690},
volume = {139},
year = {2011},
}

TY - JOUR
AU - Honda, N.
AU - Morando, G.
TI - Stratified Whitney jets and tempered ultradistributions on the subanalytic site
JO - Bulletin de la Société Mathématique de France
PY - 2011
PB - Société mathématique de France
VL - 139
IS - 3
SP - 389
EP - 435
AB - In this paper we introduce the sheaf of stratified Whitney jets of Gevrey order on the subanalytic site relative to a real analytic manifold $X$. Then, we define stratified ultradistributions of Beurling and Roumieu type on $X$. In the end, by means of stratified ultradistributions, we define tempered-stratified ultradistributions and we prove two results. First, if $X$ is a real surface, the tempered-stratified ultradistributions define a sheaf on the subanalytic site relative to $X$. Second, the tempered-stratified ultradistributions on the complementary of a $1$-regular closed subset of $X$ coincide with the sections of the presheaf of tempered ultradistributions.
LA - eng
KW - sheaves on subanalytic sites; tempered ultradistributions; Whitney jets
UR - http://eudml.org/doc/272690
ER -

References

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  1. [1] E. Bierstone & P. D. Milman – « Semianalytic and subanalytic sets », Publ. Math. I.H.É.S. 67 (1988), p. 5–42. Zbl0674.32002MR972342
  2. [2] N. Honda – « On the reconstruction theorem of holonomic modules in the Gevrey classes », Publ. Res. Inst. Math. Sci.27 (1991), p. 923–943. Zbl0807.35005MR1145670
  3. [3] J.-M. Kantor – « Classes non-quasi analytiques et décomposition des supports des ultradistributions », An. Acad. Brasil. Ci.44 (1972), p. 171–180. Zbl0261.46043MR324398
  4. [4] M. Kashiwara – « The Riemann-Hilbert problem for holonomic systems », Publ. Res. Inst. Math. Sci.20 (1984), p. 319–365. Zbl0566.32023MR743382
  5. [5] M. Kashiwara & P. Schapira – Sheaves on manifolds, Grund. Math. Wiss., vol. 292, Springer, 1990. Zbl0709.18001MR1074006
  6. [6] —, « Moderate and formal cohomology associated with constructible sheaves », Mém. Soc. Math. France (N.S.) 64 (1996), p. 76. Zbl0881.58060MR1421293
  7. [7] —, « Ind-sheaves », Astérisque 271 (2001), p. 136. Zbl0993.32009MR1827714
  8. [8] H. Komatsu – « Ultradistributions. I. Structure theorems and a characterization », J. Fac. Sci. Univ. Tokyo Sect. IA Math.20 (1973), p. 25–105. Zbl0258.46039MR320743
  9. [9] —, « Ultradistributions. II. The kernel theorem and ultradistributions with support in a submanifold », J. Fac. Sci. Univ. Tokyo Sect. IA Math.24 (1977), p. 607–628. Zbl0385.46027MR477770
  10. [10] K. Kurdyka – « On a subanalytic stratification satisfying a Whitney property with exponent 1 », in Real algebraic geometry (Rennes, 1991), Lecture Notes in Math., vol. 1524, Springer, 1992, p. 316–322. Zbl0779.32006MR1226263
  11. [11] A. Lambert – « Quelques théorèmes de décomposition des ultradistributions », Ann. Inst. Fourier (Grenoble) 29 (1979), p. 57–100. Zbl0396.46038MR552960
  12. [12] S. Łojasiewicz – « Sur le problème de la division », Studia Math.18 (1959), p. 87–136. Zbl0115.10203
  13. [13] B. Malgrange – Équations différentielles à coefficients polynomiaux, Progress in Math., vol. 96, Birkhäuser, 1991. Zbl0764.32001
  14. [14] G. Morando – « Tempered holomorphic solutions of 𝒟 -modules on curves and formal invariants », Ann. Inst. Fourier (Grenoble) 59 (2009), p. 1611–1639. Zbl1218.32015MR2566969
  15. [15] L. Prelli – « Microlocalization of subanalytic sheaves », C. R. Math. Acad. Sci. Paris345 (2007), p. 127–132. Zbl1159.14034MR2344810
  16. [16] C. Roumieu – « Ultra-distributions définies sur R n et sur certaines classes de variétés différentiables », J. Analyse Math. 10 (1962/1963), p. 153–192. Zbl0122.34802MR158261
  17. [17] P. Schapira – « Sur les ultra-distributions », Ann. Sci. École Norm. Sup.1 (1968), p. 395–415. Zbl0164.15401MR239414
  18. [18] H. Whitney – « Functions differentiable on the boundaries of regions », Ann. Math.35 (1934), p. 482–485. Zbl60.0217.03MR1503174JFM60.0217.03
  19. [19] S. Yamazaki – « Remark on division theorem of ultradistributions by Fuchsian differential operator », in Algebraic analysis and the exact WKB analysis for systems of differential equations, RIMS Kôkyûroku Bessatsu, B5, Res. Inst. Math. Sci. (RIMS), Kyoto, 2008, p. 209–223. Zbl1140.35315MR2405975

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