$L_{p,q}$-cohomology of warped cylinders

Kopylov, Yaroslav

L_{p, q}

-cohomology of warped cylinders
[Cohomologie

L_{p, q}

des cylindres tordus]

Kopylov, Yaroslav

Annales mathématiques Blaise Pascal, Tome 16 (2009), p. 321-338 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

Résumé
Abstract

On généralise quelques résultats par Gol $^{'}$ dshtein, Kuz $^{'}$ minov et Shvedov sur la cohomologie $L_{p}$ des cylindres tordus à cohomologie $L_{p, q}$ pour $p \neq q$ . Comme application, on établit des conditions suffisantes pour la non-nullité de la torsion $L_{p, q}$ d’une surface de révolution.

We extend some results by Gol $^{'}$ dshtein, Kuz $^{'}$ minov, and Shvedov about the $L_{p}$ -cohomology of warped cylinders to $L_{p, q}$ -cohomology for $p \neq q$ . As an application, we establish some sufficient conditions for the nontriviality of the $L_{p, q}$ -torsion of a surface of revolution.

Publié le : 2009-01-01

MR 2568869 | Zbl 1196.53025

DOI : https://doi.org/10.5802/ambp.270
Classification: 58A12, 46E30
Mots clés: Forme différentielle, cohomologie

L_{p, q}

, torsion

L_{p, q}

, cylindre tordu

@article{AMBP_2009__16_2_321_0,
     author = {Kopylov, Yaroslav},
     title = {$L\_{p,q}$-cohomology of warped cylinders},
     journal = {Annales math\'ematiques Blaise Pascal},
     volume = {16},
     year = {2009},
     pages = {321-338},
     doi = {10.5802/ambp.270},
     zbl = {1196.53025},
     mrnumber = {2568869},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AMBP_2009__16_2_321_0}
}

Kopylov, Yaroslav. $L_{p,q}$-cohomology of warped cylinders. Annales mathématiques Blaise Pascal, Tome 16 (2009) pp. 321-338. doi : 10.5802/ambp.270. http://gdmltest.u-ga.fr/item/AMBP_2009__16_2_321_0/

Bibliographie

[1] Batuev, E. N.; Stepanov, V. D. Weighted inequalities of Hardy type, Siberian Math. J., Tome 30 (1989) no. 1, pp. 8-16 | Article | MR 995015 | Zbl 0729.42007

[2] Cheeger, J. On the Hodge theory of Riemannian pseudomanifolds, Geometry of the Laplace Operator, Amer. Math. Soc., Providence (Proc. Sympos. Pure Math.) Tome 36 (1980), pp. 91-146 | MR 573430 | Zbl 0461.58002

[3] GolʼDshtein, V. M.; KuzʼMinov, V. I.; Shvedov, I. A. Differential forms on Lipschitz manifolds, Siberian Math. J., Tome 23 (1982) no. 2, pp. 151-161 | Article | Zbl 0522.58001

[4] GolʼDshtein, V. M.; KuzʼMinov, V. I.; Shvedov, I. A. Integration of differential forms of the classes $W_{p, q}^{*}$ , Siberian Math. J., Tome 23 (1982) no. 5, pp. 640-653 | Article | Zbl 0522.58002

[5] GolʼDshtein, V. M.; KuzʼMinov, V. I.; Shvedov, I. A. Wolfe’s theorem for differential forms of classes $W_{p, q}^{*}$ , Siberian Math. J., Tome 24 (1983) no. 5, pp. 672-681 | Article | Zbl 0547.58003

[6] GolʼDshtein, V. M.; KuzʼMinov, V. I.; Shvedov, I. A. A property of the de Rham regularization operator, Siberian Math. J., Tome 25 (1984) no. 2, pp. 251-257 | Article | Zbl 0602.58002

[7] GolʼDshtein, V. M.; KuzʼMinov, V. I.; Shvedov, I. A. The integral representation of the integral of a differential form, Functional Analysis and Mathematical Physics, Collect. Sci. Works, Inst. Mat. Sib. Otd. Akad Nauk SSSR, Novosibirsk (1985), pp. 53-87 | Zbl 0583.58002

[8] GolʼDshtein, V. M.; KuzʼMinov, V. I.; Shvedov, I. A. Normal and compact solvability of linear operators, Siberian Math. J., Tome 30 (1989) no. 5, pp. 704-712 | Article | MR 1025289 | Zbl 0706.47005

[9] GolʼDshtein, V. M.; KuzʼMinov, V. I.; Shvedov, I. A. $L_{p}$ -cohomology of warped cylinders, Siberian Math. J., Tome 31 (1990) no. 6, pp. 919-925 | Article | MR 1097955 | Zbl 0732.53029

[10] GolʼDshtein, V. M.; KuzʼMinov, V. I.; Shvedov, I. A. Reduced $L_{p}$ -cohomology of warped cylinders, Siberian Math. J., Tome 31 (1990) no. 5, pp. 716-727 | Article | MR 1088912 | Zbl 0722.53034

[11] GolʼDshtein, V. M.; Troyanov, M. The $L_{p q}$ -cohomology of SOL, Ann. Fac. Sci. Toulouse Math. (6), Tome 7 (1998) no. 4, pp. 687-698 | Numdam | MR 1693577 | Zbl 0962.53031

[12] GolʼDshtein, V. M.; Troyanov, M. Sobolev inequalities for differential forms and $L_{q, p}$ -cohomology, J. Geom. Anal., Tome 16 (2006) no. 4, pp. 597-632 | MR 2271946 | Zbl 1105.58008

[13] GolʼDshtein, V. M.; Troyanov, M. A conformal de Rham complex (2007) (Preprint arXiv:0711.1286v2. [math.CV])

[14] GolʼDshtein, V. M.; Troyanov, M. Distortion of mappings and $L_{q, p}$ -cohomology, Math. Z. (2009), pp. 264-279 | Article

[15] GolʼDshtein, V. M.; Troyanov, M. $L_{q, p}$ -cohomology of Riemannian manifolds with negative curvature, Sobolev spaces in mathematics. II, Springer, New York (Int. Math. Ser. (N. Y.)) Tome 9 (2009), pp. 199-208 | MR 2484626 | Zbl pre05553252

[16] Kopylov, Ya. A. Some properties of the operator of exterior derivation on surfaces of revolution and $L_{p}$ -cohomology, Complex Geometry of Groups, Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 240 (1999), pp. 247-257 | MR 1703564 | Zbl 0953.58001

[17] Kopylov, Ya. A. $L_{p, q}$ -cohomology and normal solvability, Arch. Math., Tome 89 (2007) no. 1, pp. 87-96 | Article | MR 2322785 | Zbl 1130.58003

[18] Kopylov, Ya. A.; KuzʼMinov, V. I. Exactness of the cohomology sequence corresponding to a short exact sequence of complexes in a semiabelian category, Siberian Adv. Math., Tome 13 (2003) no. 3, pp. 72-80 | MR 2028407 | Zbl 1046.18010

[19] KuzʼMinov, V. I.; Shvedov, I. A. On normal solvability of the operator of exterior derivation on warped products, Siberian Math. J., Tome 37 (1996) no. 2, pp. 276-287 | Article | MR 1425341 | Zbl 0893.58004

[20] KuzʼMinov, V. I.; Shvedov, I. A. Homological aspects of the theory of Banach complexes, Siberian Math. J., Tome 40 (1999) no. 4, pp. 754-763 | Article | MR 1721681 | Zbl 0939.58001

[21] MazʼYa, V. G. Sobolev Spaces, Springer-Verlag, New York (1985) | MR 817985 | Zbl 0692.46023

[22] Troyanov, M. On the Hodge decomposition in $ℝ^{n}$ (2007) (Preprint arXiv:0710.5414v1 [math.FA])