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

Yaroslav Kopylov

doi:10.5802/ambp.270

L_{p, q}

-cohomology of warped cylinders
[Cohomologie

L_{p, q}

des cylindres tordus]

Yaroslav Kopylov ¹

¹ Sobolev Institute of Mathematics Pr. Akademika Koptyuga 4 630090, Novosibirsk RUSSIA

Annales mathématiques Blaise Pascal, Tome 16 (2009) no. 2, pp. 321-338.

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.

MR Zbl

DOI : 10.5802/ambp.270

Classification : 58A12, 46E30
Keywords: Differential form, $L_{p,q}$-cohomology, $L_{p,q}$-torsion, warped cylinder
Mot clés : Forme différentielle, cohomologie $L_{p,q}$, torsion $L_{p,q}$, cylindre tordu

Affiliations des auteurs :

Yaroslav Kopylov ¹

¹ Sobolev Institute of Mathematics Pr. Akademika Koptyuga 4 630090, Novosibirsk RUSSIA

@article{AMBP_2009__16_2_321_0,
     author = {Yaroslav Kopylov},
     title = {$L_{p,q}$-cohomology of warped cylinders},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {321--338},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {16},
     number = {2},
     year = {2009},
     doi = {10.5802/ambp.270},
     mrnumber = {2568869},
     zbl = {1196.53025},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.270/}
}

TY  - JOUR
AU  - Yaroslav Kopylov
TI  - $L_{p,q}$-cohomology of warped cylinders
JO  - Annales mathématiques Blaise Pascal
PY  - 2009
SP  - 321
EP  - 338
VL  - 16
IS  - 2
PB  - Annales mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.270/
DO  - 10.5802/ambp.270
LA  - en
ID  - AMBP_2009__16_2_321_0
ER  -

%0 Journal Article
%A Yaroslav Kopylov
%T $L_{p,q}$-cohomology of warped cylinders
%J Annales mathématiques Blaise Pascal
%D 2009
%P 321-338
%V 16
%N 2
%I Annales mathématiques Blaise Pascal
%U https://ambp.centre-mersenne.org/articles/10.5802/ambp.270/
%R 10.5802/ambp.270
%G en
%F AMBP_2009__16_2_321_0

Yaroslav Kopylov. $L_{p,q}$-cohomology of warped cylinders. Annales mathématiques Blaise Pascal, Tome 16 (2009) no. 2, pp. 321-338. doi : 10.5802/ambp.270. https://ambp.centre-mersenne.org/articles/10.5802/ambp.270/

Bibliographie
Cité par

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

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

[3] V. M. Golʼdshtein; V. I. Kuzʼminov; I. A. Shvedov Differential forms on Lipschitz manifolds, Siberian Math. J., Volume 23 (1982) no. 2, pp. 151-161 | DOI | Zbl

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

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

[6] V. M. Golʼdshtein; V. I. Kuzʼminov; I. A. Shvedov A property of the de Rham regularization operator, Siberian Math. J., Volume 25 (1984) no. 2, pp. 251-257 | DOI | Zbl

[7] V. M. Golʼdshtein; V. I. Kuzʼminov; I. A. Shvedov 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

[8] V. M. Golʼdshtein; V. I. Kuzʼminov; I. A. Shvedov Normal and compact solvability of linear operators, Siberian Math. J., Volume 30 (1989) no. 5, pp. 704-712 | DOI | MR | Zbl

[9] V. M. Golʼdshtein; V. I. Kuzʼminov; I. A. Shvedov $L_{p}$ -cohomology of warped cylinders, Siberian Math. J., Volume 31 (1990) no. 6, pp. 919-925 | DOI | MR | Zbl

[10] V. M. Golʼdshtein; V. I. Kuzʼminov; I. A. Shvedov Reduced $L_{p}$ -cohomology of warped cylinders, Siberian Math. J., Volume 31 (1990) no. 5, pp. 716-727 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

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

[21] V. G. Mazʼya Sobolev Spaces, Springer-Verlag, New York, 1985 | MR | Zbl

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

Cité par Sources :