Sobolev spaces on multiple cones
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. 3-4, pp. 707-733.

The purpose of this note is to discuss how various Sobolev spaces defined on multiple cones behave with respect to density of smooth functions, interpolation and extension/restriction to/from n . The analysis interestingly combines use of Poincaré inequalities and of some Hardy type inequalities.

L’objet de cet article est de décrire le comportement de certaines familles d’espaces de Sobolev en ce qui concerne la densité des fonctions régulières, l’interpolation, les propriétés d’extension et de restriction. Les méthodes combinent de façon intéressante les inégalités de Poincaré et des inégalités de type Hardy.

DOI: 10.5802/afst.1264
Auscher, P. 1; Badr, N. 2

1 Paris-Sud, Laboratoire de Mathématiques, UMR 8628, Orsay, F-91405 ; CNRS, Orsay, F-91405
2 Université de Lyon; CNRS; Université Lyon 1, Institut Camille Jordan, 43 boulevard du 11 Novembre 1918, F-69622 Villeurbanne Cedex, France
@article{AFST_2010_6_19_3-4_707_0,
     author = {Auscher, P. and Badr, N.},
     title = {Sobolev spaces on multiple cones},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {707--733},
     publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 19},
     number = {3-4},
     year = {2010},
     doi = {10.5802/afst.1264},
     zbl = {1219.46031},
     mrnumber = {2790816},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/afst.1264/}
}
TY  - JOUR
AU  - Auscher, P.
AU  - Badr, N.
TI  - Sobolev spaces on multiple cones
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2010
SP  - 707
EP  - 733
VL  - 19
IS  - 3-4
PB  - Université Paul Sabatier, Institut de mathématiques
PP  - Toulouse
UR  - http://www.numdam.org/articles/10.5802/afst.1264/
DO  - 10.5802/afst.1264
LA  - en
ID  - AFST_2010_6_19_3-4_707_0
ER  - 
%0 Journal Article
%A Auscher, P.
%A Badr, N.
%T Sobolev spaces on multiple cones
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2010
%P 707-733
%V 19
%N 3-4
%I Université Paul Sabatier, Institut de mathématiques
%C Toulouse
%U http://www.numdam.org/articles/10.5802/afst.1264/
%R 10.5802/afst.1264
%G en
%F AFST_2010_6_19_3-4_707_0
Auscher, P.; Badr, N. Sobolev spaces on multiple cones. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. 3-4, pp. 707-733. doi : 10.5802/afst.1264. http://www.numdam.org/articles/10.5802/afst.1264/

[1] Adams (R.).— Sobolev spaces. Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London (1975). | MR | Zbl

[2] Badr (N.).— Ph.D Thesis, Université Paris-Sud (2007).

[3] Badr (N.).— Real interpolation of Sobolev Spaces, Math. Scand., volume 105, issue 2, p. 235-264 (2009). | MR | Zbl

[4] Bennett (C.), Sharpley (R.).— Interpolation of operators, Academic Press (1988). | MR | Zbl

[5] Bergh (J.), Löfström (J.).— Interpolation spaces, An introduction, Springer (Berlin) (1976). | MR | Zbl

[6] Chavel (I.).— Eigenvalues in Riemannian geometry. Academic Press (1984). | MR | Zbl

[7] Coifman (R.), Weiss (G.).— Analyse harmonique sur certains espaces homogènes, Lecture notes in Math., Springer (1971). | MR

[8] Costabel (M.), Dauge (M.), Nicaise (S.).— Singularities of Maxwell interface problems. M2AN Math. Model. Numer. Anal. 33, no. 3 (1999). | Numdam | MR | Zbl

[9] Devore (R.) Scherer (K.).— Interpolation of linear operators on Sobolev spaces, Ann. of Math., 109, p. 583-599 (1979). | MR | Zbl

[10] Evans (L.C.), Gariepy (R. F.).— Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL. viii+268 pp (1992). | MR | Zbl

[11] Hajlasz (P.), Koskela (P.).— Sobolev met Poincaré, Mem. Amer. Math. Soc., 145, (688), p. 1-101 (2000). | MR | Zbl

[12] Hajlasz (P.).— Sobolev spaces on a arbitrary metric space, Potential Anal., 5, p. 403-415 (1996). | MR | Zbl

[13] Hajlasz (P.), Koskela (P.), Tuominen (H.).— Sobolev embeddings, extensions and measure density condition, J. Funct. Anal., 254, p. 1217-1234 (2008). | MR | Zbl

[14] Heinonen (J.).— Lectures on analysis on metric spaces, Springer-Verlag (2001). | MR | Zbl

[15] Michael (J. H), Simon (L. M.).— Sobolev and Mean-Value Inequalities on Generalized Submanifolds of n , Comm. Pure and Appl. Math., vol. 3 26, p. 361-379 (1973). | MR | Zbl

[16] Maz’ya (V.).— Sobolev spaces. Springer-Verlag, Berlin. xix+486 pp (1985). | MR | Zbl

[17] Maz’ya (V.), Poborchi (S.).— Differentiable functions on bad domains. World Scientific Publishing Co., Inc., River Edge, NJ (1997). | MR | Zbl

[18] Rychkov (V.S.), Linear extension operators for restrictions of function spaces to irregular open sets, Studia Math. 140, p. 141-162 (2000). | MR | Zbl

[19] Semmes (S.), Finding Curves on General Spaces through Quantitative Topology, with Applications to Sobolev and Poincaré Inequalities.Selecta Mathematica, New Series Vol. 2, No. 2, p. 155-295 (1996). | MR | Zbl

[20] Shvartsman (P.), On extensions of Sobolev functions defined on regular subsets of metric measure spaces. J. Approx. Theory 144, no. 2, p. 139-161 (2007). | MR | Zbl

[21] Stein (E.M.), Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. (1970). | MR | Zbl

[22] Stein (E. M.), Weiss (G.), Introduction to Fourier Analysis in Euclidean spaces, Princeton University Press (1971). | MR | Zbl

[23] Ziemer (W.), Weakly differentiable functions. Sobolev spaces and functions of bounded variation. Graduate Texts in Mathematics, 120. Springer-Verlag, New York (1989). | MR | Zbl

Cited by Sources: