Sobolev spaces on multiple cones

Auscher, P.; Badr, N.

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 19 (2010) no. 3-4, pp. 707-733.

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.

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 ${ℝ}^n$. The analysis interestingly combines use of Poincaré inequalities and of some Hardy type inequalities.

[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 ${ℝ}^n$, 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