Extending Lipschitz mappings continuously

• Autorzy: Kopecka E.
• Języki publikacji: EN

Abstrakty

EN

We consider short mappings from a bounded subset of a Euclidean space into that space, that is, mappings which do not increase distances between points. By Kirszbraun's theorem any such mapping can be extended to the entire space to be short again. In general, the extension is not unique. We show that there are single-valued extension operators continuous in the supremum norm. The multivalued extension operator is lower semicontinuous.

Słowa kluczowe

EN

Lipschitz mapping; extension; Hilbert space

PL

odwzorowanie Lipschitza; przestrzeń Hilberta

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2012
• Tom: Vol. 18, nr 2
• Strony: 167--177
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Kopecka E.

• Mathematical Institute, Czech Academy of Sciences, Zitna 25, 11567 Prague, Czech Republic,

Bibliografia

• [1] Y. Benyamini and J. Lindenstrauss, Geometric Non-Linear Functional Analysis, Amer. Math. Soc. Colloq. Publ. 48, American Mathematical Society, Providence, 2000.
• [2] U. Brehm, Extensions of distance reducing mappings to piecewise congruent mappings on Rm, J. Geom. 16 (1981), no. 2, 187-193.
• [3] B. Dacorogna and P. Marcellini, Implicit Partial Differential Equations, Progr. Nonlinear Differential Equations Appl. 37, Birkhäuser, Boston, 1999.
• [4] B. Kirchheim, E. N. Spadaro and L. Szekelyhidi Jr., Lipschitz equidimensional isometries: The Baire category method, in preparation.
• [5] M. D. Kirszbraun, Über die zusammenziehende und Lipschitzsche Transformationen, Fund. Math. 22 (1934), 77-108.
• [6] N. H. Kuiper, On C’-isometric imbeddings. I, II, Indag. Math. 17 (1955), 545-556, 683-689.
• [7] E. J. Mickle, On the extension of a transformation, Bull. Amer. Math. Soc. 55 (1949), 160-164.
• [8] J. van Mill, The Infinite-Dimensional Topology of Function Spaces, North-Holland, Amsterdam, 2001.
• [9] J. Nash, C1 isometric imbeddings, Ann. of Math. (2) 60 (1954), 383-396.
• [10] I. J. Schoenberg, On a theorem of Kirszbraun and Valentine, Amer. Math. Monthly 60 (1953), 620-622.
• [11] J. H. Wells and L. R. Williams, Embeddings and Extensions in Analysis, Ergeb. Math. Grenzgeb. 84, Springer, Heidelberg, 1975.