The Lusin Theorem and Horizontal Graphs in the Heisenberg Group

• Autorzy: Piotr Hajłasz , Jacob Mirra
• Języki publikacji: EN

Abstrakty

EN

In this paper we prove that every collection of measurable functions fα , |α| = m, coincides a.e. withmth order derivatives of a function g ∈ Cm−1 whose derivatives of order m − 1 may have any modulus of continuity weaker than that of a Lipschitz function. This is a stronger version of earlier results of Lusin, Moonens-Pfeffer and Francos. As an application we construct surfaces in the Heisenberg group with tangent spaces being horizontal a.e.

Słowa kluczowe

EN

Lusin theorem; Heisenberg group; characteristic points

Kategorie tematyczne

• 46E30: Spaces of measurable functions ( L p -spaces, Orlicz spaces, K\"othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
• 46E35: Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems

• Wydawca: De Gruyter Open
• Czasopismo: Analysis and Geometry in Metric Spaces
• Rocznik: 2013
• Tom: 1
• Strony: 295-301

Daty

otrzymano

2013-05-24

zaakceptowano

2013-10-21

online

2013-11-12

Twórcy

autor

Piotr Hajłasz

• Department of Mathematics, University of Pittsburgh,
  301 Thackeray Hall, Pittsburgh, PA 15260, USA

autor

Jacob Mirra

• Department of Mathematics, University of Pittsburgh,
  301 Thackeray Hall, Pittsburgh, PA 15260, USA

Bibliografia

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• [2] Z. M. Balogh, Size of characteristic sets and functions with prescribed gradient. J. Reine Angew. Math. 564 (2003), 63–83.
• [3] Z. M. Balogh, R. Hoefer-Isenegger, J. T. Tyson, Lifts of Lipschitz maps and horizontal fractals in the Heisenberg group. Ergodic Theory Dynam. Systems 26 (2006), 621–651.
• [4] L. Capogna, D. Danielli, S. D. Pauls, J. T. Tyson, An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem. Progress in Mathematics, 259. Birkhäuser Verlag, Basel, 2007.
• [5] B. Franchi, R.L. Wheeden, Compensation couples and isoperimetric estimates for vector fields. Colloq. Math. 74 (1997), 9–27.
• [6] G. Francos, The Luzin theorem for higher-order derivatives. Michigan Math. J. 61 (2012), 507–516.
• [7] M. Gromov, Carnot-Carathéodory spaces seen from within. Sub-Riemannian geometry, pp. 79–323, Progr. Math., 144, Birkhäuser, Basel, 1996.
• [8] N. Lusin, Sur la notion de l’integrale. Ann. Mat. Pura Appl. 26 (1917), 77-129.
• [9] L. Moonens, W. F. Pfeffer, The multidimensional Luzin theorem. J. Math. Anal. Appl. 339 (2008), 746–752.

Identyfikatory

DOI

10.2478/agms-2013-0008