Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Limiting fractional and Lorentz space estimates of differential forms

Author(s): Jean Van Schaftingen
Journal: Proc. Amer. Math. Soc. 138 (2010), 235-240.
MSC (2000): Primary 35B65; Secondary 26D10, 35F05, 42B20, 46E30, 46E35, 58A10
Posted: September 3, 2009
MathSciNet review: 2550188
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We obtain estimates in Besov, Triebel-Lizorkin and Lorentz spaces of differential forms on $\mathbf{R}^n$ in terms of their norm.

References:

1.: G. Bourdaud, Calcul fonctionnel dans certains espaces de Lizorkin-Triebel, Arch. Math. (Basel) 64 (1995), no. 1, 42-47. MR 1305659 (96a:46064)
2.: J. Bourgain and H. Brezis, New estimates for the Laplacian, the div-curl, and related Hodge systems, C. R. Math. Acad. Sci. Paris 338 (2004), no. 7, 539-543. MR 2057026 (2004m:26018)
3.: -, New estimates for elliptic equations and Hodge type systems, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 2, 277-315. MR 2293957
4.: S. Chanillo and J. Van Schaftingen, Subelliptic Bourgain-Brezis estimates on groups, Math. Res. Lett. 16 (2009), no. 3, 487-501. MR 2511628
5.: L. Lanzani and E. M. Stein, A note on div curl inequalities, Math. Res. Lett. 12 (2005), no. 1, 57-61. MR 2122730 (2005m:58001)
6.: I. Mitrea and M. Mitrea, A remark on the regularity of the div-curl system, Proc. Amer. Math. Soc. 137 (2009), 1729-1733. MR 2470831
7.: R. O'Neil, Convolution operators and spaces, Duke Math. J. 30 (1963), 129-142. MR 0146673 (26:4193)
8.: J. Peetre, Espaces d'interpolation et théorème de Soboleff, Ann. Inst. Fourier (Grenoble) 16 (1966), fasc. 1, 279-317. MR 0221282 (36:4334)
9.: T. Runst and W. Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, de Gruyter Series in Nonlinear Analysis and Applications, vol. 3, Walter de Gruyter & Co., Berlin, 1996. MR 1419319 (98a:47071)
10.: H. Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983. MR 781540 (86j:46026)
11.: J. Van Schaftingen, A simple proof of an inequality of Bourgain, Brezis and Mironescu, C. R. Math. Acad. Sci. Paris 338 (2004), no. 1, 23-26. MR 2038078 (2004k:26033)
12.: -, Estimates for $L\sp 1$ -vector fields, C. R. Math. Acad. Sci. Paris 339 (2004), no. 3, 181-186. MR 2078071 (2005b:35018)
13.: -, Function spaces between BMO and critical Sobolev spaces, J. Funct. Anal. 236 (2006), no. 2, 490-516. MR 2240172 (2007e:46028)
14.: -, Estimates for $L\sp 1$ vector fields under higher-order differential conditions, J. Eur. Math. Soc. (JEMS) 10 (2008), no. 4, 867-882. MR 2443922
15.: W. P. Ziemer, Weakly differentiable functions, Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. MR 1014685 (91e:46046)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35B65, 26D10, 35F05, 42B20, 46E30, 46E35, 58A10

Retrieve articles in all Journals with MSC (2000): 35B65, 26D10, 35F05, 42B20, 46E30, 46E35, 58A10

Additional Information:

Jean Van Schaftingen
Affiliation: Département de Mathématique, Université Catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium
Email: Jean.VanSchaftingen@uclouvain.be

DOI: 10.1090/S0002-9939-09-10005-9
PII: S 0002-9939(09)10005-9
Keywords: Differential forms, div-curl system, Hodge decomposition, exterior differential, Besov spaces, Triebel--Lizorkin spaces, Lorentz-Sobolev spaces, regularity, limiting embedding
Received by editor(s): March 12, 2009,
Received by editor(s) in revised form: April 20, 2009
Posted: September 3, 2009
Additional Notes: The author is supported by the Fonds de la Recherche Scientifique-FNRS
Communicated by: Michael T. Lacey
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|