Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-05-06T04:39:28.411Z Has data issue: false hasContentIssue false
  • English
  • Français

A study of Jacobians in Hardyd–Orlicz spaces

Published online by Cambridge University Press:  14 November 2011

Tadeusz Iwaniec  and
Anna Verde
Tadeusz Iwaniec
Affiliation:
Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA
Anna Verde
Affiliation:
Dipartimento di Matematica e Applicazioni, R. Caccioppoli, Via Cintia, 80126 Napoli, Italy
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the Jacobian determinants J = det(∂fi/∂xj) of mappings f: Ω ⊂ ℝn → ℝn in a Sobolev–Orlicz space W1,Φ (Ω,ℝn). Their natural generalizations are the wedge products of differential forms. These products turn out to be in the Hardy–Orlicz spaces ℌp (Ω). Other nonlinear quantities involving the Jacobian, such as J log |J|, are also studied. In general, the Jacobians may change sign and in this sense our results generalize the existing ones concerning positive Jacobians.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 3 , 1999 , pp. 539 - 570
Copyright
Copyright © Royal Society of Edinburgh 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Brezis, H., Fusco, N. and Sbordone, C.. Integrability for the Jacobian of orientation preserving mappings. J. Fund. Analysis 115 (1993), 425–431.CrossRefGoogle Scholar
2Bonami, A., Iwaniec, T., Jones, P. and Zinsmeister, M.. Note on the product ℌl BMO. Preprint.Google Scholar
3Bonami, A. and Madan, S.. Balayage of Carleson measures and Hankel operators on generalized Hardy spaces. Math. Nachr. 153 (1991), 237–245.CrossRefGoogle Scholar
4Bagby, R. and Parson, D.. Orlicz spaces and rearranged maximal functions. Math. Nachr. 132 (1987), 15–27.CrossRefGoogle Scholar
5Coifman, R. R., Lions, P. L., Meyer, Y. and Semmes, S.. Compensated compactness and Hardy spaces. J. Math. Pures Appl. 72 (1993), 247–286.Google Scholar
6Coifman, R. and Grafakos, L.. Hardy space estimates for multilinear operators. I. Revista Matematica Iberoamericana 8 (1992), 45–67.CrossRefGoogle Scholar
7Coifman, R. and Weiss, G.. Extensions of Hardy spaces and their use in analysis. Bull. AMS 83 (1997), 569–645.CrossRefGoogle Scholar
8Greco, L.. A remark on the equality det Df = Det Df. Dig. Int. Eqns 6 (1993), 1089–1100.Google Scholar
9Greco, L. and Iwaniec, T.. New inequalities for the Jacobian. Ann. Inst. H. Poincaré 11 (1994), 17–35.Google Scholar
10Greco, L., Iwaniec, T. and Moscariello, G.. Limits of the improved integrability of the volume forms. Indiana Univ. Math. J. 44 (1995), 305–339.CrossRefGoogle Scholar
11Greco, L., Iwaniec, T. and Milman, M.. A cancellation phenomenon in nonlinear commutators in Orlicz spaces. Preprint.Google Scholar
12Iwaniec, T.. Integrability theory of the Jacobians. Lipschitz Lectures in Bonn, Preprint no. 36 Sonderforschungsbereich 256 (1995), 1–68.Google Scholar
13Iwaniec, T. and Lutoborski, A.. Integral estimates for null-Lagrangians. Arch. Ration. Mech. Analysis 125 (1993), 25–79.CrossRefGoogle Scholar
14Iwaniec, T. and Sbordone, C.. On the integrability of the Jacobian under minimal hypothesis. Arch. Ration. Mech. Analysis 119 (1992), 129–143.CrossRefGoogle Scholar
15Iwaniec, T., Scott, C. and Stroffolini, B.. Nonlinear Hodge theory on manifolds with boundary. Preprint, Syracuse University (1994).Google Scholar
16Iwaniec, T. and Verde, A.. On the operator L(f) = f log ∖f∖. J. Fund. Analysis. (In the press.)Google Scholar
17Janson, S.. Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation. Duke Math. J. 47 (1980), 959–982.CrossRefGoogle Scholar
18Janson, S., Peetre, J. and Semmes, S.. On the action of Hankel and Toeplitz operators on some function spaces. Duke Math. J. 51 (1984), 937–958.CrossRefGoogle Scholar
19Milman, M.. Inequalities for Jacobians; interpolation techniques. Preprint.Google Scholar
20Müller, S.. A surprising higher integrability property of mappings with positive determinant. Bull. AMS 21 (1989), 245–248.CrossRefGoogle Scholar
21Müller, S.. Det = det, a remark on the distributional determinant. C. R. Acad. Sd. Paris 1311 (1990), 13–17.Google Scholar
22Murat, F.. Compacité par compensation. Ann. Sc. Norm. Sup. Pisa 5 (1978), 489–507.Google Scholar
23Rao, M. M. and Ren, Z. D.. Theory of Orlicz spaces, pure and applied mathematics, vol. 146 (New York: M. Dekker, 1991).Google Scholar
24Robbin, J. W., Roger, R. C. and Temple, B.. On weak continuity and the Hodge decomposition. Trans. AMS 303 (1987), 609–618.CrossRefGoogle Scholar
25Rochberg, R. and Weiss, G.. Derivatives of analytic families of Banach spaces. Ann. Math. 118 (1983), 315–347.CrossRefGoogle Scholar
26Stein, E. M.. Note on the class L log L (theorem 2). Studia Math. 32 (1969), 305–310.CrossRefGoogle Scholar
27Stein, E. M.. Harmonic analysis (Princeton University Press, 1993).Google Scholar
28Stromberg, J. O.. Bounded mean oscillation with Orlicz norms and duality of Hardy spaces. Indiana Univ. Math. J. 28 (1979), 511–544.CrossRefGoogle Scholar
29Tartar, L.. Compensated compactness and applications to partial differential equations. In Nonlinear Analysis and Mechanics, Heriot-Watt Symp. IV (ed. Knops, R. J.), vol. 39, pp. 136–212. Research Notes in Mathematics (London: Pitman, 1979).Google Scholar
30Wu, S.. On the integrability of nonnegative Jacobians. Preprint.Google Scholar
31Coifman, R., Rochberg, R. and Weiss, G.. Factorization for Hardy spaces in several variables. Ann. Math. 103 (1976), 569–645.CrossRefGoogle Scholar