Limits of the Improved Integrability of the Volume Forms

We identify the exact degree of integrability of nonnegative volume forms and the Jacobians of orientation preserving mappings from various Orlicz-Sobolev classes. An improvement takes place when the Jacobian belongs to the Orlicz space Lψ(Ω), where ψ grows almost linearly, that is, t1−ε ≺ ψ(t) ≺ t1...

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Bibliographic Details
Published in:Indiana University mathematics journal 1995, Vol.44 (2), p.305-339
Main Authors: Greco, Luigi, Iwaniec, Tadeusz, Moscariello, Gioconda
Format: Article
Language:English
Subjects:
College mathematics
Cubes
Increasing functions
Jacobians
Mathematical functions
Mathematical inequalities
Mathematical integrals
Mathematics
Orlicz space
Preprints
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Summary:We identify the exact degree of integrability of nonnegative volume forms and the Jacobians of orientation preserving mappings from various Orlicz-Sobolev classes. An improvement takes place when the Jacobian belongs to the Orlicz space Lψ(Ω), where ψ grows almost linearly, that is, t1−ε ≺ ψ(t) ≺ t1+ε for ε > 0. Our results amount to the principle: the further the Jacobian is from $L^{1}_{\text{loc}}(\Omega)$, the less is the improvement of integrability. In fact, as shown in [LZ], [Wu], [GIM], the largest improvement happens when the Jacobian is precisely in the space $L^{1}_{\text{loc}}(\Omega)$.
ISSN:0022-2518
1943-5258
DOI:10.1512/iumj.1995.44.1990