Limits of the improved integrability of the volume forms

Abstract

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 L Ψ (Ω), where Ψ grows almost linearly, that is, t 1−ε ≺ Ψ(t) ≺ t 1+ε for ε > 0. Our results amount to the principle: the further the Jacobian is from L 1 loc (Ω), 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 loc (Ω). Contents.

A study of Jacobians in Hardyd–Orlicz spaces

“…Under the above assumptions on V we have 'P~l(t) ^ t l^n V~1(t)(4)(5)(6)(7)(8)(9)(10) or, equivalently,…”

Mappings of finite distortion: decay of the Jacobian in the plane

“…Taking this normalization into account, and applying Proposition 3.1 in [6] witĥ .s/ D p s 2 eCs log p 1 .e C s/, we conclude that …”

Composition of bi-Sobolev homeomorphisms

“…As a matter of fact, (4.7) and (4.6) are equivalent to each other, for all q 1. Indeed, (4.7) implies (4.6) with no conditions on the distortion, assuming merely J f −1 0 a.e., by higher integrability of the Jacobian determinant [8]. On the other hand, (4.6) is equivalent to J f −1 log q (e + |Df −1 |) ∈ L 1 loc by (2.3), and hence implies (4.7) by (4.15) (for α = q).…”