Global Integrability of the Jacobian of a Composite Mapping

Carl Gustav Jacob Jacobi (1804–1851), one of the nineteenth century Germany’s most accomplished scientists, developed the theory of determinants and transformations into a powerful tool for evaluating multiple integrals and solving differential equations. Since then, the Jacobian (determinant) has played a critical role in multidimensional analysis and related fields, including nonlinear elasticity, weakly differentiable mappings, continuum mechanics, nonlinear PDEs, and calculus of variations. The integrability of Jacobians has become a rather important topic in the study of Jacobians because one of the major applications of Jacobians is to evaluate multiple integrals. Higher integrability properties of the Jacobian first showed up in [2], where Gehring invented reverse Hölder inequalities and used these inequalities to establish the L1+ε-integrability of the Jacobian of a quasiconformal mapping, ε > 0. Recently, the integrability of Jacobians of orientation-preserving mappings of Sobolev class W loc (Ω,R n) has attracted the attention of mathematicians, see [1, 3–7], for instance. The purpose of this paper is to study the Lp(logL)α(Ω)-integrability of the Jacobian of a composite mapping. Let 0 < p <∞ and α≥ 0 be real numbers and let E be any subset of Rn. We define the functional on a measurable function f over E by