Split points in Musielak-Orlicz spaces: Geometric criteria and duality Mapping

This paper is an in-depth examination of split points in Musielak-Orlicz spaces LΦ , giving a full description of their geometry, and answering various outstanding questions about the geometry of these spaces. The discussion is based on the classical duality arguments together with the recent concepts of the theory of non-uniformly convex spaces and variable-exponent analysis. The key achievements of the paper may be concluded as follows. To begin with, we create conditions which are such as to ensure that split points exist. They coincide with the strict convexity of the modular function Φ(x,·), the Δ₂-condition of validity of Φ and the conjugated version, along with some assumptions of spatial symmetry. Second, we obtain a blame-sharing characterization in the form of introducing a correction term D(x,g). This expression is a generalization of the popular system of Giles that includes ideas of the modular fixed-point theory. Third, we examine the computational factors of determining split points. We find that at L(p) > 1 the instability index is computationally difficult. Specifically, we explain the cases where the identification process is unsuccessful in the Lebesgue spaces of variable-exponent Lp(x). Lastly, we demonstrate the applicability of the theory to an application to variable-exponent partial differential equations. Specifically, we consider the use of the property of split points in the norm-attainment of solutions of the equation: -Δ p(x) u = f These results imply a number of possible extensions, such as additional research studies in the context of noncommutative Musielak-Orlicz spaces.