Read–Bajraktarević functional in higher-order Orlicz–Sobolev space

In this talk, we consider a class of local Iterated Function Systems (IFSs) whose attractors are the graphs of local fractal functions both in a real Orlicz space and in a real Orlicz–Sobolev space over a domain of $\mathbb{R}^N$. We prove that these maps are in correspondence with the fixed points of the Read–Bajraktarević operator and justify a step in the proof of a theorem recently published in the literature. This result states that local fractal function of an Orlicz–Sobolev class of order $m>1$ ($m$ is an integer) appear naturally as fixed points of the restriction of the Read–Bajraktarević functional. This has been well-known since the works of Massopust et al. in the context of Lebesgue and Sobolev spaces. Our method extends a number of known theorems on the existence of local fractal functions to more general function spaces (where the role of the norm is now played by a Young function, also known as an N function) and to higher orders and dimensions. In this talk, we will relax a condition (inequality) that ensures contractivity of the Read–Bajraktarević operator, and we will prove that an asymptotic quantity appearing in the proof is not required to be greater than 1. The existence of local fractal functions of the Orlicz and of the Orlicz–Sobolev classes is demonstrated through an intermediary result. The realization of a contractive IFS in the (previously untreated) multidimensional case is obtained via a stronger version of the mean-value theorem. Our results demonstrate that it is natural to extend the Read–Bajraktarević operator to other function spaces on subdomains of differentiable and real analytic manifolds. Other questions, such as the existence of fixed points in higher orders, remain open, as well.