Local fractal functions on Orlicz-Sobolev spaces - ARRIAGADA, Waldo (Studia Universitatis Babeș-Bolyai Mathematica, 2025)
In these notes we consider a class of iterated function systems whose attractors are the graphs of local fractal functions belonging to Orlicz and to Orlicz-Sobolev spaces. We prove that these maps are in correspondence with the fixed points of the Read-Bajraktarević operator. Our method extends a number of outcomes on the existence of local fractal functions of the Lebesgue and Sobolev classes, to more general function spaces where the role of the norm is now played by a Young function. The existence of local fractal functions of the Orlicz and of the Orlicz-Sobolev classes is demonstrated through an intermediate result. The realization of a contractive iterated function system in the (previously untreated) multidimensional case is obtained via a stronger version of the finite increments theorem. Our results somewhat show that it would be natural to extend the Read-Bajraktarević functional to other function spaces on subdomains of differentiable and real analytic manifolds. Other questions, such as the existence of fixed points in higher-orders etc., remain open as well. Our generalizations may be useful in applications. Mathematics Subject Classification (2010): 28A80, 35J60, 37C70, 37G35, 46E30. Received 26 February 2025; Accepted 5 April 2025.