Local fractal functions in Besov and Triebel-Lizorkin spaces

TL;DR

This paper introduces local fractal functions derived from local iterated function systems and establishes conditions for their inclusion in various Besov and Triebel-Lizorkin spaces, linking fractal geometry with functional analysis.

Contribution

It extends the concept of fractal functions using local IFSs and provides explicit criteria for their membership in important function spaces.

Findings

Derived explicit conditions for local fractal functions to belong to Besov and Triebel-Lizorkin spaces.

Connected fractal functions with classical function spaces like Sobolev and Hardy spaces.

Enhanced understanding of the role of fractal functions in interpolation theory.

Abstract

Within the new concept of a local iterated function system (local IFS), we consider a class of attractors of such IFSs, namely those that are graphs of functions. These new functions are called local fractal functions and they extend and generalize those that are currently found in the fractal literature. For a class of local fractal functions, we derive explicit conditions for them to be elements of Besov and Triebel--Lizorkin spaces. These two scales of functions spaces play an important role in interpolation theory and for certain ranges of their defining parameters describe many classical function spaces (in the sense of equivalent norms). The conditions we derive provide immediate information about inclusion of local fractal functions in, for instance, Lebesgue, Sobolev, Slobodeckij, H\"older, Bessel potential, and local Hardy spaces.