Singular dimension of spaces of real functions

Abstract

Let X be a space of measurable real functions defined on a fixed open set Ω ⊆ R^N . It is natural to define the singular dimension of X as the supremum of Hausdorff dimension of singular sets of all functions in X.We say that f ∈ X is a maximally singular function in X if the Hausdorff dimension of its singular set is the largest possible. The paper discusses recent results about singular dimension of Banach spaces of functions, existence and density of maximally singular functions, and provides some open problems.

Keywords

• Singular set
• Fractal set
• Singular dimension
• Maximally singular function
• Function spaces