Some questions related to fractals Stephen Semmes

A fundamental aspect of harmonic analysis on Euclidean spaces deals with various classes of real or complex-valued functions, such as the C classes of functions which are continuous and continuously differentiable up to order k, where k is a nonnegative integer. If k = 0, this simply means that the function is continuous. Of course continuity of a function makes sense on any metric space, or on topological spaces more generally, while the notion of derivatives entails more structure. Compare with [34]. We can be more precise and consider C classes of functions, where k is a nonnegative integer and α is a real number, 0 ≤ α ≤ 1. If α = 0, then C is taken to mean the same as C; when α > 0, C consists of the C functions with the extra property that their kth order derivatives are locally Hölder continuous of order α. For this let us recall that a function h(x) on a metric space (M, d(x, y)) is Hölder continuous of order α if there is a nonnegative real number A such that