Some Prevalent Sets in Multifractal Analysis: How Smooth is Almost Every Function in $$T_p^\alpha (x)$$

Abstract

We present prevalent results concerning generalized versions of the \(T_p^\alpha \) spaces, initially introduced by Calderón and Zygmund. We notably show that the logarithmic correction appearing in the quasi-characterization of such spaces is mandatory for almost every function; it is in particular true for the Hölder spaces, for which the existence of the correction was showed necessary for a specific function. We also show that almost every function from \(T_p^\alpha (x_{0})\) has \(\alpha \) as generalized Hölder exponent at \(x_0\).