Some prevalent results about strongly monoHölder functions

We study the typical behaviour of strongly monoHölder functions from the prevalence point of view. To this end we first prove wavelet-based criteria for strongly monoHölder functions. We then use the notion of prevalence to show that the functions of $C^\alpha (R^d)$ are almost surely strongly monoHölder with Hölder exponent $\alpha$. Finally, we prove that for any $\alpha\in (0, 1)$ on a prevalent set of $C^\alpha (R^d)$ the Hausdorff dimension of the graph is equal to $d +1-\alpha$.