Critical Smoothness on Besov Spaces $B^s_{p}$: how does it evolved with $p$?

Question

We denote by $B_{p}^s(\mathbb{T}) := B_{p,p}^s(\mathbb{T})$ the Besov space over the circle $\mathbb{T}$ with parameters $p=q \in (0, \infty]$ and smoothness $s \in \mathbb{R}$.

For $p>0$ fixed and $f \in \mathcal{S}'(\mathbb{T})$ a generalized function, set \begin{equation} s_p(f) = \sup \{s \in \mathbb{R} , \ f \in B_p^s(\mathbb{T})\} \in \mathbb{R} \cup \{+\infty\}, \end{equation} which can be interpreted as the critical smoothness of $s$ for the $L_p$-scale.

The function $p \mapsto s_p(f)$ is obtained by characterizing in which Besov spaces $f$ is and is not. It is known for some functions. For instance, if $f = \delta$ is the Dirac distribution, we have $s_p(\delta) = \frac{1}{p} - 1$, from which we deduce $s_p ( 1_{[0,1]} ) = \frac{1}{p}$. The Besov regularity of the Gaussian white noise $W$ is also known (see for instance this paper), from which we deduce that $s_p(W) = - 1/2$. Considering other classes of random processes or fractal-type functions, it is possible to find functions for which \begin{equation} s_p(f) = \min ( 1/ p , a) + b \end{equation} for any $a , b \in \mathbb{R}$.

Question: Is there some functions $f \in \mathcal{S}'(\mathbb{T})$ for which $s_p(f)$ can take a different form? All the examples I encountered so far suggest that it is not the case.

Answer 1

The answer to my question is actually no: there exists generalized functions such that $s_p(f)$ is not of the proposed form. Stéphane Jaffard discusses the possible forms of the functions $s_p(f)$, denoted by $\eta(p)$, in his paper On the Frisch-Parisi Conjecture. This question has also been addressed in this paper, which provides some other references. For instance, it includes functions such that $$s_p(f) = \frac{\alpha}{p} + b$$ for any $\alpha \in (0,1)$ and $ b\in \mathbb{R}$.