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](https://arxiv.org/abs/1010.6219)), 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.



Critical Smoothness on Besov Spaces $B^s_{p}$: how does it evolved with $p$?
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*](https://www.sciencedirect.com/science/article/pii/S0021782400001616).
This question has also been addressed in [this paper](https://arxiv.org/abs/2009.12491), 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}$.


