The prevalent singularity spectrum conjecture for inhomogeneous Besov spaces

Let $\mu$ be an almost doubling capacity on $[0,1]^d$ satisfying $(P)$ and verifying the SMF, and let $p,q\in[1,+\infty]$. Define

$$\tau_{\mu,p}(t)=\begin{cases}\frac{p-t}{p}\,\tau_{\mu}\left(\frac{p}{p-t}t\right)&\text{if }t\in(-\infty,p),\d&\text{if }t\in[p,+\infty),\end{cases}$$

and

$$\theta_p:\alpha\in[\tau'_{\mu}(+\infty),\tau'_{\mu}(-\infty)]\longmapsto \alpha+\frac{\tau^*_{\mu}(\alpha)}{p}.$$

For $f\in\widetilde{b}^{\mu,\Psi}_{p,q}([0,1]^d)$, let $\sigma_f$ denote its singularity spectrum, and let $\sigma_\mu$ and $\tau^*_{\mu}$ denote the corresponding multifractal spectrum and Legendre transform.

Prevalent singularity spectrum conjecture. For every $f\in\widetilde{b}^{\mu,\Psi}_{p,q}([0,1]^d)$,

$$\sigma_f(h)\leq\begin{cases}\sigma_\mu(\theta^{-1}(h))=\tau^*_{\mu,p}(h)&\text{if }h\leq\tau'_{\mu,p}(0),\d&\text{if }h>\tau'_{\mu,p}(0),\end{cases}$$

and there exists a prevalent set of functions $f\in\widetilde{b}^{\mu,\Psi}_{p,q}([0,1]^d)$ satisfying

$$\sigma_f=\sigma_\mu(\theta^{-1}(\cdot))=\tau^*_{\mu,p}.$$

This is the conjectured finite-$p$ analogue of the prevalent singularity-spectrum result established in the paper for $p=\infty$, corresponding to results of Barral and Seuret for Baire-typical functions. The notation $(P)$, SMF, the capacity spectra, and the function $\theta$ are inherited from the paper; the source statement uses $\theta^{-1}$ although only $\theta_p$ is explicitly defined, so that notation should be checked.