I found a lemma in [this paper](https://www.jstor.org/stable/24899147) of Constantin and Wu, stated with no proof:



> 
> **Lemma
> 3.2.** Let $b=\chi\_{D}$ be the characteristic function of a bounded domain $D\subset\mathbb R^2$ whose boundary has box-counting (fractal) dimension not larger than $d<2:$
> $
> d\_{F}(\partial D) \leq d.
> $
> Then
> $b\in B^{\frac{2-d}p }\_{p,\infty}$
> $
> \text {for } 1 \leq p<\infty.
> $
> 
> 
> 


Is it easy to prove? For sufficiently smooth curves (so $d\_F(\partial D) = 1$) I have proven (by estimating the Gagliardo seminorm) that $b\in H^s$ for $s<1/2$, and I suspect using Theorem 2.36 of Bahouri, Chemin, and Danchin's [book](http://dx.doi.org/10.1007/978-3-642-16830-7) "Fourier analysis and nonlinear partial differential equations":



> 
> **Theorem 2.36** . Let $s$ be in $( 0,1)$ and $(p, r) \in[1, \infty]^{2}$. A constant $C$ exists such that, for any $u$ in $\mathcal{S}\_{h}^{\prime}$
> $$
> C^{-1}\|u\|\_{\dot{B}\_{p, r}^{s}} \leq\left\|\frac{\left\|\tau\_{-y} u-u\right\|\_{L^{p}}}{|y|^{s}}\right\|\_{L^{r}\left(\mathbb{R}^{d} ; \frac{d y}{|y|^{d}}\right)} \leq C\|u\|\_{\dot{B}\_{p, r}^{s}}.
> $$
> 
> 
> 


I can get a similar result for Besov spaces. But how can I prove the result in the general case?



Box counting dimension and Besov spaces on $\mathbb R^2$
The modulus of continuity in direction $v$ is $\omega\_p(t,v) := \lVert 1\_D - 1\_D(\cdot-tv)\rVert\_p$. Since $\lvert 1\_D(x) - 1\_D(x-tv)\rvert \le 1\_{N(\partial D,t)}$, where $N(\partial D,t)$ is a $t$-neighborhood of $\partial D$, then by the assumption on the dimension of the boundary
$$
\omega\_p(t,v) \le C t^\frac{2-d}{p},
$$
so $1\_D\in B^{\frac{2-d}{p}}\_{p,\infty}$.


