# Box counting dimension and Besov spaces on $\mathbb R^2$

I found a lemma in this paper 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 "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?

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 $$1_D\in B^{\frac{2-d}{p}}_{p,\infty}$$.