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

Question

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?

Answer 1

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