The relationship between wavelets and smoothness spaces is characterized by "Jackson" and "Bernstein" inequalities. Jackson inequalities look something like $$||f-\pi_k f||_{L_p} \le C_1(k,p) |f|_{\text{smoothness seminorm}},$$ where $\pi_k$ is a projector onto the $k$'th level approximation space (space spanned by all wevelets wavelets up to resolution level $k$) and $1 \le p \le \infty$.
Bernstein inequalities look something like $$|u|_{\text{smoothness seminorm}} \le C_2(k,p)||u|| _{L_p},$$ where $u$ is any function in the $k$'th level approximation space.
The exact results vary based on the domain, choice of wavelets, and smoothness space of interest. Usually $C_1 \approx 2^{-k \alpha}$, d}$, where $\alpha$ d$ is the level of smoothness in the seminorm, and $C_2 \approx 2^{k/p}$, but you'll have to look to the literature for your situation. These sort of results are often proved by breaking space up into a bunch of dyadic cubes and then applying polynomial approximation theory on them.
For a well written introduction, see DeVore's explanatory paper.
