Constructive description of Hölder spaces on a chord arc curve in $\mathbb R^3$

Let $L$ be a chord-arc curve in $\mathbb R^3$. We introduce a functional class $H^{r+\omega}(L)$ where a modulus of continuity $\omega$ satisfies the Dini condition and $r\geq1$. We define neighborhoods of $L$ $\Omega_\delta(L)=\bigcup_M\in L B_\delta(M)$, $B_\delta(M)=\big\{X\in \mathbb R^3\,:\, \|XM\|<\delta\big\}$ and set $\mathrm{Harm}\,\Omega_\delta(L)$ for harmonic functions in $\Omega_\delta(L)$. The Theorem 1 states that if $f\in H^{\omega+r}(L)$ then there exist functions $v_\delta \in \mathrm{Harm}\,\Omega_\delta(L)$ such that $\big|f(X)-v_\delta(M)\big|\leq c_{f} \delta^r \omega(\delta)$, $M\in L$, and $\big|\partial^\alpha v_\delta(M)\big|\leq c_{f}\frac{\omega(\delta)}{\delta}$, $M\in \Omega_\delta(L)$, $|\alpha|=r+1$. The Theorem 2 states that if a function $f$ defined on $L$ satisfies claim of Theorem 1 then $f\in H^{\omega+r}(L)$.