Hölder classes in the $L^p$ norm on a chord-arc curve in $\mathbb R^3$

On a chord-arc curve $L$ in $\mathbb R^3$, the function class $L_p^{\alpha}\left(L\right)$ is introduced. This class consists of functions that satisfy an $\alpha$-Hölder type condition in the $L^p\left(L\right)$-norm with respect to the arc length on $L$. Our purpose is to describe the functions in $L_p^{\alpha}\left(L\right)$ in terms of the rate of approximation by harmonic functions defined in shrinking neighborhoods of the curve. A statement about possible rate of approximation is proved for a certain subclass of $L_p^{\alpha}\left(L\right)$, a statement ensuring the smootheness of a function approximable with the rate in question is proved for the entire class $L_p^{\alpha}\left(L\right)$.