Hölder classes in the 𝐿^{𝑝} norm on a chord-arc curve in ℝ³

Abstract

The Hölder classes L p α ( L ) L_p^{\alpha } (L) in the L p ( L ) L_p(L) norm on a chord-arc curve L L in R 3 \mathbb {R}^3 are defined and direct and inverse approximation theorems are proved for functions from these classes by functions harmonic in a neighborhood of the curve. The approximation is estimated in the L p ( L ) L^p(L) norm, the direct theorem is proved for a certain subclass of L p α ( L ) L^\alpha _p(L) and the inverse theorem covers the entire Hölder class.