H{\"o}lder classes in $L^p$ norm on a chord arc curve in $\mathbb R^3$

TL;DR

This paper introduces H{"o}lder classes in the $L^p$ norm on a chord-arc curve in three-dimensional space and establishes approximation theorems for harmonic functions near the curve, with accuracy improving as the neighborhood shrinks.

Contribution

It defines new H{"o}lder classes in the $L^p$ norm on curves in $ eal^3$ and proves approximation theorems for harmonic functions in these classes.

Findings

Established direct approximation theorems in $L^p$ norm

Proved inverse approximation theorems for harmonic functions

Showed approximation accuracy improves with smaller neighborhoods

Abstract

We define H{\"o}lder classes in the $L^p$ norm on a chord-arc curve in $\mathbb{R}^3$ and prove direct and inverse approximation theorems for functions from these classes by functions harmonic in a neighborhood of the curve. The approximation is estimated in the $L^p$ norm, and the smaller the neighborhood, the more accurate the approximation.