An extension of harmonic functions along fixed direction

TL;DR

This paper investigates conditions under which harmonic functions defined on a product domain can be extended along a fixed direction, generalizing classical extension results in harmonic analysis.

Contribution

It extends the theory of harmonic function extension along fixed directions to broader classes of domains and sets, providing new criteria for such extensions.

Findings

Harmonic functions extend along fixed directions under specific set conditions.

Extension criteria depend on the properties of the set E in the domain D.

Results generalize classical harmonic extension theorems.

Abstract

Let a function $u(x,y)$ be harmonic in the domain $$ D\times V_r=D\times \{y\in \mathbb{R}^m: |y|<r\}\subset \mathbb{R}^n\times \mathbb{R}^m $$ and for each fixed point $x^0$ from some a set $E\subset D$, %which is not embedded in countable association of $N$-sets of $ Lh_0(D)$, the function $u(x^0,y)$, as a function of variable $y$, can be extended to a harmonic function on the whole $\mathbb{R}^m$. Then $u(x,y)$ harmonically extends to the domain $D\times \mathbb{R}^m$ as a function of variables $x$ and $y$.