An extension of harmonic functions along fixed direction

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 $y$.