Continuous extension of functions from a segment to functions in $\mathbb{R}^n$ with zero ball means

Let $\mathbb{R}^n$ be an Euclidean space of dimension $n\geq 2$. For a domain $G\subset \mathbb{R}^n$, we denote by $V_r(G)$ the set of functions $f\in L_{\mathrm{loc}}(G)$ having zero integrals over all closed balls of radius $r$ contained in $G$ (if the domain $G$ does not contain such balls, then we set $V_r(G)=L_{\mathrm{loc}}(G)$). Let $E$ be a nonempty subset of $\mathbb{R}^n$. In this paper we study the following questions related to with the extension problem.
1) Under what conditions given on $E$ continuous function can be extended to the whole space $\mathbb{R}^n$ to a continuous function of class $V_r(\mathbb{R}^n)$?
2) If the above extension exists, obtain growth estimates continued function at infinity.
Theorem 1 of this paper shows that for a wide class of continuous functions on segment $E$ defined in terms of the modulus of continuity there exists extension to a bounded function of class $(V_r\cap C)(\mathbb{R}^n)$ regardless of the length of segment $E$. A similar result is not true for open sets $E$ with a diameter greater than $2r$ even without conditions for extension growth. Theorem 1 also contains an estimate of the velocity decrease of the extended function at infinity in directions orthogonal to the segment $E$.
As Theorem 2 shows, in the case of a space with odd dimension $n$ Theorem 1 holds for any function continuous on $E$ with another growth estimate. The method of proving Theorems 1 and 2 allows one to obtain similar results for functions with zero integrals over all spheres of fixed radius (in this case, an analog of Theorem 2 holds for spaces with even dimension).