Continuous Extension of Functions from a Segment to Functions in $\mathbb{R}^n$ with Zero Ball Means

Abstract

Let $\mathbb{R}^n$ be a 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 domain G does not contain such balls, 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 the extension problem.

1) Which conditions guarantee the extension of a continuous function defined on E to a continuous function of class $V_r(\mathbb{R}^n)$ defined on the whole $\mathbb{R}^n$?

2) If the above extension exists, obtain growth estimates of the extended 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 an 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.