Functions with Zero Averages Over Balls on Subsets of the Space ℝⁿ

Abstract

Let r > 0 be a fixed number and let $\mathcal{U}$ be a domain in ℝⁿ containing a closed ball of radius r. Denote by V _r $\mathcal{U}$ the set of functions f ∈ L _loc $\mathcal{U}$ with zero averages over all closed balls of radius r lying in $\mathcal{U}$. For s ∈ ℕ₊ or s = ∞ we set $V_r^s \left( \mathcal{U} \right) = \left( {V_r \cap C^s } \right)\left( \mathcal{U} \right)$.