On the Besicovitch property for parabolic balls (Q1852458)

Let \(U = \{U_{\alpha} : \alpha \in \mathbb{R}^{+} \}\) be an increasing family of bounded and open neighborhoods of the origin in \(\mathbb{R}^n\). We say that the family \(U\) satisfies the Besicovitch property if there exists a constant \(C\), depending only on \(U\) and \(n\), such that for each family \(\{x_i + U_{\alpha_i} : i \in I \}\) of translated members of \(U\), such that \(\{ x_i : i \in I \}\) is a bounded set, there is a subset \(J\) of \(I\) such that \[ \{x_i : i \in I \} \subset \bigcup_{j \in J} ( x_j + U_{\alpha_j}) \quad (\text{ covering property}), \] and \[ \sum_{j \in J} \chi_{ \{ x_j + U_{\alpha_j} \} } (x) \leq C \quad ( \text{ overlapping property}). \] Let \(p \geq 1\) and \(0< a_1 \leq a_2 \leq \cdots \leq a_n\) and consider the following solids of \(\mathbb{R}^n\) given by \[ B_r = \left\{ ( x_1, \ldots , x_n) \in \mathbb{R}^n : \left( \frac{|x_1 |}{r^{a_1}} \right)^p + \cdots + \left( \frac{|x_n |}{r^{a_n}} \right)^p <1 \right\}. \] The author proves the following: Let \(U\) be the family of all \(B_r\), \(r>0\). Then \(U\) satisfies the Besicovitch property if and only if \(p \geq a_n / a_1 \).