On the maximal function for rotation invariant measures in ${\displaystyle {ℝ}^n}$

Given a positive measure μ in ${\displaystyle {ℝ}^n}$, there is a natural variant of the noncentered Hardy-Littlewood maximal operator ${\displaystyle M_{\mu }f\left(x\right)={\supset }_{x\in B}\frac{1}{\mu }\left(B\right){ʃ}_B\left|f\right|d\mu }$, where the supremum is taken over all balls containing the point x. In this paper we restrict our attention to rotation invariant, strictly positive measures μ in ${\displaystyle {ℝ}^n}$. We give some necessary and sufficient conditions for ${\displaystyle M_{\mu }}$ to be bounded from ${\displaystyle L^1(d\mu )}$ to ${\displaystyle L^{1,\infty }(d\mu )}$.