Weak type limiting estimates for maximal functions

We consider weak type limiting estimates for non-tangential maximal function N_α and radial maximal function R when λ→0．To be precise, we demonstrate that, for any given 0 \text＜ \alpha \text＜ 2^1/n - 1 and any 0 \text≤ \rmf \in L^1\left( \mathbbR^n \right), there exists a constant of 1\text＜N\text＜\infty such that \dfracV_n\varPhi \left( \alpha \right)2N^n\left\| f \right\|_1 \text≤ \mathop \rmlim\limits_\lambda \to 0 \lambda \left| \left\ x \in \mathbbR^n:N_\alpha \left( f \right)\left( x \right) \text＞ \lambda \right\ \right| \text≤ V_n\left( 2 - \dfrac\varPhi \left( \alpha \right)2N^n \right)\left\| f \right\|_1, \dfracV_n2N^n\left\| f \right\|_1 \text≤ \mathop \rmlim\limits_\lambda \to 0 \lambda \left| \left\ x \in \rmR^n:\mathbbR\left( f \right)\left( x \right) \text＞ \lambda \right\ \right|\text≤ V_n\left( 2 - \dfrac12N^n \right)\left\| f \right\|_1, where V_n denotes volume of a unit ball of \mathbbR^n．