Two presentations of a weak type inequality for geometric maximal operators

Let Φ : [ 0 , ∞ ) → [ 0 , ∞ ) {\Phi:[0,\infty)\rightarrow[0,\infty)} be a Young’s function satisfying the Δ 2 {\Delta_{2}} -condition and let M ℬ {M_{\mathcal{B}}} be the geometric maximal operator associated to a homothecy invariant basis ℬ {\mathcal{B}} acting on measurable functions on ℝ n {\mathbb{R}^{n}} . Let Q be the unit cube in ℝ n {\mathbb{R}^{n}} and let L Φ ⁢ ( Q ) {L^{\Phi}(Q)} be the Orlicz space associated to Φ with the norm given by ∥ f ∥ L Φ ⁢ ( Q ) := inf ⁡ { c > 0 : ∫ Q Φ ⁢ ( | f | c ) ≤ 1 } . \|f\|_{L^{\Phi}(Q)}:=\inf\Biggl{\{}c>0:\int_{Q}\Phi\bigg{(}\frac{|f|}{c}\bigg{% )}\leq 1\Bigg{\}}. We show that M ℬ {M_{\mathcal{B}}} satisfies the weak type estimate | { x ∈ ℝ n : M ℬ ⁢ f ⁢ ( x ) > α } | ≤ C 1 ⁢ ∫ ℝ n Φ ⁢ ( | f | α ) |\{x\in\mathbb{R}^{n}:M_{\mathcal{B}}\kern 1.422638ptf(x)>\alpha\}|\leq C_{1}% \int_{\mathbb{R}^{n}}\Phi\bigg{(}\frac{|f|}{\alpha}\bigg{)} for all measurable functions f on ℝ n {\mathbb{R}^{n}} and α > 0 {\alpha>0} if and only if M ℬ {M_{\mathcal{B}}} satisfies the weak type estimate | { x ∈ Q : M ℬ ⁢ f ⁢ ( x ) > α } | ≤ C 2 ⁢ ∥ f ∥ L Φ ⁢ ( Q ) α |\{x\in Q:M_{\mathcal{B}}\kern 1.422638ptf(x)>\alpha\}|\leq C_{2}\frac{\|f\|_{% L^{\Phi}(Q)}}{\alpha} for all measurable functions f supported on Q and α > 0 {\alpha>0} . As a consequence of this equivalence, we prove that if Φ satisfies the above conditions and ℬ {\mathcal{B}} is a homothecy invariant basis differentiating integrals of all measurable functions f on ℝ n {\mathbb{R}^{n}} such that ∫ ℝ n Φ ⁢ ( | f | ) < ∞ {\int_{\mathbb{R}^{n}}\Phi(|f|)<\infty} , then the associated maximal operator M ℬ {M_{\mathcal{B}}} satisfies both of the above weak type estimates.