A Sharp Estimate About The Constant C In The Hardy-Littlewood Maximal Inequality

Hardy-Littlewood maximal operators M and M were first introduced by Hardy and Littlewood. These tools are useful in harmonic analysis. While, sharp estimate about the constant C in the Hardy-Littlewood maximal function is more useful.The aim of this paper is to solve the following three questions. Firstly, introduce the Lortenz L(p,q) spaces and Hardy-Littlewood maximal inequalities on it. Then, using the relation between L(p,q) spaces and L~p spaces, we obtain the Hardy-Littlewood maximal inequalities on L~p spaces. Lastly, researching the special function , give the lower bound of the constant C, when 1 < p <∞.