Abstract
Let 
be a family of normalized (quasi-) Banach function spaces and µ a doubling measure. In Section 1, we give a brief introduction of this paper. In Section 2, we introduce some basic definitions about this article and extend the definitions of some maximal functions and some notions about them with Lebesgue
measure to the case of generalized measure µ in Rn . In Section 3, we consider
the inequality in [1,Theorem1.1] with doubling measure and in (quasi-) Banach
function spaces. We find that the properties of Banach function spaces and the
condition are used to get the conclusion, which are just the same as the
conditions of the estimate of the BMO (dµ) functions in [2,Theorem1.1]. Particularly, the condition could be seen as a separation condition in Banach
function spaces after using Calderon-Zygmund decomposition and may be available to solve other similar problems. Then we apply it to the localized Lorentz
spaces with 
. This section is the main part of the article. In section4,
we focused on the application of the Variable exponent Lp -spaces.
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