  The aim of this paper is to establish an abstract theory based on the
so-called fractional-maximal distribution functions (FMDs). From the rough
ideas introduced in~\cite{AM2007}, we develop and prove some abstract results
related to the level-set inequalities and norm-comparisons by using the
language of such FMDs. Particularly interesting is the applicability of our
approach that has been shown in regularity and Calder\'on-Zygmund type
estimates. In this paper, due to our research experience, we will establish
global regularity estimates for two types of general quasilinear problems
(problems with divergence form and double obstacles), via fractional-maximal
operators and FMDs. The range of applications of these abstract results is
large. Apart from these two examples of the regularity theory for elliptic
equations discussed, it is also promising to indicate further possible
applications of our approach for other special topics.
