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

Level-set inequalities on fractional maximal distribution functions and applications to regularity theory