  We construct an efficient approach to deal with the global regularity
estimates for a class of elliptic double-obstacle problems in Lorentz and
Orlicz spaces. The motivation of this paper comes from the study on an abstract
result in the viewpoint of the fractional maximal distributions and this work
also extends some regularity results proved in \cite{PN_dist} by using the
weighted fractional maximal distributions (WFMDs). We further investigate a
pointwise estimates of the gradient of weak solutions via fractional maximal
operators and Riesz potential of data. Moreover, in the setting of the paper,
we are led to the study of problems with nonlinearity is supposed to be
partially weak BMO condition (is measurable in one fixed variable and only
satisfies locally small-BMO seminorms in the remaining variables).
