Fractional maximal functions in Lorentz spaces

This thesis consist of two chapters. The first chapter devoted to the fundamentalconcepts and main theorems.In this thesis, fractional maximal functions in Lorentz Spaces are studied. Acomplete characterization of the weights for which the inequalities in question are valid isgiven in the study. Besides that the one weight problem in Lorentz Spaces are solved forthe strong type inequalities.Then this study devoted to the solution of this problem in theweak type setting.The class of couples ( w, β ) for which the two weight inequalitiy for the operatorM υ on the set of all characteristic functions of all measurable subsets is larger then thepssclass of those couples for which this inequalitiy holds ps on the whole space Lw . Aquestion arises whether the validity of the weak type inequalitiy for M υ on the set ofcharacteristic function implies its boundedness on this set. The answer is negative as anexample given.Keywords: Lorentz spaces, Maximal Functions.