91The one-sided classical fractional maximal function and the one-sided generalized fractional Morrey-type space

Recently, Shi and Fu defined a new Morrey-type space to prove some boundedness property of one-sided maximal functions. Following the work of Shi and Fu, Barman and Haloi obtained a boundedness property for the one-sided generalized maximal function on the space defined by Shi and Fu for a couple of weight functions in scalar setting. Barman et al. obtained a vector-valued analogue of the result of Shi and Fu. In this paper, we derive some norm inequalities for the one-sided classical fractional maximal function on the one-sided generalized fractional Morrey-type space. We also derive some weighted norm inequalities similar to the well-known Fefferman–Stein inequality for the one-sided classical fractional maximal function on the one-sided generalized fractional Morrey-type space.