New fractional type weights and the boundedness of some operators

Two classes of fractional type variable weights are established in this paper. The first kind of weights ${A_{\vec p( \cdot ),q( \cdot )}}$ are variable multiple weights, which are characterized by the weighted variable boundedness of multilinear fractional type operators, called multilinear Hardy--Littlewood--Sobolev theorem on weighted variable Lebesgue spaces. Meanwhile, the weighted variable boundedness for the commutators of multilinear fractional type operators are also obtained. This generalizes some known work, such as Moen (2009), Bernardis--Dalmasso--Pradolini (2014), and Cruz-Uribe--Guzm\'an (2020). Another class of weights ${{\mathbb{A}}_{p( \cdot ),q(\cdot)}}$ are variable matrix weights that also characterized by certain fractional type operators. This generalize some previous results on matrix weights ${{\mathbb{A}}_{p( \cdot )}}$.