Sharp Estimates and Extrapolation for Multilinear Weight Classes

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The subject of this thesis is the study of the multilinear Muckenhoupt weight classes and the quantitative boundedness of operators with respect to these weights in both the scalar-valued and the vector-valued setting. This includes the study of multisublinear Hardy-Littlewood maximal operators, sparse forms, and multilinear Rubio de Francia extrapolation methods.