The class of Muckenhoupt weights has been extended to the multilinear setting in several contexts of real and Fourier analysis. In this work, we provide new characterizations of these multilinear weight classes through a unifying principle based on pointwise, rather than averaged, evaluation of the weights.
. We introduce the concept of the weighted weak group inverse for elements in a ring with involution. This notion naturally extends the weak group inverse for complex matrices and the weighted weak group inverse for Hilbert operators. We characterize this generalized inverse through a novel decomposition (referred to as the w-group decomposition) that involves weighted group inverses and nilpotent elements. Additionally, the interrelationships among weighted weak group inverses, weighted Drazin inverses, and weighted core-EP core inverses are thoroughly investigated.
