Abstract
The study introduces different categories of Lipschitz operators linked with weakly $p$-compact and unconditionally $p$-compact sets. It explores some properties of these operator classes derived from linear operators associated with these sets and examines their interconnections. Additionally, it denotes that these classes are extensions of the related linear operators. Moreover, the study evaluates the concept of majorization by scrutinizing both newly obtained and pre-existing results and draws some conclusions based on these findings. The primary method used to obtain the results in the study is the linearization of Lipschitz operators through the Lipschitz-free space constructed over a pointed metric space.
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