The ideal of Lipschitz classical p-compact operators and its injective hull

We introduce and investigate the injective hull of the strongly Lipschitz classical p-compact operator ideal defined between a pointed metric space and a Banach space. As an application we extend some characterizations of the injective hull of the strongly Lipschitz classical p-compact from the linear case to the Lipschitz case. Also, we introduce the ideal of Lipschitz unconditionally quasi p-nuclear operators between pointed metric spaces and show that it coincides with the Lipschitz injective hull of the ideal of Lipschitz classical p-compact operators.