Lipschitz $(p,\sigma ,q,\eta )$-dominated operators

Given $1\leq p,q \lt \infty $ and $0\leq \sigma ,\eta \lt 1$ such that $\frac{1-\sigma}{p}+\frac{1-\eta }{q}\leq 1,$ we continue the study of the Banach Lipschitz ideal of $(p,\sigma ,q,\eta )$-dominated operators initiated by Saleh. Among other results, we introduce an associated Lipschitz tensor norm, implying that it is a maximal Banach Lipschitz operator ideal. Also, we show that this ideal extends the Banach operator ideal of $(p,\sigma ,q,\eta )$-dominated operators. Finally, we exhibit other Lipschitz operator ideals which also extend this operator ideal and we compare them with each other.