K-monotone weighted couples of Banach lattices

We give a criterion for the uniform relative K-monotonicity of weighted couples $(X,X(w_1))$ and $(X,X(w_2))$, where $X$ is some Banach lattice of measurable functions with the Fatou property while $w_1$ and $w_2$ are weight functions. Using the criterion, we prove some corollaries for sequence spaces and arbitrary Banach lattices.