A note on disjointness preserving linear operators between vector-valued Lipschitz function spaces

For a given compact metric space $(X,d)$, a Banach space $E$ and a real number $\al\in (0, 1)$, let $\Lip_\al(X,E)$ be the space of all functions $f : X \rightarrow E$ for which $$p_\al(f) = \sup\left\{\frac{\|f(x) - f(y)\| }{d^\al(x, y)}: x, y\in X, x\neq y\right\} \infty,$$ endowed with norm $\|f\|_\al =\|f\|_X + p_\al(f)$, where $\|f\|_X=\sup\{\|f(x)\| : x\in X\}$. The little Lipschitz space $\lip_\al (X,E)$ is the closed subspace of $\Lip_\al (X,E)$ formed by all those functions $f$ such that $ \frac{\|f(x)-f(y)\|}{d^\al(x,y)}\rightarrow 0$ as $d(x,y)\rightarrow 0$. A linear operator $T: \Lip_\al(X,E)\rightarrow \Lip_\al(Y,F)$ is said to be disjointness preserving, if $\|f(x)\|\|g(x)\|=0$ for all $x\in X$ implies $\|Tf(y)\|\|Tg(y)\|=0$ for all $y\in Y$, whenever $f,g\in\Lip_\al(X,E)$. In this note we present conditions under which every bounded disjointness preserving linear operator between vector-valued Lipschitz function spaces could be a weighted composition operator.