On phase-isometries between the unit spheres of the Banach space of continuous real-valued functions

For a locally compact Hausdorff space $L$, we denote by $C_0(L,\mathbb{R})$ the Banach space of all continuous real-valued functions on $L$ vanishing at infinity, endowed with the supremum norm. In this paper, we prove that every surjective phase-isometry $T\text{\hspace{0.17em}⁣}:S(C_0(X,\mathbb{R}))\to S(C_0(Y,\mathbb{R}))$ between the unit spheres of $C_0(X,\mathbb{R})$ and $C_0(Y,\mathbb{R})$ is a variant of a weighted composition operator in the following sense: there exist a function $\epsilon \text{\hspace{0.17em}⁣}:S(C_0(X,\mathbb{R}))\to \{-1,1\}$,a continuous function $\alpha \text{\hspace{0.17em}⁣}:Y\to \{-1,1\}$ and a homeomorphism $\sigma \text{\hspace{0.17em}⁣}:Y\to X$ such that $T(f)(q)=\epsilon (f)\alpha (q)f(\sigma (q))$ for every $f\in S(C_0(X,\mathbb{R}))$ and $q\in Y$.