Variants of 2-local maps on function algebras

We study several variants of 2-local isometries (or algebra isomorphisms) on some function algebras, e.g., Lipschitz algebras, algebras of differential functions, algebras of absolutely continuous functions and algebras of continuous functions with bounded variation. A typical result is this: if \(\phi \) $\varphi$ is surjective map between function algebra mentioned above with the property that for any pair f, g there is an algebra isomorphism \(\phi _{f,g}\) ${\varphi }_{f,g}$ such that \(\phi (f)\phi (g)=\phi _{f,g}(fg)\) $\varphi (f)\varphi (g)={\varphi }_{f,g}(fg)$ , then \(\phi \) $\varphi$ can be written as a weighted composition operator.