A variant of Tingley's problem on ordered Banach spaces of absolutely continuous functions

For each $1\le p\le \mathrm{\infty }$ and $j=1,2$, let $A{C}^p({\mathrm{\Omega }}_j)$ denote the Banach space of complex-valued absolutely continuous functions on a closed unit interval ${\mathrm{\Omega }}_j=[x_j,x_j+1]$. We equip $A{C}^p({\mathrm{\Omega }}_j)$ with the $p$–norm $\mathrm{\parallel }f{\mathrm{\parallel }}_{AC,p}$, and the order ${\ge }_{AC}$ defined by $f(x_j)\ge 0$ and $f^{\mathrm{\prime }}\ge 0$ a.e. Set $$S(A{C}^p({\mathrm{\Omega }}_j){)}^{+}=\{f\in A{C}^p({\mathrm{\Omega }}_j):\mathrm{\parallel }f{\mathrm{\parallel }}_{AC,p}=1,f{\ge }_{AC}0\}\mathrm{.}$$ We prove that, for each $1\le p\le \mathrm{\infty }$, every surjective isometry $S(A{C}^p({\mathrm{\Omega }}_1){)}^{+}\to S(A{C}^p({\mathrm{\Omega }}_2){)}^{+}$ extends uniquely to a complex–linear isometric order isomorphism from $A{C}^p({\mathrm{\Omega }}_1)$ onto $A{C}^p({\mathrm{\Omega }}_2)$. As an application, we obtain a corresponding extension theorem for surjective phase–isometries.