Extension of surjective isometries between unit spheres of $\ell^p$-direct sums

Abstract

We study surjective isometries between the unit spheres of Banach spaces of the form $A\oplus_p B$, where $A$ is nonzero and strictly convex, $1<p<\infty$, and every unit vector of $B\neq\{0\}$ has at least two distinct unit vectors at distance $2$. For two direct sums $A_i\oplus_{p_i}B_i$, $i=1,2$, satisfying these hypotheses, we prove that every surjective isometry $$Δ:S_{A_1\oplus_{p_1}B_1}\to S_{A_2\oplus_{p_2} B_2}$$ between their unit spheres forces equality of the direct-sum exponents $p_i$ and extends uniquely to a surjective real-linear isometry $$\widetildeΔ:A_1\oplus_p B_1\to A_2\oplus_p B_2$$ preserving the corresponding components, where $p:=p_1=p_2$. The proof identifies the component unit spheres through diametral sets and distance formulas, and then constructs isometries between the component unit balls. As a principal application, we obtain extension and representation results for absolutely continuous functions equipped with norms combining evaluation at a fixed point and the $L^1$-norm of the derivative. We also discuss Lipschitz and differentiable function spaces, distinguishing the consequences of our theorem from those already covered by existing extension results. Endpoint conclusions are derived separately, including the Mazur-Ulam property for the real absolutely continuous space with the sum norm.