关于某些函数空间的Tingley问题的一个变型
On a variant of Tingley's problem for some function spaces
摘要：
设$(Ω, \mathfrak{A}, μ)$和$(Γ, \mathfrak{B}, ν)$是两个任意测度空间，$p\in [1,\infty]$ .设置$$L^p(μ)_+^\mathrm{sp}:= \{f\in L^p(μ): \|f\|_p =1; f\geq 0\ μ\text{-a.e.} \}$$，即，$L^p(μ)$的单位球面的正部分。我们证明了每一个保持度量的双射$Φ: L^p(μ)_+^\mathrm{sp} \to L^p(ν)_+^\mathrm{sp}$都可以（必然唯一地）扩展为从$L^p(μ)$到$L^p(ν)$的等距阶同构。一种兰珀蒂形式，即，当$(Γ, \mathfrak{B}, ν)$是可局部化的（特别是当它是$σ$有限的）时，提供了$Φ$的加权合成类形式。另一方面，我们证明了对于紧Hausdorff空间$X$和$Y$，如果$Φ$是从$C(X)$的单位球面的正部分到$C(Y)$的单位球面的保度量双射，则存在一个同胚$τ:Y\to X$满足$Φ(f)(y) = f(τ(y))$（$f\in C(X)_+^\mathrm{sp}; y\in Y$）.
英文摘要：
Let $(Ω, \mathfrak{A}, μ)$ and $(Γ, \mathfrak{B}, ν)$ be two arbitrary measure spaces, and $p\in [1,\infty]$. Set $$L^p(μ)_+^\mathrm{sp}:= \{f\in L^p(μ): \|f\|_p =1; f\geq 0\ μ\text{-a.e.} \}$$ i.e., the positive part of the unit sphere of $L^p(μ)$. We show that every metric preserving bijection $Φ: L^p(μ)_+^\mathrm{sp} \to L^p(ν)_+^\mathrm{sp}$ can be extended (necessarily uniquely) to an isometric order isomorphism from $L^p(μ)$ onto $L^p(ν)$. A Lamperti form, i.e., a weighted composition like form, of $Φ$ is provided, when $(Γ, \mathfrak{B}, ν)$ is localizable (in particular, when it is $σ$-finite). On the other hand, we show that for compact Hausdorff spaces $X$ and $Y$, if $Φ$ is a metric preserving bijection from the positive part of the unit sphere of $C(X)$ to that of $C(Y)$, then there is a homeomorphism $τ:Y\to X$ satisfying $Φ(f)(y) = f(τ(y))$ ($f\in C(X)_+^\mathrm{sp}; y\in Y$).
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