论文标题
在某些功能空间的Tingley问题的变体上
On a variant of Tingley's problem for some function spaces
论文作者
论文摘要
令$（ω，\ mathfrak {a}，μ）和$（γ，\ mathfrak {b}，ν）$为两个任意度量空间，而$ p \ in [1，\ infty] $。集$$ l^p（μ）_+^\ mathrm {sp}：= \ {f \ in l^p（μ）：\ | f \ | _p = 1; f \ geq 0 \μ\ text {-a.e。} \} $$，即$ l^p（μ）$的单位球的正部分。我们表明，每个保存双重保留的指标$φ：l^p（μ）_+^\ mathrm {sp} \ to l^p（ν）_+^\ mathrm {sp} $可以扩展（必然是独特的），从$ l^p（μ）$ l^p（un）$ l^p（n p（uniove）$ l^p（n limemorplist）。提供$φ$的lamperti表单，即加权形式，当$（γ，\ mathfrak {b}，ν）$提供$φ$的形式（尤其是$σ$ -finite时）。 另一方面，我们表明，对于紧凑的Hausdorff Spaces $ x $和$ y $，如果$φ$是从$ c（x）$的$ c（y）$的单位球的积极部分中保存的二号双射击，则有一个同型$τ： c（x）_+^\ mathrm {sp}; y \ in y $）。
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 $Φ(f)(y) = f(τ(y))$ ($f\in C(X)_+^\mathrm{sp}; y\in Y$).
