Orbits in symmetric spaces (Q1028339)

Either of the spaces \(L^1(0,1)\) or \(L^1(0,\infty)\) of Lebesgue integrable functions is denoted by \(L_1\), and either of the spaces \(L^\infty(0,1)\) or \(L^\infty(0,\infty)\) of essentially bounded functions is denoted by \(L_\infty\), unless specifically stated. With \(m\) denoting Lebesgue measure, the distribution function \(\lambda_f\) and non-increasing rearrangement \(f^*\) of a measurable function \(f\) are defined by \(\lambda_f(s)= m(\{x:|f(x)|> s\})\), \(s> 0\); \(f^*(t)= \text{inf}\{s: \lambda_f(s)\leq t\}\). The space \(\Sigma\) of absolute contractions on \(L_1+L_\infty\) is defined by \[ \Sigma= \{T: L_1+ L_\infty\to L_1+L_\infty:\| T\|_{1\to 1}\leq 1;\| T\|_{\infty\to\infty}\leq 1\}, \] where \(\| T\|_{p\to p}= \sup\{\| T(f)\|_p/\| f\|_p, f\in L_p\}\), and the subspaces \(\Sigma_+\) and \(\Sigma'\) of \(\Sigma\) are defined by \(\Sigma_+= \{T\in\Sigma: T(f)\geq 0\) for \(f\geq 0\}\); \[ \Sigma'= \Biggl\{T\in \Sigma_+: \int^1_0 T(f)(t)\,dt= \int^1_0 f(t)\,dt,\,f\geq 0;\,T(1)= 1\Biggr\}. \] The Banach function space \(E\) is said to be fully symmetric if and only if \(x\in E\), \(y\in L_1+ L_\infty\) and \[ \int^t_0 y^*(s)\,ds\leq \int^t_0 x^*(s)\,ds,\;t> 0,\tag{\(*\)} \] imply that \(y\in E\) and \(\| y\|_E\leq\| x\|_E\). If \(x\in E\), then the orbits of \(x\) in the spaces \(\Sigma\), \(\Sigma_+\), \(\Sigma'\) are denoted, respectively, by \(\Omega(x)\), \(\Omega_+(x)\), \(\Omega'(x)\), and have representations in terms of elements satisfying \((*)\), according to results of Calderón-Mityagin stated in [\textit{A. P. Calderón}, Stud. Math. 26, 273--299 (1966; Zbl 0149.09203)]. The convex hulls of the extreme points of \(\Omega(x)\), \(\Omega_+(x)\), \(\Omega'(x)\) are denoted, respectively, by \(Q(x)\), \(Q_+(x)\), \(Q'(x)\). The main results of this paper include: Let \(\sigma_b\) be the translation operator and let \(\varphi\) be defined by \(\varphi(x)= \lim_{s\to\infty} s^{-1}\|\sigma_s(x^*)\|_E\). (1) If \(x\in E= E(0,1)\), then \(\Omega'(x)= Q'(x)\) if and only if \(\varphi(x)= 0\); (2) if \(x\in E= E(0,1)\), and if \(\varphi(x)= 0\), then \(\Omega_+(x)= Q_+(x)\); (3) if \(x\in E= E(0,\infty)\), \(E\not\subset L_1\), and if \(\varphi(x)= 0\), then \(\Omega_+(x)= Q_+(x)\).