Estimates projector norms on hyperplanes in rearrangement-invariant spaces (Q1595530)

Let \(E\) be a given rearrangement-invariant (RI) space and let \(\Phi\) denotes the set of special functions \(\varphi(t)\) on \([0,1]\). \(Px(t)\) is an orthogonal projector and it is well known that \(\|P\|_E= 1\) for any RI space \(E\), and \(I-P\) is an orthogoal projector onto the hyperplane. The first part of the paper contains a very important theorem which states that the condition \[ x\prec y\Rightarrow\|x\|_E\leq\|y\|_E\tag{2} \] is equivalent to the condition that \(E\) is interpolational with respect to \(L_1\) and \(L_\infty\) with interpolation constant \(1\). The main results of this paper are the following conditions: \(\|I-P\|_{\Lambda(\varphi)}> 1+{1\over 24}\) for any function \(\varphi\in \Phi\) (Theorem 3), \(\|I-P\|_{\Lambda(\varphi_0)}< 1+\varepsilon\) for any \(\varepsilon> 0\) and \(\varphi_0(t)= t^{{1\over 2}}\) (Theorem 4), and the very valuable implication \[ \|I-P\|_{\Lambda(\varphi)}\geq 2-\varepsilon\Rightarrow \max_{0\leq t\leq 1} (\varphi(t)- t)\leq 2\varepsilon\quad\text{or} \quad \max_{2\varepsilon\leq t\leq 1} (1- \varphi(t))\leq 5\varepsilon \] (for \(\varphi\in\Phi\), \(0<\varepsilon< {1\over 5}\)). Very interesting is Theorem 6 which characterizes \(L_1\) and \(L_\infty\) in the class of RI spaces satisfying the condition (2).