Haar series and spaces determined by the Paley function
Let \((E,\|.\|_E)\) be a Banach space of measurable functions on the real interval \([0,1]\) with norm \(\|.\|_E\). If \(m\) denotes the Lebesgue measure on \([0,1]\), then the distribution function \(dst_x(u)\), \(u> 0\), of \(x\in E\) is defined by \(dst_x(u)= m(\{t,0\leq t\leq 1:|x(t)|> u\})\). \(E\) is said to be symmetric or rearrangement invariant if \(|x(t)|\leq|y(t)|\), \(t\in [0,1]\), and \(y\in E\) imply that \(x\in E\) and \(\| x\|_E\leq\| y\|_E\); and \(x\), \(y\) are equi-measurable if \(dst_y(u)= dst_x(u)\), \(u> 0\), implies \(\| x\|_E=\| y\|_E\). If \(s_\tau(x)(t)= x(t/\tau)\chi_{(0,\min(1,\tau))}(t)\), where \(\chi_A\) denotes the characteristic function of \(A\), and if \(\gamma_E(t)= (\ln\| s_\tau\|_E)/\ln\tau\), then the Boyd indices of \(E\), \(\alpha_E\), \(\beta_E\), are defined respectively as \(\lim\gamma_E(t)\) (as \(t\to 0\)) and (as \(t\to\infty\)). The Haar functions \(\chi^k_n (t)\) (\(=2^{n/2}\) for \(t\in\Delta^{2k-1}_{n+1}\); \(=-2^{n/2}\) for \(t\in\Delta^{2k}_{n+1}\)) are defined on the system intervals \(\Delta^k_n= ((k-1)2^{-n},k2^{-n})\), \(k= 1,2,\dots, 2^n\), to constitute a complete orthonormal system. If the Paley function \(P(x)\) is defined by \[ P(x)(t)= \Biggl(\sum^\infty_{j=1} (c_j\chi_j(t))^2\Biggr)^{1/2}, \] where \[ x(t)= \sum^\infty_{j=0} c_j\chi_j(t), \] \(\{\chi_j, j= 0,1,2,\dots\}= \{\chi^k_n, k= 1,2,\dots, 2^n, n=1,2,\dots\}\) then \(\| x\|_{P(E)}\) is defined to be \(\| x\|_E\). The main theorems stated in this paper relate especially to the spaces \(P(E)\). In particular, statements indicate that (*1) if \(E\) is symmetric then \(P(E)\) is symmetric if and only if \(E= L_2\) with \[ \| x\|_E= \Biggl(\int^1_0 |x(t)|^2\,dt\Biggr)^{1/2}, \] and (*2) if \(E\) is a symmetric space then \(P(E)\) is symmetric up to the equivalence of norm if and only if \(0<\alpha_E\leq \beta_E< 1\). A further statement indicates the equivalence of norms between \(P(L_\infty)\) and \(\exp(L_2)\).
scientific article; zbMATH DE number 6106509