On cones of step functions in rearrangement invariant spaces
The author gives conditions on a family of sets \(\{e_k\}\) and on a space \(X\) for which norm equivalence on the cone generated by \(\{\chi_{e_k}\}\) implies norm equivalence. If \(X\) is a rearrangement invariant space (r.i.) of functions on \([0, 1]\), it is known that \(X\) is uniquely determined by the cone of functions \(y\in X\) of the form \[ y(t)= \sum_{k=1}^\infty c_k \chi_{(2^{-k}, 2^{-k+ 1}]} (t), \] where \(c_k\geq 0\), \(c_k\leq c_{k+1}\). If \(\Phi\) is a class of rearrangement invariant spaces and \(X_0\in \Phi\), we say that the system \(\{ e_k \}^\infty_{k =1}\) of subsets of \([0, 1]\) determines \(X_0\) in the class \(\Phi\) if the equivalence of the norms of \(X_0\) and of an arbitrary r.i. space \(X\in \Phi\) on the cone of functions of the form \[ y= \sum_{k=1 }^\infty c_k \chi_{e_k} \qquad (c_k\geq 0,\;c_k\leq c_{k+1}) \] implies that \(X= X_0\) (isomorphism). The author works with the rearrangement invariant Lorentz space \(\Lambda (\varphi)\), and the Marcinkiewicz space, \(M(\varphi)\): \[ |x|_{\Lambda (\varphi)}= \int_0^1 x^* (s) d\varphi (s), \qquad |x|_{M (\varphi)}= \sup_{0< t\leq 1} \int_0^t x^* (s) ds/ \varphi (t), \] where \(\varphi(t)\) is an increasing concave function on \([0, 1]\), \(\varphi (0)=0\). Take a collection of sets \(\{e_k\}\), subsets of \([0, 1]\), whose measures \(\mu (e_k)= a_k\) satisfy \(0< a_{k+1}\leq a_k\), \(\sum_{k=1 }^\infty a_k\leq 1\), and \(\sup_{k=1, 2,\dots} {1\over {a_k}} \sum_{n=k}^\infty a_n< +\infty\). Let \(M\) denote the dilation function of \(\varphi\), and \(\gamma_\varphi\), \(\delta_\varphi\) the upper and lower dilation exponents of \(\varphi\), where \[ M_\varphi (t) = \sup_{0< s\leq \min (1, 1/t)} \varphi (st)/ \varphi (s),\qquad t>0, \] \[ \gamma_\varphi = \lim_{t\to 0+} \ln M_\varphi (t)/\ln (t), \qquad \delta_\varphi = \lim_{t\to \infty} \ln M_\varphi (t)/ \ln (t). \] In the cases considered by the author, \(\varphi\) can be taken as the fundamental function of the r.i. space \(X\), namely, \(\varphi_X (t)= |\chi_{(0, t)} |_X\), which is e.g. \(t^{1/p}\) for \(L^p\) and is \(\varphi\) for \(\Lambda (\varphi)\), and is \(\widehat {\varphi} (t)= t/ \varphi(t)\) for \(M(\varphi)\). Finally, recall that a space \(X\) is an intermediate space between \(X_0\) and \(X_1\) if every linear operator bounded over \(X_0\) and over \(X_1\) is bounded over \(X\). The author's main result is that if \(\psi(t)\) is an increasing concave function on \([0,1]\), with \(0< \gamma_\psi\leq \delta_\psi< 1\), and if \(X\) is an intermediate space between \(\Lambda (\psi)\) and \(M (\widehat {\psi})\) and if \(\Delta_1= \{e_k\}^\infty_{k =1}\), \(\Delta_2= \{f_k\}^\infty_{k =1}\), where the measures of the sets in \(\Delta_1\), \(\Delta_2\) satisfy the two conditions on measures given above, then equivalence of the norms on all sequences of nonnegative numbers implies equivalence of the norms for all sequences. By applying this result to \(f_k= (2^{-k}, 2^{-k +1}]\), he shows that if a system of sets \(\{e_k \}^\infty_{k=1}\) satisfies the two conditions on its measure and if \(X_0\) is an intermediate space between \(\Lambda (\psi)\) and \(M(\widehat {\psi})\), then the system \(\{e_k \}^\infty_{k =1}\) determines \(X_0\) with respect to all spaces intermediate between \(\Lambda (\psi)\) and \(M(\widehat {\psi})\). He gives an example to show that the result fails if you slightly widen the class of r.i. spaces.
upper and lower dilation exponents of \(\varphi\)
scientific article; zbMATH DE number 652166