Fully symmetric function spaces without an equivalent Fatou norm
This paper deals with the relation of Fatou-type properties for the class of the symmetric Banach function spaces and the monotonicity properties of the lattice norms with respect to the order given by the decreasing rearrangements of the functions. This allows to define fully symmetric function spaces and strongly symmetric function spaces. Let us explain briefly these notions. Consider a Banach function space \(E\) over \([0,1]\) or \((0, \infty)\). For a function \(x \in E\), we write \(x^*\) for its decreasing rearrangement. Then \(E\) is said to be symmetric if, for \(x \in E\) and a measurable function \(y\), if \(y^*=x^*\), then \(y \in E\) and \(\|x\|_E=\|y\|_E\). It is known that every symmetric space is intermediate for the Banach couple \((L_1,L_\infty)\). Let \(x,y \in L_1 + L_\infty\); these functions satisfy the Hardy-Littlewood-Polya submajorization \(x \prec \prec y\) if \[ \int_0^t x^*(s) \, ds \leq \int_0^t y^*(s) \, ds. \] A symmetric space \(E\) is fully symmetric if for all \(y \in E\), \[ x \prec \prec y \Rightarrow x \in E,\, \| x\|_E \leq \|y\|_E. \] The space \(E\) is called strongly symmetric if \[ x,y \in E, \, x \prec \prec y \Rightarrow \| x\|_E \leq \|y\|_E. \] In this paper, it is proved that there is a wide class of symmetric spaces that are \(1\)-interpolation spaces for the couple \((L_1,L_\infty)\), but do not admit an equivalent Fatou norm (Theorem 6); the complement of this result was already known: there is a symmetric Banach function space that is not such an interpolation space but has the Fatou property, showing in this way the independence among these properties. The second relevant result that is proved in the paper is Theorem 9, which exhibits a class of symmetric spaces on \([0,1]\) that are not closed subspaces of any interpolation space. Concretely, it is shown that there is a symmetric space \(E\) which does not admit a fully symmetric norm; in particular, it is not a closed subspace of any fully symmetric space, and thus it is not a strongly symmetric space. The paper provides interesting new information on symmetric Banach function spaces. Some comments relating these results with the corresponding ``sequence space versions'' are presented, too, for closing the paper: the authors claim and justify that a similar result can also be obtained in this case.
scientific article; zbMATH DE number 6484275
scientific article