We study the optimal domain for the Hardy operator considered with values in a rearrangement invariant space. In particular, this domain can be represented as the space of integrable functions with respect to a vector measure defined on a δ-ring. A precise description is given for the case of the minimal Lorentz spaces.

No abstract

Given a vector measure ν defined on a δ-ring with values in a Banach space, we study the relation between the analytic properties of the measure ν and the lattice properties of the space L 1 (ν) of real functions which are integrable with respect to ν.Introduction. The classical theory of integration of scalar functions with respect to a vector measure (defined on a σ -algebra) was created by Bartle, Dunford and Schwartz for studying the vector extension of the Riesz representation theorem [1]. The corresponding space of integrable functions has been thoroughly studied and is now well understood; see [6], [7], [8], [11], [16], [17]. An application is the study of operators T between function spaces through the vector measure ν(A) = T (χ A ) and its space L 1 (ν) of integrable functions; see [1]. A crucial role in this study is played by the "good" properties of the space L 1 (ν) namely, it is an order continuous Banach lattice with weak order unit.There are, however, important operators which cannot be directly studied via this classical integration procedure. This happens, for example, with the Hilbert transform on the real line. In this case, the vector measure associated to the operator is defined only for Lebesgue measurable sets of finite measure, which do not constitute a σ -algebra. Thus, we are naturally lead to consider vector measures which are defined on structures weaker than σ -algebras. The extension of the integration theory to vector measures defined on δ-rings was done by Lewis [12] and Masani and Niemi [14], [15].In this paper we consider a vector measure ν defined on a δ-ring R of sets of , taking values in a Banach space. We analyse the differences with vector measures defined on σ -algebras; in particular, L 1 (ν) is an order continuous Banach lattice which may not have a weak order unit if ν is only defined on a δ-ring. We study the effect on the space L 1 (ν) of certain properties of ν: i.e., strong additivity and σ -finiteness. Namely, we show that L 1 (ν) has a weak unit g if and only if ν is σ -finite (Theorem 3.3), and in this case L 1 (ν) is order isometric to L 1 (ν g ), where ν g is the vector measure defined on the σ -algebra of sets locally in R by ν g (A) =

In order to extend the theory of optimal domains for continuous operators on a Banach function space X(μ) over a finite measure μ, we consider operators T satisfying other type of inequalities than the one given by the continuity which occur in several well-known factorization theorems (for instance, Pisier Factorization Theorem through Lorentz spaces, pth-power factorable operators . . . ). We prove that such a T factorizes through a space of multiplication operators which can be understood in a certain sense as the optimal domain for T . Our extended optimal domain technique does not need necessarily the equivalence between μ and the measure defined by the operator T and, by using δ-rings, μ is allowed to be infinite. Classical and new examples and applications of our results are also given, including some new results on the Hardy operator and a factorization theorem through Hilbert spaces.

Abstract. We study some Banach lattice properties of the space L 1 w (ν) of weakly integrable functions with respect to a vector measure ν defined on a δ-ring. Namely, we analyze order continuity, order density and Fatou type properties. We will see that the behavior of L 1 w (ν) differs from the case in which ν is defined on a σ-algebra whenever ν does not satisfy certain local σ-finiteness property.