Optimal domain for the Hardy operator

Abstract

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.

Extending and factorizing bounded bilinear maps defined on order continuous Banach function spaces

“…The space L 1 (m T ) that appears above is always optimal, in the sense that it contains any order continuous µ-Banach function space to which the operator T can be extended. This result -called the Optimal Domain Theorem for operators on Banach function spaces-has been used in a series of papers as a tool for describing the optimal domain for relevant operators, mainly in harmonic analysis (see for instance [4,5,8,7,12]). Similar arguments have also been applied for obtaining the Optimal Domain of an operator that satisfies a particular domination property; the requirement is that the extension must also satisfy the same domination property.…”

Rotundity and monotonicity properties of selected Cesàro function spaces

“…Furthermore, sometimes the cases Ces p [0,1] and Ces p [0, ∞) are essentially different (see an isomorphic description of the Köthe dual of Ces àro spaces in [1,34], see also [23] for the respective isometric description). The spaces generated by the Cesàro operator (including abstract Cesàro spaces) have been considered by Curbera, Delgado, Soria, Ricker, Leśnik and Maligranda in several papers (see [13][14][15][16][34][35][36]). …”

On the Banach lattice structure of $$L^1_w$$ of a vector measure on a $$\delta $$ -ring

“…The spaces L 1 (ν) and L 1 w (ν) of integrable and weakly integrable functions respectively have been studied in depth by many authors and their behavior is well understood, see [7], [25,Chapter 3] and the references therein. However, this framework is not enough, for instance, for applications to operators on spaces which do not contain the characteristic functions of sets (see [2], [10] and [11]) or Banach lattices without weak unit (see [12]). These cases require ν to be defined on a weaker structure than σ-algebra, namely, a δ-ring.…”

“…More applications in the setting of the theory of operators on Banach function spaces can be found in [2,3]. The relevant case of the Hardy operator has been studied in [11].…”