Banach lattices with the fatou property and optimal domains of kernel operators

The Fatou Completion of a Fréchet Function Space and Applications

“…In this regard, we develop various aspects of the theory of Fréchet function spaces (especially in relation to Lorentz function seminorms), which are not available in the literature in the form needed here; this, of interest in its own right, is done in Section 2. With these techniques we are able to establish close connections between L 1 (ν) and L 1 w (ν) which are known in the Banach space setting [7]. Namely, in Section 3 it is shown that the following are equivalent: L 1 (ν) = L 1 w (ν); L 1 (ν) has the σ -Fatou property; L 1 w (ν) is order continuous; L 1 (ν) is weakly sequentially complete; and L 1 w (ν) is weakly sequentially complete.…”

“…It is important to note that (L {ρ n } ) F is not necessarily obtained from L {ρ n } in any topological sense. Indeed, if X is a Banach space and ν is an X -valued vector measure, then for the (single) function norm [7]. For certain ν, the space L 1 (ν) can be a proper closed subspace of L 1 w (ν); [7], [12, p. 31].…”

“…Then the Banach space L 1 (ν) of all ν-integrable functions is a closed subspace (typically proper) of the Banach space L 1 w (ν) of all scalarly ν-integrable functions [20]. Both L 1 (ν) and L 1 w (ν) are Banach function spaces (relative to a control measure for ν and the pointwise almost everywhere order), with the distinction that L 1 (ν) has order continuous norm (that is, a Lebesgue topology), whereas L 1 w (ν) always has the σ -Fatou property [6,7]. Actually, L 1 (ν) is the order continuous part of L 1 w (ν), that is, the largest ideal inside L 1 w (ν) with each element having order continuous norm, and L 1 w (ν) is the σ -Fatou completion of L 1 (ν), that is, the minimal Banach function space which has the σ -Fatou property and contains L 1 (ν) [7].…”

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

“…It was proved in [5,Theorem 8] that every order continuous Banach lattice with a weak unit is order isometric to an space L 1 (ν) for a vector measure ν defined on a σ-algebra. This result allows to represent any Banach lattice E with the σ-Fatou property with a weak unit belonging to E a as an space L 1 w (ν) with ν defined on a σ-algebra, since in this case the order isometry between E a and L 1 (ν) can be extended to E and turns out to be an order isometry between E and L 1 w (ν), see [6,Theorem 2.5]. So, we have the following equivalences between classes of spaces:…”

Optimal Domains for L0-valued Operators Via Stochastic Measures

“…In the case when T : E → X with X (and also E) a Banach space, this problem has been considered in [7], [9], [12]. For the particular case of T the operator associated with Sobolev inequality and X a rearrangement invariant space, this study has been done in [8], [10]; and for T a convolution operator and X = L p (T), in [23].…”