Orlicz spaces associated to a quasi-Banach function space: applications to vector measures and interpolation

Abstract

The Orlicz spaces X associated to a quasi-Banach function space X are defined by replacing the role of the space L 1 by X in the classical construction of Orlicz spaces. Given a vector measure m, we can apply this construction to the spaces L 1 w (m), L 1 (m) and L 1 (‖m‖) of integrable functions (in the weak, strong and Choquet sense, respectively) in order to obtain the known Orlicz spaces L w (m) and L (m) and the new ones L (‖m‖). Therefore, we are providing a framework where dealing with different kind of Orlicz spaces in a unified way. Some applications to complex interpolation are also given.

Hardy spaces associated with ball quasi‐Banach function spaces on spaces of homogeneous type: Characterizations of maximal functions, decompositions, and dual spaces

“…(ii) Observe that, in Definition 2.5, if we replace any ball 𝐵 by any bounded 𝜇-measurable set 𝐸, we obtain its another equivalent formulation. (iii) By [15,Theorem 2], we know that both (ii) and (iii) of Definition 2.4 imply that any ball quasi-Banach function space is indeed complete.…”

“…Remark (i)Let $$Y({\mathcal {X}})$$ be a ball quasi‐Banach function space on $${\mathcal {X}}$$. By an elementary calculation, we know that, if $$f\in M({\mathcal {X}})$$, then $$\Vert f\Vert _{Y({\mathcal {X}})}=0$$ if and only if $$f=0$$ almost everywhere in $${\mathcal {X}}$$. (ii)Observe that, in Definition 2.5, if we replace any ball B by any bounded μ‐measurable set E , we obtain its another equivalent formulation. (iii)By [15, Theorem 2], we know that both (ii) and (iii) of Definition 2.4 imply that any ball quasi‐Banach function space is indeed complete. …”

“…Indeed, for any 𝑔 ∈ 𝐿 𝑞 0 (𝐵), ∫ 𝐵 [ F(𝑥) − 𝐹(𝑥) ] 𝑔(𝑥) 𝑑𝜇(𝑥) = 0, which, together with the fact that ℎ − 𝑚 𝐵 (ℎ) ∈ 𝐿 𝑞 0 (𝐵) for any ℎ ∈ 𝐿 𝑞 (𝐵), further implies that, for any ℎ ∈ 𝐿 𝑞 (𝐵),∫ 𝐵 [ F(𝑥) − 𝐹(𝑥) ] [ℎ(𝑥) − 𝑚 𝐵 (ℎ)] 𝑑𝜇(𝑥) = 0. (6.15)Moreover, for any 𝐻 ∈ 𝐿 𝑞 () and 𝐺 ∈ 𝐿 𝑞 ′ (), by the definitions of both 𝑚 𝐵 (𝐻) and 𝑚 𝐵 (𝐺), we obtain∫ 𝐵 [𝑚 𝐵 (𝐻)𝐺(𝑥) − 𝑚 𝐵 (𝐺)𝐻(𝑥)] 𝑑𝜇(𝑥) = ∫ 𝐵 {𝑚 𝐵 (𝐻)[𝐺(𝑥) − 𝑚 𝐵 (𝐺)] + 𝑚 𝐵 (𝐺)[𝑚 𝐵 (𝐻) − 𝐻(𝑥)]} 𝑑𝜇(𝑥) = 0,(6.16)which, together with ℎ ∈ 𝐿 𝑞 (𝐵) and 𝐹, F ∈ 𝐿 𝑞 ′ (𝐵), further implies that∫ 𝐵 𝑚 𝐵 (ℎ) [ F(𝑥) − 𝐹(𝑥) ] 𝑑𝜇(𝑥) = ∫ 𝐵 ℎ(𝑥)𝑚 𝐵(15) and (6.17), we conclude that, for any ℎ ∈ 𝐿 𝑞 (𝐵),∫ 𝐵 ℎ(𝑥) [ F(𝑥) − 𝐹(𝑥) − 𝑚 𝐵 ( F − 𝐹 )] 𝑑𝜇(𝑥) = 0,which further implies that, for almost every 𝑥 ∈ 𝐵,F(𝑥) − 𝐹(𝑥) = 𝑚 𝐵 ( F − 𝐹 ) (𝑥).Now, we take a sequence {𝐵 𝑗 } 𝑗∈ℕ of balls such that 𝐵 1 ⊂ 𝐵 2 ⊂ ⋯ ⊂ 𝐵 𝑗 ⊂ ⋯ and ⋃ 𝑗∈ℕ 𝐵 𝑗 = . Then, by (6.14), we know that there exists a sequence {𝐹 𝑗 } 𝑗∈ℕ of measurable functions such that, for any 𝑗 ∈ ℕ, 𝐹 𝑗 ∈ 𝐿 𝑞 ′ (𝐵 𝑗 ) and, for any 𝑔 ∈ 𝐿𝑞 0 (𝐵 𝑗 ), 𝑇(𝑔) = ∫  𝐹 𝑗 (𝑥)𝑔(𝑥) 𝑑𝜇(𝑥).…”

Orlicz Algebras Associated to a Banach Function Space

“…H. Hudzik also presented some conditions in [8] for an Orlicz space to be a Banach algebra with the pointwise product. Recently in [5] the authors introduced Orlicz spaces X Φ associated to a Banach function space X, where Φ is a Young function; see also [26]. This structure is a huge generalization of the classical Orlicz spaces.…”