Wavelet Bases in Banach Function Spaces
- PDF / 443,024 Bytes
- 21 Pages / 439.37 x 666.142 pts Page_size
- 97 Downloads / 191 Views
Wavelet Bases in Banach Function Spaces Alexei Yu. Karlovich1 Received: 10 February 2020 / Revised: 3 August 2020 / Accepted: 9 September 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract We show that if the Hardy–Littlewood maximal operator is bounded on a separable Banach function space X (R) and on its associate space X (R), then the space X (R) has an unconditional wavelet basis. This result extends previous results by Soardi (Proc Am Math Soc 125:3669–3673, 1997) for rearrangement-invariant Banach function spaces with nontrivial Boyd indices and by Fernandes et al. (Banach Center Publ 119:157–171, 2019) for reflexive Banach function spaces. We specify our result to the case of Lorentz spaces L p,q (R, w), 1 < p < ∞, 1 ≤ q < ∞ with Muckenhoupt weights w ∈ A p (R). Keywords Banach function spaces · Associate space · Unconditional wavelet basis · Hardy–Littlewood maximal operator Mathematics Subject Classification 46E30 · 42C40
1 Introduction Recall that a function ψ ∈ L 2 (R) is called an orthonormal wavelet if the family ψ j,k (x) := 2 j/2 ψ(2 j x − k), x ∈ R,
j, k ∈ Z,
forms an orthonormal basis in L 2 (R). It is well known that, under certain conditions, the system {ψ j,k : j, k ∈ Z} forms an unconditional basis in many classical function
Communicated by Sorina Barza. This work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the Projects UIDB/MAT/ 00297/2020 (Centro de Matemática e Aplicações).
B 1
Alexei Yu. Karlovich [email protected] Centro de Matemática e Aplicações, Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829–516 Caparica, Portugal
123
A. Yu. Karlovich
spaces (see, e.g., [9, Chaps. 5–6], [15, Chap. 7], [18, Chap. 6] and also [10,22], to mention a few closely related works). The aim of this work is to prove that a so-called Banach function space X (R) admits an unconditional wavelet basis under minimal assumptions on the space. For simplicity, we restrict ourselves to the one-dimensional case. Let us recall the necessary definitions. The set of all Lebesgue measurable complexvalued functions on R is denoted by M(R). Let M+ (R) be the subset of functions in M(R) whose values lie in [0, ∞]. For a measurable set E ⊂ R, its Lebesgue measure and the characteristic function are denoted by |E| and χ E , respectively. Following [2, Chap. 1, Definition 1.1], a mapping ρ : M+ (R) → [0, ∞] is called a Banach function norm if, for all functions f , g, f n (n ∈ N) in M+ (R), for all constants a ≥ 0, and for all measurable subsets E of R, the following properties hold: (A1) (A2)
ρ( f ) = 0 ⇔ f = 0 a.e., ρ(a f ) = aρ( f ), ρ( f + g) ≤ ρ( f ) + ρ(g), 0 ≤ g ≤ f a.e. ⇒ ρ(g) ≤ ρ( f ) (the lattice property),
(A3) (A4)
0 ≤ f n ↑ f a.e. ⇒ ρ( f n ) ↑ ρ( f ) (the Fatou property), |E| < ∞ ⇒ ρ(χ E ) < ∞, |E| < ∞ ⇒ f (x) dx ≤ C E ρ( f ),
(A5)
E
where C E ∈ (0, ∞) may depend on E and ρ but is independent of f . When
Data Loading...
