New Hardy Spaces of Musielak–Orlicz Type and Boundedness of Sublinear Operators

By Luong Dang Ky. 2011

176 citations recorded by the source corpus.

We introduce a new class of Hardy spaces $${H^{\varphi(\cdot, \cdot)}(\mathbb{R}^{n})}$$Hφ(·,·)(Rn), called Hardy spaces of Musielak–Orlicz type, which generalize the Hardy–Orlicz spaces of Janson and the weighted Hardy spaces of García-Cuerva, Strömberg, and Torchinsky. Here, $${\varphi : \mathbb{R}^{n} \times [0, \infty) \to [0, \infty)}$$φ:Rn×[0,∞)→[0,∞) is a function such that $${\varphi(x, \cdot)}$$φ(x,·) is an Orlicz function and $${\varphi(\cdot, t)}$$φ(·,t) is a Muckenhoupt $${A_{\infty}}$$A∞ weight. A function f belongs to $${H^{\varphi(\cdot, \cdot)}(\mathbb{R}^{n})}$$Hφ(·,·)(Rn) if and only if its maximal function f* is so that $${x \mapsto \varphi(x, |f^{*}(x)|)}$$x↦φ(x,|f∗(x)|) is integrable. Such a space arises naturally for instance in the description of the product of functions in $${H^{1}(\mathbb{R}^{n})}$$H1(Rn) and $${BMO(\mathbb{R}^{n})}$$BMO(Rn) respectively (see Bonami et al. in J Math Pure Appl 97:230–241, 2012). We characterize these spaces via the grand maximal function and establish their atomic decomposition. We characterize also their dual spaces. The class of pointwise multipliers for $${BMO(\mathbb{R}^{n})}$$BMO(Rn) characterized by Nakai and Yabuta can be seen as the dual of $${L^{1}(\mathbb{R}^{n}) + H^{\rm log}(\mathbb{R}^{n})}$$L1(Rn)+Hlog(Rn) where $${H^{\rm log}(\mathbb{R}^{n})}$$Hlog(Rn) is the Hardy space of Musielak–Orlicz type related to the Musielak–Orlicz function $${\theta(x, t) = \frac{t}{{\rm log}(e + |x|) + {\rm log}(e + t)}}$$θ(x,t)=tlog(e+|x|)+log(e+t). Furthermore, under additional assumption on $${\varphi(\cdot, \cdot)}$$φ(·,·) we prove that if T is a sublinear operator and maps all atoms into uniformly bounded elements of a quasi-Banach space $${\mathcal{B}}$$B, then T uniquely extends to a bounded sublinear operator from $${H^{\varphi(\cdot,\cdot)}(\mathbb{R}^{n})}$$Hφ(·,·)(Rn) to $${\mathcal{B}}$$B. These results are new even for the classical Hardy–Orlicz spaces on $${\mathbb{R}^{n}}$$Rn.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3e969172bae40cae7b14360b6ee2c0bf67b9eea8

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Citations in the corpus

New real-variable characterizations of Musielak-Orlicz Hardy spaces (2012, methodology): “with p∈ (0,1], or its weighted variant. To be precise, this paper is organized as follows. In Section 2, we recall some notions concerning growth functions and some of their properties established in [24]. Then via some skillful applications of these properties on growth functions, such as their equ”
Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates (2013, methodology): “namely, ϕmay not be of uniformly upper type I(ϕ) and uniformly lower type i(ϕ) (see below for some examples). Let ϕ: X × [0,∞) → [0,∞) satisfy that x→ ϕ(x,t) is measurable for all t∈ [0,∞). Following [49], ϕ(·,t) is called uniformly locally integrable if, for all bounded sets Kin X, ˆ K sup t∈(0,∞) ”
Besov-Type Spaces with Variable Smoothness and Integrability (2015, unknown): “s ∈ [0, 1]), φ(x, st) ≤ Csφ(x, t) (see [32]).”
On Function Spaces with Mixed Norms — A Survey (2019, unknown): “and, for any ψ, φ∈Y , ‖T(ψ)−T(φ)‖Bγ ≤C‖T(ψ−φ)‖Bγ (see [42, 56, 81–83]).”
Real-variable characterizations of Orlicz-slice Hardy spaces (2018, background): “Next we recall the notion of γ-quasi-Banach spaces (see, for example, [27], [44], [47] and [46]).”
Lusin Area Function and Molecular Characterizations of Musielak-Orlicz Hardy Spaces and Their Applications (2012, background): “ion ϕ(x,t) := t log(e+|x|) +log(e+t) for all x∈ Rn and t∈ [0,∞). Musielak-Orlicz functions are the natural generalization of Orlicz functions that may vary in the spatial variables; see, for example, [11, 12, 27, 34]. Recall that the motivation to study function spaces of Musielak-Orlicz type comes ”
Intrinsic Square Function Characterizations of Hardy Spaces with Variable Exponents (2014, methodology): “However, a general Musielak-Orlicz function φ satisfying all the assumptions in [23] (and hence [18, 26]) may not have the form as in (1.”
Weak Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces II: Littlewood–Paley Characterizations and Real Interpolation (2019, background): “In recent decades, various variants of classical Hardy spaces have been introduced and their real-variable theories have been well developed; these variants include weighted Hardy spaces (see [68]), (weighted) Herz–Hardy spaces (see, for instance, [18, 30, 31, 56, 57]), (weighted) Hardy–Morrey space”
Bilinear decompositions and commutators of singular integral operators (2011, background): “s a small variation of a standard statement which can be found in [21], Section 6.5. Remark that the interest of dealing with ﬁnite atomic decompositions has been underlined recently, for instance in [34, 23]. Now, we denote by H1 ﬁn(R n) the vector space of all ﬁnite linear combinations of ψ-atoms,”
Variable Anisotropic Hardy Spaces and Their Applications (2017, unknown): “In addition, for any given γ ∈ (0, 1], a quasi-Banach space Bγ with quasi-norm ‖·‖Bγ is called a γ-quasi-Banach space if there exists a constant κ ∈ [1,∞) such that, for any m ∈ N and {fi}i=1 ⊂ Bγ , it holds true that ∥∥∑m i=1 fi ∥∥γ Bγ ≤ κ ∑m i=1 ‖fi‖ γ Bγ (see [37,78,79]).”