A Survey on Some Anisotropic Hardy-Type Function Spaces

Abstract

Let A be a general expansive matrix on R n. The aims of this article are twofold. The first one is to give a survey on the recent developments of anisotropic Hardy-type function spaces on R n , including anisotropic Hardy-Lorentz spaces, anisotropic variable Hardy spaces and anisotropic variable Hardy-Lorentz spaces as well as anisotropic Musielak-Orlicz Hardy spaces. The second one is to correct some errors and seal some gaps existing in the known articles. Some unsolved problems are also presented.

Musielak–Orlicz Hardy space estimates for commutators of Calderón–Zygmund operators

“…The following is due to M. Liao, J. Li, B. Li, and B. Li [18, Proof of Lemma 3.7] (see also [7, 19]). Lemma Let $$q\in (1,\infty )$$, $$p\in (0,1]$$, $$\varepsilon \in (\frac{nq}{p},\infty )$$, and let $$\varphi \in \mathbb {A}_q(\mathbb {R}^n)$$ be of uniformly lower type p .…”

“…In order to prove Theorem 1.4, we need to recall the notion of molecules (see [18, 19]) as follows. Definition Let φ, q be as in Definition 2.4, and let $$\varepsilon \in \left(\frac{n q(\varphi )}{i(\varphi )},\infty \right)$$.…”

“…) for all 𝜆 > 0 since 𝔭(𝑛 + 𝛿) − 𝑛𝔮 > 0 (see Remark 1.5(ii)). Thus, by [16, Lemma 4.3(i)], The following is due to M. Liao, J. Li, B. Li, and B. Li [18, Proof of Lemma 3.7] (see also [7,19]).…”

“…In order to improve the result of Bonami–Iwaniec–Jones–Zinsmeister, in 2012, Bonami, Grellier, and Ky [3] (see also [2]) proposed the space $$H^{\log }(\mathbb {R}^n)$$, which is the best space for this decomposition in the sense of that it could not be replaced by a smaller space. The space $$H^{\log }(\mathbb {R}^n)$$ is a natural example of very general Hardy spaces of Musielak–Orlicz–type $$H^\varphi (\mathbb {R}^n)$$, which is introduced and investigated by the second author in [16] (see also [7, 19, 23]). We also want to emphasize additionally that the Musielak–Orlicz Hardy space $$H^\varphi (\mathbb {R}^n)$$ consists of the well‐known classical Hardy‐type spaces such as the weighted Hardy spaces of García–Cuerva [8], Strömberg and Torchinsky [21] ($$\varphi (x,t)=w(x)t^p$$ with $$p\in (0,1]$$ and $$w\in A_\infty (\mathbb {R}^n)$$ in this context), and the Hardy–Orlicz spaces of Janson [13] ($$\varphi (x,t)=\Phi (t)$$ with Φ being an Orlicz function in this context).…”