Let $({\mathcal X},\rho,\mu)$ be a space of homogeneous type in the sense of Coifman and Weiss, and $Y({\mathcal X})$ a ball quasi-Banach function space on ${\mathcal X}$, which supports both a Fefferman--Stein vector-valued maximal inequality and the boundedness of the powered Hardy--Littlewood maximal operator on its associate space. The authors first introduce the Hardy space $H_{Y}({\mathcal X})$ associated with $Y({\mathcal X})$, via the Lusin-area function, and then establish its various equivalent characterizations, respectively, in terms of atoms, molecules, and Littlewood--Paley $g$-functions and $g_{\lambda}^*$-functions. As an application, the authors obtain the boundedness of Calder\'on--Zygmund operators from $H_{Y}({\mathcal X})$ to $Y({\mathcal X})$, or to $H_{Y}({\mathcal X})$ via first establishing a boundedness criterion of linear operators on $H_{Y}({\mathcal X})$. All these results have a wide range of generality and, particularly, even when they are applied to variable Hardy spaces, the obtained results are also new. The major novelties of this article exist in that, to escape the reverse doubling condition of $\mu$ and the triangle inequality of $\rho$, the authors subtly use the wavelet reproducing formula, originally establish an admissible molecular characterization of $H_{Y}({\mathcal X})$, and fully apply the geometrical properties of ${\mathcal X}$ expressed by dyadic reference points or dyadic cubes.

Let $\mathcal{L}=-\Delta+\mathit{V}$ be a Schrödinger operator, where $\Delta$ is the Laplacian operator on $\mathbb{R}^{d}$ $(d\geq 3)$, while the nonnegative potential $\mathit{V}$ belongs to the reverse Hölder class $B_{q}, q>d/2$. In this paper, we study weighted compactness of commutators of some Schrödinger operators, which include Riesz transforms, standard Calderón--Zygmund operators and Littlewood--Paley functions. These results substantially generalize some well-known results.

In this article, the authors first establish the pointwise characterizations of Besov and Triebel-Lizorkin spaces with generalized smoothness on $\mathbb{R}^n$ via the Hajłlasz gradient sequences, which serve as a way to extend these spaces to more general metric measure spaces. Moreover, on metric spaces with doubling measures, the authors further prove that the Besov and the Triebel-Lizorkin spaces with generalized smoothness defined via Hajłlasz gradient sequences coincide with those defined via hyperbolic fillings. As an application, some trace theorems of these spaces on Ahlfors regular spaces are established.

In this paper, carrying on with our study of the Hardy-amalgam spaces $\mathcal H^{(q,p)}$ and $\mathcal{H}_{\mathrm{loc}}^{(q,p)}$ ($0< q,p <\infty$), we give a characterization of their dual spaces whenever $0< q\leq 1$ and $q\leq p<\infty$. Moreover, when $0< q\leq p\leq 1$, these characterizations coincide with those obtained in our earlier papers.

Let T be a strongly singular Calderón-Zygmund operator and b ∈ L_loc(Rⁿ). This article finds out a class of non-trivial subspaces BMOω,p,u(Rⁿ) of BMO(Rⁿ) for certain ω ∈ A₁, 0 < p ≤ 1 and 1 < u ≤ ∞, such that the commutator[b, T] is bounded from weighted Hardy space H_ω^p(Rⁿ) to weighted Lebesgue space L_ω^p(Rⁿ) if b ∈ BMOω,p,∞(Rⁿ), and is bounded from weighted Hardy space H_ω^p(Rⁿ) to itself if T^∗1=0 and b ∈ BMOω,p,u(Rⁿ) for 1 < u < 2.