arXiv subjects

Xianjie Yan

Publications and source records attributed to Xianjie Yan.

7 recordsLinked to original sources

arXiv2026

Real-Variable Theory of Hardy--Lorentz Spaces on Quasi-Ultrametric Spaces of Homogeneous Type with Reverse-Doubling Property

Let $(X,\mathbf{q},μ)$ be an ultra-RD-space with upper dimension $n\in(0,\infty)$; i.e., it is a quasi-ultrametric space of homogeneous type whose measure $μ$ satisfies an additional reverse doubling property. Let $\mathrm{ind\,}(X,\mathbf{q})\in(0,\infty]$ denote its lower smoothness index, as introduced by Mitrea et al. In this monograph, the authors first construct a new approximation of the identity on quasi-ultrametric spaces of homogeneous type, achieving a maximal degree of smoothness $0<\varepsilon\preceq\mathrm{ind\,}(X,\mathbf{q})$. This fundamental tool is then used to derive sharp homogeneous (as well as inhomogeneous) continuous/discrete Calderón reproducing formulae on ultra-RD-spaces. As applications, the authors establish Littlewood--Paley function characterizations for both Hardy spaces and Triebel--Lizorkin spaces on ultra-RD-spaces. The authors further introduce Hardy--Lorentz spaces $H^{p,q}_\ast(X)$ via the grand maximal function, with the sharp range $p\in(\frac{n}{n+\mathrm{ind\,}(X,\mathbf{q})},\infty)$ and $q\in(0,\infty]$, and provide their real-variable characterizations using radial/non-tangential maximal functions, (finite) atoms, molecules, and various Littlewood--Paley functions. Based on these characterizations, the authors prove a duality theorem between Hardy--Lorentz spaces and Campanato--Lorentz spaces, establish a real interpolation theorem for Hardy--Lorentz spaces, and derive boundedness results for Calderón--Zygmund operators on them. It should be emphasized that many of the main results in this monograph are indeed established in the more general setting of quasi-ultrametric spaces of homogeneous type.

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arXiv2025

Time-frequency representations on Lorentz spaces over locally compact Abelian groups

Let $G$ be a locally compact Abelian group with a fixed Haar measure and, denote by $\widehat{G}$ its dual group. In this article, the authors obtain various boundedness of the short-time Fourier transform on Lorentz spaces: $$L^{p_1,u}(G)\times L^{p_2,v}(G)\to L^{q,w}(G\times\widehat{G})$$ with the indexes satisfying appropriate relations. These results are then used to prove the corresponding boundedness of $τ$-Wigner transforms and $τ$-Weyl operators. As an application, the Lieb's uncertainty principle in the context of Lorentz spaces is finally investigated. All these results are new even for the case when $G$ is finite.

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arXiv2024

The Variable Muckenhoupt Weight Revisited

Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function and $X$ a ball quasi-Banach function space. In this paper, we first study the relationship between two kinds of variable weights $\mathcal{W}_{p(\cdot)}(\mathbb{R}^n)$ and $A_{p(\cdot)}(\mathbb{R}^n)$. Then, by regarding the weighted variable Lebesgue space $L^{p(\cdot)}_ω(\mathbb{R}^n)$ with $ω\in\mathcal{W}_{p(\cdot)}(\mathbb{R}^n)$ as a special case of $X$ and applying known results of the Hardy-type space $H_{X}(\mathbb{R}^n)$ associated with $X$, we further obtain several equivalent characterizations of the weighted variable Hardy space $H^{p(\cdot)}_ω(\rn)$ and the boundedness of some sublinear operators on $H^{p(\cdot)}_ω(\rn)$. All of these results coincide with or improve existing ones, or are completely new.

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arXiv2023

Fourier Transform of Anisotropic Hardy Spaces Associated with Ball Quasi-Banach Function Spaces and Its Applications to Hardy--Littlewood Inequalities

Let $A$ be a general expansive matrix and $X$ be a ball quasi-Banach function space on $\mathbb R^n$, whose certain power (namely its convexification) supports a Fefferman--Stein vector-valued maximal inequality and the associate space of whose other power supports the boundedness of the powered Hardy--Littlewood maximal operator. Let $H_X^A(\mathbb{R}^n)$ be the anisotropic Hardy space associated with $A$ and $X$. The authors first prove that the Fourier transform of $f\in H^A_{X}(\mathbb{R}^n)$ coincides with a continuous function $F$ on $\mathbb{R}^n$ in the sense of tempered distributions. Moreover, the authors obtain a pointwise inequality that the function $F$ is less than the product of the anisotropic Hardy space norm of $f$ and a step function with respect to the transpose matrix of the expansive matrix $A$. Applying this, the authors further induce a higher order convergence for the function $F$ at the origin and give a variant of the Hardy--Littlewood inequality in $H^A_{X}(\mathbb{R}^n)$. All these results have a wide range of applications. Particularly, the authors apply these results, respectively, to classical (variable and mixed-norm) Lebesgue spaces, Morrey spaces, Lorentz spaces, Orlicz spaces, Orlicz-slice spaces, and local generalized Herz spaces and, even on the last five function spaces, the obtained results are completely new.

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arXiv2023

Anisotropic Ball Campanato-Type Function Spaces and Their Applications

Let $A$ be a general expansive matrix and let $X$ be a ball quasi-Banach function space on $\mathbb R^n$, 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 some anisotropic ball Campanato-type function spaces associated with both $A$ and $X$, prove that these spaces are dual spaces of anisotropic Hardy spaces $H_X^A(\mathbb R^n)$ associated with both $A$ and $X$, and obtain various anisotropic Littlewood--Paley function characterizations of $H_X^A(\mathbb R^n)$. Also, as applications, the authors establish several equivalent characterizations of anisotropic ball Campanato-type function spaces, which, combined with the atomic decomposition of tent spaces associated with both $A$ and $X$, further induces their Carleson measure characterizations. All these results have a wide range of generality and, particularly, even when they are applied to Morrey spaces and Orlicz-slice spaces, some of the obtained results are also new. The novelties of this article are reflected in that, to overcome the essential difficulties caused by the absence of both an explicit expression and the absolute continuity of quasi-norm $\|\cdot\|_X$, the authors embed $X$ under consideration into the anisotropic weighted Lebesgue space with certain special weight and then fully use the known results of this weighted Lebesgue space.

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arXiv2021

Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Characterizations of Maximal Functions, Decompositions, and Dual Spaces

Let $({\mathcal X},ρ,μ)$ 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 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 grand maximal function, and then establish its various real-variable characterizations, respectively, in terms of radial or non-tangential maximal functions, atoms or finite atoms, and molecules. As an application, the authors give the dual space of $H_{Y}^*({\mathcal X})$, which proves to be a ball Campanato-type function space associated with $Y({\mathcal X})$. 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 $μ$ and the triangle inequality of $ρ$, the authors cleverly construct admissible sequences of balls, and fully use the geometrical properties of ${\mathcal X}$ expressed by dyadic reference points or dyadic cubes and, to overcome the difficulty caused by the lack of the good dense subset of $H_{Y}^*({\mathcal X})$, the authors further prove that $Y({\mathcal X})$ can be embedded into the weighted Lebesgue space with certain special weight, and then can fully use the known results of the weighted Lebesgue space.

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arXiv2016

Variable Weak Hardy Spaces and Their Applications

Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors first introduce the variable weak Hardy space on $\mathbb R^n$, $W\!H^{p(\cdot)}(\mathbb R^n)$, via the radial grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain various equivalent characterizations of $W\!H^{p(\cdot)}(\mathbb R^n)$, respectively, by means of atoms, molecules, the Lusin area function, the Littlewood-Paley $g$-function or $g_λ^\ast$-function. As an application, the authors establish the boundedness of convolutional $δ$-type and non-convolutional $γ$-order Calderón-Zygmund operators from $H^{p(\cdot)}(\mathbb R^n)$ to $W\!H^{p(\cdot)}(\mathbb R^n)$ including the critical case $p_-={n}/{(n+δ)}$, where $p_-:=\mathop\mathrm{ess\,inf}_{x\in \rn}p(x).$

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