arXiv subjects

Ciqiang Zhuo

Publications and source records attributed to Ciqiang Zhuo.

17 recordsLinked to original sources

arXiv2026

The distance from functions in BMO to BLO

Let BMO and BLO denote the spaces of all locally integrable real-valued functions on $\mathbb{R}^n$ with bounded mean oscillation and bounded lower oscillation, respectively. It is well known that $$L^\infty(\mathbb{R}^n)\subsetneqq {\rm BLO}\subsetneqq {\rm BMO}.$$ In 1978, Garnett and Jones gave distance formulas of $f\in {\rm BMO}$ to $L^\infty(\mathbb{R}^n)$ and recently, Angrisani studied the distance of $f\in {\rm BLO}$ to $L^\infty(\mathbb{R}^n)$. In this paper, we characterize the distance from any given function $f \in {\rm BMO}$ to BLO via the Muckenhoupt weight class $A_p$ as follows \begin{center} dist\,($f$,\ BLO)\,$\sim\inf\left\{ξ\in(0,\infty):\ e^{-\frac fξ}\in A_p\ \mathrm{for\ some\ }p\in(1,\infty)\right\}$. \end{center} Two equivalent representations of this distance are also established in terms of exponential form and the infimum of the constant in a variant of John--Nirenberg inequality, respectively.

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arXiv2026

Capacitary Muckenhoupt Weight, BMO and BLO Spaces with Hausdorff Content, Factorization Theorems and Applications

Let $δ\in(0,n]$, $p\in[1,\infty)$, $\mathcal H_{\infty}^δ$ denote the Hausdorff content on $\mathbb R^n$, and $\mathcal A_{p,δ}$ be the capacitary Muckenhoupt weight class. We are interested in understanding the relationship between the capacitary Muckenhoupt weight class $\mathcal A_{p,δ}$ and ${\rm{BMO}}(\mathbb R^n, \mathcal H_{\infty}^δ)$ or ${\rm{BLO}}(\mathbb R^n, \mathcal H_{\infty}^δ)$ spaces for all dimension $δ\in(0,n]$, and further to comprehend the structure of these two spaces. Our main result shows that $\mathcal A_{p,δ}$ for $p\in(1,\infty)$ is equivalent to the BMO spaces, while $\mathcal A_{1,δ}$ is equivalent to the BLO spaces, and consequently yields the factorization theorems for these BMO and BLO spaces via capacitary Hardy--Littlewood maximal operators, which essentially extend main results of Coifman and Rochberg in 1980 beyond measure theory. As applications, by establishing some capacitary weighted John--Nirenberg inequalities, we obtain the equivalence between capacitary weighted BMO or BLO spaces and ${\rm{BMO}}(\mathbb R^n, \mathcal H_{\infty}^δ)$ or ${\rm{BLO}}(\mathbb R^n, \mathcal H_{\infty}^δ)$ respectively. These results reveal deep connections between $\mathcal A_{p,δ}$ and BMO or BLO spaces with Hausdorff content, beyond the classical measure-theoretic settings. We develop some approaches in the proofs and using a new observation, that is, the additivity of measures and linearity of integrals are superfluous for the corresponding classical theory.

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arXiv2025

Capacitary Muckenhoupt Weights and Weighted Norm Inequalities for Hardy-Littlewood Maximal Operators

Let $\mathcal H_{\infty}^δ$ denote the Hausdorff content of dimension $δ\in(0,n]$ defined on subsets of $\mathbb R^n$. The principal problem, considered in this paper, is to characterize the non-negative function $w$ for which the weighted $L^p$-norm inequality with $p\in(1,\infty)$ and the weighted weak $L^1$-norm inequality on Hardy-Littlewood maximal operators associated with Hausdorff contents hold true. To achieve this, we introduce a class of capacitary Muckenhoupt weights depending on the dimension $δ$, denoted as $\mathcal A_{p,δ}$, which enjoys the strict monotonicity on the dimension index $δ$. Then we show that, for any $p\in(1,\infty)$ and $δ\in(0,n]$, the weighted $L^p$-norm inequality holds true if and only if $w\in\mathcal A_{p,δ}$, and the weighted weak $L^1$-norm inequality holds true if and only if $w\in\mathcal A_{1,δ}$ by a new approach developed in this paper. As the second objective, applying this new approach, the seminal properties of classical Muckenhoupt $A_p$ weights, such as the reverse Hölder inequality [R. R. Coifman and C. Fefferman, Studia Math. 51 (1974), 241-250], the self-improving property [B. Muckenhoupt, Trans. Amer. Math. Soc. 165 (1972), 207-226], and Jones' factorization theorem [P. W. Jones, Ann. of Math. (2) 111 (1980), 511-530], are all established within the framework of capacitary Muckenhoupt weight class $\mathcal A_{p,δ}$. Finally, we also show that the maximal operator is bounded on the weak weighted Choquet-Lebesgue space $L_w^{p,\infty}(\mathbb R^n,{\mathcal H}_\infty^δ)$ if and only if $w\in\mathcal A_{p,δ}$ with $p\in(1,\infty)$ and $δ\in(0,n]$.

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arXiv2024

Sharp Poincaré--Sobolev Inequalities of Choquet--Lorentz Integrals with Respect to Hausdorff Contents on Bounded John Domains

Let $Ω$ be a bounded John domain in $\mathbb R^n$ with $n\ge 2$, and let $\mathcal{H}_{\infty }^δ$ denote the Hausdorff content of dimension $δ\in (0,n]$. In this article, the authors prove the Poincaré and the Poincaré--Sobolev inequalities, with sharp ranges of indices, on Choquet--Lorentz integrals with respect to $\mathcal{H}_{\infty }^δ$ for all continuously differentiable functions on $Ω$. These results not only extend the recent Poincaré and Poincaré--Sobolev inequalities to the Choquet--Lorentz integrals, but also provide some endpoint estimates (weak type) in the critical case. One of the main novelties exists in that, to achieve the goals, the authors develop some new tools associated with Choquet--Lorentz integrals on $\mathcal{H}_{\infty }^δ$, such as the fractional Hardy--Littlewood maximal inequality and the Hedberg-type pointwise estimate on the Riesz potential. As an application, the authors obtain the sharp boundedness of the Riesz potential on Choquet--Lorentz integrals. Moreover, even for classical Lorentz integrals, these Poincaré and Poincaré--Sobolev inequalities are also new.

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arXiv2024

The Molecular Characterizations of Variable Triebel-Lizorkin Spaces Associated with the Hermite Operator and Its Applications

In this article, we introduce inhomogeneous variable Triebel-Lizorkin spaces, $F_{p(\cdot),q(\cdot)}^{α(\cdot),H}(\mathbb R^n)$, associated with the Hermite operator $H:=-Δ+|x|^2$, where $Δ$ is the Laplace operator on $\mathbb R^n$, and mainly establish the molecular characterization of this space. As applications, we obtain some regularity results to fractional Hermite equations $$(-Δ+|x|^2)^σu=f,\quad (-Δ+|x|^2+I)^σu=f,$$ and the boundedness of spectral multiplier associated to the operator $H$ on the variable Triebel-Lizorkin space $F_{p(\cdot),q(\cdot)}^{α(\cdot),H}(\mathbb R^n)$. Furthermore, we explain the relationship between $F_{p(\cdot),q(\cdot)}^{α(\cdot),H}(\mathbb R^n)$ and the variable Triebel-Lizorkin spaces $F_{p(\cdot),q(\cdot)}^{α(\cdot)}(\mathbb R^n)$ (introduced in Diening t al. J. Funct. Anal. 256(2009), 1731-1768.) via the atomic decomposition.

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arXiv2021

Matrix weighted Kolmogorov-Riesz's compactness theorem

In this paper, several versions of the Kolmogorov-Riesz compactness theorem in weighted Lebesgue spaces with matrix weights are obtained. In particular, when the matrix weight $W$ is in the known $A_p$ class, a characterization of totally bounded subsets in $L^p(W)$ with $p\in(1, \infty)$ is established.

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arXiv2020

Complex Interpolation of Lizorkin-Triebel-Morrey Spaces on Domains

In this article the authors study complex interpolation of Sobolev-Morrey spaces and their generalizations, Lizorkin-Triebel-Morrey spaces. Both scales are considered on bounded domains. Under certain conditions on the parameters the outcome belongs to the scale of the so-called diamond spaces.

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arXiv2018

Variable Weak Hardy Spaces $W\!H_L^{p(\cdot)}({\mathbb R}^n)$ Associated with Operators Satisfying Davies-Gaffney Estimates

Let $p(\cdot):\ \mathbb R^n\to(0,1]$ be a variable exponent function satisfying the globally log-Hölder continuous condition and $L$ a one to one operator of type $ω$ in $L^2({\mathbb R}^n)$, with $ω\in[0,\,π/2)$, which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. In this article, the authors introduce the variable weak Hardy space $W\!H_L^{p(\cdot)}(\mathbb R^n)$ associated with $L$ via the corresponding square function. Its molecular characterization is then established by means of the atomic decomposition of the variable weak tent space $W\!T^{p(\cdot)}(\mathbb R^n)$ which is also obtained in this article. In particular, when $L$ is non-negative and self-adjoint, the authors obtain the atomic characterization of $W\!H_L^{p(\cdot)}(\mathbb R^n)$. As an application of the molecular characterization, when $L$ is the second-order divergence form elliptic operator with complex bounded measurable coefficient, the authors prove that the associated Riesz transform $\nabla L^{-1/2}$ is bounded from $W\!H_L^{p(\cdot)}(\mathbb R^n)$ to the variable weak Hardy space $W\!H^{p(\cdot)}(\mathbb R^n)$. Moreover, when $L$ is non-negative and self-adjoint with the kernels of $\{e^{-tL}\}_{t>0}$ satisfying the Gauss upper bound estimates, the atomic characterization of $W\!H_L^{p(\cdot)}(\mathbb R^n)$ is further used to characterize the space via non-tangential maximal functions.

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arXiv2017

Variable Hardy Spaces Associated with Operators Satisfying Davies-Gaffney Estimates

Let $L$ be a one-to-one operator of type $ω$ in $L^2(\mathbb{R}^n)$, with $ω\in[0,\,π/2)$, which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. Let $p(\cdot):\ \mathbb{R}^n\to(0,\,1]$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors introduce the variable Hardy space $H^{p(\cdot)}_L(\mathbb{R}^n)$ associated with $L$. By means of variable tent spaces, the authors establish the molecular characterization of $H^{p(\cdot)}_L(\mathbb{R}^n)$. Then the authors show that the dual space of $H^{p(\cdot)}_L(\mathbb{R}^n)$ is the BMO-type space ${\rm BMO}_{p(\cdot),\,L^\ast}(\mathbb{R}^n)$, where $L^\ast$ denotes the adjoint operator of $L$. In particular, when $L$ is the second order divergence form elliptic operator with complex bounded measurable coefficients, the authors obtain the non-tangential maximal function characterization of $H^{p(\cdot)}_L(\mathbb{R}^n)$ and show that the fractional integral $L^{-α}$ for $α\in(0,\,\frac12]$ is bounded from $H_L^{p(\cdot)}(\mathbb{R}^n)$ to $H_L^{q(\cdot)}(\mathbb{R}^n)$ with $\frac1{p(\cdot)}-\frac1{q(\cdot)}=\frac{2α}{n}$ and the Riesz transform $\nabla L^{-1/2}$ is bounded from $H^{p(\cdot)}_L(\mathbb{R}^n)$ to the variable Hardy space $H^{p(\cdot)}(\mathbb{R}^n)$.

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arXiv2017

Interpolation between $H^{p(\cdot)}(\mathbb R^n)$ and $L^\infty(\mathbb R^n)$: Real Method

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 obtain a decomposition for any distribution of the variable weak Hardy space into "good" and "bad" parts and then prove the following real interpolation theorem between the variable Hardy space $H^{p(\cdot)}(\mathbb R^n)$ and the space $L^{\infty}(\mathbb R^n)$: \begin{equation*} (H^{p(\cdot)}(\mathbb R^n),L^{\infty}(\mathbb R^n))_{θ,\infty} =W\!H^{p(\cdot)/(1-θ)}(\mathbb R^n),\quad θ\in(0,1), \end{equation*} where $W\!H^{p(\cdot)/(1-θ)}(\mathbb R^n)$ denotes the variable weak Hardy space. As an application, the variable weak Hardy space $W\!H^{p(\cdot)}(\mathbb R^n)$ with $p_-:=\mathop\mathrm{ess\,inf}_{x\in\rn}p(x)\in(1,\infty)$ is proved to coincide with the variable Lebesgue space $W\!L^{p(\cdot)}(\mathbb R^n)$.

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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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arXiv2016

Maximal Function Characterizations of Variable Hardy Spaces Associated with Non-negative Self-adjoint Operators Satisfying Gaussian Estimates

Let $p(\cdot):\ \mathbb R^n\to(0,1]$ be a variable exponent function satisfying the globally $\log$-Hölder continuous condition and $L$ a non-negative self-adjoint operator on $L^2(\mathbb R^n)$ whose heat kernels satisfying the Gaussian upper bound estimates. Let $H_L^{p(\cdot)}(\mathbb R^n)$ be the variable exponent Hardy space defined via the Lusin area function associated with the heat kernels $\{e^{-t^2L}\}_{t\in (0,\infty)}$. In this article, the authors first establish the atomic characterization of $H_L^{p(\cdot)}(\mathbb R^n)$; using this, the authors then obtain its non-tangential maximal function characterization which, when $p(\cdot)$ is a constant in $(0,1]$, coincides with a recent result by Song and Yan [Adv. Math. 287 (2016), 463-484] and further induces the radial maximal function characterization of $H_L^{p(\cdot)}(\mathbb R^n)$ under an additional assumption that the heat kernels of $L$ have the Hölder regularity.

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arXiv2015

Molecular Characterizations and Dualities of Variable Exponent Hardy Spaces Associated with Operators

Let $L$ be a linear operator on $L^2(\mathbb R^n)$ generating an analytic semigroup $\{e^{-tL}\}_{t\ge0}$ with kernels having pointwise upper bounds and $p(\cdot):\ \mathbb R^n\to(0,1]$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors introduce the variable exponent Hardy space associated with the operator $L$, denoted by $H_L^{p(\cdot)}(\mathbb R^n)$, and the BMO-type space ${\mathrm{BMO}}_{p(\cdot),L}(\mathbb R^n)$. By means of tent spaces with variable exponents, the authors then establish the molecular characterization of $H_L^{p(\cdot)}(\mathbb R^n)$ and a duality theorem between such a Hardy space and a BMO-type space. As applications, the authors study the boundedness of the fractional integral on these Hardy spaces and the coincidence between $H_L^{p(\cdot)}(\mathbb R^n)$ and the variable exponent Hardy spaces $H^{p(\cdot)}(\mathbb R^n)$.

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arXiv2015

Characterizations of Variable Exponent Hardy Spaces via Riesz Transforms

Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function satisfying that there exists a constant $p_0\in(0,p_-)$, where $p_-:=\mathop{\mathrm {ess\,inf}}_{x\in \mathbb R^n}p(x)$, such that the Hardy-Littlewood maximal operator is bounded on the variable exponent Lebesgue space $L^{p(\cdot)/p_0}(\mathbb R^n)$. In this article, via investigating relations between boundary valued of harmonic functions on the upper half space and elements of variable exponent Hardy spaces $H^{p(\cdot)}(\mathbb R^n)$ introduced by E. Nakai and Y. Sawano and, independently, by D. Cruz-Uribe and L.-A. D. Wang, the authors characterize $H^{p(\cdot)}(\mathbb R^n)$ via the first order Riesz transforms when $p_-\in (\frac{n-1}n,\infty)$, and via compositions of all the first order Riesz transforms when $p_-\in(0,\frac{n-1}n)$.

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arXiv2015

Triebel-Lizorkin-Type Spaces with Variable Exponents

In this article, the authors first introduce the Triebel-Lizorkin-type space $F_{p(\cdot),q(\cdot)}^{s(\cdot),ϕ}(\mathbb R^n)$ with variable exponents, and establish its $φ$-transform characterization in the sense of Frazier and Jawerth, which further implies that this new scale of function spaces is well defined. The smooth molecular and the smooth atomic characterizations of $F_{p(\cdot),q(\cdot)}^{s(\cdot),ϕ}(\mathbb R^n)$ are also obtained, which are used to prove a trace theorem of $F_{p(\cdot),q(\cdot)}^{s(\cdot),ϕ}(\mathbb R^n)$. The authors also characterize the space $F_{p(\cdot),q(\cdot)}^{s(\cdot),ϕ}(\mathbb R^n)$ via Peetre maximal functions.

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arXiv2015

Besov-Type Spaces with Variable Smoothness and Integrability

In this article, the authors introduce Besov-type spaces with variable smoothness and integrability. The authors then establish their characterizations, respectively, in terms of $φ$-transforms in the sense of Frazier and Jawerth, smooth atoms or Peetre maximal functions, as well as a Sobolev-type embedding. As an application of their atomic characterization, the authors obtain a trace theorem of these variable Besov-type spaces.

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arXiv2014

Intrinsic Square Function Characterizations of Hardy Spaces with Variable Exponents

Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a measurable function satisfying some decay condition and some locally log-Hölder continuity. In this article, via first establishing characterizations of the variable exponent Hardy space $H^{p(\cdot)}(\mathbb R^n)$ in terms of the Littlewood-Paley $g$-function, the Lusin area function and the $g_λ^\ast$-function, the authors then obtain its intrinsic square function characterizations including the intrinsic Littlewood-Paley $g$-function, the intrinsic Lusin area function and the intrinsic $g_λ^\ast$-function. The $p(\cdot)$-Carleson measure characterization for the dual space of $H^{p(\cdot)}(\mathbb R^n)$, the variable exponent Campanato space $\mathcal{L}_{1,p(\cdot),s}(\mathbb R^n)$, in terms of the intrinsic function is also presented.

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