arXiv subjects

Dinghuai Wang

Publications and source records attributed to Dinghuai Wang.

18 recordsLinked to original sources

arXiv2026

On the Two-Weight Problem for One-Sided Maximal Operators in Higher Dimensions

For the one-sided Hardy--Littlewood maximal operator $M_d^+$ on $\mathbb R^d$, the natural two-weight Muckenhoupt condition was shown by Sawyer in 1986 to characterize the weak $(p,p)$ inequality in dimension one, and by Forzani, Martín-Reyes and Ombrosi in 2011 in dimension two. At the endpoint $p=1$ in dimension three, Ombrosi and Nazarov have recently given negative answers to both the Fefferman--Stein-type question and the related weighted weak-type $(1,1)$ question \cite{OmbrosiNazarov} (personal communication). In this paper, we prove that the two-weight characterization fails for every $d\geq 3$ and $p>1$. More precisely, for every $1 3$.

math.CA↗

arXiv2026

Properties and applications of Lorentz--Muckenhoupt classes

In this paper, through the introduction of Lorentz--Muckenhoupt classes, we systematically investigate the boundedness of maximal operators on multiplier weighted Lorentz spaces. As applications, we give the characterization of the commutators of fractional integrals, which yields a partial answer to an open question proposed by D. Cruz-Uribe. Second, Hardy inequalities in Lorentz spaces are established, with the critical case $p=d$, in which the classical Hardy inequality fails. Finally, we apply the Lorentz estimates to fractional Schrödinger equations with singular potentials.

math.CA↗

arXiv2026

Pointwise endpoint limits for nonlocal operators

We establish two pointwise endpoint limits for nonlocal operators: one based on normalized integrals and a distributional limit based on weighted weak-type norms. As applications, the limits yield pointwise Bourgain--Brezis--Mironescu, Maz'ya--Shaposhnikova, Brezis--Seeger--Van Schaftingen--Yung and Gu--Yung formulas, together with higher-order and mean-oscillation variants. We then investigate endpoint limits for fractional powers generated by semigroups and for approximation processes, and derive sharp strong and weak endpoint estimates for operators in harmonic analysis. Finally, while Domínguez and Milman obtained weak-type estimates on product spaces for $p>1$ and left the endpoint $p=1$ open (see [Adv.~Math.~411 (2022), Paper~No.~108774, p.~22]), we prove a sharp pointwise limit in the parameter variable and obtain two--sided iterated weak-type estimates at $p=1$.

math.AP↗

arXiv2026

Caffarelli-Kohn-Nirenberg Inequalities in Weak Lebesgue Spaces

By employing harmonic analysis techniques, we derive weak-type Caffarelli-Kohn-Nirenberg inequalities under natural parameter conditions. A key feature of these weak-type versions is that they remain valid even at critical parameter values where the classical inequalities fail. As an important corollary, we obtain weak-type Hardy inequalities that hold true even in the critical dimension $d = p$. The methods developed here are sufficiently flexible to handle homogeneous, non-homogeneous and anisotropic weights, providing a unified approach to various endpoint cases in interpolation theory.

math.CA↗

arXiv2023

Endpoint theory for the compactness of commutators

In this paper, we establish a Minkowski-type inequality for weak Lebesgue space, which allows us to obtain a characterization of relative compactness in these spaces. Furthermore, we are the first to investigate the compactness results of commutators at the endpoint. The paper provides a comprehensive study of the compactness properties of commutators of Calderón-Zygmund operators in Hardy and $L^{1}(\mathbb{R}^n)$ type spaces. Additionally, we provide factorization theorems for Hardy spaces in terms of singular integral operators in the $L^1(\mathbb{R}^n)$ space.

math.FA↗

arXiv2021

Weak factorizations of the Hardy space in terms of multilinear fractional integral operator

We give a constructive proof of the factorization theorem for the classical Hardy space in terms of fractional integral operator. Moreover, the result is extended to the multilinear case and weighted case. As an application, we obtain the characterization of $BMO$ via the weighted boundedness of commutators of the multilinear fractional integral operator, without individual conditions on the weights class.

math.FA↗

arXiv2021

The factorizations of $H^ρ(\mathbb{R}^n)$ via multilinear Calderón-Zygmund operators on weighted Lebesgue spaces

We extend the recently much-studied Hardy factorization theorems to the weight case. The key point of this paper is to establish the factorization theorems without individual condition on the weight functions. As a direct application, we obtain the characterizations of $BMO(\mathbb{R}^n)$ space and Lipschitz spaces via the weighted boundedness of commutators of multilinear Calderón-Zygmund operators with the genuinely multilinear weights.

math.FA↗

arXiv2021

New function classes of Morrey-Campanato type and their applications

The aim of this paper is to introduce and investigative some new function classes of Morrey-Campanato type. Let $0 0}ρ^{-λ}\int_{Ω(x_{0},ρ)}\big|f(x)-|f|_{Ω(x_{0},ρ)}\big|^pdx<\infty,$$ where $Ω(x_{0},ρ)=Q(x_{0},ρ)\cap Ω$ and $Q(x,ρ)$ is denote the cube of $\mathbb{R}^n$. Some basic properties and characterizations of these classes are presented. If $0\leq λ<n$, the space is equivalent to related Morrey space. If $λ=n$, then $f \in \mathcal{\bar{L}}^{p,n}(Ω)$ if and only if $f\in BMO(Ω)$ with $f^{-}\in L^{\infty}(Ω)$, where $f^{-}=-\min\{0,f\}$. If $n<λ\leq n+p$, the $\mathcal{\bar{L}}^{p,λ}(Ω)$ functions establish an integral characterization of the nonnegative Hölder continue functions. As applications, this paper gives unified criterions on the necessity of bounded commutators of maximal functions.

math.FA↗

arXiv2021

The necessity theory for commutators of multilinear singular integral operators: the weighted case

In this paper, the necessity theory for commutators of multilinear singular integral operators on weighted Lebesgue spaces is investigated. The results relax the restriction of the weights class to the general multiple weights, which can be regarded as an essential improvement of \cite{ChafCruz2018,GLW2020}. Our approach elaborates on a commonly expanding the kernel locally by Fourier series, recovering many known results but yielding also numerous new ones. In particular, we answer the question about the necessity theory of the iterated commutators of the multilinear singular integral operators.

math.FA↗

arXiv2017

A note on commutator in the multilinear setting

Let $m\in \mathbb{N}$ and $\vec{b}=(b_{1},\cdots,b_{m})$ be a collection of locally integrable functions. It is proved that $b_{1},b_{2},\cdots, b_{m}\in BMO$ if and only if $$\sup_{Q}\frac{1}{|Q|^{m}}\int_{Q^{m}}\Big|\sum_{i=1}^{m}\big(b_{i}(x_{i})-(b_{i})_{Q}\big)\Big|d\vec{x}<\infty,$$ where $(b_{i})_{Q}=\frac{1}{|Q|}\int_{Q}b_{i}(x)dx$. As an application, we show that if the linear commutator of certain multilinear Calderón-Zygmund operator $[Σ\vec{b},T]$ is bounded from $L^{p_{1}}\times\cdots\times L^{p_{m}}$ to $L^{p}$ with $\sum_{i=1}^{m}1/p_{i}=1/p$ and $1<p,p_{1},\cdots,p_{m}<\infty$, then $b_{1},\cdots,b_{m}\in BMO$. Therefore, the result of Chaffee \cite{C} (or Li and Wick \cite{LW}) is extended to the general case.

math.CA↗

arXiv2017

Necessary and sufficient conditions for boundedness of commutators of bilinear Hardy-Littlewood maximal function

Let $\mathcal{M}$ be the bilinear Hardy-Littlewood maximal function and $\vec{b}=(b,b)$ be a collection of locally integrable functions. In this paper, the authors establish characterizations of the weighted {\rm BMO} space in terms of several different commutators of bilinear Hardy-Littlewood maximal function, respectively; these commutators include the maximal iterated commutator $\mathcal{M}_{Π\vec{b}}$, the maximal linear commutator $\mathcal{M}_{Σ\vec{b}}$, the iterated commutator $[Π\vec{b},\mathcal{M}]$ and the linear commutator $[Σ\vec{b},\mathcal{M}]$.

math.FA↗

arXiv2017

Central BMO spaces with variable exponent

In this paper, the central BMO spaces with variable exponent are introduced. As an application, we characterize these spaces by the boundedness of commutators of Hardy operator and its dual operator on variable Lebesgue spaces. The boundedness of vector-valued commutators on Herz spaces with variable exponent are also considered.

math.FA↗

arXiv2017

Characterizations of weighted BMO space and its application

In this paper, we prove that the weighted BMO space as follows $${\rm BMO}^{p}(ω)=\Big\{f\in L^{1}_{\rm loc}:\sup_{Q}\|χ_{Q}\|^{-1}_{L^{p}(ω)}\big\|(f-f_{Q})ω^{-1}χ_{Q}\big\|_{L^{p}(ω)}<\infty\Big\}$$ is independent of the scale $p\in (0,\infty)$ in sense of norm when $ω\in A_{1}$. Moreover, we can replace $L^{p}(ω)$ by $L^{p,\infty}(ω)$. As an application, we characterize this space by the boundedness of the bilinear commutators $[b,T]_{j} (j=1,2)$, generated by the bilinear convolution type Calderón-Zygmund operators and the symbol $b$, from $L^{p_{1}}(ω)\times L^{p_{2}}(ω)$ to $L^{p}(ω^{1-p})$ with $1<p_{1},p_{2}<\infty$, $1/p=1/p_{1}+1/p_{2}$ and $ω\in A_{1}$. Thus we answer the open problem proposed in \cite{C} affirmatively.

math.FA↗

arXiv2017

Boundedness and Compactness of commutator of Hardy-Littlewood maximal operator

We study the mapping property of the commutator of bilinear Hardy-Littlewood maximal operator in homogeneous Triebel-Lizorkin space. We also show that the commutator of bilinear Hardy-Littlewood maximal operator is a compact operator acting on product of Lebesgue spaces. The results are new even in the linear case.

math.FA↗

arXiv2017

Sharp estimates for commutators of bilinear operators on Morrey type spaces

Denote by $T$ and $I_α$ the bilinear Calderón-Zygmund operators and bilinear fractional integrals, respectively. In this paper, it is proved that if $b_{1},b_{2}\in {\rm CMO}$ (the {\rm BMO}-closure of $C^{\infty}_{c}(\mathbb{R}^n)$), $[Π\vec{b},T]$ and $[Π\vec{b},I_α]$ $(\vec{b}=(b_{1},b_{2}))$ are all the compact operators from $\mathcal{M}^{p_{0}}_{\vec{P}}$ (the norm of $\mathcal{M}^{p_{0}}_{\vec{P}}$ is strictly smaller than $2-$fold product of the Morrey norms) to $M^{q_{0}}_{q}$ for some suitable indexes $p_{0},p_{1},p_{2}$ and $q_{0},q$. Specially, we also show that if $b_{1}=b_{2}$, then $b_{1}, b_{2}\in {\rm CMO}$ is necessary for the compactness of $[Π\vec{b},I_α]$ on Morrey space.

math.FA↗

arXiv2016

Characterizations of the BMO and Lipschitz spaces via commutators on weak Lebesgue and Morrey spaces

We prove that the weak Morrey space $WM^{p}_{q}$ is contained in the Morrey space $M^{p}_{q_{1}}$ for $1\leq q_{1}< q\leq p<\infty$. As applications, we show that if the commutator $[b,T]$ is bounded from $L^p$ to $L^{p,\infty}$ for some $p\in (1,\infty)$, then $b\in \mathrm{BMO}$, where $T$ is a Calderón-Zygmund operator. Also, for $1<p\leq q<\infty$, $b\in \mathrm{BMO}$ if and only if $[b,T]$ is bounded from $M^{p}_{q}$ to $WM_{q}^{p}$. For $b$ belonging to Lipschitz class, we obtain similar results.

math.FA↗

arXiv2016

Characterization of CMO via compactness of the commutators of bilinear fractional integral operators

Let $I_α$ be the bilinear fractional integral operator, $B_α$ be a more singular family of bilinear fractional integral operators and $\vec{b}=(b,b)$. Bényi et al. in \cite{B1} showed that if $b\in {\rm CMO}$, the {\rm BMO}-closure of $C^{\infty}_{c}(\mathbb{R}^n)$, the commutator $[b,B_α]_{i}(i=1,2)$ is a separately compact operator. In this paper, it is proved that $b\in {\rm CMO}$ is necessary for $[b,B_α]_{i}(i=1,2)$ is a compact operator. Also, the authors characterize the compactness of the {\bf iterated} commutator $[Π\vec{b},I_α]$ of bilinear fractional integral operator. More precisely, the commutator $[Π\vec{b},I_α]$ is a compact operator if and only if $b\in {\rm CMO}$.

math.FA↗

arXiv2016

Another characterizations of Muckenhoupt $A_{p}$ class

This manuscript addresses Muckenhoupt $A_{p}$ weight theory in connection to Morrey and BMO spaces. It is proved that $ω$ belongs to Muckenhoupt $A_{p}$ class, if and only if Hardy-Littlewood maximal function $M$ is bounded from weighted Lebesgue spaces $L^{p}(ω)$ to weighted Morrey spaces $M^{p}_{q}(ω)$ for $1<q< p<\infty$. As a corollary, if $M$ is (weak) bounded on $M^{p}_{q}(ω)$, then $ω\in A_{p}$. The $A_{p}$ condition also characterizes the boundedness of the Riesz transform $R_{j}$ and convolution operators $T_ε$ on weighted Morrey spaces. Finally, we show that $ω\in A_{p}$ if and only if $ω\in \mathrm{BMO}^{p'}(ω)$ for $1\leq p< \infty$ and $1/p+1/p'=1$.

math.FA↗