arXiv subjects

Xiangxing Tao

Publications and source records attributed to Xiangxing Tao.

11 recordsLinked to original sources

arXiv2026

Quantitative Bounds and Compactness for the Commutators of Area Integrals Associated with Self-adjoint Operators on Weighted $L^p$ and Morrey Spaces

Let $L$ be a non-negative self-adjoint operator, we consider some commutators generated by the BMO function $b$ and the area integral operator $S_H$ associated with the heat semigroup $\{e^{-tL}\}_{t>0}$ or the area integral operator $S_P$ associated with the Poisson semigroup $\{e^{-t\sqrt{L}}\}_{t>0}$. The strong-type estimates of these commutators on weighted $L^p$ spaces and weighted Morrey spaces are established. At the same time, we verified that these commutators are compact operators on weighted Morrey spaces.

math.CA↗

arXiv2024

Weak Factorizations of the Hardy Spaces in Terms of Multilinear Calderón-Zygmund Operators on Ball Banach Function Spaces

In this paper, our main purpose is to establish a weak factorization of the classical Hardy spaces in terms of a multilinear Calderón-Zygmund operator on the ball Banach function spaces. Furthermore, a new characterization of the BMO space via the boundedness of the commutator generated by the multilinear Calderón-Zygmund operator is also obtained. The results obtained in this paper have generality. As examples, we apply the above results to weighted Lebesgue space, variable Lebesgue space, Herz space, mixed-norm Lebesgue space, Lorentz space and so on.

math.FA↗

arXiv2024

An endpoint estimate for the maximal Calderón commutator with rough kernel

In this paper, the authors consider the endpoint estimates for the maximal Calderón commutator defined by $$T_{Ω,\,a}^*f(x)=\sup_{ε>0}\Big|\int_{|x-y|>ε}\frac{Ω(x-y)}{|x-y|^{d+1}} \big(a(x)-a(y)\big)f(y)dy\Big|,$$ where $Ω$ is homogeneous of degree zero, integrable on $S^{d-1}$ and has vanishing moment of order one, $a$ be a function on $\mathbb{R}^d$ such that $\nabla a\in L^{\infty}(\mathbb{R}^d)$. The authors prove that if $Ω\in L\log L(S^{d-1})$, then $T^*_{Ω,\,a}$ satisfies an endpoint estimate of $L\log\log L$ type.

math.CA↗

arXiv2023

A bilinear sparse domination for the maximal singular integral operators with rough kernels

Let $Ω$ be homogeneous of degree zero, integrable on $S^{d-1}$ and have mean value zero, $T_Ω$ be the homogeneous singular integral operator with kernel $\frac{Ω(x)}{|x|^d}$ and $T_Ω^*$ be the maximal operator associated to $T_Ω$. In this paper, the authors prove that if $Ω\in L^{\infty}(S^{d-1})$, then for all $r\in (1,\,\infty)$, $T_Ω^*$ enjoys a $(L^Φ,\,L^r)$ bilinear sparse domination with bound $Cr'\|Ω\|_{L^{\infty}(S^{d-1})}$, where $Φ(t)=t\log\log ({\rm e}^2+t)$.

math.CA↗

arXiv2022

$L^p(\mathbb{R}^d)$ boundedness for the Calderón commutator with rough kernel

Let $k\in\mathbb{N}$, $Ω$ be homogeneous of degree zero, integrable on $S^{d-1}$ and have vanishing moment of order $k$, $a$ be a function on $\mathbb{R}^d$ such that $\nabla a\in L^{\infty}(\mathbb{R}^d)$, and $T_{Ω,\,a;k}$ be the $d$-dimensional Calderón commutator defined by $$T_{Ω,\,a;k}f(x)={\rm p.\,v.}\int_{\mathbb{R}^d}\frac{Ω(x-y)}{|x-y|^{d+k}}\big(a(x)-a(y)\big)^kf(y){d}y.$$ In this paper, the authors prove that if $$\sup_{ζ\in S^{d-1}}\int_{S^{d-1}}|Ω(θ)|\log ^β \big(\frac{1}{|θ\cdotζ|}\big)dθ<\infty,$$ with $β\in(1,\,\infty]$, then for $\frac{2β}{2β-1}<p<2β$, $T_{Ω,\,a;\,k}$ is bounded on $L^p(\mathbb{R}^d)$.

math.CA↗

arXiv2022

Bilinear integral operator on Morrey-Banach spaces and its application

In this paper, we give the definability of bilinear singular and fractional integral operators on Morrey-Banach space, as well as their commutators and we prove the boundedness of such operators on Morrey-Banach spaces. Moreover, the necessary condition for BMO via the bounedness of bilinear commutators on Morrey-Banach space is also given. As a application of our main results, we get the necessary conditions for BMO via the bounedness of bilinear integral operators on weighted Morrey space and Morrey space with variable exponents. Finally, we obtain the boundedness of bilinear C-Z operator on Morrey space with variable exponents.

math.FA↗

arXiv2022

On the boundedness of non-standard rough singular integral operators

Let $Ω$ be homogeneous of degree zero, have vanishing moment of order one on the unit sphere $\mathbb {S}^{d-1}$($d\ge 2$). In this paper, our object of investigation is the following rough non-standard singular integral operator $$T_{Ω,\,A}f(x)={\rm p.\,v.}\int_{\mathbb{R}^d}\frac{Ω(x-y)}{|x-y|^{d+1}}\big(A(x)-A(y)-\nabla A(y)(x-y)\big)f(y){\rm d}y,$$ where $A$ is a function defined on $\mathbb{R}^d$ with derivatives of order one in ${\rm BMO}(\mathbb{R}^d)$. We show that $T_{Ω,\,A}$ enjoys the endpoint $L\log L$ type estimate and is $L^p$ bounded if $Ω\in L(\log L)^{2}(\mathbb{S}^{d-1})$. These resuts essentially improve the previous known results given by Hofmann for the $L^p$ boundedness of $T_{Ω,\,A}$ under the condition $Ω\in L^{q}(\mathbb {S}^{d-1})$ $(q>1)$, Hu and Yang for the endpoint weak $L\log L$ type estimates when $Ω\in {\rm Lip}_α(\mathbb{S}^{d-1})$ for some $α\in (0,\,1]$. Quantitative weighted strong and endpoint weak $L\log L$ type inequalities are proved whenever $Ω\in L^{\infty}(\mathbb {S}^{d-1})$. The analysis of the weighted results relies heavily on two bilinear sparse dominations of $T_{Ω,\,A}$ established herein.

math.CA↗

arXiv2021

An endpoint estimate for the commutators of singular integral operators with rough kernels

Let $Ω$ be homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{d-1}$, $T_Ω$ be the homogeneous singular integral operator with kernel $\frac{Ω(x)}{|x|^d}$ and $T_{Ω,\,b}$ be the commutator of $T_Ω$ with symbol $b$. In this paper, we prove that if $Ω\in L(\log L)^2(S^{d-1})$, then for $b\in {\rm BMO}(\mathbb{R}^d)$, $T_{Ω,\,b}$ satisfies an endpoint estimate of $L\log L$ type.

math.CA↗

arXiv2020

Weak Type Endpoint Estimates for the Commutators of Rough Singular Integral Operators

Let $Ω$ be homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{n-1}$, $T_Ω$ be the convolution singular integral operator with kernel $\frac{Ω(x)}{|x|^n}$. For $b\in{\rm BMO}(\mathbb{R}^n)$, let $T_{Ω,\,b}$ be the commutator of $T_Ω$. In this paper, by establishing suitable sparse dominations, the authors establish some weak type endpoint estimates of $L\log L$ type for $T_{Ω,\,b}$ when $Ω\in L^q(S^{n-1})$ for some $q\in (1,\,\infty]$.

math.CA↗

arXiv2019

A sparse domination for the Marcinkiewicz integral with rough kernel and applications

Let $Ω$ be homogeneous of degree zero, have mean value zero and integrable on the unit sphere, and $μ_Ω$ be the higher-dimensional Marcinkiewicz integral defined by $$μ_Ω(f)(x)= \Big(\int_0^\infty\Big|\int_{|x-y|\leq t}\frac{Ω(x-y)}{|x-y|^{n-1}}f(y)dy\Big|^2\frac{dt}{t^3}\Big)^{1/2}. $$ In this paper, the authors establish a bilinear sparse domination for $μ_Ω$ under the assumption $Ω\in L^{\infty}(S^{n-1})$. As applications, some quantitative weighted bounds for $μ_Ω$ are obtained.

math.CA↗

arXiv2018

Some estimates for the bilinear fractional integrals on the Morrey space

In this paper, we are interested in the following bilinear fractional integral operator $B\mathcal{I}_α$ defined by \[ B\mathcal{I}_α({f,g})(x)=\int_{% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{n}}\frac{f(x-y)g(x+y)}{|y|^{n-α}}dy, \] with $0< α<n$. We prove the weighted boundedness of $B\mathcal{I}_α$ on the Morrey type spaces. Moreover, an Olsen type inequality for $B\mathcal{I}_α$ is also given.

math.CA↗