arXiv subjects

Qingying Xue

Publications and source records attributed to Qingying Xue.

At least 37 records · Page 2Linked to original sources

arXiv2021

The limiting weak type behaviors and The lower bound for a new weak $L\log L$ type norm of strong maximal operators

It is well known that the weak ($1,1$) bounds doesn't hold for the strong maximal operators, but it still enjoys certain weak $L\log L$ type norm inequality. Let $Φ_n(t)=t(1+(\log^+t)^{n-1})$ and the space $L_{Φ_n}({\mathbb R^{n}})$ be the set of all measurable functions on ${\mathbb R^{n}}$ such that $\|f\|_{L_{Φ_n}({\mathbb R^{n}})} :=\|Φ_n(|f|)\|_{L^1({\mathbb R^{n}})}<\infty$. In this paper, we introduce a new weak norm space $L_{Φ_n}^{1,\infty}({\mathbb R^{n}})$, which is more larger than $L^{1,\infty}({\mathbb R^{n}})$ space, and establish the correspondng limiting weak type behaviors of the strong maximal operators. As a corollary, we show that $ \max\{{2^n}{((n-1)!)^{-1}},1\}$ is a lower bound for the best constant of the $L_{Φ_n}\to L_{Φ_n}^{1,\infty}$ norm of the strong maximal operators. Similar results have been extended to the multilinear strong maximal operators.

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arXiv2021

On weighted Compactness of Commutator of semi-group maximal function associated to Schrödinger operators

Let $\mathcal{T}^*$ be the semi-group maximal function associated to the Schrödinger operator $-Δ+V(x)$ with $V$ satisfying an appropriate reverse Hölder inequality. In this paper, we show that the commutator of $\mathcal{T}^*$ is a compact operator on $L^p(w)$ for $1<p<\infty$ if $b\in \text{CMO}_θ(ρ)(\mathbb{R}^n)$ and $w\in A_p^{ρ,θ}(\mathbb{R}^n)$. Here $\text{ CMO}_θ(ρ)(\mathbb{R}^n)$ denotes the closure of $\mathcal{C}_c^\infty(\mathbb{R}^n)$ in the $\text{BMO}_θ(ρ)(\mathbb{R}^n)$ (which is larger than the classical $\text{BMO}(\mathbb{R}^n)$ space) topology. The space where $b$ belongs and the weighs class $w$ belongs are more larger than the usual $\text{CMO}(\mathbb{R}^n)$ space and the Muckenhoupt $A_p$ weights class, respectively.

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arXiv2020

On weighted Compactness of commutators of bilinear maximal Calderón-Zygmund singular integral operators

Let $T$ be a bilinear Calderón-Zygmund singular integral operator and $T^*$ be its corresponding truncated maximal operator. For any $b\in\text{BMO}(\mathbb {R}^n)$ and $\vec{b}=(b_1,\ b_2)\in\text{BMO}(\mathbb {R}^n)\times\text {BMO}(\mathbb{R}^n)$, let $T^*_{b,j}$ (j=1,2), $T^*_{\vec{b}}\ $ be the commutators in the j-th entry and the iterated commutators of $T^*$, respectively. In this paper, for all $1<p_1,p_2<\infty$, $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$, we show that $T^*_{b,j}$ and $T^*_{\vec{b}}$ are compact operators from $L^{p_1}(w_1)\times L^{p_2}(w_2)$ to $L^p(v_{\vec{w}})$, if $b,b_1,b_2\in{\rm CMO}(\mathbb{R}^n)$ and $\vec{w}=(w_1,w_2)\in A_{\vec{p}}$, $v_{\vec{w}}=w_1^{p/p_1}w_2^{p/p_2}$. Here ${\rm CMO}(\mathbb{R}^n)$ denotes the closure of $\mathcal{C}_c^\infty(\mathbb{R}^n)$ in the ${\rm BMO}(\mathbb{R}^n)$ topology and $A_{\vec{p}}$ is the multiple weights class.

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arXiv2020

Limiting weak-type behavior for rough bilinear operators

Let $Ω_1,Ω_2$ be functions of homogeneous of degree $0$ and $\vecΩ=(Ω_1,Ω_2)\in L\log L(\mathbb{S}^{n-1})\times L\log L(\mathbb{S}^{n-1})$. In this paper, we investigate the limiting weak-type behavior for bilinear maximal function $M_{\vecΩ}$ and bilinear singular integral $T_{\vecΩ}$ associated with rough kernel $\vecΩ$. For all $f,g\in L^1(\mathbb{R}^n)$, we show that $$\lim_{λ\to 0^+}λ|\big\{ x\in\mathbb{R}^n:M_{\vecΩ}(f_1,f_2)(x)>λ\big\}|^2 = \frac{\|Ω_1Ω_2\|_{L^{1/2}(\mathbb{S}^{n-1})}}{ω_{n-1}^2}\prod\limits_{i=1}^2\| f_i\|_{L^1}$$ and $$\lim_{λ\to 0^+}λ|\big\{ x\in\mathbb{R}^n:| T_{\vecΩ}(f_1,f_2)(x)|>λ\big\}|^{2} = \frac{\|Ω_1Ω_2\|_{L^{1/2}(\mathbb{S}^{n-1})}}{n^2}\prod\limits_{i=1}^2\| f_i\|_{L^1}.$$ As consequences, the lower bounds of weak-type norms of $M_{\vecΩ}$ and $T_{\vecΩ}$ are obtained. These results are new even in the linear case. The corresponding results for rough bilinear fractional maximal function and fractional integral operator are also discussed.

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arXiv2020

Sparse domination and weighted estimates for rough bilinear singular integrals

Let $r>\frac{4}{3}$ and let $Ω\in L^{r}(\mathbb{S}^{2n-1})$ have vanishing integral. We show that the bilinear rough singular integral $$T_Ω(f,g)(x)= \textrm{p.v.} \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{Ω((y,z)/|(y,z)|)}{|(y,z)|^{2n}}f(x-y)g(x-z)\,dydz,$$ satisfies a sparse bound by $(p,p,p)$-averages, where $p$ is bigger than a certain number explicitly related to $r$ and $n$. As a consequence we deduce certain quantitative weighted estimates for bilinear homogeneous singular integrals associated with rough homogeneous kernels.

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arXiv2020

Multilinear Littlewood-Paley-Stein Operators on Non-homogeneous Spaces

Let $κ\ge 2, λ> 1$ and define the multilinear Littlewood-Paley-Stein operators by $$g_{λ,μ}^*(\vec{f})(x) = \bigg(\iint_{\mathbb{R}^{n+1}_{+}} \vartheta_t(x, y) \bigg|\int_{\mathbb{R}^{n κ}} s_t(y,\vec{z}) \prod_{i=1}^κ f_i(z_i) \ dμ(z_i)\bigg|^2 \frac{dμ(y) dt}{t^{m+1}}\bigg)^{\frac12}, $$ where $\vartheta_t(x, y)=\big(\frac{t}{t + |x - y|}\big)^{m λ}$. In this paper, our main aim is to investigate the boundedness of $g_{λ,μ}^*$ on non-homogeneous spaces. By means of probabilistic and dyadic techniques, together with non-homogeneous analysis, we show that $g_{λ,μ}^*$ is bounded from $L^{p_1}(μ) \times \cdots \times L^{p_κ}(μ)$ to $L^p(μ)$ under certain weak type assumptions. The multilinear non-convolution type kernels $s_t$ only need to satisfy some weaker conditions than the standard conditions of multilinear Calderón-Zygmund type kernels and the measures $μ$ are only assumed to be upper doubling measures (non-doubling). The above results are new even under Lebesgue measures. This was done by considering first a sufficient condition for the strong type boundedness of $g_{λ,μ}^*$ based on an endpoint assumption, and then directly deduce the strong bound on a big piece from the weak type assumptions.

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arXiv2020

XMO and Weighted Compact Bilinear Commutators

To study the compactness of bilinear commutators of certain bilinear Calderón--Zygmund operators which include (inhomogeneous) Coifman--Meyer bilinear Fourier multipliers and bilinear pseudodifferential operators as special examples, Torres and Xue [Rev. Mat. Iberoam. 36 (2020), 939--956] introduced a new subspace of BMO$\,(\mathbb{R}^n)$, denoted by XMO$\,(\mathbb{R}^n)$, and conjectured that it is just the space VMO$\,(\mathbb{R}^n)$ introduced by D. Sarason. In this article, the authors give a negative answer to this conjecture by establishing an equivalent characterization of XMO$\,(\mathbb{R}^n)$, which further clarifies that XMO$\,(\mathbb{R}^n)$ is a proper subspace of VMO$\,(\mathbb{R}^n)$. This equivalent characterization of XMO$\,(\mathbb{R}^n)$ is formally similar to the corresponding one of CMO$\,(\mathbb{R}^n)$ obtained by A. Uchiyama, but its proof needs some essential new techniques on dyadic cubes as well as some exquisite geometrical observations. As an application, the authors also obtain a weighted compactness result on such bilinear commutators, which optimizes the corresponding result in the unweighted setting.

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arXiv2019

On the composition for rough singular integral operators

In this paper, we investigate the behavior of the bounds of the composition for rough singular integral operators on the weighted space. More precisely, we obtain the quantitative weighted bounds of the composite operator for two singular integral operators with rough homogeneous kernels on $L^p(\mathbb{R}^d,\,w)$, $p\in (1,\,\infty)$, which is smaller than the product of the quantitative weighted bounds for these two rough singular integral operators. Moreover, at the endpoint $p=1$, the $L\log L$ weighted weak type bound is also obtained, which has interests of its own in the theory of rough singular integral even in the unweighted case.

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arXiv2019

Weighted Fréchet-Kolmogorov theorem and compactness of vector-valued multilinear operators

In this paper, we gave a weighted compactness theory for the generalized commutators of vecotor-valued multilinear Calderón-Zygmund operators. This was done by establishing a weighted Fréchet-Kolmogorov theorem, which holds for weights not merely in $A_\infty$. This weighted theory also extends the previous known unsatisfactory results in the terms of relaxing the index to the natural range. As consequences, we not only obtained the weighted compactness theory for the commutators of multilinear Calderón-Zygmund operators, but also extended the same results to the commutators of multilinear Littlewood-Paley type operators. In addition, the generalized commutators contain almost all the commutators formerly considered in this literature.

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arXiv2018

Regularity and continuity of local Multilinear Maximal type operator

This paper will be devoted to study the regularity and continuity properties of the following local multilinear fractional type maximal operators, $$\mathfrak{M}_{α,Ω}(\vec{f})(x)=\sup\limits_{0<r<{\rm dist}(x,Ω^c)}\frac{r^α}{|B(x,r)|^m}\prod\limits_{i=1}^m\int_{B(x,r)}|f_i(y)|dy,\quad \hbox{for \ }0\leqα<mn,$$ where $Ω$ is a subdomain in $\mathbb{R}^n$, $Ω^c=\mathbb{R}^n\backslashΩ$ and $B(x,r)$ is the ball in $\mathbb{R}^n$ centered at $x$ with radius $r$. Several new pointwise estimates for the derivative of the local multilinear maximal function $\mathfrak{M}_{0,Ω}$ and the fractional maximal functions $\mathfrak{M}_{α,Ω}$ $(0<α< mn)$ will be presented. These estimates will not only enable us to establish certain norm inequalities for these operators in Sobolev spaces, but also give us the opportunity to obtain the bounds of these operators on the Sobolev space with zero boundary values.

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arXiv2018

Regularity and continuity of the multilinear strong maximal operators

Let $m\ge 1$, in this paper, our object of investigation is the regularity and and continuity properties of the following multilinear strong maximal operator $${\mathscr{M}}_{\mathcal{R}}(\vec{f})(x)=\sup_{\substack{R \ni x R\in\mathcal{R}}}\prod\limits_{i=1}^m\frac{1}{|R|}\int_{R}|f_i(y)|dy,$$ where $x\in\mathbb{R}^d$ and $\mathcal{R}$ denotes the family of all rectangles in $\mathbb{R}^d$ with sides parallel to the axes. When $m=1$, denote $\mathscr{M}_{\mathcal{R}}$ by $\mathcal {M}_{\mathcal{R}}$.Then, $\mathcal {M}_{\mathcal{R}}$ coincides with the classical strong maximal function initially studied by Jessen, Marcinkiewicz and Zygmund. We showed that ${\mathscr{M}}_{\mathcal{R}}$ is bounded and continuous from the Sobolev spaces $W^{1,p_1}(\mathbb{R}^d)\times \cdots\times W^{1,p_m}(\mathbb{R}^d)$ to $W^{1,p} (\mathbb{R}^d)$, from the Besov spaces $B_{s}^{p_1,q} (\mathbb{R}^d)\times\cdots\times B_s^{p_m,q}(\mathbb{R}^d)$ to $B_s^{p,q}(\mathbb{R}^d)$, from the Triebel-Lizorkin spaces $F_{s}^{p_1,q}(\mathbb{R}^d)\times\cdots\times F_s^{p_m,q}(\mathbb{R}^d)$ to $F_s^{p,q}(\mathbb{R}^d)$. As a consequence, we further showed that ${\mathscr{M}}_{\mathcal{R}}$ is bounded and continuous from the fractional Sobolev spaces $W^{s,p_1}(\mathbb{R}^d)\times \cdots\times W^{s,p_m}(\mathbb{R}^d)$ to $W^{s,p}(\mathbb{R}^d)$ for $0<s\leq 1$ and $1<p<\infty$. As an application, we obtain a weak type inequality for the Sobolev capacity, which can be used to prove the $p$-quasicontinuity of $\mathscr{M}_{\mathcal{R}}$. The discrete type of the strong maximal operators has also been considered. We showed that this discrete type of the maximal operators enjoys somewhat unexpected regularity properties.

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arXiv2017

A characterization of two weight norm inequality for Littlewood-Paley $g_λ^{*}$-function

Let $n\ge 2$ and $g_λ^{*}$ be the well-known high dimensional Littlewood-Paley function which was defined and studied by E. M. Stein, \begin{align*} g_λ^{*}(f)(x) =\bigg(\iint_{\mathbb R^{n+1}_{+}} \Big(\frac{t}{t+|x-y|}\Big)^{nλ} |\nabla P_tf(y,t)|^2 \frac{dy dt}{t^{n-1}}\bigg)^{1/2}, \ \quad λ> 1, \end{align*} where $P_tf(y,t)=p_t*f(y)$, $p_t(y)=t^{-n}p(y/t)$ and $p(x) = (1+|x|^2)^{-(n+1)/2}$, $\nabla =(\frac{\partial}{\partial y_1},\ldots,\frac{\partial}{\partial y_n},\frac{\partial}{\partial t})$. In this paper, we give a characterization of two-weight norm inequality for $g_λ^{*}$-function. We show that, $\big\| g_λ^{*}(f σ) \big\|_{L^2(w)} \lesssim \big\| f \big\|_{L^2(σ)}$ if and only if the two-weight Muchenhoupt $A_2$ condition holds, and a testing condition holds : \begin{align*} \sup_{Q : cubes \ in \mathbb R^n} \frac{1}{σ(Q)} \int_{\mathbb R^n} \iint_{\widehat{Q}} \Big(\frac{t}{t+|x-y|}\Big)^{nλ}|\nabla P_t(\mathbf{1}_Q σ)(y,t)|^2 \frac{w dx dt}{t^{n-1}} dy < \infty, \end{align*} where $\widehat{Q}$ is the Carleson box over $Q$ and $(w, σ)$ is a pair of weights. We actually prove this characterization for $g_λ^{*}$-function associated with more general fractional Poisson kernel $p^α(x) = (1+|x|^2)^{-{(n+α)}/{2}}$. Moreover, the corresponding results for intrinsic $g_λ^*$-function are also presented.

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arXiv2017

The Fefferman-Stein type inequalities for the multilinear strong maximal functions

Let $\vecω=( ω_{1},...,ω_{m})$ be a multiple weight and $\{Ψ_{j}\}^{m}_{j=1}$ be a sequence of Young functions. Let $\mathcal{M}_{\mathcal{R}}^{\vecΨ}$ be the multilinear strong maximal function with Orlicz norms which is defined by $$\mathcal{M}_{\mathcal{R}}^{\vecΨ}(\vec{f})(x)=\sup_{R\in \mathcal{R},R\ni x}\prod^{m}_{j=1}\|f_{j}\|_{Ψ_{j},R}$$ where the supremum is taken over all rectangles with sides parallel to the coordinate axes. If $Ψ_j(t)=t$, then $\mathcal{M}_{\mathcal{R}}^{\vec{t}}$ coincides with the multilinear strong mximal function $\mathcal{M}_{\mathcal{R}}$ defined and studied by Grafakos et al. In this paper, we first investigated the Fefferman-Stein type inequality for $\mathcal{M}_{\mathcal{R}}^{\vecΨ}$ when $\vecω$ satisfies the $A_{\infty,\mathcal{R}}$ condition. Then, for arbitrary $\vecω\geq 0$( each $ ω_{j}\ge 0$), the Fefferman-Stein type inequality for the multilinear strong maximal function $\mathcal{M}_{\mathcal{R}} $ associated with rectangles will be given.

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arXiv2017

On Some properties of dyadic operators

In this paper, the objects of our investigation are some dyadic operators, including dyadic shifts, multilinear paraproducts and multilinear Haar multipliers. We mainly focus on the continuity and compactness of these operators. First, we consider the continuity properties of these operators. Then, by the Fréchet-Kolmogorov-Riesz-Tsuji theorem, the non-compactness properties of these dyadic operators will be studied. Moreover, we show that their commutators are compact with \textit{CMO} functions, which is quite different from the non-compaceness properties of these dyadic operators. These results are similar to those for Calderón-Zygmund singular integral operators.

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arXiv2016

$L^p$ boundedness of non-homogeneous Littlewood-Paley $g^*_{λ,μ}$-function with non-doubling measures

It is well-known that the $L^p$ boundedness and weak $(1,1)$ estiamte $(λ>2)$ of the classical Littlewood-Paley $g_λ^{*}$-function was first studied by Stein, and the weak $(p,p)$ $(p>1)$ estimate was later given by Fefferman for $λ=2/p$. In this paper, we investigated the $L^p(μ)$ boundedness of the non-homogeneous Littlewood-Paley $g_{λ,μ}^{*}$-function with non-convolution type kernels and a power bounded measure $μ$: $$ g_{λ,μ}^*(f)(x) = \bigg(\iint_{{\mathbb R}^{n+1}_{+}} \Big(\frac{t}{t + |x - y|}\Big)^{m λ} |θ_t^μf(y)|^2 \frac{dμ(y) dt}{t^{m+1}}\bigg)^{1/2},\ x \in {\mathbb R}^n,\ λ> 1, $$ where $θ_t^μf(y) = \int_{{\mathbb R}^n} s_t(y,z) f(z) dμ(z)$, and $s_t$ is a non-convolution type kernel. Based on a big piece prior boundedness, we first gave a sufficient condition for the $L^p(μ)$ boundedness of $g_{λ,μ}^*$. This was done by means of the non-homogeneous good lambda method. Then, using the methods of dyadic analysis, we demonstrated a big piece global $Tb$ theorem. Finally, we obtaind a sufficient and necessary condition for $L^p(μ)$ boundedness of $g_{λ,μ}^*$-function. It is worth noting that our testing conditions are weak $(1,1)$ type with respect to measures.

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arXiv2016

Boundedness of Bi-parameter Littlewood-Paley operators on product Hardy space

Let $n_1,n_2\ge 1, λ_1>1$ and $λ_2>1$. For any $x=(x_1,x_2) \in \mathbb {R}^n\times\mathbb{R}^m$, let $g$ and $g_{\vecλ}^*$ be the bi-parameter Littlewood-Paley square functions defined by \begin{align*} g(f)(x)= \Big(\int_0^{\infty}\int_0^{\infty}|θ_{t_1,t_2} f(x_1,x_2)|^2 \frac{dt_1}{t_1} \frac{dt_2}{t_2} \Big)^{1/2}, \hbox{and} \end{align*} $$ g_{\vecλ}^*(f)(x) = \Big(\iint_{\mathbb{R}^{m+1}_{+}} \iint_{\mathbb{R}^{n+1}_{+}} \prod_{i=1}^2\Big(\frac{t_1}{t_i + |x_i - y_i|}\Big)^{n_i λ_i} |θ_{t_1,t_2} f(y_1,y_2)|^2 \frac{dy_1 dt_1}{t_1^{n+1}} \frac{dy_2 dt_2}{t_2^{m+1}} \Big)^{1/2}, $$ \noindent where $θ_{t_1,t_2} f(x_1, x_2) = \iint_{\mathbb{R}^n\times\mathbb{R}^m} s_{t_1,t_2}(x_1,x_2,y_1,y_2)f(y_1,y_2) dy_1dy_2$. It is known that the $L^2$ boundedness of bi-parameter $g$ and $g_{\vecλ}^*$ have been established recently by Martikainen, and Cao, Xue, respectively. In this paper, under certain structure conditions assumed on the kernel $s_{t_1,t_2},$ we show that both $g$ and $g_{\vecλ}^*$ are bounded from product Hardy space $H^1(\mathbb{R}^n\times\mathbb{R}^m)$ to $L^1(\mathbb{R}^n\times\mathbb{R}^m)$. As consequences, the $L^p$ boundedness of $g$ and $g_{\vecλ}^*$ will be obtained for $1<p<2$.

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arXiv2016

A Revisit on Commutators of linear and bilinear Fractional Integral Operator

Let $I_α$ be the linear and $\mathcal{I}_α$ be the bilinear fractional integral operators. In the linear setting, it is known that the two-weight inequality holds for the first order commutators of $I_α$. But the method can't be used to obtain the two weighted norm inequality for the higher order commutators of $I_α$. In this paper, we first give an alternative proof for the first order commutators of $I_α$. This new approach allows us to consider the higher order commutators. This was done by showing that the commutator $[b,I_α]$ can be represented as a finite linear combination of some paraproducts. Then, by using the Cauchy integral theorem, we show that the two-weight inequality holds for the higher order commutators of $I_α$. In the bilinear setting, we present a dyadic proof for the characterization between $BMO$ and the boundedness of $[b,\mathcal{I}_α]$. Moreover, some bilinear paraproducts are also treated in order to obtain the boundedness of $[b,\mathcal{I}_α]$.

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arXiv2016

Non-homogeneous $Tb$ Theorem for Bi-parameter $g$-Function

The main result of this paper is a bi-parameter $Tb$ theorem for Littlewood-Paley $g$-function, where $b$ is a tensor product of two pseudo-accretive functions. Instead of the doubling measure, we work with a product measure $μ= μ_n \times μ_m$, where the measures $μ_n$ and $μ_m$ are only assumed to be upper doubling. The main techniques of the proof include a bi-parameter $b$-adapted Haar function decomposition and an averaging identity over good double Whitney regions. Moreover, the non-homogeneous analysis and probabilistic methods are used again.

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