arXiv subjects

Qianjun He

Publications and source records attributed to Qianjun He.

11 recordsLinked to original sources

arXiv2023

New fractional type weights and the boundedness of some operators

Two classes of fractional type variable weights are established in this paper. The first kind of weights ${A_{\vec p( \cdot ),q( \cdot )}}$ are variable multiple weights, which are characterized by the weighted variable boundedness of multilinear fractional type operators, called multilinear Hardy--Littlewood--Sobolev theorem on weighted variable Lebesgue spaces. Meanwhile, the weighted variable boundedness for the commutators of multilinear fractional type operators are also obtained. This generalizes some known work, such as Moen (2009), Bernardis--Dalmasso--Pradolini (2014), and Cruz-Uribe--Guzm\'an (2020). Another class of weights ${{\mathbb{A}}_{p( \cdot ),q(\cdot)}}$ are variable matrix weights that also characterized by certain fractional type operators. This generalize some previous results on matrix weights ${{\mathbb{A}}_{p( \cdot )}}$.

math.CA↗

arXiv2023

Characterizations for multilinear fractional maximal and integral operators and their commutators on generalized weighted Morrey spaces and applications

This paper is devoted to studying the boundedness of multilinear operartors and their commutators on generalized weighted Morrey spaces, which includes multilinear fractional maximal operator and multilinear fractional integral operator. Moreover, we show that two different characterizations for the boundedness of multilinear fractional maximal operators and their commutators on generalized weighted Morrey spaces under different conditions. As some inportant applications, we give the boundedness of multilinear fractional integral operator on generalized weighted Besov-Morrey spaces and also obtain two embedding theorems as well as apriori estimates for the sub-Laplacian $\mathcal L$.

math.CA↗

arXiv2021

On weighted compactness of commutators of Schr\"{o}dinger operators

Let $\mathcal{L}=-\Delta+\mathit{V}(x)$ be a Schr\"{o}dinger operator, where $\Delta$ is the Laplacian operator on $\mathbb{R}^{d}$ $(d\geq 3)$, while the nonnegative potential $\mathit{V}(x)$ belongs to the reverse H\"{o}lder class $B_{q}, q>d/2$. In this paper, we study weighted compactness of commutators of some Schr\"{o}dinger operators, which include Riesz transforms, standard Calder\'{o}n-Zygmund operatos and Littlewood-Paley functions. These results generalize substantially some well-know results.

math.CA↗

arXiv2021

Two-weight Norm Inequalities for Local Fractional Integrals on Gaussian Measure Spaces

In this paper, the authors establish the two-weight boundedness of the local fractional maximal operators and local fractional integrals on Gaussian measure spaces associated with the local weights. More precisely, the authors first obtain the two-weight weak-type estimate for the local-$a$ fractional maximal operators of order $\alpha$ from $L^{p}(v)$ to $L^{q,\infty}(u)$ with $1\leq p\leq q<\infty$ under a condition of $(u,v)\in \bigcup_{b'>a} A_{p,q,\alpha}^{b'}$, and then obtain the two-weight weak-type estimate for the local fractional integrals. In addition, the authors obtain the two-weight strong-type boundedness of the local fractional maximal operators under a condition of $(u,v)\in\mathscr{M}_{p,q,\alpha}^{6a+9\sqrt{d}a^2}$ and the two-weight strong-type boundedness of the local fractional integrals. These estimates are established by the radialization method and dyadic approach.

math.CA↗

arXiv2021

Some Weighted Estimates on Gaussian Measure Spaces

In this paper, we obtain the weighted boundedness for the local multi(sub)linear Hardy-Littlewood maximal operators and local multilinear fractional integral operators associated with the local Muckenhoupt weights on Gaussian measure spaces. We deal with these problems by introducing a new pointwise equivalent "radial" definitions of these local operators. Moreover using a similar approach, we also get the weighted boundedness for the local fractional maximal operators with rough kernel and local fractional integral operators with rough kernel on Gaussian measure spaces.

math.CA↗

arXiv2019

Weighted estimates for bilinear fractional integral operators and their commutators on Morrey spaces

This paper mainly dedicates to prove a plethora of weighted estimates on Morrey spaces for bilinear fractional integral operators and their general commutators with BMO functions of the form $$B_{\alpha}(f,g)(x)=\int_{\mathbb{R}^{n}}\frac{f(x-y)g(x+y)}{|y|^{n-\alpha}}dy,\qquad 0<\alpha<n.$$ We also prove some maximal function control theorems for these operators, that is, the weighted Morrey norm is bounded by the weighted Morrey norm of a natural maximal operator when the weight belongs to $A_{\infty}$. As a corollary, some new weighted estimates for the bilinear maximal function associated to the bilinear Hilbert transform are obtained. Furthermore, we formulate a bilinear version of Stein-Weiss inequality on Morrey spaces for fractional integrals.

math.CA↗

arXiv2018

Bilinear fractional integral operators on Morrey spaces

We prove a plethora of boundedness property of the Adams type for bilinear fractional integral operators of the form $$B_{\alpha}(f,g)(x)=\int_{\mathbb{R}^{n}}\frac{f(x-y)g(x+y)}{|y|^{n-\alpha}}dy,\qquad 0<\alpha<n.$$ For $1<t\leq s<\infty$, we prove the non-weighted case through the known Adams type result. And we show that these results of Adams type is optimal. For $0<t\leq s<\infty$ and $0<t\leq1$, we obtain new result of a weighted theory describing Morrey boundedness of above form operators if two weights $(v,\vec{w})$ satisfy $$ [v,\vec{w}]_{t,\vec{q}/{a}}^{r,as}=\mathop{\sup_{Q,Q^{\prime}\in\mathscr{D}}}_{Q\subset Q^{\prime}}\left(\frac{|Q|}{|Q^{\prime}|}\right)^{\frac{1-s}{as}}|Q^{\prime}|^{\frac{1}{r}}\left(\fint_{Q}v^{\frac{t}{1-t}}\right)^{\frac{1-t}{t}}\prod_{i=1}^{2}\left(\fint_{Q^{\prime}}w_{i}^{-(q_{i}/a)^{\prime}}\right)^{\frac{1}{(q_{i}/a)^{\prime}}}<\infty,\,\,\, 0<t<s<1 $$ and $$ [v,\vec{w}]_{t,\vec{q}/{a}}^{r,as}:=\mathop{\sup_{Q,Q^{\prime}\in\mathscr{D}}}_{Q\subset Q^{\prime}}\left(\frac{|Q|}{|Q^{\prime}|}\right)^{\frac{1-as}{as}}|Q^{\prime}|^{\frac{1}{r}}\left(\fint_{Q}v^{\frac{t}{1-t}}\right)^{\frac{1-t}{t}}\prod_{i=1}^{2}\left(\fint_{Q^{\prime}}w_{i}^{-(q_{i}/a)^{\prime}}\right)^{\frac{1}{(q_{i}/a)^{\prime}}}<\infty, \,\,\,s\geq1 $$ where $\|v\|_{L^{\infty}(Q)}=\sup_{Q}v$ when $t=1$, $a$, $r$, $s$, $t$ and $\vec{q}$ satisfy proper conditions. As some applications we formulate a bilinear version of the Olsen inequality, the Fefferman-Stein type dual inequality and the Stein-Weiss inequality on Morrey spaces for fractional integrals.

math.CA↗

arXiv2018

Sharp off-diagonal weighted weak type estimates for sparse operators

We prove sharp weak type weighted estimates for a class of sparse operators that includes majorants of standard $\alpha$-fractional singular integrals, fractional integral operators, Marcinkiewicz integral operators, and square functions. These bounds are knows to be sharp in many cases, and our main new result is the optimal bound $$[w]_{A_{p,q}}^{\frac{1}{q}}[w^{q}]_{A_{\infty}}^{\frac{1}{2}-\frac{1}{p}}\lesssim[w]_{A_{p,q}}^{\frac{1}{2}-\frac{\alpha}{d}}$$ for proper conditions which satisfy that three index $p$, $q$ and $\alpha$ ensure weak type norm of fractional square functions on $L^{q}(w^{q})$ with $p>2$.

math.AP↗

arXiv2018

Sharp weak bounds and limiting weak-type behavior for Hardy type operators

In this paper, Hardy type operator $H_{\beta}$ on $\bR^{n}$ and its adjoint operator $H_{\beta}^{*}$ are investigated. We use novel methods to obtain two main results. One is that we obtain the operators $H_{\beta}$ and $H_{\beta}^{*}$ being bounded from $L^{p}(|x|^{\alpha})$ to $L^{q,\infty}(|x|^{\gamma})$, and the bounds of the operators $H_{\beta}$ and $H_{\beta}^{*}$ are sharp worked out. In particular, when $\alpha=\gamma=0$, the norm of $H_{\beta}$ is equal to $1$. The other is that we study limiting weak-type behavior for the operator $H_{\beta}$ and its optimal form was obtained.

math.CA↗

arXiv2018

Sharp bounds for Hardy type operators on higher-dimensional product spaces

In this paper, we investigate a class of fractional Hardy type operators $\mathscr{H}_{\beta_{1},\cdots,\beta_{m}}$ defined on higher-dimensional product spaces $\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}}$. We use novel methods to obtain two main results. One is that we obtain the operator $\mathscr{H}_{\beta_{1},\cdots,\beta_{m}}$ is bounded from $L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}},|x|^{\gamma})$ to $L^{q}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}},|x|^{\alpha})$ and the bounds of the operator $\mathscr{H}_{\beta_{1},\cdots,\beta_{m}}$ is sharp worked out. The other is that when $\alpha=\gamma=(0,\cdots,0)$, the norm of the operator $\mathscr{H}_{\beta_{1},\cdots,\beta_{m}}$ is obtained.

math.CA↗

arXiv2018

Weighted boundedness of Hardy and P\'{o}lya-Knopp type operators with variable limits on product space

We characterize the sufficient conditions which three weight functions $u$ and $v_{1}, v_{2}$ satisfy ensure the boundedness of the Hardy operator with variable limits on product space. The corresponding bound is explicitly worked out. Moreover, as application, we can obtain an explicit scale of bound for the P\'{o}lya-Knopp type operator with certain weights.

math.CA↗