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The purpose of this study is twofold. On the one hand, we establish a new class of fractional-type multiple weights ${A_{\vec p( \cdot ),q( \cdot )}}$, which can be regarded as a variable exponents generalization of ${A_{\vec p,q}}$, a multilinear generalization of ${A_{p( \cdot ),q( \cdot )}}$, and a fractional-type generalization of ${A_{\vec p( \cdot )}}$. Moreover, it can be characterized by multilinear fractional-type operators ${\mathscr T}_\alpha$. This generalizes the work by Moen (2009), Cruz-Uribe and Wang (2017), and Cruz-Uribe and Guzm\'an (2020), which is also called multilinear Hardy-Littlewood-Sobolev theorem on weighted variable exponents Lebesgue spaces. On the other hand, we also set up a new class of fractional-type variable matrix weights ${{\mathbb{A}}_{p( \cdot ),q( \cdot )}}$ that are characterized by fractional-type averaging operators ${\mathscr A}_{\alpha,Q}$ and fractional Christ-Goldberg maximal operator ${{\mathcal M}_{\alpha ,W}}$., Comment: 41 pages; 1 figure

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 $\mathcal L$., Comment: 25 pages. arXiv admin note: text overlap with arXiv:2305.13133

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., Comment: 25pages

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., Comment: 25 pages, 2 figures

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., Comment: 17 pages

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

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

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 $p>2$.

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.

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., Comment: This paper appear fatal error so that some results not holds

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.