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Xi Cen

Publications and source records attributed to Xi Cen.

12 recordsLinked to original sources
arXiv2025

The multilinear fractional bounded mean oscillation operator theory I: sparse domination, sparse $T1$ theorem, off-diagonal extrapolation, quantitative weighted estimate -- for generalized commutators

This paper introduces and studies a class of multilinear fractional bounded mean oscillation operators (denoted {\rm $m$-FBMOOs}) defined on ball-basis measure spaces $(X, \mu, \mathcal{B})$. These operators serve as a generalization of canonical classes, such as the multilinear fractional maximal operators, the multilinear fractional Ahlfors-Beurling operators, the multilinear pseudo-differential operators with multi-parameter H\"ormander symbol, and some multilinear operators admitting $\mathbb{V}$-valued $m$-linear fractional Dini-type Calder\'on-Zygmund kernel representation. Crucially, the definition utilized here, incorporating the notion of "bounded mean oscillation," provides greater generality compared to those in Karagulyan (2019) and Cao et al. (2023). Our investigation systematically examines the properties of these operators and their generalized commutators through the lens of modern harmonic analysis, focusing on two principal directions: 1.We establish Karagulyan-type sparse domination for the generalized commutators. Subsequently, by developing bespoke dyadic representation theorems pertinent to this setting, we prove a corresponding multilinear fractional sparse $T1$ theorem for these generalized commutators. 2.With sparse bounds established, we obtain weighted estimates in multiple complementary methods: (1) Under a novel class of multilinear fractional weights, we prove four types of weighted inequalities: sharp-type, Bloom-type, mixed weak-type, and local decay-type.(2) We develop a multilinear non-diagonal extrapolation framework for these weights, which transfers boundedness flexibly among weighted spaces and establishes corresponding vector-valued inequalities.

math.CA↗
arXiv2025

The multilinear fractional sparse operator theory II: refining weighted estimates via multilinear fractional sparse forms

This paper refines the main results from our previous study on sparse bounds of generalized commutators of multilinear fractional singular integral operators in \cite{CenSong2412}. The key improvements are: 1. We replace pointwise domination with the $(m+1)$-linear fractional sparse form ${\mathcal A}_{\eta,\mathcal{S},\tau,{\vec{r}},s'}^\mathbf{b,k,t}$, advancing the vector-valued multilinear fractional sparse form domination principle, and relax conditions from multilinear weak type boundedness to multilinear locally weak type boundedness $W_{\vec{p}, q}(X)$. 2. We introduce a multilinear fractional $\vec{r}$-type maximal operator $\mathscr{M}_{\eta,\vec{r}}$ and develop a new class of weights $A_{(\vec{p},q),(\vec{r}, s)}(X)$ to characterize it, establishing norm equivalence with the sparse forms. 3. This norm equivalence provides sharp quantitative weighted estimates for $(m+1)$-linear fractional sparse form, removing exponent parameter limitations and achieving sharp operator norm bounds. 4. We demonstrate applications in two ways: (1) Providing sharp or Bloom type estimates for generalized commutators of multilinear fractional Calder\'on--Zygmund operators and multilinear fractional rough singular integral operators. (2) Investigating sparse form type weighted Lebesgue $L^p(\omega)$ and weighted Sobolev $W^{s,p}(\omega)$ regularity estimates for solutions of fractional Laplacian equations with higher-order commutators.

math.CA↗
arXiv2024

The multilinear fractional sparse operator theory I: pointwise domination and weighted estimate

How to establish some specific quantitative weighted estimates for the generalized commutator of multilinear fractional singular integral operator $\mathcal{T}_{\eta}^{{\bf b}}$ is the focus of this paper, which is defined by $$\mathcal{T}_{\eta}^{{\bf b}}(\vec{f})(x):= \mathcal{T}_{\eta}\left((b_1(x) - b_1)^{\beta_1}f_1,\ldots,(b_m(x) - b_m)^{\beta_m}f_m\right)(x),$$ where $\mathcal{T}_{\eta}$ is a multilinear fractional singular integral operator, ${\bf b}:=({b_1}, \cdots ,{b_m})$ is a set of symbol functions, and $({\beta_1}, \cdots ,{\beta_m}) \in {\mathbb{N}_0^m}$. Pointwise dominating the aforementioned commutator leads us to consider a class of higher order multi-symbol multilinear fractional sparse operator ${\mathcal A}_{\eta ,\mathcal{S},\tau}^\mathbf{b,k,t}$ to achieve this long-cherished wish. Therefore, it suffices to construct its quantitative weighted estimates, which firstly include the characterization of several types of multilinear weighted conditions $A_{\vec p,q}^*$, $W_{\vec p,q}^\infty$, and $H_{\vec p,q}^\infty$. Within the scope of this work, Bloom type estimate for first order multi-symbol multilinear fractional sparse operator is established herein. Moreover, we derive two distinct Bloom type estimates for higher order multi-symbol multilinear fractional sparse operator by using "maximal weight method" and "iterated weight method" respectively, which not only refines some of Lerner's methods but greatly enhances the generality of our conclusions. Endpoint quantitative estimates for multilinear fractional singular integral operators and their first order commutators are also obtained as the last main result. It is also worthy of highlighting that some important multilinear fractional operators are applicable to our results as applications.

math.CA↗
arXiv2024

New variable weighted conditions for fractional maximal operators over spaces of homogeneous type

Based on the rapid development of dyadic analysis and the theory of variable weighted function spaces over the spaces of homogeneous type $(X,d,\mu)$ in recent years, we systematically consider the quantitative variable weighted characterizations for fractional maximal operators. On the one hand, a new class of variable multiple weight $A_{\vec{p}(\cdot),q(\cdot)}(X)$ is established, which enables us to prove the strong and weak type variable multiple weighted estimates for multilinear fractional maximal operators ${{{\mathscr M}_{\eta }}}$. More precisely, \[ {\left[ {\vec \omega } \right]_{{A_{\vec p( \cdot ),q( \cdot )}}(X)}} \lesssim {\left\| \mathscr{M}_\eta \right\|_{\prod\limits_{i = 1}^m {{L^{p_i( \cdot )}}({X,\omega _i})} \to {L^{q( \cdot )}}(X,\omega )({WL^{q( \cdot )}}(X,\omega ))}} \le {C_{\vec \omega ,\eta ,m,\mu ,X,\vec p( \cdot )}}. \] On the other hand, on account of the classical Sawyer's condition $S_{p,q}(\mathbb{R}^n)$, a new variable testing condition $C_{{p}(\cdot),q(\cdot)}(X)$ also appears in here, which allows us to obtain quantitative two-weighted estimates for fractional maximal operators ${{{M}_{\eta }}}$. To be exact, \begin{align*} \|M_{\eta}\|_{L^{p(\cdot)}(X,\omega)\rightarrow L^{q(\cdot)}(X,v)} \lesssim \sum\limits_{\theta = \frac{1}{{{p_{\rm{ - }}}}},\frac{1}{{{p_{\rm{ + }}}}}} {{{\left( {{{[\omega ,v]}_{C_{p( \cdot ),q( \cdot )}^2(X)}} + {{[\omega ]}_{C_{p( \cdot ),q( \cdot )}^1(X)}}{{[\omega ,v]}_{C_{p( \cdot ),q( \cdot )}^2(X)}}} \right)}^\theta }}. \end{align*} The implicit constants mentioned above are independent on the weights.

math.CA↗
arXiv2024

Fractional maximal operators on weighted variable Lebesgue spaces over the spaces of homogeneous type

Let $(X,d,\mu)$ is a space of homogeneous type, we establish a new class of fractional-type variable weights $A_{p(\cdot), q(\cdot)}(X)$. Then, we get the new weighted strong-type and weak-type characterizations for fractional maximal operators $M_\eta$ on weighted variable Lebesgue spaces over $(X,d,\mu)$. This study generalizes the results by Cruz-Uribe-Fiorenza-Neugebauer (2012), Bernardis-Dalmasso-Pradolini (2014), Cruz-Uribe-Shukla (2018), and Cruz-Uribe-Cummings (2022).

math.CA↗
arXiv2024

Extrapolation to product Morrey-Herz spaces and applications

The purpose of this paper is threefold. First, we introduce product Morrey-Herz spaces and product block-Herz spaces, establish their duality, and prove the boundedness of the strong maximal operator on product block-Herz spaces; these results provide the foundation for extrapolation. Second, using the Rubio de Francia iteration method, we establish extrapolation results on product Morrey-Herz spaces. Finally, we give applications to Fefferman-Stein vector-valued strong maximal inequalities, the John-Nirenberg inequality, a characterization of little bmo in terms of product Morrey-Herz spaces, and the boundedness of bi-parameter Calder\'on-Zygmund operators and their commutators.

math.FA↗
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↗
arXiv2023

Characterizations for multi-sublinear operators and their commutators on three kinds of generalized weighted Morrey spaces and applications

The main questions raised in this paper are to find the sufficient conditions that make multi-sublinear operators $T$ and their commutators ${T_{\prod \vec b }}$, ${T_{\sum {\vec b} }}$ to be bounded on three kinds of generalized weighted Morrey spaces. We give the main theorems of this paper to solve the above related questions. As corollaries of the main theorems, we give sufficient and necessary conditions for a class of multi-sublinear operators which are bounded on three kinds of generalized weighted Morrey spaces. As some inportant applications, we apply the main results to the multilinear vector-valued Calder\'on-Zygmund operators, multilinear Littlewood-Paley square operators, multilinear pseudo-differential operators and multilinear paraproducts.

math.FA↗
arXiv2023

Boundedness of the multilinear integral operators on Heisenberg group

In this paper, we will obtain the sharp constant for multilinear integral operator on Heisenberg group Lebesgue space which is based on the Stein-Weiss lemma, the boundedness for multilinear integral operator on Heisenberg group $A_p$ weighted Morrey space and the sharp constant for multilinear integral operator on Heisenberg group two power weight Morrey space.

math.CA↗
arXiv2020

HyperFaceNet: A Hyperspectral Face Recognition Method Based on Deep Fusion

Face recognition has already been well studied under the visible light and the infrared,in both intra-spectral and cross-spectral cases. However, how to fuse different light bands, i.e., hyperspectral face recognition, is still an open research problem, which has the advantages of richer information retaining and all-weather functionality over single band face recognition. Among the very few works for hyperspectral face recognition, traditional non-deep learning techniques are largely used. Thus, we in this paper bring deep learning into the topic of hyperspectral face recognition, and propose a new fusion model (termed HyperFaceNet) especially for hyperspectral faces. The proposed fusion model is characterized by residual dense learning, a feedback style encoder and a recognition-oriented loss function. During the experiments, our method is proved to be of higher recognition rates than face recognition using either visible light or the infrared. Moreover, our fusion model is shown to be superior to other general-purposed image fusion methods including state-of-the-arts, in terms of both image quality and recognition performance.

cs.CV↗