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Wen Yuan

Publications and source records attributed to Wen Yuan.

At least 55 records · Page 3Linked to original sources
arXiv2025

Matrix-Weighted Besov-Triebel-Lizorkin Spaces of Optimal Scale: Real-Variable Characterizations, Invariance on Integrable Index, and Sobolev-Type Embedding

In this article, using growth functions we introduce generalized matrix-weighted Besov-Triebel-Lizorkin-type spaces with matrix $\mathcal{A}_{\infty}$ weights. We first characterize these spaces, respectively, in terms of the $φ$-transform, the Peetre-type maximal function, and the Littlewood-Paley functions. Furthermore, after establishing the boundedness of almost diagonal operators on the corresponding sequence spaces, we obtain the molecular and the wavelet characterizations of these spaces. As applications, we find the sufficient and necessary conditions for the invariance of those Triebel-Lizorkin-type spaces on the integrable index and also for the Sobolev-type embedding of all these spaces. The main novelty exists in that these results are of wide generality, the growth condition of growth functions is not only sufficient but also necessary for the boundedness of almost diagonal operators and hence this new framework of Besov-Triebel-Lizorkin-type is optimal, some results either are new or improve the known ones even for known matrix-weighted Besov-Triebel-Lizorkin spaces, and, furthermore, even in the scalar-valued setting, all the results are also new.

math.FA↗
arXiv2025

Parabolic Extrapolation and Its Applications to Characterizing Parabolic BMO Spaces via Parabolic Fractional Commutators

In this article, we establish the parabolic version of the celebrated Rubio de Francia extrapolation theorem. As applications, we obtain new characterizations of parabolic BMO-type spaces in terms of various commutators of parabolic fractional operators with time lag. The key tools to achieve these include to establish the appropriate form in the parabolic setting of the parabolic Rubio de Francia iteration algorithm, the Cauchy integral trick, and a modified Fourier series expansion argument adapted to the parabolic geometry. The novelty of these results lies in the fact that, for the first time, we not only introduce a new class of commutators associated with parabolic fractional integral operators with time lag, but also utilize them to provide a characterization of the parabolic BMO-type space in the high-dimensional case.

math.FA↗
arXiv2025

Maximal Function and Atomic Characterizations of Matrix-Weighted Hardy Spaces with Their Applications to Boundedness of Calderón--Zygmund Operators

Let $p\in(0,1]$ and $W$ be an $A_p$-matrix weight, which in scalar case is exactly a Muckenhoupt $A_1$ weight. In this article, we introduce matrix-weighted Hardy spaces $H^p_W$ via the matrix-weighted grand non-tangential maximal function and characterize them, respectively, in terms of various other maximal functions and atoms, both of which are closely related to matrix weights under consideration and their corresponding reducing operators. As applications, we first establish the finite atomic characterization of $H^p_W$, then using it we give a criterion on the boundedness of sublinear operators from $H^p_W$ to any $γ$-quasi-Banach space, and finally applying this criterion we further obtain the boundedness of Calderón--Zygmund operators on $H^p_W$. The main novelty of these results lies in that the aforementioned maximal functions related to reducing operators are new even in the scalar weight case and we characterize these matrix-weighted Hardy spaces by a fresh and natural variant of classical weighted atoms via first establishing a Calderón--Zygmund decomposition which is also new even in the scalar weight case.

math.FA↗
arXiv2025

Besov--Triebel--Lizorkin-Type Spaces with Matrix $A_\infty$ Weights

Introduced by A. Volberg, matrix $A_{p,\infty}$ weights provide a suitable generalization of Muckenhoupt $A_\infty$ weights from the classical theory. In our previous work, we established new characterizations of these weights. Here, we use these results to study inhomogeneous Besov-type and Triebel--Lizorkin-type spaces with such weights. In particular, we characterize these spaces, in terms of the $φ$-transform, molecules, and wavelets, and obtain the boundedness of almost diagonal operators, pseudo-differential operators, trace operators, pointwise multipliers, and Calderón--Zygmund operators on these spaces. This is the first systematic study of inhomogeneous Besov--Triebel--Lizorkin-type spaces with $A_{p,\infty}$-matrix weights, but some of the results are new even when specialized to the scalar unweighted case.

math.FA↗
arXiv2024

Regularity of Characteristic Functions in Besov-Type and Triebel--Lizorkin-Type Spaces

In this article, the authors determine the optimal regularity of characteristic functions in Besov-type and Triebel--Lizorkin-type spaces under restrictions on the measure of the $δ$-neighborhoods of the boundary. In particular, the necessary and sufficient conditions for the membership in these spaces of characteristic functions of the snowflake domain and also some spiral type domains are obtained.

math.FA↗
arXiv2024

Parabolic Muckenhoupt Weights Characterized by Parabolic Fractional Maximal and Integral Operators with Time Lag

In this article, motivated by the regularity theory of the solutions of doubly nonlinear parabolic partial differential equations the authors introduce the off-diagonal two-weight version of the parabolic Muckenhoupt class with time lag. Then the authors introduce the uncentered parabolic fractional maximal operator with time lag and characterize its two-weighted boundedness (including the endpoint case) via these weights under an extra mild assumption (which is not necessary for one-weight case). The most novelty of this article exists in that the authors further introduce a new parabolic shaped domain and its corresponding parabolic fractional integral with time lag and, moreover, applying the aforementioned two-weighted boundedness of the uncentered parabolic fractional maximal operator with time lag, the authors characterize the (two-)weighted boundedness (including the endpoint case) of these parabolic fractional integrals in terms of the off-diagonal (two-weight) parabolic Muckenhoupt class with time lag; as applications, the authors further establish a parabolic weighted Sobolev embedding and a priori estimate for the solution of the heat equation. The key tools to achieve these include the parabolic Calderón--Zygmund-type decomposition, the chaining argument, and the parabolic Welland inequality which is obtained by making the utmost of the geometrical relation between the parabolic shaped domain and the parabolic rectangle.

math.AP↗
arXiv2024

Matrix-Weighted Besov-Type and Triebel--Lizorkin-Type Spaces II: Sharp Boundedness of Almost Diagonal Operators

This article is the second one of three successive articles of the authors on the matrix-weighted Besov-type and Triebel--Lizorkin-type spaces. In this article, we obtain the sharp boundedness of almost diagonal operators on matrix-weighted Besov-type and Triebel--Lizorkin-type sequence spaces. These results not only possess broad generality but also improve several existing related results in various special cases covered by this family of spaces. This improvement depends, on the one hand, on the notion of $A_p$-dimensions of matrix weights and their properties introduced in the first article of this series and, on the other hand, on a careful direct analysis of sequences of averages avoiding maximal operators. While a recent matrix-weighted extension of the Fefferman--Stein vector-valued maximal inequality would provide an alternative route to some of our results in the restricted range of function space parameters $p,q\in(1,\infty)$, our approach covers the full scale of exponents $p\in(0,\infty)$ and $q\in(0,\infty]$ that is relevant in the theory of function spaces.

math.FA↗
arXiv2024

New approach to affine Moser-Trudinger inequalities via Besov polar projection bodies

We extend the affine inequalities on $\mathbb{R}^n$ for Sobolev functions in $W^{s,p}$ with $1 \leq p < n/s$ obtained recently by Haddad-Ludwig [16, 17] to the remaining range $p \geq n/s$. For each value of $s$, our results are stronger than affine Moser-Trudinger and Morrey inequalities. As a byproduct, we establish the analog of the classical $L^p$ Bourgain-Brezis-Mironescu inequalities related to the Moser-Trudinger case $p=n$. Our main tool is the affine invariant provided by Besov polar projection bodies.

math.MG↗
arXiv2023

Matrix-Weighted Besov-Type and Triebel--Lizorkin-Type Spaces III: Characterizations of Molecules and Wavelets, Trace Theorems, and Boundedness of Pseudo-Differential Operators and Calderón--Zygmund Operators

This is the last one of three successive articles by the authors on matrix-weighted Besov-type and Triebel--Lizorkin-type spaces $\dot B^{s,τ}_{p,q}(W)$ and $\dot F^{s,τ}_{p,q}(W)$. In this article, the authors establish the molecular and the wavelet characterizations of these spaces. Furthermore, as applications, the authors obtain the optimal boundedness of trace operators, pseudo-differential operators, and Calderón--Zygmund operators on these spaces. Due to the sharp boundedness of almost diagonal operators on their related sequence spaces obtained in the second article of this series, all results presented in this article improve their counterparts on matrix-weighted Besov and Triebel--Lizorkin spaces $\dot B^{s}_{p,q}(W)$ and $\dot F^{s}_{p,q}(W)$. In particular, even when reverting to the boundedness of Calderón--Zygmund operator on unweighted Triebel--Lizorkin spaces $\dot F^{s}_{p,q}$, these results are still better.

math.FA↗
arXiv2023

Matrix-Weighted Besov-Type and Triebel--Lizorkin-Type Spaces I: $A_p$-Dimensions of Matrix Weights and $φ$-Transform Characterizations

Let $s\in{\mathbb R}$, $q\in (0,\infty]$, and $τ\in[0,\infty)$. It is well known that Besov-type spaces $\dot B^{s,τ}_{p,q}$ with $p\in (0,\infty]$ and Triebel--Lizorkin-type spaces $\dot F^{s,τ}_{p,q}$ with $p\in (0,\infty)$ when $τ\in [0,\infty)$ or with $p\in (0,\infty]$ when $τ=0$ on $\mathbb{R}^n$ consist of a general family of function spaces that cover not only the well-known Besov and Triebel--Lizorkin spaces $\dot B^{s}_{p,q}$ and $\dot F^{s}_{p,q}$ (when $τ=0$) but also several other function spaces of interest, such as Morrey spaces and $Q$ spaces. In three successive articles, the authors develop a complete real-variable theory of matrix-weighted Besov-type spaces $\dot B^{s,τ}_{p,q}(W)$ and matrix-weighted Triebel--Lizorkin-type spaces $\dot F^{s,τ}_{p,q}(W)$ on $\mathbb{R}^n$, where $W$ is a matrix-valued Muckenhoupt $A_p$ weight. This article is the first one, whose main novelty exists in that the authors introduce the new concept, $A_p$-dimensions of matrix weights, and intensively study their properties, especially those elaborate properties expressed via reducing operators. The authors then introduce the spaces $\dot B^{s,τ}_{p,q}(W)$ and $\dot F^{s,τ}_{p,q}(W)$ and, using $A_p$-dimensions and their nice properties, the authors establish the $φ$-transform characterization of $\dot B^{s,τ}_{p,q}(W)$ and $\dot F^{s,τ}_{p,q}(W)$. The $A_p$-dimensions of matrix weights and their properties also enable the authors to obtain the sharp boundedness of almost diagonal operators on related sequence spaces in the subsequent second article and the optimal characterizations of molecules and wavelets, trace theorems, and the optimal boundedness of pseudo-differential operators and Calderón--Zygmund operators in the subsequent third article.

math.FA↗
arXiv2023

New Characterizations and Properties of Matrix $A_\infty$ Weights

We provide several new characterizations of $A_{p,\infty}$-matrix weights, originally introduced by A. Volberg as matrix-valued substitutes of the classical $A_\infty$ weights. In analogy with the notion of $A_p$-dimension of matrix weights introduced in our previous work, we introduce the concepts of the lower and the upper dimensions of $A_{p,\infty}$-matrix weights, which enable us to obtain sharp estimates related to their reducing operators. In a follow-up work, these results will play a key role in the study of function spaces with $A_{p,\infty}$-matrix weights, which extends earlier results in the more restricted class of $A_p$-matrix weights.

math.FA↗
arXiv2023

Learning Unified Representations for Multi-Resolution Face Recognition

In this work, we propose Branch-to-Trunk network (BTNet), a representation learning method for multi-resolution face recognition. It consists of a trunk network (TNet), namely a unified encoder, and multiple branch networks (BNets), namely resolution adapters. As per the input, a resolution-specific BNet is used and the output are implanted as feature maps in the feature pyramid of TNet, at a layer with the same resolution. The discriminability of tiny faces is significantly improved, as the interpolation error introduced by rescaling, especially up-sampling, is mitigated on the inputs. With branch distillation and backward-compatible training, BTNet transfers discriminative high-resolution information to multiple branches while guaranteeing representation compatibility. Our experiments demonstrate strong performance on face recognition benchmarks, both for multi-resolution identity matching and feature aggregation, with much less computation amount and parameter storage. We establish new state-of-the-art on the challenging QMUL-SurvFace 1: N face identification task. Our code is available at https://github.com/StevenSmith2000/BTNet.

cs.CV↗
arXiv2023

Unveiling Optimal SDG Pathways: An Innovative Approach Leveraging Graph Pruning and Intent Graph for Effective Recommendations

The recommendation of appropriate development pathways, also known as ecological civilization patterns for achieving Sustainable Development Goals (namely, sustainable development patterns), are of utmost importance for promoting ecological, economic, social, and resource sustainability in a specific region. To achieve this, the recommendation process must carefully consider the region's natural, environmental, resource, and economic characteristics. However, current recommendation algorithms in the field of computer science fall short in adequately addressing the spatial heterogeneity related to environment and sparsity of regional historical interaction data, which limits their effectiveness in recommending sustainable development patterns. To overcome these challenges, this paper proposes a method called User Graph after Pruning and Intent Graph (UGPIG). Firstly, we utilize the high-density linking capability of the pruned User Graph to address the issue of spatial heterogeneity neglect in recommendation algorithms. Secondly, we construct an Intent Graph by incorporating the intent network, which captures the preferences for attributes including environmental elements of target regions. This approach effectively alleviates the problem of sparse historical interaction data in the region. Through extensive experiments, we demonstrate that UGPIG outperforms state-of-the-art recommendation algorithms like KGCN, KGAT, and KGIN in sustainable development pattern recommendations, with a maximum improvement of 9.61% in Top-3 recommendation performance.

cs.IR↗
arXiv2023

A unified approach to self-improving property via K-functionals

In this paper we obtain new quantitative estimates that improve the classical inequalities: Poincaré-Ponce, Gaussian Sobolev, and John-Nirenberg. Our method is based on the K-functionals and allows one to derive self-improving type inequalities. We show the optimality of the method by obtaining new Bourgain-Brezis-Mironescu and Maz'ya-Shaposhnikova limiting formulas. In particular, we derive these formulas for fractional powers of infinitesimal generators of operator semigroups on Banach spaces.

math.FA↗
arXiv2023

Brezis--Seeger--Van Schaftingen--Yung-Type Characterization of Homogeneous Ball Banach Sobolev Spaces and Its Applications

Let $γ\in\mathbb{R}\setminus\{0\}$ and $X(\mathbb{R}^n)$ be a ball Banach function space satisfying some extra mild assumptions. Assume that $Ω=\mathbb{R}^n$ or $Ω\subset\mathbb{R}^n$ is an $(\varepsilon,\infty)$-domain for some $\varepsilon\in(0,1]$. In this article, the authors prove that a function $f$ belongs to the homogeneous ball Banach Sobolev space $\dot{W}^{1,X}(Ω)$ if and only if $f\in L_{\mathrm{loc}}^1(Ω)$ and $$ \sup_{λ\in(0,\infty)}λ\left\|\left[\int_{\{y\inΩ:\ |f(\cdot)-f(y)|>λ|\cdot-y|^{1+\fracγ{p}}\}} \left|\cdot-y\right|^{γ-n}\,dy \right]^\frac{1}{p}\right\|_{X(Ω)}<\infty, $$ where $p\in[1,\infty)$ is related to $X(\mathbb{R}^n)$. This result is of wide generality and can be applied to various specific Sobolev-type function spaces, including Morrey [Bourgain--Morrey-type, weighted (or mixed-norm or variable) Lebesgue, local (or global) generalized Herz, Lorentz, and Orlicz (or Orlicz-slice)] Sobolev spaces, which is new even in all these special cases; in particular, it coincides with the well-known result of H. Brezis, A. Seeger, J. Van Schaftingen, and P.-L. Yung when $X(Ω):=L^q(\mathbb{R}^n)$ with $1<p=q<\infty$, while it is still new even when $X(Ω):=L^q(\mathbb{R}^n)$ with $1\leq p<q<\infty$.

math.FA↗
arXiv2023

Extension Theorem and Bourgain--Brezis--Mironescu-Type Characterization of Ball Banach Sobolev Spaces on Domains

Let $Ω\subset\mathbb{R}^n$ be a bounded $(\varepsilon,\infty)$-domain with $\varepsilon\in(0,1]$, $X(\mathbb{R}^n)$ a ball Banach function space satisfying some extra mild assumptions, and $\{ρ_ν\}_{ν\in(0,ν_0)}$ with $ν_0\in(0,\infty)$ a $ν_0$-radial decreasing approximation of the identity on $\mathbb{R}^n$. In this article, the authors establish two extension theorems, respectively, on the inhomogeneous ball Banach Sobolev space $W^{m,X}(Ω)$ and the homogeneous ball Banach Sobolev space $\dot{W}^{m,X}(Ω)$ for any $m\in\mathbb{N}$. On the other hand, the authors prove that, for any $f\in\dot{W}^{1,X}(Ω)$, $$ \lim_{ν\to0^+} \left\|\left[\int_Ω\frac{|f(\cdot)-f(y)|^p}{ |\cdot-y|^p}ρ_ν(|\cdot-y|)\,dy \right]^\frac{1}{p}\right\|_{X(Ω)}^p =\frac{2π^{\frac{n-1}{2}}Γ(\frac{p+1}{2})}{Γ(\frac{p+n}{2})} \left\|\,\left|\nabla f\right|\,\right\|_{X(Ω)}^p, $$ where $Γ$ is the Gamma function and $p\in[1,\infty)$ is related to $X(\mathbb{R}^n)$. Using this asymptotics, the authors further establish a characterization of $W^{1,X}(Ω)$ in terms of the above limit. To achieve these, the authors develop a machinery via using a method of the extrapolation, two extension theorems on weighted Sobolev spaces, and some recently found profound properties of $W^{1,X}(\mathbb{R}^n)$ to overcome those difficulties caused by that the norm of $X(\mathbb{R}^n)$ has no explicit expression and that $X(\mathbb{R}^n)$ might be neither the reflection invariance nor the translation invariance. This characterization has a wide range of generality and can be applied to various Sobolev-type spaces, such as Morrey [Bourgain--Morrey-type, weighted (or mixed-norm or variable), local (or global) generalized Herz, Lorentz, and Orlicz (or Orlicz-slice)] Sobolev spaces, all of which are new.

math.FA↗
arXiv2023

Hardy Spaces Associated with Non-Negative Self-Adjoint Operators and Ball Quasi-Banach Function Spaces on Doubling Metric Measure Spaces and Their Applications

Let $(\mathcal{X},d,μ)$ be a doubling metric measure space in the sense of R. R. Coifman and G. Weiss, $L$ a non-negative self-adjoint operator on $L^2(\mathcal{X})$ satisfying the Davies--Gaffney estimate, and $X(\mathcal{X})$ a ball quasi-Banach function space on $\mathcal{X}$ satisfying some mild assumptions. In this article, the authors introduce the Hardy type space $H_{X,\,L}(\mathcal{X})$ by the Lusin area function associated with $L$ and establish the atomic and the molecular characterizations of $H_{X,\,L}(\mathcal{X}).$ As an application of these characterizations of $H_{X,\,L}(\mathcal{X})$, the authors obtain the boundedness of spectral multiplies on $H_{X,\,L}(\mathcal{X})$. Moreover, when $L$ satisfies the Gaussian upper bound estimate, the authors further characterize $H_{X,\,L}(\mathcal{X})$ in terms of the Littlewood--Paley functions $g_L$ and $g_{λ,\,L}^\ast$ and establish the boundedness estimate of Schrödinger groups on $H_{X,\,L}(\mathcal{X})$. Specific spaces $X(\mathcal{X})$ to which these results can be applied include Lebesgue spaces, Orlicz spaces, weighted Lebesgue spaces, and variable Lebesgue spaces. This shows that the results obtained in the article have extensive generality.

math.FA↗
arXiv2023

Generalized Brezis--Van Schaftingen--Yung Formulae and Their Applications in Ball Banach Sobolev Spaces

Let $X$ be a ball Banach function space on $\mathbb{R}^n$. In this article, under some mild assumptions about both $X$ and the boundedness of the Hardy--Littlewood maximal operator on both $X$ and the associate space of its convexification, the authors successfully recover the homogeneous ball Banach Sobolev semi-norm $\|\,|\nabla f|\,\|_X$ via the functional $$ \sup_{λ\in(0,\infty)}λ\left\|\left[\int_{\{y\in\mathbb{R}^n:\ |f(\cdot)-f(y)|>λ|\cdot-y|^{1+\fracγ{q}}\}} \left|\cdot-y\right|^{γ-n}\,dy\right]^\frac{1}{q}\right\|_X $$ for any distributions $f$ with $|\nabla f|\in X$, as well as the corresponding limiting identities with the limit for $λ\to\infty$ when $γ\in(0,\infty)$ or the limit for $λ\to0^+$ when $γ\in(-\infty,0)$, where $γ\in\mathbb{R}\setminus\{0\}$ and where $q\in(0,\infty)$ is related to $X$. In particular, some of these results are still new even when $X:=L^p(X)$ with $p\in[1,\infty)$. As applications, the authors obtain some fractional Sobolev-type and some fractional Gagliardo--Nirenberg-type inequalities in the setting of $X$. All these results are of quite wide generality and are applied to various specific function spaces, including Morrey, mixed-norm (or variable or weighted) Lebesgue, Lorentz, and Orlicz (or Orlicz-slice) spaces, some of which are new even in all these special cases. The novelty of this article is to use both the method of the extrapolation and the boundedness of the Hardy--Littlewood maximal operator on both $X$ and the associate space of its convexification to overcome the essential difficulties caused by the deficiency of both the translation and the rotation invariance and an explicit expression of the norm of $X$.

math.FA↗