Wen Yuan

Full-Range $Γ$ -Convergence of Weak-Type Nonlocal Functionals

Xiaosheng Lin, Dachun Yang, Sibei Yang +2

Let $N \geq 1$ , $p \in [1, \infty)$ , and $Ω \subset R^{N}$ be a bounded open interval when $N = 1$ or a bounded Lipschitz domain when $N \geq 2$ . For any given $γ\in\mathbb R\setminus\{0\…

Capacitary Trapping and Brownian Escape: A Sharp Hardy Principle for Absorbing Media

Yiqun Chen, Jie Xiao, Dachun Yang +2

We study Brownian motion killed upon exiting arbitrary open sets $Ω \subset R^{n}$ , $n \in [3, \infty) \cap N$ . Let $α \in (0, 1]$ . The infimal radius $d_{α} (x)$ at which the abs…

$Γ$ -Convergence of Weak-Type Nonlocal Functionals on Bounded Domains

Xiaosheng Lin, Dachun Yang, Sibei Yang +2

Let $N \geq 1$ , $p \in [1, \infty)$ , $γ \in (0, \infty)$ , and $Ω \subset R^{N}$ be a bounded open interval when $N = 1$ or a bounded Lipschitz domain when $N \geq 2$ . For any $λ\in(0,\infty)…

Parabolic BMO Spaces, Muckenhoupt Weights, and Reverse Hölder Classes with Time Lag: Equivalence and Characterizations

Weiyi Kong, Dachun Yang, Wen Yuan

For any given time lag $γ \in (0, 1)$ , we prove that the one-sided parabolic BMO space $BMO^{+} (γ)$ coincides with the parabolic BMO space $PBMO^{-} (γ)$ with equivalent…

Sobolev--Morrey Spaces and Divergence-Form Degenerate Second-Order Elliptic Equations on Domains with Higher Co-Dimensional Boundaries

Weiyi Kong, Yoshihiro Sawano, Dachun Yang +2

In this article, we study the weighted homogeneous Sobolev--Morrey spaces on domains in $R^{n}$ with higher co-dimensional boundaries. Precisely, we systematically establish…

Compactness and Its Applications of Sobolev Spaces Associated with Ball Banach Function Spaces

Xiaosheng Lin, Dachun Yang, Wen Yuan +1

Let $N \in N \cap [2, \infty),$ $Ω$ be a bounded Lipschitz domain in $R^{N}$ , and $X (Ω)$ be a ball Banach function space on $Ω.$ In this article, under some mild assumpt…

A Counterexample to the Necessity of the Vanishing Carleson Condition for VMO Poisson Kernels

Xiaosheng Lin, Dachun Yang, Sibei Yang +2

In [Problem 3.2.23, CBMS Regional Conference Series in Mathematics 83, 1994], Kenig asked whether the vanishing Carleson condition is the necessary and sufficient for the logarithm…

A Counterexample to Kenig's Interpolation Problem for Sobolev Spaces with Zero Boundary Conditions

Xiaosheng Lin, Dachun Yang, Sibei Yang +2

Let $n \in N \cap [2, \infty)$ . In this article, we show that there exists a bounded $C^{1}$ domain $Ω \subset R^{n}$ such that, for any given $s\in(1,2)\setminus\{\frac32\}…

Atomic Characterization and Its Applications of Matrix-Weighted Variable Hardy Spaces

Yiqun Chen, Dachun Yang, Wen Yuan +1

In this article, by means of the matrix-weighted grand maximal function we first introduce the variable Hardy space $H_{W}^{p (\cdot)}$ on $R^{n}$ with the $\mathscr{A}_{p(\cdo…

Variable Muckenhoupt $A_{\infty}$ Weights

Dachun Yang, Wen Yuan, Zongze Zeng

In this article, with introducing concepts of variable scalar $A_{p (\cdot), \infty}$ weights and variable matrix $A_{p (\cdot), \infty}$ weights, we seek a compreh…

Real-variable theory of function spaces with operator-valued $A_{p}$ weights in Banach spaces

Tuomas P. Hytönen, Yinqin Li, Dachun Yang +1

While the theory of matrix-weighted function spaces is well established, the majority of previous results in the infinite-dimensional operator-valued setting deal with "no go" theo…

Real-Variable Theory of Hardy--Lorentz Spaces on Quasi-Ultrametric Spaces of Homogeneous Type with Reverse-Doubling Property

Chenfeng Zhu, Ryan Alvarado, Xianjie Yan +2

Let $(X, q, μ)$ be an ultra-RD-space with upper dimension $n \in (0, \infty)$ ; i.e., it is a quasi-ultrametric space of homogeneous type whose measure $μ$ satisfies an addition…

Real-variable theory of matrix-weighted multi-parameter Besov--Triebel--Lizorkin-type spaces

Fan Bu, Yiqun Chen, Tuomas Hytönen +2

We develop a comprehensive theory for a general class of multi-parameter function spaces of Besov-Triebel-Lizorkin type, with a matrix weight. We prove the equivalence of different…

Matrix-Weighted Poincaré-Type Inequalities with Applications to Logarithmic Hajłasz--Besov Spaces on Spaces of Homogeneous Type

Ziwei Li, Dachun Yang, Wen Yuan

Let $X$ be a space of homogeneous type. In this article, based on the reducing operators of matrix $A_{p}$ -weights, the authors introduce the vector-valued Hajłasz gradi…

One-Sided and Parabolic BLO Spaces with Time Lag and Their Applications to Muckenhoupt $A_{1}$ Weights and Doubly Nonlinear Parabolic Equations

Weiyi Kong, Dachun Yang, Wen Yuan

In this article, we first introduce the one-sided BLO space $BLO^{+} (R)$ and characterize it, respectively, in terms of the one-sided Muckenhoupt class $A_1^+(\math…

A Sharp Localized Weighted Inequality Related to Gagliardo and Sobolev Seminorms and Its Applications

Pingxu Hu, Yinqin Li, Dachun Yang +1

In this article, we establish a nearly sharp localized weighted inequality related to Gagliardo and Sobolev seminorms, respectively, with the sharp $A_{1}$ -weight constant or with th…

Maz'ya--Shaposhnikova Representation of Quasi-Norms of Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type with Weak Reverse Doubling Property

Eiichi Nakai, Menghao Tang, Dachun Yang +2

Let $Y (X)$ be a ball quasi-Banach function space on the space of homogeneous type $(X, ρ, μ)$ satisfying some mild additional assumptions, $q \in (0, \infty)$ , and $…

Variable Matrix-Weighted Besov Spaces

Dachun Yang, Wen Yuan, Zongze Zeng

In this article, using variable matrix $A_{p (\cdot), \infty}$ weights, we introduce the matrix-weighted variable Besov space $B_{p (\cdot), q (\cdot)}^{s (\cdot)} (W)$ and th…

Fractional Heat Semigroup Characterization of Distances from Functions in Lipschitz Spaces to Their Subspaces

Feng Dai, Eero Saksman, Dachun Yang +2

Let $Λ_{s}$ denote the inhomogeneous Lipschitz space of order $s \in (0, \infty)$ on $R^{n}$ . This article characterizes the distance $d(f, V)_{Λ_s}: = \inf_{g\in V} \|f-g\|_{Λ_s…

Matrix-Weighted Campanato Spaces: Duality and Calderón--Zygmund Operators

Yiqun Chen, Dachun Yang, Wen Yuan

Let $p \in (0, \infty)$ , $q \in [1, \infty)$ , $s \in Z_{+}$ , and $W$ be an $A_{p}$ -matrix weight, which in the scalar case is exactly a Muckenhoupt $A_{m a x {1, p}}$ weight. In this ar…

Difference and Wavelet Characterizations of Distances from Functions in Lipschitz Spaces to Their Subspaces

Feng Dai, Eero Saksman, Dachun Yang +2

Let $Λ_{s}$ denote the Lipschitz space of order $s \in (0, \infty)$ on $R^{n}$ , which consists of all $f \in C \cap L^{\infty}$ such that, for some constant $L\in(0,\infty)…

Sharp Brezis--Seeger--Van Schaftingen--Yung Formulae for Higher-Order Gradients in Ball Banach Function Spaces

Pingxu Hu, Yinqin Li, Dachun Yang +2

Let $X$ be a ball Banach function space on $R^{n}$ , $k \in N$ , $h \in R^{n}$ , and $Δ_{h}^{k}$ denote the $k$ {\rm th} order difference. In this article, under some mi…

Matrix-Weighted Besov--Triebel--Lizorkin Spaces of Optimal Scale: Boundedness of Pseudo-Differential, Trace, and Calderón--Zygmund Operators

Dachun Yang, Wen Yuan, Mingdong Zhang

This article is a continuation of our work on generalized matrix-weighted Besov--Triebel--Lizorkin-type spaces with matrix $A_{\infty}$ weights. In this article, we estab…