arXiv subjects

Wen Yuan

Publications and source records attributed to Wen Yuan.

At least 37 records · Page 2Linked to original sources

arXiv2026

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

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 of the Poisson kernel of a perturbation of the Laplacian on the unit ball in $\mathbb{R}^n$ belonging to the space VMO. The sufficiency was proved by Escauriaza [Israel J. Math. 1996] and extended by Milakis, Pipher, and Toro [Contemp. Math. 2014] to more general setting. In this article, using the technique of bi-Lipschitz mappings, we construct a counterexample to show that the vanishing Carleson condition is not necessary and hence give a negative answer to the aforementioned problem.

math.AP↗

arXiv2026

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

Let $n\in \mathbb N\cap[2,\infty)$. In this article, we show that there exists a bounded $C^1$ domain $Ω\subset \mathbb R^n$ such that, for any given $s\in(1,2)\setminus\{\frac32\}$, \begin{align*} \left[H_0^1(Ω),H^2(Ω)\cap H_0^1(Ω)\right]_{s-1} =H^s(Ω)\cap H_0^1(Ω)=H_0^s(Ω) \end{align*} with equivalent norms, but \begin{align*} \left[H_0^1(Ω),H^2(Ω)\cap H_0^1(Ω)\right]_{\frac12} \subsetneqq H^{\frac32}(Ω)\cap H_0^1(Ω), \end{align*} which provides a counterexample to Problem 3.3.19 of Kenig in [CBMS Regional Conf. Ser. in Math. 83, 1994]. As applications, we prove that for such a domain $Ω$ \begin{align*} H^2(Ω)\cap H_0^1(Ω)\subsetneqq D(-Δ_D) \end{align*} (the domain of the Dirichlet Laplacian operator $-Δ_D$ on $Ω$) and construct a solution of the homogeneous heat equation with zero Dirichlet boundary condition, which does not belong to $L^2((0,T);H^2(Ω)\cap H_0^1(Ω))$ for any given $T\in(0,\infty)$.

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arXiv2026

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

In this article, by means of the matrix-weighted grand maximal function we first introduce the variable Hardy space $H^{p(\cdot)}_W$ on $\mathbb{R}^n$ with the $\mathscr{A}_{p(\cdot),\infty}$ matrix weight $W$ and with the variable exponent $p(\cdot)$ having globally log-Hölder continuity, and then via using several different convex body valued maximal functions we establish its various maximal function equivalent characterizations. Combining a refined Whitney decomposition with both the convex body valued maximal function and its corresponding convex-body reducing operator, we obtain the atomic characterization of $H^{p(\cdot)}_W$. As applications, we give its dual space and establish the boundedness of Calderón--Zygmund operators from $H^{p(\cdot)}_W$ to the matrix-weighted variable Lebesgue space $L^{p(\cdot)}_W$ and to itself. This approach to establishing atomic characterization differs from all previous ones.

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arXiv2026

Variable Muckenhoupt $A_\infty$ Weights

In this article, with introducing concepts of variable scalar $\mathcal{A}_{p(\cdot),\infty}$ weights and variable matrix $\mathscr{A}_{p(\cdot),\infty}$ weights, we seek a comprehensive theory of $A_\infty$ weights within the framework of variable exponent spaces. We first show that a weight belongs to $\mathcal{A}_{p(\cdot),\infty}$ if and only if its $p(\cdot)$-th power is an $A_\infty$ weight. Using this, we characterize the $\mathcal{A}_{p(\cdot),\infty}$ condition by the minimal operator. Then we establish the reverse Hölder's inequality for $\mathcal{A}_{p(\cdot),\infty}$ weights in variable Lebesgue spaces with explicit constants and, combining this with the previously established relationship between $\mathcal{A}_{p(\cdot),\infty}$ weights and $A_\infty$ weights, we prove that, for any weight $w$, the reverse Hölder's inequality holds in variable Lebesgue spaces if and only if $w$ is an $\mathcal{A}_{p(\cdot),\infty}$ weight. For the matrix $\mathscr{A}_{p(\cdot),\infty}$ weights, we first show the existence of the reducing operators for matrix $\mathscr{A}_{p(\cdot),\infty}$ weights and then, combining the matrix $\mathscr{A}_{p(\cdot),\infty}$ weights with the scalar $\mathcal{A}_{p(\cdot),\infty}$ weights, we establish the reverse Hölder's inequality for $\mathscr{A}_{p(\cdot),\infty}$ weights in variable Lebesgue spaces. Finally, for further applications to variable matrix-weighted function spaces, we introduce the upper and the lower dimensions for $\mathscr{A}_{p(\cdot),\infty}$ weights and use these concepts to establish the sharp estimate involving reducing operators.

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arXiv2026

Real-variable theory of function spaces with operator-valued $A_p$ weights in Banach spaces

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" theorems, showing the impossibility of some prospective generalizations. However, we show that a complete real-variable theory of Besov and Triebel-Lizorkin spaces with operator-valued Muckenhoupt $A_p$ weights can still be developed, once correctly formulated. This covers operator-weighted extensions of results like the $φ$-transform characterization in terms of discrete sequence spaces, the boundedness of almost diagonal operators, and applications to the $T(1)$ theorem and trace/extension theorems. A key tool is a version of the reverse Hölder inequality, which is weak enough to follow from the operator-valued $A_p$ condition (unlike a variant that had to be imposed as an additional assumption in some previous works), yet strong enough to be used much like its classical counterpart. In contrast to the established scalar and matrix-weighted theories, our approach cannot build on operator-weighted $L^p$ results, as these fail in a very definite way. We also strengthen the existing "no go" statements in Hilbert spaces, showing (among other counterexamples) that every infinite-dimensional Banach space has an operator-valued $A_p$ weight $V$ for which the Hilbert transform is unbounded on $L^p(V)$. This is nontrivial, since the lack of Hilbert space structure also complicates the construction of $A_p$ weights. Building on results from Banach space theory, we achieve this unboundedness by combining two distinct methods in two different classes of spaces (so-called $K$-convex ones and those that are {\em not} UMD), whose union covers all Banach spaces.

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arXiv2026

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

Let $(X,\mathbf{q},μ)$ 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 additional reverse doubling property. Let $\mathrm{ind\,}(X,\mathbf{q})\in(0,\infty]$ denote its lower smoothness index, as introduced by Mitrea et al. In this monograph, the authors first construct a new approximation of the identity on quasi-ultrametric spaces of homogeneous type, achieving a maximal degree of smoothness $0<\varepsilon\preceq\mathrm{ind\,}(X,\mathbf{q})$. This fundamental tool is then used to derive sharp homogeneous (as well as inhomogeneous) continuous/discrete Calderón reproducing formulae on ultra-RD-spaces. As applications, the authors establish Littlewood--Paley function characterizations for both Hardy spaces and Triebel--Lizorkin spaces on ultra-RD-spaces. The authors further introduce Hardy--Lorentz spaces $H^{p,q}_\ast(X)$ via the grand maximal function, with the sharp range $p\in(\frac{n}{n+\mathrm{ind\,}(X,\mathbf{q})},\infty)$ and $q\in(0,\infty]$, and provide their real-variable characterizations using radial/non-tangential maximal functions, (finite) atoms, molecules, and various Littlewood--Paley functions. Based on these characterizations, the authors prove a duality theorem between Hardy--Lorentz spaces and Campanato--Lorentz spaces, establish a real interpolation theorem for Hardy--Lorentz spaces, and derive boundedness results for Calderón--Zygmund operators on them. It should be emphasized that many of the main results in this monograph are indeed established in the more general setting of quasi-ultrametric spaces of homogeneous type.

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arXiv2026

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

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 quasi-norms, the identification of function and sequence spaces via the $φ$-transform, the boundedness of almost diagonal operators and multi-parameter singular integrals under minimal assumptions, molecular and wavelet characterisations, and Sobolev-type embedding theorems. We identify matrix-weighted $L^p$ spaces, Sobolev spaces, and multi-parameter BMO spaces as examples of our general scale of spaces. Thus, our result on the boundedness of multi-parameter singular integrals on these spaces is seen as an extension, with a different method, of a recent theorem of Domelevo et al. [J. Math. Anal. Appl. 2024] on matrix-weighted $L^p$ spaces. For this theory, we develop several tools of independent interest. Many previous results were restricted to integrability exponents $p\in(1,\infty)$, while Besov-Triebel-Lizorkin spaces naturally involve the full range $p\in(0,\infty)$. We extend the definition of multi-parameter $A_p$ matrix weights to $p\in(0,1]$ and establish their basic properties, culminating in the $L^p$-boundedness of a matrix-weighted strong maximal operator (suitably rescaled when $p\in(0,1]$) for all $p\in(0,\infty)$. For $p\in(1,\infty)$, this is due to Vuorinen [Adv. Math. 2024] by convex-set-valued techniques of Bownik and Cruz-Uribe [arXiv 2022; Math. Ann. (to appear)]; the lack of convexity requires us to develop a new approach that works for all $p\in(0,\infty)$. We also need and prove a multi-parameter extension of Carleson-type embeddings from Frazier and Roudenko [Math. Ann. 2021] but attributed by them to F. Nazarov. We prove the necessity of the conditions of the new embedding using a nontrivial elaboration of Carleson's classical counterexample [Mittag-Leffler Rep. 1974].

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arXiv2026

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

Let ${\mathcal {X}}$ 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 gradient sequences and establish some related matrix-weighted Poincaré-type inequalities on ${\mathcal {X}}$. As an application, the authors introduce the matrix-weighted logarithmic Besov spaces on ${\mathcal {X}}$ and establish their pointwise characterization via Hajłasz gradient sequences. The novelty of this article lies in that, by means of both the $A_p$ dimension and its properties of matrix $A_p$-weights and the wavelet reproducing formula with exponential decay of P. Auscher and T. Hytönen, all the main results get rid of the dependence on the reverse doubling conditions of both weights and ${\mathcal {X}}$ under consideration and these results are also completely new even for unweighted logarithmic Besov spaces on ${\mathcal {X}}$.

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arXiv2026

Variable Matrix-Weighted Besov Spaces

In this article, using variable matrix ${\mathscr{A}}_{p(\cdot),\infty}$ weights, we introduce the matrix-weighted variable Besov space $B^{s(\cdot)}_{p(\cdot),q(\cdot)}(W)$ and the corresponding averaging variable Besov space $B^{s(\cdot)}_{p(\cdot),q(\cdot)}(\mathbb{A})$ and prove that they are equivalent. Applying this, we establish the $φ$-transform characterization of $B^{s(\cdot)}_{p(\cdot),q(\cdot)}(W)$. By this and via first establishing the boundedness of $α$-convexification $η$-type operators on variable Lebesgue spaces, we obtain the boundedness of almost diagonal operators on the sequence space $b^{s(\cdot)}_{p(\cdot),q(\cdot)}(W)$ related to $B^{s(\cdot)}_{p(\cdot),q(\cdot)}(W)$, which is further used to establish various decomposition characterizations of $B^{s(\cdot)}_{p(\cdot),q(\cdot)}(W)$, respectively, in terms of molecules, wavelets, and atoms. Applying the wavelet decomposition of $B^{s(\cdot)}_{p(\cdot),q(\cdot)}(W)$, we obtain the trace theorem and the extension properties of $B^{s(\cdot)}_{p(\cdot),q(\cdot)}(W)$, and, applying the molecular characterization, we obtain the boundedness of Calderón--Zygmund operators on $B^{s(\cdot)}_{p(\cdot),q(\cdot)}(W)$.

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arXiv2026

One-Sided and Parabolic BLO Spaces with Time Lag and Their Applications to Muckenhoupt $A_1$ Weights and Doubly Nonlinear Parabolic Equations

In this article, we first introduce the one-sided BLO space $\mathrm{BLO}^+(\mathbb{R})$ and characterize it, respectively, in terms of the one-sided Muckenhoupt class $A_1^+(\mathbb{R})$ and the one-sided John--Nirenberg inequality. Using these, we establish the Coifman--Rochberg type decomposition of $\mathrm{BLO}^+(\mathbb{R})$ functions and show that $\mathrm{BLO}^+(\mathbb{R})$ is independent of the distance between the two intervals, which further induces the characterization of this space in terms of the one-sided BMO space $\mathrm{BMO}^+(\mathbb{R})$ (the Bennett type lemma). As applications, we prove that any $\mathrm{BMO}^+(\mathbb{R})$ function can split into the sum of two $\mathrm{BLO}^+(\mathbb{R})$ functions and we provide an explicit description of the distance from $\mathrm{BLO}^+(\mathbb{R})$ functions to $L^\infty(\mathbb{R})$. Finally, as a higher-dimensional analogue we introduce the parabolic BLO space $\mathrm{PBLO}_γ^-(\mathbb{R}^{n+1})$ with time lag, and we extend all the above one-dimensional results to $\mathrm{PBLO}_γ^-(\mathbb{R}^{n+1})$; furthermore, as applications, we not only establish the relationships between $\mathrm{PBLO}_γ^-(\mathbb{R}^{n+1})$ and the solutions of doubly nonlinear parabolic equations, but also provide a necessary condition for the negative logarithm of the parabolic distance function to belong to $\mathrm{PBLO}_γ^-(\mathbb{R}^{n+1})$ in terms of the weak porosity of the set.

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arXiv2026

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

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 the specific $A_p$-weight constant when $p\in (1,\infty)$. As applications, we further obtain a new characterization of Muckenhoupt weights and, in the framework of ball Banach function spaces, an inequality related to Gagliardo and Sobolev seminorms on cubes, a Gagliardo--Nirenberg interpolation inequality, and a Bourgain--Brezis--Mironescu formula. All these obtained results have wide generality and are proved to be (nearly) sharp. The original version of this article was published in [Adv. Math. 481 (2025), Paper No. 110537]. In this revised version, we correct an error appeared in Theorem 1.1 in the case where $p=1$, which was pointed out to us by Emiel Lorist.

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arXiv2025

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

Let $Y(\mathcal{X})$ be a ball quasi-Banach function space on the space of homogeneous type $(\mathcal{X},ρ,μ)$ satisfying some mild additional assumptions, $q\in(0,\infty)$, and $\dot{W}^{s,q}_Y(\mathcal{X})$ with $s\in(0,1)$ be the homogeneous fractional Sobolev space associated with $Y(\mathcal{X})$. In this article, we show that, for any $f\in Y(\mathcal{X})\cap\bigcup_{s\in(0,1)} \dot{W}^{s,q}_Y(\mathcal{X})$, \begin{align*} \|f\|_{Y(\mathcal{X})} &\lesssim\varliminf_{s \to 0^+} s^{\frac{1}{q}}\left\| \left\{\int_{\mathcal{X}} \frac{|f(\cdot)-f(y)|^q}{U(\cdot,y)[ρ(\cdot,y)]^{sq}} \, dμ(y) \right\}^{\frac{1}{q}}\right\|_{{Y(\mathcal{X})}}\\ &\leq \varlimsup_{s \to 0^+} s^{\frac{1}{q}}\left\|\left\{\int_{\mathcal{X}} \frac{|f(\cdot)-f(y)|^q}{U(\cdot,y)[ρ(\cdot,y)]^{sq}} \, dμ(y) \right\}^{\frac{1}{q}}\right\|_{{Y(\mathcal{X})}} \lesssim\|f\|_{Y(\mathcal{X})}, \end{align*} where $U(x,y):=\min\{μ(B(x,ρ(x,y))),\,μ(B(y,ρ(x,y)))\}$ for any $x,y\in\mathcal{X}$ and the implicit positive constants are independent of $f$, which is applied to ten specific ball quasi-Banach function spaces and hence is of wide generality. In particular, when $Y(\mathcal{X})=L^q(\mathbb{R}^n)$ with $q\in[1,\infty)$, the above formula is closely related to the celebrated result of Maz'ya and Shaposhnikova in 2002. We also establish the above representation formula on domains of $\mathcal{X}$. The main novelty lies in proposing two new concepts, namely the weak reverse doubling condition (for $\mathcal{X}$) and the weak measure density condition (for domains of $\mathcal{X}$), which are proved to be necessary in some sense. In addition, we find an interesting fact that, when the underlying space under consideration is bounded, the above Maz'ya--Shaposhnikova-type limit always tends to zero.

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arXiv2025

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

Let $Λ_s$ denote the inhomogeneous Lipschitz space of order $s\in(0,\infty)$ on $\mathbb{R}^n$. This article characterizes the distance $d(f, V)_{Λ_s}: = \inf_{g\in V} \|f-g\|_{Λ_s}$ from a function $f\in Λ_s$ to a non-dense subspace $V\subset Λ_s$ via the fractional semigroup $\{T_{α, t}: =e^{-t (-Δ)^{α/2}}: t\in (0, \infty)\}$ for any $α\in(0,\infty)$. Given an integer $ r >s/α$, a uniformly bounded continuous function $f$ on $\mathbb{R}^n$ belongs to the space $Λ_s$ if and only if there exists a constant $λ\in(0,\infty)$ such that \begin{align*} \left|(-Δ)^{\frac {αr}2} (T_{α, t^α} f)(x) \right|\leq λt^{s -rα}\ \ \text{for any $x\in\mathbb{R}^n$ and $t\in (0, 1]$}.\end{align*} The least such constant is denoted by $λ_{ α, r, s}(f)$. For each $f\in Λ_s$ and $0<\varepsilon< λ_{α,r, s}(f)$, let $$ D_{α, r}(s,f,\varepsilon):=\left\{ (x,t)\in \mathbb{R}^n\times (0,1]:\ \left| (-Δ)^{\frac {αr}2} (T_{α, t^α} f)(x) \right|> \varepsilon t^{s -r α}\right\}$$ be the set of ``bad'' points. To quantify its size, we introduce a class of extended nonnegative \emph{admissible set functions} $ν$ on the Borel $σ$-algebra $\mathcal{B}(\mathbb{R}^n\times [0, 1])$ and define, for any admissible function $ν$, the \emph{critical index} $ \varepsilon_{α, r, s,ν}(f):=\inf\{\varepsilon\in(0,\infty):\ ν(D_{α, r}(s,f,\varepsilon))<\infty\}.$ Our result shows that, for a broad class of subspaces $V\subset Λ_s$, including intersections of $Λ_s$ with Sobolev, Besov, Triebel--Lizorkin, and Besov-type spaces, there exists an admissible function $ν$ depending on $V$ such that $\varepsilon_{α, r, s,ν}(f)\sim \mathrm{dist}(f, V)_{Λ_s}.$

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arXiv2025

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

Let $p\in(0,\infty)$, $q\in[1,\infty)$, $s\in\mathbb Z_+$, and $W$ be an $A_p$-matrix weight, which in the scalar case is exactly a Muckenhoupt $A_{\max\{1,p\}}$ weight. In this article, by using the reducing operators of $W$, we introduce matrix-weighted Campanato spaces $\mathcal L_{p,q,s,W}$. When $p\in(0,1]$, applying the atomic and the finite atomic characterizations of the matrix-weighted Hardy space $H^p_W$, we prove that the dual space of $H^p_W$ is precisely $\mathcal L_{p,q,s,W}$, which further induces several equivalent characterizations of $\mathcal L_{p,q,s,W}$. In addition, we obtain a necessary and sufficient condition for the boundedness of modified Calderón--Zygmund operators on $\mathcal L_{p,q,s,W}$ with $p\in(0,\infty)$, which, combined with the duality, further gives a necessary and sufficient condition for the boundedness of Calderón--Zygmund operators on $H^p_W$ with $p\in(0,1]$.

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arXiv2025

Sharp Weighted Cohen--Dahmen--Daubechies--DeVore Inequality with Applications to (Weighted) Critical Sobolev Spaces, Gagliardo--Nirenberg Inequalities, and Muckenhoupt Weights

In this article, we establish a quantitative weighted variant of a far-reaching inequality obtained by A. Cohen, W. Dahmen, I. Daubechies, and R. DeVore in 2003, whose dependence on the $A_p$-weight constant for any $p\in[1,\infty)$ is sharp. As applications, we obtain the almost characterization of the critical weighted Sobolev space in terms of wavelets, a sharp real interpolation between this weighted Sobolev space and weighted Besov spaces, and three new Gagliardo--Nirenberg type inequalities in the framework of ball Banach function spaces. Moreover, we apply this sharp weighted inequality to extend the famous Brezis--Seeger--Van Schaftingen--Yung formula in ball Banach function spaces, which gives an affirmative answer to the question in page 29 of [Calc. Var. Partial Differential Equations 62 (2023), Paper No. 234]. Notably, we further establish two new characterizations of Muckenhoupt weights related to the inequality of Cohen et al.\ and the formula of Brezis et al. The most novelty of this article exists in applying and further developing the good cube method introduced by Cohen et al.\ to trace the sharp dependences on weight constants.

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arXiv2025

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

Let $X$ be a ball Banach function space on $\mathbb{R}^n$, $k\in\mathbb{N}$, $h\in\mathbb{R}^n$, and $Δ^k_h$ denote the $k${\rm th} order difference. In this article, under some mild extra assumptions about $X$, the authors prove that, for both parameters $q$ and $γ$ in \emph{sharp} ranges which are related to $X$ and for any locally integrable function $f$ on ${\mathbb{R}^n}$ satisfying $|\nabla^k f|\in X$, $$ \sup_{λ\in(0,\infty)}λ\left\|\left[\int_{\{h\in\mathbb{R}^n:\ |Δ_h^k f(\cdot)|>λ|h|^{k+\fracγ{q}}\}} \left|h\right|^{γ-n}\,dh\right]^\frac{1}{q}\right\|_X \sim \left\|\,\left|\nabla^k f\right|\,\right\|_{X} $$ with the positive equivalence constants independent of $f$. As applications, the authors establish the Brezis--Seeger--Van Schaftingen--Yung (for short, BSVY) characterization of higher-order homogeneous ball Banach Sobolev spaces and higher-order fractional Gagliardo--Nirenberg and Sobolev type inequalities in critical cases. All these results are of quite wide generality and can be applied to various specific function spaces; moreover, even when $X:= L^{q}$, these results when $k=1$ coincide with the best known results and when $k\ge 2$ are completely new. The first novelty is to establish a sparse characterization of dyadic cubes in level sets related to the higher-order local approximation, which, together with the well-known Whitney inequality in approximation theory, further induces a higher-order weighted variant of the remarkable inequality obtained by A. Cohen, W. Dahmen, I. Daubechies, and R. DeVore; the second novelty is to combine this weighted inequality neatly with a variant higher-order Poincaré inequality to establish the desired upper estimate of BSVY formulae in weighted Lebesgue spaces.

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arXiv2025

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

Let $Λ_s$ denote the Lipschitz space of order $s\in(0,\infty)$ on $\mathbb{R}^n$, which consists of all $f\in\mathfrak{C}\cap L^\infty$ such that, for some constant $L\in(0,\infty)$ and some integer $r\in(s,\infty)$, \begin{equation*} \label{0-1}Δ_r f(x,y): =\sup_{|h|\leq y} |Δ_h^r f(x)|\leq L y^s, \ x\in\mathbb{R}^n, \ y \in(0, 1]. \end{equation*} Here (and throughout the article) $\mathfrak{C}$ refers to continuous functions, and $Δ_h^r$ is the usual $r$-th order difference operator with step $h\in\mathbb{R}^n$. For each $f\in Λ_s$ and $\varepsilon\in(0,L)$, let $ S(f,\varepsilon):= \{ (x,y)\in\mathbb{R}^n\times [0,1]: \frac {Δ_r f(x,y)}{y^s}>\varepsilon\}$, and let $μ: \mathcal{B}(\mathbb{R}_+^{n+1})\to [0,\infty]$ be a suitably defined nonnegative extended real-valued function on the Borel $σ$-algebra of subsets of $\mathbb{R}_+^{n+1}$. Let $\varepsilon(f)$ be the infimum of all $\varepsilon\in(0,\infty)$ such that $μ(S(f,\varepsilon))<\infty$. The main target of this article is to characterize the distance from $f$ to a subspace $V\cap Λ_s$ of $Λ_s$ for various function spaces $V$ (including Sobolev, Besov--Triebel--Lizorkin, and Besov--Triebel--Lizorkin-type spaces) in terms of $\varepsilon(f)$, showing that \begin{equation*} \varepsilon(f)\sim \mathrm{dist} (f, V\cap Λ_s)_{Λ_s}: = \inf_{g\in Λ_s\cap V} \|f-g\|_{Λ_s}.\end{equation*} Moreover, we present our results in a general framework based on quasi-normed lattices of function sequences $X$ and Daubechies $s$-Lipschitz $X$-based spaces.

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arXiv2025

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

This article is a continuation of our work on generalized matrix-weighted Besov--Triebel--Lizorkin-type spaces with matrix $\mathcal{A}_{\infty}$ weights. In this article, we establish the boundedness of pseudo-differential, trace, and Calderón--Zygmund operators on these spaces. The main tools involved in this article are the molecular and the wavelet characterizations of these spaces. Since generalized matrix-weighted Besov--Triebel--Lizorkin-type spaces include many classical function spaces such as matrix-weighted Besov--Triebel--Lizorkin spaces, all the results in this article are of wide generality.

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