arXiv subjects

Wen Yuan

Publications and source records attributed to Wen Yuan.

At least 19 recordsLinked to original sources

arXiv2026

Sharp Nonlocal Poincaré Inequalities and Liminf Lower Bounds for Sobolev and BV Energies on Metric Measure Spaces

Let $p\in[1,\infty)$, and let $(X,ρ,μ)$ be a complete doubling $q$-Poincaré space for some $q\in[1,p]$ with a lower dimension $d$ and an upper dimension $D$. In this article, we establish Poincaré inequalities for two families of nonlocal nonconvex functionals defined using distances or volumes. More precisely, for any ball $B$ with its radius $r_B\in(0,\operatorname{diam} X)$, any $λ\in(0,\infty)$, and any measurable function $f:X\to\mathbb R$, \begin{align*} &\frac{1}{μ(B)^2}\int_B\int_B|f(x)-f(y)|^p\,dμ(x)\,dμ(y)\\ &\quad\lesssimλ^pr_B^pϕ(B)^{γp}+\frac{λ^pr_B^p}{μ(B)}\mathop{\int_{ΛB}\int_{ΛB}}\limits_{\genfrac{}{}{}{2}{|f(x)-f(y)|}{ρ(x,y)ϕ(x,y)^γ}>λ}\frac{ϕ(x,y)^{γp}}{V(x,y)}\,dμ(x)\,dμ(y), \end{align*} where $V(x,y):=μ(B(x,ρ(x,y)))$ and either $ϕ=ρ$ with $ϕ(B)=r_B$ and $γ\in(-\infty,-\frac{\min\{q,D\}}{p}]$ or $ϕ=V$ with $ϕ(B)=μ(B)$ and $γ\in(-\infty,-\frac{q}{p\max\{q,d\}}]$. Here both ranges of $γ$ are sharp over the class of $q$-Poincaré spaces with the prescribed lower and upper dimensions, and the constant $Λ\in[1,\infty)$ and the implicit positive constant depend only on $p$, $γ$, and the geometric constants of $X$. We also obtain lower bounds for Sobolev energy and total variation in terms of the limit inferior of these functionals, for $γ$ in either the above negative ranges or $(0,\infty)$. Our proofs use the curve characterization of Poincaré spaces and an adaptive dyadic refinement argument based on medians.

math.AP↗

arXiv2026

Sharp Quantitative Weighted Estimates for Lifted Maximal Operators with Applications

In this article, we establish sharp quantitative weighted estimates for the lifted Hardy--Littlewood (resp Sobolev sharp) maximal operators $\{\mathcal M_θ\}_{θ\in(0,\infty)}$ (resp $\{\mathcal M_θ^\sharp\}_{θ\in(0,\infty)}$). Specifically, for $p\in(1,\infty)$, $γ\in\mathbb R\setminus\{0\}$, $\upsilon\in A_p$, and $f\in L^p_\upsilon$, we prove that \begin{align*} \sup_{θ,λ\in(0,\infty)}λ^p \underset{\mathcal M_θ(f)(x,t)>λt^γ} {\int_{\mathbb R^n}\int_0^\infty} t^{γp-1}\,dt\,\upsilon(x)\,dx \lesssim [\upsilon]_{A_p}^{α_{p,γ}} \|f\|_{L^p_\upsilon}^p, \end{align*} where $α_{p,γ}:=\frac{p}{p-1}$ for $γ\in[-n,0)$ and $α_{p,γ}:=1$ otherwise and where the implicit positive constant depends only on $n$, $p$, and $γ$. For $\mathcal M_θ^\sharp$, the corresponding estimate holds with $\|f\|_{L^p_\upsilon}^p$ replaced by $\|\nabla f\|_{L^p_\upsilon}^p$ for any $f\in\dot W^{1,p}_\upsilon$, and with $α_{p,γ}:=\frac{p}{p-1}$ for $γ\in[-1,0)$ and $α_{p,γ}:=1$ otherwise. We show that, for an $A_1$ weight $\upsilon$, the weight constant in the above estimates becomes $[\upsilon]_{A_1}$ for the sharp ranges $p\in(1,\infty)$ and $γ\in\mathbb R\setminus\{0\}$, or $p=1$ and $γ\in\mathbb R\setminus[-n,0]$ (resp. $γ\in\mathbb R\setminus[-1,0]$). We also prove that $α_{p,γ}$ is sharp, and we establish exact limiting formulae and characterize Muckenhoupt weights via weak-type bounds. As applications, we characterize the weighted Hardy(--Sobolev) space $H^1_\upsilon$ with $\upsilon\in A_1$ in terms of the truncated Riesz transforms with optimal range $γ\in \mathbb R\setminus\{0\}$. More applications include the sharp weighted Caffarelli--Silvestre extension and the generalization to ball Banach function spaces.

math.CA↗

arXiv2026

A Sharp Planar Fractional Isoperimetric Inequality

In this article, we prove the sharp form of the fractional isoperimetric inequality in the plane, originally posed by Maz'ya [Problem 1, Integral Equations Operator Theory, 2018]. More precisely, we show that, for any given $s\in(0,1)$ and any bounded domain $Ω\subset \mathbb{R}^2$ with $C^1$ boundary, $$ P_s(Ω) \le \frac{π^{s-\frac{1}{2}}Γ\left(\frac{3-s}{2}\right)}{s(1-s)Γ\left(\frac{4-s}{2}\right)}\left[\mathcal{H}^1(\partialΩ)\right]^{2-s}, $$ where $P_s$ denotes the fractional $s$-perimeter, $Γ$ denotes the Gamma function, and $\mathcal{H}^{1}$ denotes the $1$-dimensional Hausdorff measure on $\mathbb{R}^2$. The constant is sharp and equality is attained by the disk. The proof proceeds in three steps: reducing the fractional perimeter to a chord functional, reducing connected domains to convex bodies, and proving the sharp convex chord inequality via a fractional Willmore-type inequality, a variational formula along outer parallel bodies, and asymptotic analysis.

math.FA↗

arXiv2026

Sharp Truncated Difference Estimates for Nonlocal Nonconvex Functionals Defined by Superlevel Sets

Let $N\in\mathbb N$ and $p\in[1,\infty)$. In this article, we consider a family $\{Φ_{λ,p}^γ\}_{λ\in(0,\infty)}$ of nonlocal nonconvex functionals with a real parameter $γ$ and establish two truncated difference estimates bounded by these functionals with the sharp range of $γ$; that is, for any $γ\in(-\infty,-1]$, any $λ\in(0,\infty)$, any cube $Q\subset\mathbb R^N$, and any measurable function $g:Q\to\mathbb R$, \begin{align*} &\int_Q\int_Q\left[\frac{|g(x)-g(y)|-λ|x-y|^{1+γ/p}}{\ell(Q)^{1+γ/p}}\right]_+^p\,dx\,dy\\ &\quad\lesssim λ^p\mathop{\int_Q\int_Q}\limits_{\frac{|g(x)-g(y)|}{|x-y|^{1+γ/p}}>λ}\left[\frac{|x-y|}{\ell(Q)}\right]^{γ-N}\,dx\,dy=:\frac{Φ_{λ,p}^γ(g;Q)}{\ell(Q)^{γ-N}}, \end{align*} where the implicit positive constant depends only on $N$, $p$, and $γ$. For $γ\in(0,\infty)$, the above estimate holds with the left-most integrand replaced by $\frac{\min\{|g(x)-g(y)|^p,λ^p|x-y|^{p+γ}\}}{\ell(Q)^{p+γ}}$. As applications, these estimates yield related nonlocal Poincaré inequalities with the sharp range of $γ$, and also recover the known $Γ$-liminf lower bounds and characterizations of Sobolev and BV spaces in terms of the liminf of the functionals $\{Φ^γ_{λ,p}\}_{λ\in(0,\infty)}$ for all $γ\in\mathbb{R}\setminus(-1,0]$. We investigate further applications to compactness and variational problems as well as a new VMO criterion. Inspired by a recent work of R. L. Frank, we develop an elementary approach based on dyadic iteration of truncated differences along line segments, which allows us to treat the aforementioned problems in a unified way.

math.AP↗

arXiv2026

Endpoint Lorentz Inequality for Riesz Potentials with Respect to General Measures

Let $n\ge2$, $α\in(0,n)$, and $μ$ be a locally finite nonnegative Borel measure on $\mathbb{R}^n$. We prove that, for any $f\in C_{\rm c}^\infty(\mathbb{R}^n)$, \begin{align*} \|I_αf\|_{L^{\frac{n}{n-α},1}(μ)} \le C_{n,α} \int_{\mathbb{R}^n} |Rf(x)|[Mμ(x)]^{1-\fracα{n}}\,dx, \end{align*} where $I_α$ is the Riesz potential, $R$ is the Riesz transform, $Mμ$ is the Hardy--Littlewood maximal function of $μ$, and $C_{n,α}$ is a positive constant depending only on $n$ and $α$. This answers a question posed by Bortz, Moen, Olivo, Pérez, and Rela, with the case $α=1$ already known, and extends Spector's optimal unweighted Lorentz inequality to general measures. As applications, we obtain a Lorentz estimate for curl-free vector fields, Sobolev--Lorentz and Hardy inequalities with respect to general measures, and an $A_1$-weighted Lorentz estimate with a sharp exponent of the $A_1$ characteristic.

math.CA↗

arXiv2026

Sharp Matrix-Weighted Lower Bounds for Non-Degenerate Convolution Calderón--Zygmund Operators

Let $T$ be a convolution Calderón--Zygmund operator satisfying Hytönen's non-degeneracy condition. We prove that, for every $p\in(1,\infty)$ and $m\ge3$, there exists a positive constant $C$ such that, for any $t\in[1,\infty)$, \begin{equation*} \sup_{[W]_{\mathcal A_p}\le t} \|T\|_{L^p(W)\to L^p(W)} \ge C t^{1+\frac{1}{p(p-1)}}, \end{equation*} where the supremum is taken over all matrix weights $W:\ \mathbb R^n\to\mathbb C^{m\times m}$ with their weight constants $[W]_{\mathcal A_p}\le t$. If, in addition, the odd part of the kernel is non-degenerate (such as the Hilbert transform and the Riesz transforms), the same conclusion holds for every $m\ge2$.

math.CA↗

arXiv2026

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

We study Brownian motion killed upon exiting arbitrary open sets $Ω\subset\mathbb R^n$, $n\in[3,\infty)\cap\mathbb N$. Let $α\in(0,1]$. The infimal radius $d_α(x)$ at which the absorber near $x$ carries an $α$-fraction of the Newtonian capacity of the ball of the same radius provides a local measure of electrostatic trapping. We prove that this capacitary loading forces an absorption probability of at least $c_nα$ within a dimension-dependent multiple of the diffusive time scale $d_α(x)^2$. Independently, if $p\in (0,1)$ and $τ_p(x)$ is the time by which Brownian motion started at $x$ has been absorbed with probability at least $p$, then \begin{align*} \int_Ω\frac{|u\left(x\right)|^2}{τ_p\left(x\right)}\,\mathrm d x \le\frac{2}{p^2}\int_Ω|\nabla u\left(x\right)|^2\,\mathrm d x \end{align*} for every $u\in C_{\mathrm{c}}^\infty(Ω)$. Together these statements give a capacitary Hardy inequality with constant $C_nα^{-2}$ on every open set, answering a problem of Maz'ya, where $C_n$ is a positive constant and, moreover, the exponent of $α^{-2}$ is sharp. As applications, we derive corresponding estimates for survival probabilities, a corresponding spectral lower bound (including a recovery of the Maz'ya--Shubin lower bound), heat dissipation, Schrödinger forms, and Fisher information.

math.CA↗

arXiv2026

Sharp Quantitative Matrix-Weighted Bounds for Fractional Integrals and Sobolev Inequalities

For $α\in(0,n)$, $p\in(1,\frac{n}α)$, $q:=\frac{np}{n-αp}$, and $W\in\mathscr A_{p,q}$, we obtain the following matrix-weighted boundedness of the fractional integral $I_α$: for any $\vec{f}\in L^p(W^p)$, \begin{align*} \left\|I_α\vec f\right\|_{L^q(W^q)} \lesssim[W]_{\mathscr A_{p,q}}^{ (1-\fracα{n})\max\{1,\frac{p'}{q}\}} \left\|\vec f\right\|_{L^p(W^p)}, \end{align*} where the implicit positive constant is independent of $W$ and $\vec{f}$. For integer $n\geq2$, $p\in[1,n)$, $q:=\frac{np}{n-p}$, and $W\in\mathscr A_{p,q}$, we establish the following matrix-weighted Sobolev inequality: for any smooth $\mathbb{C}^d$-valued function $\vec{f}$ with compact support, \begin{align*} \left\|\vec{f}\right\|_{L^{q}(W^q)} \lesssim[W]_{\mathscr A_{p,q}}^{\frac{n-1}{n}} \left\|WD\vec{f}\right\|_{L^p(\mathbb R^n,\mathbb C^{d\times n})}, \end{align*} where the implicit positive constant is independent of $W$ and $\vec{f}$ and where $D\vec{f}$ is the Jacobian matrix of $\vec{f}$. Surprisingly, in both bounds above the exponents of $[W]_{\mathscr A_{p,q}}$ coincide with the corresponding known optimal scalar exponents, and hence are optimal, which is different from the Calderón--Zygmund operator case. A key idea to obtain the above matrix-weighted boundedness of the fractional integrals is to establish the off-diagonal chain-packing principle of dyadic cubes, which is inspired by a recent work of A. K. Lerner.

math.CA↗

arXiv2026

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

Let $N\ge1$, $p\in[1,\infty)$, and $Ω\subset\mathbb R^N$ be a bounded open interval when $N=1$ or a bounded Lipschitz domain when $N\ge2$. For any given $γ\in\mathbb R\setminus\{0\}$ and $λ\in(0,\infty)$, consider the weak-type nonlocal functional \begin{align*} F_{λ,p,γ}(u;Ω):=λ^p \underset{|u(x)-u(y)|>λ|x-y|^{1+\fracγ{p}}} {\int_Ω\int_Ω}|x-y|^{γ-N}\,dy\,dx. \end{align*} In this article, we prove that, for $γ\in(0,\infty)$ as $λ\to\infty$ or for $γ\in(-\infty,-1]$ as $λ\to0^+$, the family $F_{λ,p,γ}(\cdot;Ω)$ converges, in the sense of $Γ$-convergence in $L^p(Ω)$, to the functional \begin{align*} Ψ_{p,γ}^Ω(u) :=\begin{cases} C_{N,p,γ}^{\rm cell}\displaystyle\int_Ω|\nabla u|^p\,dx, &p\in(1,\infty)\ \hbox{and}\ u\in W^{1,p}(Ω),\\[2mm] C_{N,1,γ}^{\rm cell}|Du|(Ω), &p=1\ \hbox{and}\ u\in BV(Ω),\\[1mm] \infty,&\hbox{otherwise}. \end{cases} \end{align*} where the positive constants $C_{N,p,γ}^{\mathrm{cell}}$ are independent of $Ω$. As applications, we further show that, for $γ\in(0,\infty)$ as $λ\to\infty$ or for $γ\in(-\infty,-1]$ as $λ\to0^+$, the weak-type nonlocal functional $F_{λ,p,γ}(\cdot;\mathbb R^N)$ converges, in the sense of $Γ$-convergence in $L^1_{\rm loc}(\mathbb R^N)$, to a similar functional; this gives an complete answer to the problem posed by Brezis et al. in [Section~7C, Anal. PDE 2024].

math.CA↗

arXiv2026

A Solution of Problem 2.5 by Brezis on the Planar Ginzburg--Landau Equation

Brezis, Merle, and Riviére [Arch. Rational Mech. Anal. 1994] proved that, if a smooth solution $u:\mathbb{R}^2\to\mathbb{C}$ of the planar entire Ginzburg--Landau equation $$ -Δu = u(1-|u|^2)\quad \text{in}\quad \mathbb{R}^2 $$ satisfies the finite potential energy estimate that $$ \int_{\mathbb{R}^2}\left[1-|u(x)|^2\right]^2\,dx < \infty, $$ then it has the asymptotic property that $$ |u(x)|\to 1\quad \text{as}\quad |x|\to\infty. $$ The converse problem whether the asymptotic property implies the finite potential energy estimate was originally posed in their work and later formulated by Brezis as \emph{Open Problem 2.5} in [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 2023]. In this article, we give an affirmative answer to this question. Our proof relies on the Kelvin inversion, a Morrey-type energy decay for the translation Jacobi system, the $L^4$-integrability of the phase form, and a linearised amplitude equation; all these tools together imply the $L^2$-integrability of the function $1-|u|$ on an exterior domain, without any a priori integrability assumption.

math.AP↗

arXiv2026

On Two Questions by Brezis et al Concerning the Critical Difference Quotient Characterization of First-Order Sobolev Spaces

Let $N\in\mathbb N$ and $γ\in[-1,0)$. In [Anal. PDE 17 (2024)], Brezis, Seeger, Van~Schaftingen, and Yung asked how, in the exceptional range $γ\in[-1,0)$, $\dot{\mathrm{BV}}(γ)$ and $\dot W^{1,1}(γ)$ on ${\mathbb R}^N$ are related to other function spaces, especially to Hardy--Sobolev spaces, and whether these spaces are normable. In this article, we prove that the homogeneous Hardy--Sobolev space is strictly embedded, respectively, into $\dot{\mathrm{BV}}(γ)$ and $\dot W^{1,1}(γ)$, and neither $\dot{W}^{1,1}(γ)/\mathbb R$ nor $\dot{BV}(γ)/\mathbb R$ is normable, which answers the above two questions.

math.FA↗

arXiv2026

Local Matrix Muckenhoupt Weights and Quantitative Weighted Inequalities Achieving Global Best Known Exponents

In this article, we give various real-variable properties of local Muckenhoupt matrix weights $W$ and establish the quantitative boundedness of several operators on local matrix-weighted Lebesgue spaces $L^p(W)$, including local (fractional) maximal operators, local fractional integral operators, local Haar square functions, and Calderón--Zygmund operators with exponential decay. For the local maximal operators, we obtain the sharp quantitative bounds when $p\in(1,2]$, while, for local fractional integral operators, we obtain the quantitative bounds matching the global best known exponents, whose scalar case is known to be sharp. The key used strategies include giving a new extension property (which can clarify their relationships with global ones) and an optimal scale lifting property (which can balance the locality of weights and operators under consideration) of local matrix weights. As an application, we establish the quantitative boundedness on $L^p(W)$ of the Riesz transform associated with Schrödinger operators $-Δ+m^2I$ with $m\in(0,\infty)$ being large enough, whose quantitative bound when $p=2$ is precisely $[W]_{\mathscr{A}^{\operatorname{loc}}_{2}(r)}^{\frac 32}$.

math.FA↗

arXiv2026

Hardy--Littlewood Maximal Operator and Two-Layer Muckenhoupt Weights on Infinite Rooted $k$-Ary Trees

Let $k\geq 2$ be an integer, $T$ an infinite rooted $k$-ary tree, and $M$ the Hardy--Littlewood maximal operator on $T$. For any $p\in(0,\infty)$, we characterize the weight $w$ such that $M$ is bounded on $L^p(w)$. To this end, we introduce a two-layer Muckenhoupt weight class $\mathscr A_p$ and prove that, for any $p\in(\frac{1}{2},\infty)$, the boundedness of $M$ on $L^p(w)$, $w\in\mathscr A_p$, and the exponential decay boundedness of spherical averaging operators on $L^p(w)$ are mutually equivalent, and that, when $p\in(0,\frac{1}{2}]$, there exists no weight $w$ such that $M$ is bounded on $L^p(w)$. Moreover, for any $p\in(\frac{1}{2},\infty)$, we establish the quantitative estimate, with the optimal exponent $\frac{1}{p}$ of the weight constant, for the boundedness of $M$ on $L^p(w)$. For any $p\in(1,\infty)$, we also obtain two further equivalent characterizations of the boundedness of $M$ on $L^p(w)$, respectively, in terms of a global Sawyer-type testing condition and an estimate for the weighted product measure of distance incidence sets. As applications, for any $p\in(\frac{1}{2},\infty)$, under the assumption that $M$ is bounded on $L^p(w)$, we establish the boundedness of exponentially decaying kernel operators on $L^p(w)$ and weighted Fefferman--Stein vector-valued inequalities.

math.CA↗

arXiv2026

Weak-type characterizations of Sobolev and bounded variation spaces on metric measure spaces

Given a complete doubling metric measure space $(X,ρ,μ)$ supporting a Poincaré inequality, we prove weak-type characterizations of the Sobolev space $\dot{W}^{1,p}(μ)$ and the space of functions of bounded variation, achieving a full analogy in general Poincaré spaces with the Euclidean results of Brezis et al. [Anal. PDE 17 (2024), 943-979]. The main novelty is that the finiteness of a weak-type norm, which only refers to differences or mean oscillations of $f$ without assuming any smoothness a priori, already guarantees the membership of $f$ in the relevant Sobolev or BV space. This distinguishes our contribution from the recent work of F. Dai et al. [Adv. Math. 502 (2026), Paper No. 111153], where the related norm-equivalence was obtained under the a priori Lipschitz assumption on $f$. A key intermediate step in our approach is a new localized Bourgain-Brezis-Mironescu type characterization. More precisely, we prove that, if $p\in(1,\infty)$ and $γ\in\mathbb R\setminus\{0\}$, then, for any $f\in L^1_{\mathrm{loc}}(μ)$, \begin{equation*}\tag{$*$} \|f\|_{\dot W^{1,p}(μ)} \sim \|ρ^{-1}ϕ^{-γ}F\|_{L^{p,\infty}(ϕ^{γp}V^{-1})}, \qquad F\in\{Δf,m_f\},\quad ϕ\in\{ρ,V\}, \end{equation*} where the homogeneous Sobolev space $\dot{W}^{1,p}(μ)$ is defined by the minimal $p$-weak upper gradient and, for any $x,y\in X$, we denote $V(x,y):=μ(B(x,ρ(x,y)))$ and $Δf(x,y):=|f(x) - f(y)|$, and $m_f(x,y)$ is the mean oscillation of $f$ on the ball $B(x,ρ(x,y))$. For $p=1$, the equivalence $(*)$ holds after replacing $\|f\|_{\dot W^{1,1}(μ)}$ by a bounded variation norm and restricting the parameters to the optimal ranges $γ\in(-\infty,-1)\cup(0,\infty)$ for $ϕ=ρ$ or $γ\in (-\infty,-\frac1d)\cup(0,\infty)$ for $ϕ=V$, where $d\in(0,\infty)$ is the lower dimension of $X$.

math.FA↗

arXiv2026

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

Let $N\ge1$, $p\in[1,\infty)$, $γ\in(0,\infty)$, and $Ω\subset\mathbb R^N$ be a bounded open interval when $N=1$ or a bounded Lipschitz domain when $N\ge2$. For any $λ\in(0,\infty)$ and any measurable function $u$, consider the weak-type nonlocal functional \begin{align*} G_{λ,p,γ}(u;Ω) :=λ\iint_{Ω\timesΩ} \mathbf 1_{\left\{(x,y)\inΩ\timesΩ:\ x\neq y,\ \frac{|u(x)-u(y)|^p}{|x-y|^{p+γ}}\geqλ\right\}} |x-y|^{γ-N}\,dx\,dy. \end{align*} In this article, we prove that, as $λ\to\infty$, the family $G_{λ,p,γ}$ converges, in the sense of $Γ$-convergence in $L^p(Ω)$, to the functional \begin{align*} Ψ_{p,γ}^{\mathrm{cell}}(u;Ω):= \begin{cases} C_{N,p,γ}^{\mathrm{cell}}\displaystyle\int_Ω|\nabla u|^p\,dx, &p\in(1,\infty)\ \hbox{and}\ u\in W^{1,p}(Ω),\\[2mm] C_{N,1,γ}^{\mathrm{cell}}|Du|(Ω), &p=1\ \hbox{and}\ u\in BV(Ω),\\[1mm] \infty,&\hbox{otherwise}, \end{cases} \end{align*} where the positive constants $C_{N,p,γ}^{\mathrm{cell}}$ are independent of $Ω$ and characterized by a cell formula. This gives an affirmative answer to the problem posed by Brezis [Open Problem~9.3, Rend. Lincei Mat. Appl. 2023].

math.CA↗

arXiv2026

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

For any given time lag $γ\in(0,1)$, we prove that the one-sided parabolic BMO space $\mathrm{BMO}^+(γ)$ coincides with the parabolic BMO space $\mathrm{PBMO}^-(γ)$ with equivalent norms, the parabolic Muckenhoupt class $A_{\infty}^+(γ)$ defined via the reverse Jensen inequality can be represented as the union of the parabolic Muckenhoupt classes $A_r^+(γ)$ with $r\in[1,\infty)$, and the parabolic reverse Hölder classes $\bigcup_{q\in(1,\infty]}RH_q^+$ coincide with the parabolic Muckenhoupt classes $\bigcup_{r\in[1,\infty)}A_r^+(γ)$, and hence give affirmative answers to Questions 4.5 and 4.6 posed by Kinnunen and Saari [Nonlinear Anal. 131 (2016)]. To show them, we establish the uniform parabolic space-time shifting property for parabolic reverse Hölder weights, and develop the one-sided stopping time argument which yields a new parabolic John--Nirenberg inequality for $\mathrm{BMO}^+(γ)$. As applications, we obtain John--Nirenberg and exponential integrability characterizations of $\mathrm{BMO}^+(γ)$, prove that $\mathrm{BMO}^+(γ)$ is independent of the positive time lag, and identify its null space.

math.CA↗

arXiv2026

Real-Variable Characterizations and Their Applications of Anisotropic Besov Spaces with Matrix $\mathcal A_\infty$ Weights

Let $α\in\mathbb{R}$, $p\in(0,\infty)$, and $q\in(0,\infty]$. In this article, we develop a theory of matrix-weighted anisotropic Besov spaces associated with an expansive matrix $A$ and an $\mathcal A_{p,\infty}$-matrix weight $W$. We first introduce the homogeneous spaces $\dot B_{p,q}^α(A,W)$ and establish their $φ$-transform characterization. Then we construct counterexamples to show that the assumption $W\in\mathcal A_{p,\infty}$ in this characterization cannot be relaxed to $W\in\bigcup_{r\in(0,\infty)}\mathcal A_r$. The same counterexamples also show that this weaker condition $W\in\bigcup_{r\in(0,\infty)}\mathcal A_r$ is insufficient to ensure the well-definedness of $\dot B_{p,q}^α(A,W)$. Next we characterize $\mathcal A_{p,\infty}$-matrix weights via the rescaled maximal operator, which leads naturally to a new concept of the critical rescaling index that quantitatively captures the self-improving behavior of matrix weights. In terms of this index, we obtain optimal boundedness for almost diagonal operators on the associated sequence spaces $\dot b_{p,q}^α(A,W)$. Based on these, we further establish the molecular characterization of $\dot B_{p,q}^α(A,W)$ and some sharp boundedness results for pseudo-differential operators on these spaces.

math.CA↗

arXiv2026

Fractional Gagliardo--Nirenberg Inequalities: Pointwise Estimates,Sharp Asymptotics, and Optimal Target Spaces

We establish two pointwise estimates for fractional difference operators, tracking explicitly the dependence of the constants on the smoothness index $s\in(0,1)$. Using these, within the framework of ball Banach function spaces we obtain two fractional Gagliardo--Nirenberg inequalities, including the BMO endpoint case. Furthermore, we establish endpoint asymptotic results as $s\to0^+$ and $s\to1^-$, proving that the asymptotic factors appearing in these inequalities have optimal order. Under the additional assumption that the underlying function space is rearrangement invariant, we show that the optimal Gagliardo--Nirenberg target spaces are precisely those given by the Calderón--Lozanovski\uı space. This completely characterizes the rearrangement invariant target spaces for which the corresponding Gagliardo--Nirenberg inequalities hold, thereby answering an open question posed by K. Leśnik, T. Roskovec, and F. Soudský. These results can be applied to various function spaces; in particular, they are completely new in the off-diagonal and BMO cases.

math.CA↗