arXiv subjects

You-Wei Benson Chen

Publications and source records attributed to You-Wei Benson Chen.

9 recordsLinked to original sources

arXiv2026

Commutators of Fractional Integrals with $\operatorname{BMO}^β$ Functions

We study commutators of the Riesz potential $I_α$ with functions $b$ in the capacitary space $\mathrm{BMO}^β(\mathbb{R}^n)$, defined through the Hausdorff content $\mathcal{H}^β_\infty$. We prove a Chanillo-type theorem characterising $\mathrm{BMO}^β(\mathbb{R}^n)$ via the boundedness of the commutator $[b,I_α]$ on capacitary Lebesgue spaces. In addition, we obtain the endpoint estimate in the form of a capacitary modular weak-type inequality. These results follow from a pointwise estimate for the $β$-dimensional sharp maximal function of the commutator, together with a capacitary Fefferman-Stein inequality recently proved in [CC24].

math.CA↗

arXiv2025

Uncentered Fractional Maximal functions and mean oscillation spaces associated with dyadic Hausdorff content

We study the action of uncentered fractional maximal functions on mean oscillation spaces associated with the dyadic Hausdorff content $\mathcal{H}_{\infty}^β$ with $0<β\leq n$. For $0 < α< n$, we refine existing results concerning the action of the Euclidean uncentered fractional maximal function $\mathcal{M}_α$ on the functions of bounded mean oscillations (BMO) and vanishing mean oscillations (VMO). In addition, for $0 < β_1 \leq β_2 \leq n$, we establish the boundedness of the $β_2$-dimensional uncentered maximal function $\mathcal{M}^{β_2}$ on the space $\text{BMO}^{β_1}(\mathbb{R}^n)$, where $\text{BMO}^{β_1}(\mathbb{R}^n)$ denotes the mean oscillation space adapted to the dyadic Hausdorff content $\mathcal{H}_{\infty}^{β_1}$ on $\mathbb{R}^n$.

math.FA↗

arXiv2025

$β$-dimensional sharp maximal function and applications

In this paper, we study $β$-dimensional sharp maximal operator defined as \begin{align*} \mathcal{M}^{\#} _βf(x) := \sup_{Q} \inf_{c \in \mathbb{R}} χ_{Q}(x) \frac{1}{\ell(Q)^β} \int_Q |f-c| \; d \mathcal{H}^β_\infty, \end{align*} where the supremum is taken over all cubes in $\mathbb{R}^d$ with sides pararell to the coordinate axes, $\ell(Q)$ is the length side of $Q$ and $\mathcal{H}^β_\infty$ is the Hausdorff content. In particular, we prove Fefferman-Stein inequality for $\mathcal{M}^{\#} _βf$ by giving a good lambda estimate for $β$-dimensional sharp maximal operator in the context of Hausdorff content. Additionally, we prove the Muckenhoupt-Wheeden inequality in this framework by establishing a good lambda inequality of independent interest.

math.FA↗

arXiv2025

An atomic decomposition of one-dimensional metric currents without boundary

This paper proves an atomic decomposition of the space of $1$-dimensional metric currents without boundary, in which the atoms are specified by closed Lipschitz curves with uniform control on their Morrey norms. Our argument relies on a geometric construction which states that for any $ε>0$ one can express a piecewise-geodesic closed curve as the sum of piecewise-geodesic closed curves whose total length is at most $(1+ε)$ times the original length and whose Morrey norms are each bounded by a universal constant times $ε^{-2}$. In Euclidean space, our results refine the state of the art, providing an approximation of divergence free measures by limits of sums of closed polygonal paths whose total length can be made arbitrarily close to the norm of the approximated measure.

math.FA↗

arXiv2025

The Capacitary John-Nirenberg Inequality Revisited

In this paper, we establish maximal function estimates, Lebesgue differentiation theory, Calderón-Zygmund decompositions, and John-Nirenberg inequalities for translation invariant Hausdorff contents. We further identify a key structural component of these results -- a packing condition satisfied by these Hausdorff contents which compensates for the non-linearity of the capacitary integrals. We prove that for any outer capacity, this packing condition is satisfied if and only if the capacity is equivalent to its induced Hausdorff content. Finally, we use this equivalence to extend the preceding theory to general outer capacities which are assumed to satisfy this packing condition.

math.FA↗

arXiv2024

$BMO$ and gradient estimates for solutions of critical elliptic equations

In this paper we explore several applications of the recently introduced spaces of functions of bounded $β$-dimensional mean oscillation for $β\in (0,n]$ to regularity theory of critical exponent elliptic equations. We first show that functions with gradient in weak-$L^n$ are in $BMO^β$ for any $β\in (0,n]$, improving the classical result $\nabla u\in L^n$ implies $u\in BMO$. We apply this result to the Poisson equation $-Δu = \operatorname*{div} F$ with zero boundary conditions in a bounded $C^1$ domain to show that $u\in BMO^β$ when $F$ is in weak-$L^n$. Next, we consider the $n$-Laplace equation \begin{align*} -\operatorname*{div}( |\nabla U|^{n-2} \nabla U) &= F \text{ in } Ω, \newline U &=0 \text{ on }\partial Ω. \end{align*} with $F\in L^1(Ω)$ and show that the classical result $u\in BMO$ can be improved to $u\in BMO^β$. Finally, we consider the $n$-Laplace equation in the case when $F \in L^1$, $\operatorname*{div} F=0$ and prove that for smooth domains $Ω$ we have the estimate \begin{align*} \|\nabla U \|_{L^n} \mathbb \leq C \, \|F\|^{1/(n-1)}_{L^1}, \end{align*} where the constant $C$ is independent of $F$.

math.AP↗

arXiv2024

A self-improving property of Riesz potentials in BMO

In this paper we prove that for non-negative measurable functions $f$, \begin{align*} I_αf \in BMO(\mathbb{R}^n) \text{ if and only if } I_αf \in BMO^β(\mathbb{R}^n) \text{ for } β\in (n-α,n]. \end{align*} Here $I_α$ denotes the Riesz potential of order $α$ and $BMO^β$ represents the space of functions of bounded $β$-dimensional mean oscillation.

math.FA↗

arXiv2023

Capacitary Maximal Inequalities and Applications

In this paper we introduce capacitary analogues of the Hardy-Littlewood maximal function, \begin{align*} \mathcal{M}_C(f)(x):= \sup_{r>0} \frac{1}{C(B(x,r))} \int_{B(x,r)} |f|\;dC, \end{align*} for $C=$ the Hausdorff content or a Riesz capacity. For these maximal functions, we prove a strong-type $(p,p)$ bound for $1<p \leq+\infty$ on the capacitary integration spaces $L^p(C)$ and a weak-type $(1,1)$ bound on the capacitary integration space $L^1(C)$. We show how these estimates clarify and improve the existing literature concerning maximal function estimates on capacitary integration spaces. As a consequence, we deduce correspondingly stronger differentiation theorems of Lebesgue-type, which in turn, by classical capacitary inequalities, yield more precise estimates concerning Lebesgue points for functions in Sobolev spaces.

math.FA↗

arXiv2022

On functions of bounded $β$-dimensional mean oscillation

In this paper, we define a notion of $β$-dimensional mean oscillation of functions $u: Q_0 \subset \mathbb{R}^d \to \mathbb{R}$ which are integrable on $β$-dimensional subsets of the cube $Q_0$: \begin{align*} \|u\|_{BMO^β(Q_0)}:= \sup_{Q \subset Q_0} \inf_{c \in \mathbb{R}} \frac{1}{l(Q)^β} \int_{Q} |u-c| \;d\mathcal{H}^β_\infty, \end{align*} where the supremum is taken over all finite subcubes $Q$ parallel to $Q_0$, $l(Q)$ is the length of the side of the cube $Q$, and $\mathcal{H}^β_\infty$ is the Hausdorff content. In the case $β=d$ we show this definition is equivalent to the classical notion of John and Nirenberg, while our main result is that for every $β\in (0,d]$ one has a dimensionally appropriate analogue of the John-Nirenberg inequality for functions with bounded $β$-dimensional mean oscillation: There exist constants $c,C>0$ such that \begin{align*} \mathcal{H}^β_\infty \left(\{x\in Q:|u(x)-c_Q|>t\}\right) \leq C l(Q)^β\exp(-ct/\|u\|_{BMO^β(Q_0)}) \end{align*} for every $t>0$, $u \in BMO^β(Q_0)$, $Q\subset Q_0$, and suitable $c_Q \in \mathbb{R}$. Our proof relies on the establishment of capacitary analogues of standard results in integration theory that may be of independent interest.

math.AP↗