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Cody B. Stockdale

Publications and source records attributed to Cody B. Stockdale.

At least 19 recordsLinked to original sources
arXiv2026

Sharp weighted weak-type bounds for commutators of singular integrals

We establish sharp weak-type bounds for commutators of Calderón--Zygmund operators and $\text{BMO}(\mathbb{R}^n)$ symbols with respect to $A_p$ weights. We prove that the bound \looseness=-1 $$ \|[b,T]\|_{L^p(w)\rightarrow L^{p,\infty}(w)}\lesssim [w]_{A_p}^{\max(2,p')} $$ for $p \in (1,\infty)$ and $w \in A_p$ is sharp for general $T \in \text{CZO}(\mathbb{R}^n)$ and $b \in \text{BMO}(\mathbb{R}^n)$. In the case of the standard logarithmic symbol, we show that the estimate improves to $$ \|[\log|\cdot|,T]\|_{L^p(w)\rightarrow L^{p,\infty}(w)}\lesssim [w]_{A_p}^{\max\big(2,\frac{1}{p}+\frac{p'}{p}\big)} $$ and that this dependence is optimal for this symbol. Our upper bound for the logarithmic symbol follows from the pointwise control of the commutator by a sum of the Hardy--Littlewood maximal operator and the squares of the radial Hardy operator and its adjoint, while our lower bounds are consequences of a general result for singular integral operators.

math.CA↗
arXiv2026

The weak-type (1,1) bound for the Hardy--Littlewood maximal function is $O(\sqrt{n} \log n)$

We prove a weak-type $(1,1)$ estimate for the centered Hardy--Littlewood maximal function with respect to Euclidean balls with dimensional dependence $O(\sqrt{n} \log n)$. This improves the order of growth in the classical $O(n)$ estimate of Stein and Strömberg. The proof goes through a pointwise bound of the Hardy--Littlewood maximal operator by the heat maximal operator with $\sqrt{n}$ loss. The key technical aspect of our result is an improvement of the weak-type bound for the heat maximal operator from $O(\sqrt{n})$ to $O(\log n)$.

math.CA↗
arXiv2026

A dimension-free weak-type $(1,1)$ bound for the vector Riesz transform on $\mathbb{R}^n$

We show that the best constant in the weak-type $(1,1)$ bound for the vector Riesz transform on $\mathbb{R}^n$ is at most $2$, independent of the dimension $n$. The proof relies on a new decomposition of the input data involving an obstacle problem for the fractional Laplacian and an associated Lewy-Stampacchia type estimate on an unbounded domain. This settles a problem posed by E. M. Stein at the 1986 International Congress of Mathematicians.

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arXiv2026

Weak-type estimates for the Bergman projection on planar domains

We investigate the relationship between the weak-type regularity of the Bergman projection, $Π_Ω$, of a simply connected domain $Ω\subset \mathbb{C}$ and the boundary geometry of $Ω$ in terms of a conformal map $ψ\colon\mathbb{D}\rightarrowΩ$. We show that $Π_Ω$ is of weak-type $(1,1)$ whenever $|ψ'|$ is in the Bekollé-Bonami class $B_1$, give a more general necessary condition for the weak-type $(p,p)$ bounds of $Π_Ω$ when $1\leq p<\infty$, and establish sharpened sufficient conditions for the weak-type bounds when $p>1$. Our results follow from a reformulation in terms of mixed-weighted weak-type inequalities for $Π_{\mathbb{D}}$. We provide several applications.

math.CV↗
arXiv2026

Local dyadic fractional Sobolev spaces: paraproducts, commutators, and the algebra property

We characterize the boundedness and compactness of dyadic paraproducts on local dyadic fractional Sobolev spaces, $H^s$. We apply this result to establish the algebra property for $H^s$ when $s \in (\frac{1}{2},1)$ and to deduce the boundedness and compactness of commutators with the Haar shift on $H^s$. Our conditions are stated in terms of new dyadic fractional $\text{BMO}^s$ and $\text{CMO}^s$ conditions involving the dyadic fractional Sobolev capacity, and our proof uses a new dyadic fractional version of the Carleson embedding theorem.

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arXiv2026

On the compactness of bi-parameter singular integrals

We establish a new $T1$ theorem for the compactness of bi-parameter Calderón-Zygmund singular integral operators. Namely, we show that if a bi-parameter CZO $T$ satisfies the product weak compactness property, the mixed weak compactness/CMO property, and $T1, T^t1,$ $T_t1, T_t^t1 \in \text{CMO}(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})$, then $T$ is compact on $L^2(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})$. We also obtain endpoint compactness results for these operators and use them to deduce the necessity of most of our hypotheses. In particular, our conditions characterize the simultaneous $L^2(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})$-compactness of a bi-parameter CZO and its partial transpose. Our assumptions improve upon previously known sufficient conditions, and our proof, which is shorter and simpler than earlier arguments, utilizes a new abstract compactness criterion for partially localized operators on tensor products of Hilbert spaces.

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arXiv2025

Weak-type bounds for the Bergman projection with Bekollé-Bonami weights

We establish weighted weak-type bounds for the Bergman projection with respect to Bekollé-Bonami characteristics. We present two proofs of an improved quantitative weak-type $(1,1)$ estimate, as well as sharp weak-type $(p,p)$ bounds for $p>1$ and mixed weighted weak-type $(1,1)$ inequalities. Our results, which hold for a wide class of simple domains in $\mathbb{C}^n$, are new even in the classical settings of the upper half-plane and the unit disk.

math.FA↗
arXiv2025

Compact pseudodifferential and Fourier integral operators via localization

We present a general framework of localized operators, i.e., operators whose matrix coefficients with respect to the Gabor frame are concentrated on the diagonal. We show that localized operators are bounded between modulation spaces, and we deduce their compactness from an easily verifiable weak compactness condition. We apply this abstract formalism to unify and extend existing theorems for pseudodifferential and Fourier integral operators, and to obtain new results for three-parameter pseudodifferential operators.

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arXiv2025

An atomic decomposition for functions of bounded variation

In this paper, we give a decomposition of the gradient measure $Du$ of an arbitrary function of bounded variation $u$ into a sum of atoms $μ=Dχ_{F}$, where $F$ is a set of finite perimeter. The atoms further satisfy the support, cancellation, normalization, and size conditions: For each $μ$, there exists a cube $Q$ such that $\operatorname*{supp}μ\subset Q$, $μ(Q)=0$, $|μ|(Q)\leq 1$, and, denoting by $p_t$ the heat kernel in $\mathbb{R}^d$, \[ \sup_{x \in \mathbb{R}^d, t>0} |t^{1/2} p_t \ast μ(x)| \leq \frac{1}{l(Q)^{d-1}}. \] Our proof relies on a sampling of the coarea formula and a new boxing identity. We present several consequences of this result, including Sobolev inequalities, dimension estimates, and trace inequalities.

math.FA↗
arXiv2024

Endpoint weak-type bounds beyond Calderón-Zygmund theory

We prove weighted weak-type $(r,r)$ estimates for operators satisfying $(r,s)$ limited-range sparse domination of $\ell^q$-type. Our results contain improvements for operators satisfying limited-range and square function sparse domination. In the case of operators $T$ satisfying standard sparse form domination such as Calderón-Zygmund operators, we provide a new and simple proof of the sharp bound $$ \|T\|_{L^1_w(\mathbf{R}^d)\rightarrow L^{1,\infty}_w(\mathbf{R}^d)} \lesssim [w]_1(1+\log [w]_{\text{FW}}). $$

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arXiv2024

Sparse domination results for compactness on weighted spaces

By means of appropriate sparse bounds, we deduce compactness on weighted $L^p(w)$ spaces, $1<p<\infty$, for all Calderón-Zygmund operators having compact extensions on $L^2(\mathbb{R}^n)$. Similar methods lead to new results on boundedness and compactness of Haar multipliers on weighted spaces. In particular, we prove weighted bounds for weights in a class strictly larger than the typical $A_p$ class.

math.CA↗
arXiv2024

Weighted weak-type bounds for multilinear singular integrals

We establish analogs of sharp weighted weak-type bounds for $m$-sublinear operators satisfying sparse form domination, including multilinear Calderón-Zygmund singular integrals. Our results, which hold for general $\vec{p} \in [1,\infty)^m$ and feature quantitative improvements, rely on new local testing conditions and good-$λ$ inequalities. We address weak-type bounds in both the change of measure and multiplier settings.

math.CA↗
arXiv2023

On the $T1$ theorem for compactness of Calderón-Zygmund operators

We give a new formulation of the $T1$ theorem for compactness of Calderón-Zygmund singular integral operators. In particular, we prove that a Calderón-Zygmund operator $T$ is compact on $L^2(\mathbb{R}^n)$ if and only if $T1,T^*1\in \text{CMO}(\mathbb{R}^n)$ and $T$ is weakly compact. Compared to existing compactness criteria, our characterization more closely resembles David and Journé's classical $T1$ theorem for boundedness, avoids technical conditions involving the Calderón-Zygmund kernel, and follows from a simpler argument.

math.CA↗
arXiv2023

Weighted theory of Toeplitz operators on the Bergman space

We study the weighted compactness and boundedness properties of Toeplitz operators on the Bergman space with respect to Békollè-Bonami type weights. Let $T_u$ denote the Toeplitz operator on the (unweighted) Bergman space of the unit ball in $\mathbb{C}^n$ with symbol $u \in L^{\infty}$. We characterize the compact Toeplitz operators on the weighted Bergman space $\mathcal{A}^p_σ$ for all $σ$ in a subclass of the Békollè-Bonami class $B_p$ that includes radial weights and powers of the Jacobian of biholomorphic mappings. Concerning boundedness, we show that $T_u$ extends boundedly on $L^p_σ$ for $p \in (1,\infty)$ and weights $σ$ in a $u$-adapted class of weights containing $B_p$, and we establish analogous weighted endpoint weak-type $(1,1)$ bounds for weights beyond $B_1$.

math.CV↗
arXiv2022

Riesz-Kolmogorov type compactness criteria in function spaces with applications

We present forms of the classical Riesz-Kolmogorov theorem for compactness that are applicable in a wide variety of settings. In particular, our theorems apply to classify the precompact subsets of the Lebesgue space $L^2$, Paley-Wiener spaces, weighted Bargmann-Fock spaces, and a scale of weighted Besov-Sobolev spaces of holomorphic functions that includes weighted Bergman spaces of general domains as well as the Hardy space and the Dirichlet space. We apply the compactness criteria to characterize the compact Toeplitz operators on the Bergman space, deduce the compactness of Hankel operators on the Hardy space, and obtain general umbrella theorems.

math.CV↗
arXiv2020

Weighted endpoint bounds for the Bergman and Cauchy-Szegő projections on domains with near minimal smoothness

We study the Bergman projection, $\mathcal{B}$, and the Cauchy-Szegő projection, $\mathcal{S}$, on bounded domains with near minimal smoothness. We prove that $\mathcal{B}$ has the weak-type $(1,1)$ property with respect to weighted measures assuming that the underlying domain is strongly pseudoconvex with $C^4$ boundary and the weight satisfies the $B_1$ condition, and the same property for $\mathcal{S}$ on domains with $C^3$ boundaries and weights satisfying the $A_1$ condition. We also obtain weighted Kolmogorov and weighted Zygmund inequalities for $\mathcal{B}$ and $\mathcal{S}$ in their respective settings as corollaries.

math.CV↗
arXiv2020

On the dimensional weak-type $(1,1)$ bound for Riesz transforms

Let $R_j$ denote the $j^{\text{th}}$ Riesz transform on $\mathbb{R}^n$. We prove that there exists an absolute constant $C>0$ such that \begin{align*} |\{|R_jf|>λ\}|\leq C\left(\frac{1}λ\|f\|_{L^1(\mathbb{R}^n)}+\sup_ν |\{|R_jν|>λ\}|\right) \end{align*} for any $λ>0$ and $f \in L^1(\mathbb{R}^n)$, where the above supremum is taken over measures of the form $ν=\sum_{k=1}^Na_kδ_{c_k}$ for $N \in \mathbb{N}$, $c_k \in \mathbb{R}^n$, and $a_k \in \mathbb{R}^+$ with $\sum_{k=1}^N a_k \leq 16\|f\|_{L^1(\mathbb{R}^n)}$. This shows that to establish dimensional estimates for the weak-type $(1,1)$ inequality for the Riesz tranforms it suffices to study the corresponding weak-type inequality for Riesz transforms applied to a finite linear combination of Dirac masses. We use this fact to give a new proof of the best known dimensional upper bound, while our reduction result also applies to a more general class of Calderón-Zygmund operators.

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