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A. Walton Green

Publications and source records attributed to A. Walton Green.

10 recordsLinked to original sources
arXiv2026

Wavelet resolution and Sobolev regularity of Calderón-Zygmund operators on domains

Given a uniform domain $Ω\subset {\mathbb R}^d$, we resolve each element of a suitably defined class of Calderòn-Zygmund (CZ) singular integrals on $Ω$ as the linear combination of Triebel wavelet operators and paraproduct terms. Our resolution formula entails a testing type characterization, loosely in the vein of the David-Journé theorem, of weighted Sobolev space bounds in terms of Triebel-Lizorkin and tree Carleson measure norms of the paraproduct symbols, which is new already in the case $Ω={\mathbb R}^d$ with Lebesgue measure. Our characterization covers the case of compressions to $Ω$ of global CZ operators, extending and sharpening past results of Prats and Tolsa for the convolution case. The weighted estimates we obtain, particularized to the Beurling operator on a Lipschitz domain with normal to the boundary in the corresponding sharp Besov class, may be used to deduce quantitative estimates for quasiregular mappings with dilatation in the Sobolev space $W^{1,p}(Ω)$, $p>2$.

math.CA↗
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↗
arXiv2025

An $A_2$ Theorem for One-Sided Calderón-Zygmund Operators

We present a proof of the one-sided $A_2$ theorem in dimension one, with a logarithmic loss. This theorem concerns one-sided Calderón-Zygmund operators (CZOs) whose kernels $K(x,y)$ vanish whenever $x < y$. These operators are bounded on $L^2(w)$ provided that the weight $w$ belongs to the one-sided class $A_2^{\uparrow}$. The argument reduces the norm estimate to testing on indicator functions via a two-weight testing theorem. By combining this with the weak-type $(1,1)$ estimate in the one-sided setting and an extrapolation theorem, we obtain the one-sided $A_2^{\uparrow}$ theorem with a logarithmic loss. We develop a localized theory on fixed intervals by introducing adapted weight classes and showing that the same quantitative bound holds locally for one-sided operators.

math.CA↗
arXiv2025

Multilinear Wavelet Compact T(1) Theorem

We prove a wavelet $T(1)$ theorem for compactness of multilinear Calderón-Zygmund (CZ) operators. Our approach characterizes compactness in terms of testing conditions and yields a representation theorem for compact CZ forms in terms of wavelet and paraproduct forms that reflect the compact nature of the operator.

math.CA↗
arXiv2024

Quantitative Sobolev regularity of quasiregular maps

We quantify the Sobolev space norm of the Beltrami resolvent $(I- μ\mathcal{B})^{-1}$, where $\mathcal B$ is the Beurling-Ahlfors transform, in terms of the corresponding Sobolev space norm of the dilatation $μ$ in the critical and supercritical ranges. Our estimate entails as a consequence quantitative self-improvement inequalities of Caccioppoli type for quasiregular distributions with dilatations in $W^{1,p}$, $p \geq 2$. Our proof strategy is then adapted to yield quantitative estimates for the resolvent $(I-μ{\mathcal B}_Ω)^{-1}$ of the Beltrami equation on a sufficiently regular domain $Ω$, with $μ\in W^{1,p}(Ω)$. Here, ${\mathcal B}_Ω$ is the compression of ${\mathcal B}$ to a domain $Ω$. Our proofs do not rely on the compactness or commutator arguments previously employed in related literature. Instead, they leverage the weighted Sobolev estimates for compressions of Calderón-Zygmund operators to domains, recently obtained by the authors, to extend the Astala-Iwaniec-Saksman technique to higher regularities.

math.AP↗
arXiv2024

On the dimension of observable sets for the heat equation

We consider the heat equation on a bounded $C^1$ domain in $\mathbb{R}^n$ with Dirichlet boundary conditions. The primary aim of this paper is to prove that the heat equation is observable from any measurable set with a Hausdorff dimension strictly greater than $n - 1$. The proof relies on a novel spectral estimate for linear combinations of Laplace eigenfunctions, achieved through the propagation of smallness for solutions to Cauchy-Riemann systems as established by Malinnikova, and uses the Lebeau-Robbiano method. While this observability result is sharp regarding the Hausdorff dimension scale, our secondary goal is to construct families of sets with dimensions less than $n - 1$ from which the heat equation is still observable.

math.AP↗
arXiv2024

Multilinear paraproducts on Sobolev spaces

Paraproducts are a special subclass of the multilinear Calderón-Zygmund operators, and their Lebesgue space estimates in the full multilinear range are characterized by the $\mathrm{BMO}$ norm of the symbol. In this note, we characterize the Sobolev space boundedness properties of multilinear paraproducts in terms of a suitable family of Triebel-Lizorkin type norms of the symbol. Coupled with a suitable wavelet representation theorem, this characterization leads to a new family of Sobolev space $T(1)$-type theorems for multilinear Calderón-Zygmund operators.

math.CA↗
arXiv2024

Weighted estimates for the Bergman projection on planar domains

We investigate weighted Lebesgue space estimates for the Bergman projection on a simply connected planar domain via the domain's Riemann map. We extend the bounds which follow from a standard change-of-variable argument in two ways. First, we provide a regularity condition on the Riemann map, which turns out to be necessary in the case of uniform domains, in order to obtain the full range of weighted estimates for the Bergman projection for weights in a Békollè-Bonami-type class. Second, by slightly strengthening our condition on the Riemann map, we obtain the weighted weak-type $(1,1)$ estimate as well. Our proofs draw on techniques from both conformal mapping and dyadic harmonic analysis.

math.CV↗
arXiv2022

Bilinear Wavelet Representation of Calderón-Zygmund Forms

We represent a bilinear Calderón-Zygmund operator at a given smoothness level as a finite sum of cancellative, complexity zero operators, involving smooth wavelet forms, and continuous paraproduct forms. This representation results in a sparse $T(1)$-type bound, which in turn yields directly new sharp weighted bilinear estimates on Lebesgue and Sobolev spaces. Moreover, we apply the representation theorem to study fractional differentiation of bilinear operators, establishing Leibniz-type rules in weighted Sobolev spaces which are new even in the simplest case of the pointwise product.

math.CA↗