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Brett D. Wick

Publications and source records attributed to Brett D. Wick.

At least 19 recordsLinked to original sources
arXiv2026

Martingale Transforms and Compensated Bellman Estimates for Dunkl Riesz Transforms

We develop a martingale-transform framework for Dunkl harmonic analysis and apply it to prove $L^p$ estimates for the Dunkl Riesz transforms using the martingale decomposition of the Dunkl process. A fundamental difference from the classical Brownian setting is that the martingale representing the Dunkl Riesz transform of a function is not, in general, differentially subordinated to the Poisson martingale of the function itself. Our main argument bypasses this obstruction by applying Burkholder's Bellman function directly. The possible positive defect of the continuous Itô drift is compensated by the negative contribution produced by the reflection jumps. This yields $L^p$ estimates for single Dunkl Riesz transforms for $1<p<\infty$, and vector-valued estimates for $p\geq2$ with a spectral dependence on the root system. For $G$-invariant functions, differential subordination can be recovered and the resulting estimates are independent of the root system.

math.CA↗
arXiv2026

The Corona Problem for Slice Holomorphic Functions via the Matrix Corona Problem

In this paper we prove the Corona theorem for slice hyperholomorphic functions in a general way that applies to functions with values in a Clifford algebra $\mathbb R_n$ for any $n\geq 2$. This general proof is based on a matrix version of Corona's theorem and then exploiting some symmetries of the matrices that allow to reconstruct the values of the functions. The crucial idea is to tie together the algebra of the functions, the Bezout identities and the Carleson-type conditions. The results we obtain rest on a new way of writing the slice product which improves the standard approach and which holds in general dimension.

math.CV↗
arXiv2026

Two-weight inequalities for the Dunkl--Poisson integrals

We prove an $L^2$ two-weight testing theorem for the Dunkl--Poisson semigroup. The difficulty is geometric. The Dunkl orbit distance has several reflected diagonals, so a single orbit-box test may mix different chamber components. We avoid this by working on one Weyl chamber and keeping the chamber indices. Under the wall-null assumption the full operator becomes a finite matrix of scalar positive Poisson-type operators. In each entry the orbit diagonal is just the ordinary diagonal in the chamber variables. The scalar proof is then a principal-cube stopping-time argument, with two Dunkl kernel comparisons as the only new estimates. The resulting forward and backward tests are necessary and sufficient for the original Dunkl--Poisson inequality.

math.CA↗
arXiv2026

Haar bases for multi-parameter twisted structures

Motivated by the Cauchy--Szegő projections on a broad class of Siegel domains and the geometric quotient structures of nilpotent Lie groups observed by Nagel, Ricci, and Stein, we develop a martingale and Haar wavelet framework for twisted multi-parameter geometries. We introduce twisted dyadic filtrations and construct adapted Haar bases on Euclidean spaces $\mathbb{R}^{2m}$. Each of the resulting dyadic systems forms a complete orthonormal basis of $L^2(\mathbb R^{2m})$, and their union yields a tight frame with frame bound $3$. We establish $L^p$-equivalences for the associated discrete twisted Littlewood--Paley square functions. Furthermore, we extend this discrete real-variable theory to the non-abelian setting of a nilpotent Lie group of step two, $\mathscr{N}$, which serves as the Shilov boundary of certain fundamental Siegel domains. By projecting product fractal tiles from a lifting group of Heisenberg products, we define twisted dyadic shards and construct twisted nilpotent Haar frames. More precisely, we first introduce raw projected shards that reflect the quotient geometry, and then pass to analytic dyadic shards which are exactly rectifiable and remain uniformly comparable to the raw quotient structure in the relevant scale regimes. This yields a discrete framework adapted to twisted quotient geometries in both the Euclidean and nilpotent settings, providing a basic dyadic infrastructure for further developments in twisted real-variable theory.

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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

Optimal Sparse Bounds and Commutator Characterizations Without Doubling

We examine dyadic paraproducts and commutators in the non-homogeneous setting, where the underlying Borel measure $μ$ is not assumed to be doubling. We first establish a pointwise sparse domination for dyadic paraproducts and related operators with symbols $b \in \textrm{BMO}(μ)$, improving upon an earlier result of Lacey, where the symbol $b$ was assumed to satisfy a stronger Carleson-type condition, that coincides with $\textrm{BMO}$ only in the doubling setting. As an application of this result, we obtain sharpened weighted inequalities for the commutator of a dyadic Hilbert transform $\mathcal{H}$ previously studied by Borges, Conde Alonso, Pipher, and the third author. We also characterize the symbols for which the commutator $[\mathcal{H},b]$ is bounded on $L^p(μ)$ for $1<p<\infty$ and provide some interesting examples to prove that this class of symbols strictly depends on $p$ and is nested between symbols satisfying the $p$-Carleson packing condition and symbols belonging to martingale BMO (even in the case of absolutely continuous measures).

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↗
arXiv2025

The Corona Problem for Slice Hyperholomorphic Functions

This paper addresses the Corona problem for slice hyperholomorphic functions for a single quaternionic variable. While the Corona problem is well-understood in the context of one complex variable, it remains highly challenging in the case of several complex variables. The extension of the theory of one complex variable to several complex variables is not the only possible extension to multi-dimensional complex analysis. Instead of functions holomorphic in each variable separately, in this paper we will consider functions in some hypercomplex algebras, in particular in the algebra of quaternions. Previously, the Corona problem had not been studied within the hypercomplex framework because of challenges posed by pointwise multiplication, which is not closed for hypercomplex-valued analytic functions. Alternative notions of multiplication that are closed often compromise other desirable properties, further complicating the analysis. In this work, we resolve the Corona problem within the quaternionic slice hyperholomorphic setting. Our approach involves reformulating the quaternionic Bezout equation with respect to the appropriate multiplication into a new system of Bezout equations on the unit disc. We solve this system by adapting Wolff's proof of the Corona theorem for bounded analytic functions. As the number of generators increases, the associated algebra grows increasingly intricate.

math.CV↗
arXiv2025

The Cauchy--Szegö Projection for domains in $\mathbb C^n$ with minimal smoothness: weighted theory

Let $D\subset\mathbb C^n$ be a bounded, strongly pseudoconvex domain whose boundary $bD$ satisfies the minimal regularity condition of class $C^2$. A 2017 result of Lanzani \& Stein states that the Cauchy--Szegö projection $S_ω$ defined with respect to a bounded, positive continuous multiple $ω$ of induced Lebesgue measure, {maps $L^p(bD, ω)$ to $L^p(bD, ω)$ continuously} for any $1<p<\infty$. Here we show that $S_ω$ satisfies explicit quantitative bounds in $L^p(bD, Ω)$, for any $1<p<\infty$ and for any $Ω$ in the maximal class of \textit{$A_p$}-measures, that is for $Ω_p = ψ_pσ$ where $ψ_p$ is a Muckenhoupt $A_p$-weight and $σ$ is the induced Lebesgue measure (with $ω$'s as above being a sub-class). Earlier results rely upon an asymptotic expansion and subsequent pointwise estimates of the Cauchy--Szegö kernel, but these are unavailable in our setting of minimal regularity {of $bD$}; at the same time, more recent techniques that allow to handle domains with minimal regularity (Lanzani--Stein 2017) are not applicable to $A_p$-measures. It turns out that the method of {quantitative} extrapolation is an appropriate replacement for the missing tools. To finish, we identify a class of holomorphic Hardy spaces defined with respect to $A_p$-measures for which a meaningful notion of Cauchy--Szegö projection can be defined when $p=2$.

math.CV↗
arXiv2025

The commutator of the Cauchy--Szegő Projection for domains in $\mathbb C^n$ with minimal smoothness: weighted regularity

Let $D\subset\mathbb C^n$ be a bounded, strongly pseudoconvex domain whose boundary $bD$ satisfies the minimal regularity condition of class $C^2$, and let $S_ω$ denote the Cauchy--Szegő projection defined with respect to (any) positive continuous multiple $ω$ of induced Lebesgue measure for the boundary of $D$. We characterize compactness and boundedness (the latter with explicit bounds) of the commutator $[b, S_ω]$ in the Lebesgue space $L^p(bD, Ω_p)$ where $Ω_p$ is any measure in the Muckenhoupt class $A_p(bD)$, $1<p<\infty$. We next fix $p =2$ and we let $S_{Ω_2}$ denote the Cauchy--Szegő projection defined with respect to (any) measure $Ω_2 \in A_2(bD)$, which is the largest class of reference measures for which a meaningful notion of Cauchy-Leray measure may be defined. We characterize boundedness and compactness in $L^2(bD, Ω_2)$ of the commutator $\displaystyle{[b,S_{Ω_2}]}$.

math.CV↗
arXiv2025

Weighted Inequalities for $t$-Haar multipliers

In this paper, we provide necessary and sufficient conditions on a triple of weights $(u,v,w)$ so that the $t$-Haar multipliers $T^t_{w,σ}$, $t\in \R$, %defined in \cite{P} when $σ=1$, are uniformly (on the choice of signs $σ$) bounded from $L^2(u)$ into $L^2(v)$. These dyadic operators have symbols $s(x,I)=σ_I\,(w(x)/\langle w\rangle_I)^t$ which are functions of the space variable $x\in\R$ and the frequency variable $I\in \mathcal{D}$, making them dyadic analogues of pseudo-differential operators. Here $\mathcal{D}$ denotes the dyadic intervals, $σ_I=\pm1$, and $\langle w\rangle_I$ denotes the integral average of $w$ on $I$. When $w\equiv 1$ we have the martingale transform and our conditions recover the known two-weight necessary and sufficient conditions of Nazarov, Treil and Volberg. %We will discuss some relations between the three weights inequality for these operators given the inequality for other dyadic operators. We also show how these conditions are simplified when $u=v$. In particular, the martingale one-weight and the $t$-Haar multiplier unsigned and unweighted (corresponding to $σ_I\equiv 1$ and $u=v\equiv 1$) known results are recovered or improved. We also obtain necessary and sufficient testing conditions of Sawyer type for the two-weight boundedness of a single variable Haar multiplier similar to those known for the martingale transform.

math.CA↗
arXiv2024

Schatten classes and commutators in the two weight setting, II. Riesz transforms

We characterize the Schatten class $S^p$ of the commutator of Riesz transforms $[b,R_j]$ in $\mathbb R^n$ ($j=1,\ldots, n$) in the two weight setting for $n< p<\infty$, by introducing the condition that the symbol $b$ being in Besov spaces associated with the given two weights. At the critical index $p=n$, the commutator $[b,R_j]$ belongs to Schatten class $S^{n}$ if and only if $b$ is a constant, and to the weak Schatten class $S^{n,\infty}$ if and only if $b$ is in an oscillation sequence space associated with the given two weights. As a direct application, we have the Schatten class estimate for A. Connes' quantised derivative in the two weight setting.

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

Weighted Estimates of Singular Integrals and Commutators in the Zygmund Dilation Setting

The main purpose of this paper is to establish weighted estimates for singular integrals associated with Zygmund dilations via a discrete Littlewood--Paley theory, and then apply it to obtain the upper bound of the norm of commutators of such singular integrals with a function in the little bmo space associated with Zygmund dilations. Examples of such singular integrals associated with Zygmund dilations include a class of singular integrals studied by Ricci--Stein and Fefferman--Pipher, as well as a singular integral along a particular surface studied by Nagel--Wainger. We show that the lower bound of the norm of this commutator is not true for any singular integral in the class considered in Ricci--Stein and Fefferman--Pipher, but does in fact hold for the specific singular integral studied in Nagel--Wainger. In particular this implies that the family of singular integrals studied in these papers is not sufficiently general to contain the operator of Nagel--Wainger, which we show is of significance in this theory.

math.CA↗
arXiv2024

The Hytönen-Vuorinen L^{p} conjecture for the Hilbert transform, with an extended energy side condition, when (4/3)<p<4 and the measures share no point masses

In the case (4/3)<p<4, and assuming a pair of locally finite positive Borel measures on the real line have no common point masses, we prove variants of two conjectures of T. Hytönen and E. Vuorinen from 2018 on two weight testing theorems for the Hilbert transform on weighted L^{p} spaces, but with extended energy side conditions. Namely, assuming the extended energy conditions, the two weight norm inequality holds (1) if and only if the global quadratic interval testing conditions hold, (2) if and only if the local quadratic interval testing, the quadratic Muckenhoupt, and the quadratic weak boundedness conditions all hold. We also give a slight improvement of the second conjecture in this setting by replacing the quadratic Muckenhoupt conditions with two smaller conditions.

math.CA↗
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

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.

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