arXiv subjects

Nathan A. Wagner

Publications and source records attributed to Nathan A. Wagner.

At least 19 recordsLinked to original sources

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

Endpoint Estimates for Bergman Commutators and New Characterizations of the Bloch Space and $H^\infty$

We prove an $\LlogL $-type distributional inequality for the commutator of the Bergman projection with a conjugate Bloch symbol function on the unit ball. Such an inequality can be seen as a Bergman version of a result due to C. Pérez for real-variable Calderón-Zygmund operators and BMO functions. We also prove that this inequality characterizes membership of analytic functions in the Bloch space and is further equivalent to a kind of modified restricted weak-type estimate, where one only tests over characteristic functions of sets comparable to Bergman balls. We also show our estimate is sharp in the sense that there exists a Bloch function $b$ so that the commutator $[\bar{b},P]$ is not weak-type $(1,1)$, and prove $[\bar{b},P]$ with $b$ analytic is weak-type $(1,1)$ if and only if $b \in H^\infty$.

math.CV↗

arXiv2025

Boundedness and compactness of Bergman projection commutators in two-weight setting

The goal of this paper is to study the boundedness and compactness of the Bergman projection commutators in two weighted settings via the weighted BMO and VMO spaces, respectively. The novelty of our work lies in the distinct treatment of the symbol b in the commutator, depending on whether it is analytic or not, which turns out to be quite different. In particular, we show that an additional weight condition due to Aleman, Pott, and Reguera is necessary to study the commutators when b is not analytic, while it can be relaxed when b is analytic. In the analytic setting, we completely characterize boundedness and compactness, while in the non-analytic setting, we provide a sufficient condition which generalizes the Euclidean case and is also necessary in many cases of interest. Our work initiates a study of the commutators acting on complex function spaces with different symbols.

math.CA↗

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

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

Matrix Weighted $L^p$ Estimates in the Nonhomogeneous Setting

We establish a modified pointwise convex body domination for vector-valued Haar shifts in the nonhomogeneous setting, strengthening and extending the scalar case developed in arXiv:2309.13943. Moreover, we identify a subclass of shifts, called $L^1$-normalized, for which the standard convex body domination holds without requiring any regularity assumption on the measure. Finally, we extend the best-known matrix weighted $L^p$ estimates for sparse forms to the nonhomogeneous setting. The key difficulty here is the lack of a reverse-Hölder inequality for scalar weights, which was used in arXiv:1710.03397 to establish $L^p$ matrix weighted estimates and only works in the doubling setting. Our approach relies instead on a generalization of the weighted Carleson embedding theorem which allows to control not only a fixed weight, but also collections of weights localized on different dyadic cubes that satisfy a certain compatibility condition.

math.CA↗

arXiv2025

A new way to express boundary values in terms of holomorphic functions on planar Lipschitz domains

We decompose $p$ - integrable functions on the boundary of a simply connected Lipschitz domain $Ω\subset \mathbb C$ into the sum of the boundary values of two, uniquely determined holomorphic functions, where one is holomorphic in $Ω$ while the other is holomorphic in $\mathbb C \setminus \overlineΩ$ and vanishes at infinity. This decomposition has been described previously for smooth functions on the boundary of a smooth domain. Uniqueness of the decomposition is elementary in the smooth case, but extending it to the $L^p$ setting relies upon a classical albeit little-known regularity theorem for the holomorphic Hardy space $h^p(bΩ)$ of planar domains for which we provide a new proof that is valid also in higher dimensions. An immediate consequence of our result will be a new characterization of the kernel of the Cauchy transform acting on $L^p(bΩ)$. These results give a new perspective on the classical Dirichlet problem for harmonic functions and the Poisson formula even in the case of the disc. Further applications are presented along with directions for future work.

math.CV↗

arXiv2024

Endpoint estimates for Haar shift operators with balanced measures

We prove $\mathrm{H}^1$ and $\mathrm{BMO}$ endpoint inequalities for generic cancellative Haar shifts defined with respect to a possibly non-homogeneous Borel measure $μ$ satisfying a weak regularity condition. This immediately yields a new, highly streamlined proof of the $L^p$-results for the same operators due to López-Sanchez, Martell, and Parcet. We also prove regularity properties for the Haar shift operators on the natural martingale Lipschitz spaces defined with respect to the underlying dyadic system, and show that the class of measures that we consider is sharp.

math.CA↗

arXiv2024

Commutator estimates for Haar shifts with general measures

We study $L^p(μ)$ estimates for the commutator $[H,b]$, where the operator $H$ is a dyadic model of the classical Hilbert transform introduced in \cite{arXiv:2012.10201,arXiv:2212.00090} and is adapted to a non-doubling Borel measure $μ$ satisfying a dyadic regularity condition which is necessary for $H$ to be bounded on $L^p(μ)$. We show that $\|[H, b]\|_{L^p(μ) \rightarrow L^p(μ)} \lesssim \|b\|_{\mathrm{BMO}(μ)}$, but to {\it characterize} martingale BMO requires additional commutator information. We prove weighted inequalities for $[H, b]$ together with a version of the John-Nirenberg inequality adapted to appropriate weight classes $\widehat{A}_p$ that we define for our non-homogeneous setting. This requires establishing reverse Hölder inequalities for these new weight classes. Finally, we revisit the appropriate class of nonhomogeneous measures $μ$ for the study of different types of Haar shift 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↗

arXiv2023

The Commutator of the Bergman Projection on Strongly Pseudoconvex Domains with Minimal Smoothness

Consider a bounded, strongly pseudoconvex domain $D\subset \mathbb C^n$ with minimal smoothness (namely, the class $C^2$) and let $b$ be a locally integrable function on $D$. We characterize boundedness (resp., compactness) in $L^p(D), p > 1$, of the commutator $[b, P]$ of the Bergman projection $P$ in terms of an appropriate bounded (resp. vanishing) mean oscillation requirement on $b$. We also establish the equivalence of such notion of BMO (resp., VMO) with other BMO and VMO spaces given in the literature. Our proofs use a dyadic analog of the Berezin transform and holomorphic integral representations going back (for smooth domains) to N. Kerzman & E. M. Stein, and E. Ligocka.

math.CV↗

arXiv2023

Balanced measures, sparse domination and complexity-dependent weight classes

We study sparse domination for operators defined with respect to an atomic filtration on a space equipped with a general measure $μ$. In the case of Haar shifts, $L^p$-boundedness is known to require a weak regularity condition, which we prove to be sufficient to have a sparse domination-like theorem. Our result allows us to characterize the class of weights where Haar shifts are bounded. A surprising novelty is that said class depends on the complexity of the Haar shift operator under consideration. Our results are qualitatively sharp.

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↗

arXiv2023

Some Results for the Szegő and Bergman Projections on Planar Domains

The purpose of this note is to prove some boundedness/compactness results of a harmonic analysis flavor for the Bergman and Szegő projections on certain classes of planar domains using conformal mappings. In particular, we prove weighted estimates for the projections, provide quantitative $L^p$ estimates and a specific example of such estimates on a domain with a sharp $p$ range, and show that the ``difference'' of the Bergman and Szegő projections is compact at the endpoints $p = 1, \infty$ for domains with sufficient smoothness. We also pose some open questions that naturally arise from our investigation.

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↗

arXiv2021

Weighted $L^p$ Estimates for the Bergman and Szegő Projections on Strongly Pseudoconvex Domains with Near Minimal Smoothness

We prove the weighted $L^p$ regularity of the ordinary Bergman and Cauchy-Szegő projections on strongly pseudoconvex domains $D$ in $\mathbb{C}^n$ with near minimal smoothness for appropriate generalizations of the $B_p/A_p$ classes. In particular, the $B_p/A_p$ Muckenhoupt type condition is expressed relative to balls in a quasi-metric that arises as a space of homogeneous type on either the interior or the boundary of the domain $D$.

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

A Békollè-Bonami Class of Weights for Certain Pseudoconvex Domains

We prove the weighted $L^p$ regularity of the ordinary Bergman projection on certain pseudoconvex domains where the weight belongs to an appropriate generalization of the Békollè-Bonami class. The main tools used are estimates on the Bergman kernel obtained by McNeal and Békollè's original approach of proving a good-lambda inequality.

math.CV↗