arXiv subjects

Brett D. Wick

Publications and source records attributed to Brett D. Wick.

At least 73 records · Page 4Linked to original sources

arXiv2019

Weak-type estimates for the Bergman projection on the polydisc and the Hartogs triangle

In this paper, we investigate the weak-type regularity of the Bergman projection. The two domains we focus on are the polydisc and the Hartogs triangle. For the polydisc we provide a proof that the weak-type behavior is of "$L\log L$" type. This result is likely known to the experts, but does not appear to be in the literature. For the Hartogs triangle we show that the operator is of weak-type $(4,4)$; settling the question of the behavior of the projection at this endpoint. At the other endpoint of interest, we show that the Bergman projection is not of weak-type $(\frac{4}{3}, \frac{4}{3})$ and provide evidence as to what the correct behavior at this endpoint might be.

math.CV↗

arXiv2019

Commutators of Cauchy--Szegő type integrals for domains in $\mathbb C^n$ with minimal smoothness

In this paper we study the commutator of Cauchy type integrals $\EuScript C$ on a bounded strongly pseudoconvex domain $D$ in $\mathbb C^n$ with boundary $bD$ satisfying the minimum regularity condition $C^{2}$ as in the recent result of Lanzani--Stein. We point out that in this setting the Cauchy type integrals $\EuScript C$ is the sum of the essential part $\EuScript C^\sharp$ which is a Calderón--Zygmund operator and a remainder $\EuScript R$ which is no longer a Calderón--Zygmund operator. We show that the commutator $[b, \EuScript C]$ is bounded on $L^p(bD)$ ($1<p<\infty$) if {\color{black}and only if}\ $b$ is in the BMO space on $bD$. Moreover, the commutator $[b, \EuScript C]$ is compact on $L^p(bD)$ ($1<p<\infty$) if {\color{black}and only if}\ $b$ is in the VMO space on $bD$. Our method can also be applied to the commutator of Cauchy--Leray integral in a bounded, strongly $\mathbb C$-linearly convex domain $D$ in $\mathbb C^n$ with the boundary $bD$ satisfying the minimum regularity $C^{1,1}$. Such a Cauchy--Leray integral is a Calderón--Zygmund operator as proved in the recent result of Lanzani--Stein. We also point out that our method provides another proof of the boundedness and compactness of commutator of Cauchy--Szeg\H o operator on a bounded strongly pseudoconvex domain $D$ in $\mathbb C^n$ with smooth boundary (first established by Krantz--Li).

math.CV↗

arXiv2019

Commutators of Hilbert transforms along monomial curves

The Hilbert transforms associated with monomial curves have a natural non-isotropic structure. We study the commutator of such Hilbert transforms and a symbol $b$ and prove the upper bound of this commutator when $b$ is in the corresponding non-isotropic BMO space by using the Cauchy integral trick. We also consider the lower bound of this commutator by introducing a new testing BMO space associated with the given monomial curve, which shows that the classical non-isotropic BMO space is contained in the testing BMO space. We also show that the non-zero curvature of such monomial curves are important, since when considering Hilbert transforms associated with lines, the parallel version of non-isotropic BMO space and testing BMO space have overlaps but do not have containment.

math.CA↗

arXiv2019

Two-Weight Tb Theorems for Well-Localized Operators

This paper first defines operators that are "well-localized" with respect to a pair of accretive functions and establishes a global two-weight Tb theorem for such operators. Then it defines operators that are "well-localized" with respect to a pair of accretive systems and establishes a local two-weight Tb theorem for them. The proofs combine recent Tb proof techniques with arguments used to prove earlier T1 theorems for well-localized operators.

math.CA↗

arXiv2019

Weighted Alpert Wavelets

In this paper we construct a wavelet basis in weighted L^2 of Euclidean space possessing vanishing moments of a fixed order for a general locally finite positive Borel measure. The approach is based on a clever construction of Alpert in the case of Lebesgue measure that is appropriately modified to handle the general measures considered here. We then use this new wavelet basis to study a two-weight inequality for a general Calderón-Zygmund operator on the real line and show that under suitable natural conditions, including a weaker energy condition, the operator is bounded from one weighted L^2 space to another if certain stronger testing conditions hold on polynomials. An example is provided showing that this result is logically different than existing results in the literature.

math.CA↗

arXiv2019

The Two-Weight Inequality for the Poisson Operator in the Bessel Setting

Fix $λ>0$. Consider the Bessel operator $Δ_λ:=-\frac{d^2}{dx^2}-\frac{2λ}{x}\frac d{dx}$ on $\mathbb{R}_+:=(0,\infty)$ and the harmonic conjugacy introduced by Muckenhoupt and Stein. We provide the two-weight inequality for the Poisson operator $\mathsf{P}^{[λ]}_t=e^{-t\sqrt{Δ_λ}}$ in this Bessel setting. In particular, we prove that for a measure $μ$ on $\mathbb{R}^2_{+,+}:=(0,\infty)\times (0,\infty)$ and $σ$ on $\mathbb{R}_+$: $$ \|\mathsf{P}^{[λ]}_σ(f)\|_{L^2(\mathbb{R}^2_{+,+};μ)} \lesssim \|f\|_{L^2(\mathbb{R}_+;σ)}, $$ if and only if testing conditions hold for the the Poisson operator and its adjoint. Further, the norm of the operator is shown to be equivalent to the best constant in the testing conditions.

math.AP↗

arXiv2019

A note on weak factorization of Meyer-type Hardy space via Cauchy integral operator

This paper provides a weak factorization for the Meyer-type Hardy space $H^1_b(\mathbb{R})$, and characterizations of its dual ${\rm BMO}_b(\mathbb{R})$ and its predual ${\rm VMO}_b(\mathbb{R})$ via boundedness and compactness of a suitable commutator with the Cauchy integral $\mathscr{C}_Γ$, respectively. Here $b(x)=1+iA'(x)$ where $A'\in L^{\infty}(\mathbb{R})$, and the Cauchy integral $\mathscr{C}_Γ$ is associated to the Lipschitz curve $Γ=\{x+iA(x)\, : \, x\in \mathbb{R}\}$.

math.CA↗

arXiv2018

Functional calculus of operators with heat kernel bounds on non-doubling manifolds with ends

Let $Δ$ be the Laplace--Beltrami operator acting on a non-doubling manifold with two ends $\mathbb R^m \sharp \mathcal R^n$ with $m > n \ge 3$. Let $\frak{h}_t(x,y)$ be the kernels of the semigroup $e^{-tΔ}$ generated by $Δ$. We say that a non-negative self-adjoint operator $L$ on $L^2(\mathbb R^m \sharp \mathcal R^n)$ has a heat kernel with upper bound of Gaussian type if the kernel $h_t(x,y)$ of the semigroup $e^{-tL}$ satisfies $ h_t(x,y) \le C \frak{h}_{αt}(x,y)$ for some constants $C$ and $α$. This class of operators includes the Schrödinger operator $L = Δ+ V$ where $V$ is an arbitrary non-negative potential. We then obtain upper bounds of the Poisson semigroup kernel of $L$ together with its time derivatives and use them to show the weak type $(1,1)$ estimate for the holomorphic functional calculus $\frak{M}(\sqrt{L})$ where $\frak{M}(z)$ is a function of Laplace transform type. Our result covers the purely imaginary powers $L^{is}, s \in \mathbb R$, as a special case and serves as a model case for weak type $(1,1)$ estimates of singular integrals with non-smooth kernels on non-doubling spaces.

math.AP↗

arXiv2018

Compactness of operators on the Bergman space of the Thullen domain

We study compact operators on the Bergman space of the Thullen domain defined by $\{(z_1,z_2)\in \mathbb C^2: |z_1|^{2p}+|z_2|^2<1\}$ with $p>0$ and $p\neq 1$. The domain need not be smooth nor have a transitive automorphism group. We give a sufficient condition for the boundedness of various operators on the Bergman space. Under this boundedness condition, we characterize the compactness of operators on the Bergman space of the Thullen domain.

math.CV↗

arXiv2018

$A_p$ weights and Quantitative Estimates in the Schrödinger Setting

Suppose $L=-Δ+V$ is a Schrödinger operator on $\mathbb{R}^n$ with a potential $V$ belonging to certain reverse Hölder class $RH_σ$ with $σ\geq n/2$. The aim of this paper is to study the $A_p$ weights associated to $L$, denoted by $A_p^L$, which is a larger class than the classical Muckenhoupt $A_p$ weights. We first establish the "exp--log" link between $A_p^L$ and $BMO_L$ (the BMO space associated with $L$), which is the first extension of the classical result to a setting beyond the Laplace operator. Second, we prove the quantitative $A_p^L$ bound for the maximal function and the maximal heat semigroup associated to $L$. Then we further provide the quantitative $A_{p,q}^L$ bound for the fractional integral operator associated to $L$. We point out that all these quantitative bounds are known before in terms of the classical $A_{p,q}$ constant. However, since $A_{p,q}\subset A_{p,q}^L$, the $A_{p,q}^L$ constants are smaller than $A_{p,q}$ constant. Hence, our results here provide a better quantitative constant for maximal functions and fractional integral operators associated to $L$.

math.CA↗

arXiv2018

Two weight commutators on spaces of homogeneous type and applications

In this paper, we establish the two weight commutator of Calderón--Zygmund operators in the sense of Coifman--Weiss on spaces of homogeneous type, by studying the weighted Hardy and BMO space for $A_2$ weight and by proving the sparse operator domination of commutators. The main tool here is the Haar basis and the adjacent dyadic systems on spaces of homogeneous type, and the construction of a suitable version of a sparse operator on spaces of homogeneous type. As applications, we provide a two weight commutator theorem (including the high order commutator) for the following Calderón--Zygmund operators: Cauchy integral operator on $\mathbb R$, Cauchy--Szegö projection operator on Heisenberg groups, Szegö projection operators on a family of unbounded weakly pseudoconvex domains, Riesz transform associated with the sub-Laplacian on stratified Lie groups, as well as the Bessel Riesz transforms (one-dimension and high dimension).

math.CA↗

arXiv2018

Ideal membership in $H^\infty$: Toeplitz corona approach

We study the ideal membership problem in $H^\infty$ on the unit disc. Thus, given functions $f,f_1,\ldots,f_n$ in $H^\infty$, we seek sufficient conditions on the size of $f$ in order for $f$ to belong to the ideal of $H^\infty$ generated by $f_1,\ldots,f_n$. We provide a different proof of a theorem of Treil, which gives the sharpest known sufficient condition. To this end, we solve a closely related problem in the Hilbert space $H^2$, which is equivalent to the ideal membership problem by the Nevanlinna-Pick property of $H^2$.

math.CV↗

arXiv2018

Commutators of multi-parameter flag singular integrals and applications

We introduce the iterated commutator for the Riesz transforms in the multi-parameter flag setting, and prove the upper bound of this commutator with respect to the symbol $b$ in the flag BMO space. Our methods require the techniques of semigroups, harmonic functions and multi-parameter flag Littlewood-Paley analysis. We also introduce the big commutator in this multi-parameter flag setting and prove the upper bound with symbol $b$ in the flag little-bmo space by establishing the "exponential-logarithmic" bridge between this flag little bmo space and the Muckenhoupt $A_p$ weights with flag structure. As an application, we establish the div-curl lemmas with respect to the appropriate Hardy spaces in the multi-parameter flag setting.

math.CA↗

arXiv2018

Lower bound of Riesz transform kernels revisited and commutators on stratified Lie groups

Let $\mathcal G$ be a stratified Lie group and $\{\X_j\}_{1 \leq j \leq n}$ a basis for the left-invariant vector fields of degree one on $\mathcal G$. Let $Δ= \sum_{j = 1}^n \X_j^2 $ be the sub-Laplacian on $\mathcal G$ and the $j^{\mathrm{th}}$ Riesz transform on $\mathcal G$ is defined by $R_j:= \X_j (-Δ)^{-\frac{1}{2}}$, $1 \leq j \leq n$. In this paper we give a new version of the lower bound of the kernels of Riesz transform $R_j$ and then establish the Bloom-type two weight estimates as well as a number of endpoint characterisations for the commutators of the Riesz transforms and BMO functions, including the $L\log^+L(\mathcal G)$ to weak $L^1(\mathcal G)$, $H^1(\mathcal G)$ to $L^1(\mathcal G)$ and $L^\infty(\mathcal G)$ to BMO$(\mathcal G)$. Moreover, we also study the behaviour of the Riesz transform kernel on a special case of stratified Lie group: the Heisenberg group, and then we obtain the weak type $(1,1)$ characterisations for the Riesz commutators.

math.CA↗

arXiv2018

Lower bound of Riesz transform kernels and Commutator Theorems on stratified nilpotent Lie groups

We provide a study of the Riesz transforms on stratified nilpotent Lie groups, and obtain a certain version of the pointwise lower bound of the Riesz transform kernel. Then we establish the characterisation of the BMO space on stratified nilpotent Lie groups via the boundedness of the commutator of the Riesz transforms and the BMO function. This extends the well-known Coifman, Rochberg, Weiss theorem from Euclidean space to the setting of stratified nilpotent Lie groups. In particular, these results apply to the well-known example of the Heisenberg group. As an application, we also study the curl operator on the Heisenberg group and stratified nilpotent Lie groups, and establish the div-curl lemma with respect to the Hardy space on stratified nilpotent Lie groups.

math.CA↗

arXiv2018

Two Weight Inequalities for the Cauchy Transform from $ \mathbb{R}$ to $ \mathbb{C}_+$

We characterize those pairs of weights $ σ$ on $ \mathbb{R}$ and $ τ$ on $ \mathbb{C}_+$ for which the Cauchy transform $\mathsf{C}_σ f (z) \equiv \int_{\mathbb{R}} \frac {f(x)} {x-z} \; σ(dx)$, $ z\in \mathbb{C}_+$, is bounded from $L ^2(\mathbb{R};σ)$ to $L ^{2}(\mathbb{C}_+; τ)$. The characterization is in terms of an $A_2$ condition on the pair of weights and testing conditions for the transform, extending the recent solution of the two weight inequality for the Hilbert transform. As corollaries of this result we derive (1) a characterization of embedding measures for the model space $K_\vartheta$, for arbitrary inner function $ \vartheta $, and (2) a characterization of the (essential) norm of composition operators mapping $K_\vartheta$ into a general class of Hardy and Bergman spaces.

math.CV↗

arXiv2018

A reproducing kernel thesis for operators on Bergman-type function spaces

In this paper we consider the reproducing kernel thesis for boundedness and compactness for various operators on Bergman-type spaces. In particular, the results in this paper apply to the weighted Bergman space on the unit ball, the unit polydisc and more generally to weighted Fock spaces.

math.CV↗

arXiv2018

Weighted little bmo and two-weight inequalities for Journé commutators

We characterize the boundedness of the commutators $[b, T]$ with biparameter Journé operators $T$ in the two-weight, Bloom-type setting, and express the norms of these commutators in terms of a weighted little $bmo$ norm of the symbol $b$. Specifically, if $μ$ and $λ$ are biparameter $A_p$ weights, $ν:= μ^{1/p}λ^{-1/p}$ is the Bloom weight, and $b$ is in $bmo(ν)$, then we prove a lower bound and testing condition $\|b\|_{bmo(ν)} \lesssim \sup \| [b, R_k^1 R_l^2]: L^p(μ) \rightarrow L^p(λ)\|$, where $R_k^1$ and $R_l^2$ are Riesz transforms acting in each variable. Further, we prove that for such symbols $b$ and any biparameter Journé operators $T$ the commutator $[b, T]:L^p(μ) \rightarrow L^p(λ)$ is bounded. Previous results in the Bloom setting do not include the biparameter case and are restricted to Calderón-Zygmund operators. Even in the unweighted, $p=2$ case, the upper bound fills a gap that remained open in the multiparameter literature for iterated commutators with Journé operators. As a by-product we also obtain a much simplified proof for a one-weight bound for Journé operators originally due to R. Fefferman.

math.CA↗