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Brett D. Wick

Publications and source records attributed to Brett D. Wick.

At least 37 records · Page 2Linked to original sources
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.

math.CA↗
arXiv2024

$L^p$ regularity of the Bergman projection on the symmetrized polydisc

We study the $L^p$ regularity of the Bergman projection $P$ over the symmetrized polydisc in $\mathbb C^n$. We give a decomposition of the Bergman projection on the polydisc and obtain an operator equivalent to the Bergman projection over anti-symmetric function spaces. Using it, we obtain the $L^p$ irregularity of $P$ for $p=\frac{2n}{n-1}$ which also implies that $P$ is $L^p$ bounded if and only if $p\in (\frac{2n}{n+1},\frac{2n}{n-1})$.

math.CV↗
arXiv2024

Two weight L^{p} inequalities for fractional vector Riesz transforms and doubling measures

If T is a fractional vector Riesz transform, 1<p<infinity, and sigma and omega are doubling measures, then the two weight L^{p} norm inequality holds if and only if the quadratic triple testing conditions of Hytönen and Vuorinen hold. We also show that these quadratic triple testing conditions can be relaxed to quadratic local testing conditions, quadratic offset Muckenhoupt conditions, and a quadratic weak boundedness property.

math.CA↗
arXiv2024

Muckenhoupt-Type Weights and Quantitative Weighted Estimate in the Bessel Setting

Part of the intrinsic structure of singular integrals in the Bessel setting is captured by Muckenhoupt-type weights. Anderson--Kerman showed that the Bessel Riesz transform is bounded on weighted $L^p_w$ if and only if $w$ is in the class $A_{p,λ}$. We introduce a new class of Muckenhoupt-type weights $\widetilde A_{p,λ}$ in the Bessel setting, which is different from $A_{p,λ}$ but characterizes the weighted boundedness for the Hardy--Littlewood maximal operators. We also establish the weighted $L^p$ boundedness and compactness, as well as the endpoint weak type boundedness of Riesz commutators. The quantitative weighted bound is also established.

math.CA↗
arXiv2024

Aharonov-Bohm effect and superoscillations

The path-integral technique in quantum mechanics provides an intuitive framework for comprehending particle propagation and scattering. Calculating the propagator for the Aharonov-Bohm potential fits into the range of potentials in multiply-connected spaces, with the propagator represented through a series expansion. In this paper, we analyze the Schrödinger evolution of superoscillations, showing the supershift properties of the solution to the Schrödinger equation for this potential. Our proof is based on the continuity of particular infinite order differential operators acting on spaces of entire functions.

math-ph↗
arXiv2024

Two weight Sobolev norm inequalities for fractional vector Riesz transforms and doubling weights

We prove a T1 theorem for fractional vector Riesz transforms mapping one weighted Sobolev space to another, where the weights are doubling measures on Euclidean space. Boundedness is characterized by the classical A_2 condition and two dual testing conditions on indicators of cubes. We also show the equivalence of various weighted Sobolev norms when the measure is doubling, something that fails in general.

math.CA↗
arXiv2024

Riesz transform and commutators in the Dunkl setting

In this paper we characterise the optimal pointwise size and regularity estimates for the Dunkl Riesz transform kernel involving both the Euclidean metric and the Dunkl metric, where these two metrics are not equivalent. We further establish a suitable version of the pointwise kernel lower bound of the Dunkl Riesz transform via the Euclidean metric only. Then we show that the lower bound of commutator of the Dunkl Riesz transform is with respect to the BMO space associated with the Euclidean metric, and that the upper bound is respect to the BMO space associated with the Dunkl metric. Moreover, the compactness and the two types of VMO are also addressed.

math.CA↗
arXiv2023

Lipschitz and Triebel--Lizorkin spaces, commutators in Dunkl setting

We first study the Lipschitz spaces $Λ_{d}^β$ associated with the Dunkl metric, $β\in(0,1)$, and prove that it is a proper subspace of the classical Lipschitz spaces $Λ^β$ on $\mathbb R^N$, as the Dunkl metric and the Euclidean metric are non-equivalent. Next, we further show that the Lipschitz spaces $Λ^β$ connects to the Triebel--Lizorkin spaces $\dot{ F}^{α,q}_{p,{\rm D}}$ associated with the Dunkl Laplacian $\triangle_{\rm D}$ in $\mathbb R^ N $ and to the commutators of the Dunkl Riesz transform and the fractional Dunkl Laplacian $\triangle_{\rm D}^{-α/2}$, $0<α<\textbf{N}$ (the homogeneous dimension for Dunkl measure), which is represented via the functional calculus of the Dunkl heat semigroup $e^{-t\triangle_{\rm D}}$. The key steps in this paper are a finer decomposition of the underlying space via Dunkl metric and Euclidean metric to bypass the use of Fourier analysis, and a discrete weak-type Calderón reproducing formula in these new Triebel--Lizorkin spaces $\dot{ F}^{α,q}_{p,{\rm D}}$.

math.FA↗
arXiv2023

The Riesz Transform and Fractional Integral Operators in the Bessel Setting

Fix $λ>0$. Consider the Bessel operator $\triangle_λ:=-\frac{d^2}{dx^2}-\frac{2λ}{x} \frac d{dx}$ on $\mathbb{R_+}$, where $\mathbb{R_+}:=(0,\infty)$ and $dm_λ:=x^{2λ}dx$ with $dx$ the Lebesgue measure. We provide a deeper study of the Bessel Riesz transform and fractional integral operator via the related Besov and Triebel--Lizorkin spaces associated with $\triangle_λ$. Moreover, we investigate some possible characterization of the commutator of fractional integral operator, which was missing in the literature of the Bessel setting.

math.CA↗
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↗
arXiv2022

The Two Weight Inequality for Poisson Semigroup on Manifold with Ends

Let $M = \mathbb R^m \sharp \mathcal R^n$ be a non-doubling manifold with two ends $\mathbb R^m \sharp \mathcal R^n$, $m > n \ge 3$. Let $Δ$ be the Laplace--Beltrami operator which is non-negative self-adjoint on $L^2(M)$. Then $Δ$ and its square root $\sqrtΔ$ generate the semigroups $e^{-tΔ}$ and $e^{-t\sqrtΔ}$ on $L^2(M)$, respectively. We give testing conditions for the two weight inequality for the Poisson semigroup $e^{-t\sqrtΔ}$ to hold in this setting. In particular, we prove that for a measure $μ$ on $M_{+}:=M\times (0,\infty)$ and $σ$ on $M$: $$ \|\mathsf{P}_σ(f)\|_{L^2(M_{+};μ)} \lesssim \|f\|_{L^2(M;σ)}, $$ with $\mathsf{P}_σ(f)(x,t):= \int_M \mathsf{P}_t(x,y)f(y) \,dσ(y)$ if and only if testing conditions hold for the Poisson semigroup and its adjoint. Further, the norm of the operator is shown to be equivalent to the best constant in these testing conditions.

math.CA↗
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↗
arXiv2022

Weighted estimates of the Bergman projection with matrix weights

We establish a weighted inequality for the Bergman projection with matrix weights for a class of pseudoconvex domains. We extend a result of Aleman-Constantin and obtain the following estimate for the weighted norm of $P$: \[\|P\|_{L^2(Ω,W)}\leq C(\mathcal B_2(W))^{2}.\] Here $\mathcal B_2(W)$ is the Bekollé-Bonami constant for the matrix weight $W$ and $C$ is a constant that is independent of the weight $W$ but depends upon the dimension and the domain.

math.CV↗
arXiv2022

Schatten classes and commutator in the two weight setting, I. Hilbert transform

We characterize the Hilbert--Schmidt class membership of commutator with the Hilbert transform in the two weight setting. The characterization depends upon the symbol of the commutator being in a new weighted Besov space. This follows from a Schatten class $S_p$ result for dyadic paraproducts, where $1< p < \infty $. We discuss the difficulties in extending the dyadic result to the full range of Schatten classes for the Hilbert transform.

math.FA↗