arXiv 1303.03672013-03-02math.APmath.CAmath.MG

The proof of non-homogeneous T1 theorem via averaging of dyadic shifts

Alexander Volberg

备注：: 11 pages

英文摘要：

We give again a proof of non-homogeneous T1 theorem. Our proof consists of three main parts: a construction of a random dyadic lattice; an estimate of matrix coefficients of a Calderón--Zygmund operator with respect to random Haar basis if a smaller Haar support is good; a clever averaging trick from Hytönen's papers which uses the averaging over dyadic lattices to decompose operator into dyadic shifts eliminating the error term that was present in the previous proofs by Nazarov--Treil--Volberg.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Singular integrals with variable kernels in dyadic settings

In this paper we explore conditions on variable symbols with respect to Haar systems, defining Calderón-Zygmund type operators with respect to the dyadic metrics associated to the Haar bases.We show that Petermichl's dyadic kernel can be seen as a variable kernel singular integral and we extend it to dyadic systems built on spaces of homogeneous type.

2021-01-26math.APmath.CA

Partial derivatives, singular integrals and Sobolev Spaces in dyadic settings

In this note we show that the general theory of vector valued singular integral operators of Calderón-Zygmund defined on general metric measure spaces, can be applied to obtain Sobolev type regularity properties for solutions of the dyadic fractional Laplacian. In doing so, we define partial derivatives in terms of Haar multipliers and dyadic homogeneous singular integral operators.

2020-04-23math.APmath.FA

On the Calderón-Zygmund structure of Petermichl's kernel. Weighted inequalities

We show that Petermichl's dyadic operator $\mathcal{P}$ (S. Petermichl (2000), Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol) is a Calderón-Zygmund type operator on an adequate metric normal space of homogeneous type. As a consequence of a general result on spaces of homogeneous type, we get weighted boundedness of the maximal operator $\mathcal{P}^*$ of truncations of the singular integral. We show that dyadic $A_p$ weights are the good weights for the maximal operator $\mathcal{P}^*$ of the scale truncations of $\mathcal{P}$.

2017-09-13math.AP

Systems of dyadic cubes in a doubling metric space

A number of recent results in Euclidean Harmonic Analysis have exploited several adjacent systems of dyadic cubes, instead of just one fixed system. In this paper, we extend such constructions to general spaces of homogeneous type, making these tools available for Analysis on metric spaces. The results include a new (non-random) construction of boundedly many adjacent dyadic systems with useful covering properties, and a streamlined version of the random construction recently devised by H. Martikainen and the first author. We illustrate the usefulness of these constructions with applications t…

2011-12-12math.AP

On the failure of lower square function estimates in the non-homogeneous weighted setting

We show that the classical $A_{\infty}$ condition is not sufficient for a lower square function estimate in the non-homogeneous weighted $L^2$ space. We also show that under the martingale $A_2$ condition, an estimate holds true, but the optimal power of the characteristic jumps from $1 / 2$ to $1$ even when considering the classical $A_2$ characteristic. This is in a sharp contrast to known estimates in the dyadic homogeneous setting as well as the recent positive results in this direction on the discrete timenon-homogeneous martingale transforms. Last, we give a sharp $A_{\infty}$ estimate f…

2017-05-23math.AP

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 …

2018-09-21math.APmath.CA

更多相关论文收起更多论文

Weighted martingale multipliers in non-homogeneous setting and outer measure spaces

We investigate the unconditional basis property of martingale differences in weighted $L^2$ spaces in the non-homogeneous situation (i.e. when the reference measure is not doubling). Specifically, we prove that finiteness of the quantity $[w]_{A_2}=\sup_I \, < w>_I < w^{-1}>_I$, defined through averages $ <\cdot >_I$ relative to the reference measure $ν$, implies that each martingale transform relative to $ν$ is bounded in $L^2(w\, dν)$. Moreover, we prove the linear in $[w]_{A_2}$ estimate of the unconditional basis constant of the Haar system. Even in the classical case of the standard dyadi…

2014-11-19math.APmath.CA

临界索伯列夫空间中空间均匀玻尔兹曼方程的唯一性

Uniqueness for the spatially homogeneous Boltzmann equation in critical Sobolev spaces

Tsinghua University（清华大学）；University of Oxford（牛津大学）；Nanjing University of Aeronautics and Astronautics（南京航空航天大学）

本文研究极软势下无角截断的空间均匀玻尔兹曼方程，在带对数修正的临界索伯列夫空间中建立解的存在性、唯一性等，结合费舍尔信息单调性实现解的全局时间延拓，采用相空间与频率二进局域化及精确交换子估计作为核心工具。

2026-08-25math.AP

Besov and Triebel-Lizorkin Spaces on Spaces of Homogeneous Type with Applications to Boundedness of Calderón-Zygmund Operators

In this article, the authors introduce Besov and Triebel-Lizorkin spaces on spaces of homogeneous type in the sense of Coifman and Weiss, prove that these (in)homogeneous Besov and Triebel-Lizorkin spaces are independent of the choices of both exp-ATIs (or exp-IATIs) and underlying spaces of distributions, and give some basic properties of these spaces. As applications, the authors show that some known function spaces coincide with certain special cases of Besov and Triebel-Lizorkin spaces and, moreover, obtain the boundedness of Calderón-Zygmund operators on these Besov and Triebel-Lizorkin s…

2020-12-24math.APmath.CAmath.FA

齐次型空间上的Banach格中的紧集

Compact sets in Banach lattices over spaces of homogeneous type

Ankara University（安卡拉大学）；Institute of Mathematics of the Academy of Sciences of the Czech Republic（捷克科学院数学研究所）；Chuo University（中央大学）

本文证明了齐次型空间上Banach格的Kolmogorov-Riesz紧性定理，用有界性、紧性和平均算子逼近刻画紧性，无需Vitali覆盖等假设，并推广至重排不变空间和Morrey空间。

2026-09-23math.APmath.CAmath.FA

Weak Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Decompositions, Real Interpolation, and Calderón--Zygmund Operators

Let $(\mathbb{X},d,μ)$ be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, and $X(\mathbb{X})$ a ball quasi-Banach function space on $\mathbb{X}$. In this article, the authors introduce the weak Hardy space $WH_X(\mathbb{X})$ associated with $X(\mathbb{X})$ via the grand maximal function, and characterize $WH_X(\mathbb{X})$ by other maximal functions and atoms. The authors then apply these characterizations to obtain the real interpolation and the boundedness of Calderón--Zygmund operators in the critical case. The main novelties of this article exist in that the authors…

2022-01-23math.APmath.CAmath.FA

Scaling effects on the periodic homogenization of a reaction-diffusion-convection problem posed in homogeneous domains connected by a thin composite layer

We study the question of periodic homogenization of a variably scaled reaction-diffusion problem with non-linear drift posed for a domain crossed by a flat composite thin layer. The structure of the non-linearity in the drift was obtained in earlier works as hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) process for a population of interacting particles crossing a domain with obstacle. Using energy-type estimates as well as concepts like thin-layer convergence and two-scale convergence, we derive the homogenized evolution equation and the corresponding effective mo…

2021-07-18math.AP