Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1204.6526

The Perfect Local $ Tb$ Theorem and Twisted Martingale Transforms

Michael T. Lacey·Antti V. Vähäkangas
View original source ↗Read source document ↗Export metadata ↓

Abstract

A local Tb Theorem provides a flexible framework for proving the boundedness of a Calderón-Zygmund operator T. One needs only boundedness of the operator T on systems of locally pseudo-accretive functions \{b_Q\}, indexed by cubes. We give a new proof of this Theorem in the setting of perfect (dyadic) models of Calderón-Zygmund operators, imposing integrability conditions on the b_Q functions that are the weakest possible. The proof is a simple direct argument, based upon a new inequality for transforms of so-called twisted martingale differences.

Explore related subjects

math.CA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Michael T. Lacey, Antti V. Vähäkangas. 2012-05-14. The Perfect Local $ Tb$ Theorem and Twisted Martingale Transforms. https://doi.org/10.1090/s0002-9939-2014-11930-7

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Weighted inequalities in ergodic theory via transference

We first extend Calderón's transfer principle to weighted spaces in various different settings under suitable assumptions. Then we apply our results for some inequalities on the real line obtained by the author to prove corresponding inequalities in ergodic theory and ergodic $H^1$ spaces as well.

math.CA↗
arXiv2026

Wavelet resolution and Sobolev regularity of Calderón-Zygmund operators on domains

Given a uniform domain $Ω\subset {\mathbb R}^d$, we resolve each element of a suitably defined class of Calderòn-Zygmund (CZ) singular integrals on $Ω$ as the linear combination of Triebel wavelet operators and paraproduct terms. Our resolution formula entails a testing type characterization, loosely in the vein of the David-Journé theorem, of weighted Sobolev space bounds in terms of Triebel-Lizorkin and tree Carleson measure norms of the paraproduct symbols, which is new already in the case $Ω={\mathbb R}^d$ with Lebesgue measure. Our characterization covers the case of compressions to $Ω$ of global CZ operators, extending and sharpening past results of Prats and Tolsa for the convolution case. The weighted estimates we obtain, particularized to the Beurling operator on a Lipschitz domain with normal to the boundary in the corresponding sharp Besov class, may be used to deduce quantitative estimates for quasiregular mappings with dilatation in the Sobolev space $W^{1,p}(Ω)$, $p>2$.

math.CA↗