On the Local Tb Theorem: A Direct Proof under the Duality Assumption. Issue 1 (13th February 2015)

• Record Type: Journal Article
• Authors: Lacey, Michael T.; Vähäkangas, Antti V.
• Is Part Of: Proceedings of the Edinburgh Mathematical Society. Volume 59:Issue 1(2016)
• Page Start: 193
• Page End: 222
• Publication Date: 2015-02-13
• Subjects: local Tb theorem, -- Tl theorem, -- corona, -- twisted martingale transform, -- stopping cubes; Primary 42B20, -- Secondary 42B25, -- 42B35; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0013091514000340
• Languages: English
• ISSNs: 0013-0915

Abstract

Abstract: We give a new direct proof of the local Tb theorem in the Euclidean setting and under the assumption of dual exponents. This theorem provides a flexible framework for proving the boundedness of a Calderón–Zygmund operator, supposing the existence of systems of local accretive functions. We assume that the integrability exponents on these systems of functions are of the form 1 /p + 1 /q ⩽ 1, the 'dual case' 1 /p + 1 /q = 1 being the most difficult one. Our proof is direct: it avoids a reduction to the perfect dyadic case unlike some previous approaches. The principal point of interest is in the use of random grids and the corresponding construction of the corona. We also use certain twisted martingale transform inequalities.