On Hörmander condition
type: Item
instance of: scholarly article
title: On Hörmander condition
zbMATH Open document ID: 0904.42011
DOI: 10.1007/BF02882861
author: Donggao Deng
author: Qixiang Yang
author: Lixin Yan
published in: Chinese Science Bulletin
publication date: 1998-12-29T00:00:00Z
review text: The famous \(T(1)\) theorem of \textit{G. David} and \textit{J.-L. Journé} [Ann. Math., II. Ser. 120, 371-397 (1984; Zbl 0567.47025)], which gives a condition for an operator \(T\) to be bounded on \(L^2\), is usually stated with the regularity assumption \[ | K(x,y)-K(x',y)| \lesssim | x-x'| ^\theta | x-y| ^{n-\theta} \] for all \(| x-x'| \leq 1/2 | x-y| \) and some \(0 < \theta < 1\), together with a similar condition for the transpose of \(K\). It is unknown whether the conclusion of the \(T(1)\) theorem continues to hold if one only assumes the Hörmander condition \[ \int_{| x-y| > 2| x-x'| } | K(x,y)-K(x',y)| \;dy \lesssim 1 \] together with the same condition on the transpose. In this note, the authors show that the Hörmander condition is not enough to imply the boundedness of \(T(1)\) on the Hardy space \(H^1\) or on the Besov space \(\dot B^{0,1}_1\). In fact, even a certain logarithmic strengthening of the Hörmander condition is insufficient, nor does it imply a natural Dini condition \(\sum_k \varepsilon(k) < \infty\), where \[ \varepsilon(k) = \sup_{u,v,r: | u| , | v| < r} \int_{| x-y| \sim 2^k r} | K(x+u,y+v) - K(x,y)| \;dy. \] It is known that \(\sum_k \sqrt{k} \varepsilon(k) < \infty\), together with the same condition on the transpose, is enough to imply the conclusion of the \(T(1)\) theorem.
Mathematics Subject Classification ID: 42B20
zbMATH DE Number: 1106407
zbMATH Keywords: \(T(1)\) theorem
zbMATH Keywords: Caldéron-Zygmund kernels
zbMATH Keywords: Hörmander condition
zbMATH Keywords: boundedness
zbMATH Keywords: Hardy space
zbMATH Keywords: Besov space
reviewed by: Terence Tao
MaRDI profile type: Publication
cites work: A boundedness criterion for generalized Calderón-Zygmund operators
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description: scientific article; zbMATH DE number 1106407