On Hörmander condition
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.
scientific article; zbMATH DE number 1106407
scientific article