Comparison of 𝑇1 conditions for multi-parameter operators

de la Herrán, Ana

doi:10.1090/proc/12891

Comparison of $T1$ conditions for multi-parameter operators
HTML articles powered by AMS MathViewer

by Ana Grau de la Herrán

Proc. Amer. Math. Soc. 144 (2016), 2437-2443

DOI: https://doi.org/10.1090/proc/12891

Published electronically: October 14, 2015

PDF | Request permission

Abstract:

When Journé established the classical multi-parameter singular integral theory in 1985, it was formulated in the language of vector-valued Calderón-Zygmund theory. In 2011, Pott and Villarroya formulated a new $T1$ Theorem for product spaces. In their work, they replaced the vector-valued formulations for several mixed type conditions. Inspired by their work, Martikainen redefined the bi-parameter operators. Originally, it was thought that this new class of operators was larger (and contained the ones defined by Journé). In this paper we prove that both classes are actually equal.

References

Michael Christ, A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), no. 2, 601–628. MR 1096400, DOI 10.4064/cm-60-61-2-601-628
Guy David and Jean-Lin Journé, A boundedness criterion for generalized Calderón-Zygmund operators, Ann. of Math. (2) 120 (1984), no. 2, 371–397. MR 763911, DOI 10.2307/2006946
Loukas Grafakos, Classical and modern Fourier analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004. MR 2449250
Steve Hofmann, Local $Tb$ theorems and applications in PDE, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1375–1392. MR 2275650
Steve Hofmann and Alan McIntosh, Boundedness and applications of singular integrals and square functions: a survey, Bull. Math. Sci. 1 (2011), no. 2, 201–244. MR 2901001, DOI 10.1007/s13373-011-0014-3
Tuomas Hytönen and Henri Martikainen, On general local $Tb$ theorems, Trans. Amer. Math. Soc. 364 (2012), no. 9, 4819–4846. MR 2922611, DOI 10.1090/S0002-9947-2012-05599-1
Tuomas Hytönen and Henri Martikainen, Non-homogeneous $T1$ theorem for bi-parameter singular integrals, Adv. Math. 261 (2014), 220–273. MR 3213300, DOI 10.1016/j.aim.2014.02.011
T. Hytönen and F. Nazarov, The local Tb Theorem with rough test functions, arxiv:1206.0907.
Jean-Lin Journé, Calderón-Zygmund operators on product spaces, Rev. Mat. Iberoamericana 1 (1985), no. 3, 55–91. MR 836284, DOI 10.4171/RMI/15
Henri Martikainen, Representation of bi-parameter singular integrals by dyadic operators, Adv. Math. 229 (2012), no. 3, 1734–1761. MR 2871155, DOI 10.1016/j.aim.2011.12.019
S. Pott and P. Villarroya, A T(1) theorem on product spaces, arxiv:1105.2516