zbMATH — the first resource for mathematics

Multi-parameter singular integral operators and representation theorem. (English) Zbl 1366.42016
The author formulates a class of singular integral operators in arbitrarily many parameters using mixed type characterizing conditions. The main result obtained for this class of operators is a multi-parameter representation theorem stating that a generic operator in this class can be represented as an average of sums of dyadic shifts, which implies a new multi-parameter \(T1\) theorem as a byproduct. This extends the representation principles of Hytönen’s and Martikainen’s to the multi-parameter setting. Furthermore, equivalence between the studied class and Journé’s class of multi-parameter operators is established, whose proof requires the multiparameter \(T1\) theorem.

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
30H35 BMO-spaces
Full Text: DOI
[1] Dalenc, L. and Ou, Y.: Upper bound for multi-parameter iterated commutators. Publ. Mat.60 (2016), no. 1, 191–220. · Zbl 1333.42018
[2] Fefferman, R.: Harmonic analysis on product spaces. Ann. of Math. (2)126 (1987), no. 1, 109–130. · Zbl 0644.42017
[3] Fefferman, R. and Stein, E. M.: Singular integrals on product spaces. Adv. Math.45 (1982), no. 2, 117–143. · Zbl 0517.42024
[4] Grau de la Herrán, A.: Comparison ofT 1 conditions for multi-parameter operators. Proc. Amer. Math. Soc.144 (2016), no 6, 2437–2443. · Zbl 1350.42024
[5] Hytönen. T: Representation of singular integrals by dyadic operators, and theA2 theorem. To appear in Expo. Math., doi: 10.1016/j.exmath.2016.09.003. · Zbl 1402.42026
[6] Hytönen, T., Pérez, C., Treil, S. and Volberg, V.: Sharp weighted estimates for dyadic shifts and theA2conjecture. J. Reine Angew. Math.687 (2014), 43–86. · Zbl 1311.42037
[7] Journé, J.-L.: Calderón–Zygmund operators on product spaces. Rev. Mat. Iberoamericana1 (1985), no. 3, 55–91.
[8] Journé, J.-L.: Two problems of Calderón–Zygmund theory on product-spaces. Ann. Inst. Fourier (Grenoble)38 (1988), no. 1, 111–132.
[9] Lacey, M. and Petermichl, S.: Personal communication.
[10] Martikainen, H.: Representation of bi-parameter singular integrals by dyadic operators. Adv. Math.229 (2012), no. 3, 1734–1761. · Zbl 1241.42012
[11] Pipher, J. and Ward, L. A.: BMO from dyadic BMO on the bidisc. J. London Math. Soc. (2)77 (2008), no. 2, 524–544. · Zbl 1143.42025
[12] Pott, S. and Villarroya, P.: AT (1) theorem on product spaces. Preprint available at arXiv: 1105.2516, 2013.
[13] Treil, S.:H1and dyadicH1. In Linear and complex analysis, 179–193. Amer. Math. Soc. Transl. Ser. 2, 226, Adv. Math. Sci 63, Amer. Math. Soc., Providence, RI, 2009.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.