Bilinear representation theorem

Li, Kangwei; Martikainen, Henri; Ou, Yumeng; Vuorinen, Emil

doi:10.1090/tran/7505

Bilinear representation theorem
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by Kangwei Li, Henri Martikainen, Yumeng Ou and Emil Vuorinen PDF

Trans. Amer. Math. Soc. 371 (2019), 4193-4214 Request permission

Abstract:

We represent a general bilinear Calderón–Zygmund operator as a sum of simple dyadic operators. The appearing dyadic operators also admit a simple proof of a sparse bound. In particular, the representation implies a so-called sparse $T1$ theorem for bilinear singular integrals.

References

José M. Conde-Alonso and Guillermo Rey, A pointwise estimate for positive dyadic shifts and some applications, Math. Ann. 365 (2016), no. 3-4, 1111–1135. MR 3521084, DOI 10.1007/s00208-015-1320-y
A. Culiuc, F. Di Plinio, and Y. Ou, Uniform sparse domination of singular integrals via dyadic shifts, arXiv:1610.01958 (2016). Math. Res. Lett. (to appear).
I. Holmes, S. Petermichl, and B. Wick, Weighted little BMO and two-weight inequalities for Journé commutators, arXiv:1701.06526 (2017). Anal. & PDE (to appear).
Tuomas P. Hytönen, The sharp weighted bound for general Calderón-Zygmund operators, Ann. of Math. (2) 175 (2012), no. 3, 1473–1506. MR 2912709, DOI 10.4007/annals.2012.175.3.9
Tuomas P. Hytönen, Representation of singular integrals by dyadic operators, and the $A_2$ theorem, Expo. Math. 35 (2017), no. 2, 166–205. MR 3654073, DOI 10.1016/j.exmath.2016.09.003
Timo S. Hänninen and Tuomas P. Hytönen, Operator-valued dyadic shifts and the $T(1)$ theorem, Monatsh. Math. 180 (2016), no. 2, 213–253. MR 3502626, DOI 10.1007/s00605-016-0891-3
Kangwei Li, Kabe Moen, and Wenchang Sun, The sharp weighted bound for multilinear maximal functions and Calderón-Zygmund operators, J. Fourier Anal. Appl. 20 (2014), no. 4, 751–765. MR 3232584, DOI 10.1007/s00041-014-9326-5
Michael T. Lacey and Darío Mena Arias, The sparse T1 theorem, Houston J. Math. 43 (2017), no. 1, 111–127. MR 3647935
A. Lerner and F. Nazarov, Intuitive dyadic calculus: The basics, arXiv:1508.05639 (2015).
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
H. Martikainen and E. Vuorinen, Dyadic-probabilistic methods in bilinear analysis, arXiv:1609.01706 (2016).
Yumeng Ou, Multi-parameter singular integral operators and representation theorem, Rev. Mat. Iberoam. 33 (2017), no. 1, 325–350. MR 3615454, DOI 10.4171/RMI/939
Yumeng Ou, Stefanie Petermichl, and Elizabeth Strouse, Higher order Journé commutators and characterizations of multi-parameter BMO, Adv. Math. 291 (2016), 24–58. MR 3459014, DOI 10.1016/j.aim.2015.12.029
Stefanie Petermichl, Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 6, 455–460 (English, with English and French summaries). MR 1756958, DOI 10.1016/S0764-4442(00)00162-2