It is well known that the Hörmander smoothness condition implies weak-type estimates for associated -bounded Calderón–Zygmund operators. It has been an open question to know whether Hörmander's condition also suffices to guarantee weak-type estimates for bilinear Calderón–Zygmund operators that are bounded at one point. In this paper, we provide a negative answer to this question.
Il est bien connu que la condition de lissage de Hörmander implique des estimations faibles de type pour les opérateurs de Calderón–Zygmund -bornés. La question s'est alors posée de savoir si cette condition de Hörmander est également suffisante pour assurer des estimations faibles de type pour les opérateurs bilinéaires de Calderón–Zygmund qui sont bornés en un point. Nous donnons ici une réponse négative à cette question.
Accepted:
Published online:
@article{CRMATH_2019__357_4_382_0, author = {Grafakos, Loukas and He, Danqing and Slav{\'\i}kov\'a, Lenka}, title = {Failure of the {H\"ormander} kernel condition for multilinear {Calder\'on{\textendash}Zygmund} operators}, journal = {Comptes Rendus. Math\'ematique}, pages = {382--388}, publisher = {Elsevier}, volume = {357}, number = {4}, year = {2019}, doi = {10.1016/j.crma.2019.04.002}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2019.04.002/} }
TY - JOUR AU - Grafakos, Loukas AU - He, Danqing AU - Slavíková, Lenka TI - Failure of the Hörmander kernel condition for multilinear Calderón–Zygmund operators JO - Comptes Rendus. Mathématique PY - 2019 SP - 382 EP - 388 VL - 357 IS - 4 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2019.04.002/ DO - 10.1016/j.crma.2019.04.002 LA - en ID - CRMATH_2019__357_4_382_0 ER -
%0 Journal Article %A Grafakos, Loukas %A He, Danqing %A Slavíková, Lenka %T Failure of the Hörmander kernel condition for multilinear Calderón–Zygmund operators %J Comptes Rendus. Mathématique %D 2019 %P 382-388 %V 357 %N 4 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2019.04.002/ %R 10.1016/j.crma.2019.04.002 %G en %F CRMATH_2019__357_4_382_0
Grafakos, Loukas; He, Danqing; Slavíková, Lenka. Failure of the Hörmander kernel condition for multilinear Calderón–Zygmund operators. Comptes Rendus. Mathématique, Volume 357 (2019) no. 4, pp. 382-388. doi : 10.1016/j.crma.2019.04.002. http://archive.numdam.org/articles/10.1016/j.crma.2019.04.002/
[1] E. Buriánková, D. He, P. Honzík, Multilinear rough singular integrals, in preparation.
[2] Multilinear weighted norm inequalities under integral type regularity conditions (Chanillo, S. et al., eds.), Harmonic Analysis, Partial Differential Equations and Applications, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Cham, Switzerland, 2017, pp. 193-216
[3] On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc., Volume 212 (1975), pp. 315-331
[4] Commutateurs d' intégrales singulières et opérateurs multilinéaires, Ann. Inst. Fourier (Grenoble), Volume 28 (1978), pp. 177-202
[5] Au-delà des opérateurs pseudo-différentiels, Astérisque, vol. 57, 1978
[6] Method of rotations for bilinear singular integrals, Commun. Math. Anal. Commun. Math. Anal. Conf., Volume 3 (2011), pp. 99-107
[7] Modern Fourier Analysis, Graduate Texts in Mathematics, vol. 250, Springer, New York, 2014
[8] Rough bilinear singular integrals, Adv. Math., Volume 326 (2018), pp. 54-78
[9] boundedness criteria, Math. Ann. (2019) | DOI
[10] Uniform bounds for the bilinear Hilbert transforms, I, Ann. of Math. (2), Volume 159 (2004), pp. 889-933
[11] Multilinear Calderón–Zygmund theory, Adv. Math., Volume 165 (2002), pp. 124-164
[12] Estimates for translation invariant operators in spaces, Acta Math., Volume 104 (1960), pp. 93-140
[13] Multilinear estimates and fractional integration, Math. Res. Lett., Volume 6 (1999), pp. 1-15
[14] Uniform bounds for the bilinear Hilbert transform II, Rev. Mat. Iberoam., Volume 22 (2006), pp. 1069-1126
[15] Sparse domination theorem for multilinear singular integral operators with -Hörmander condition, Michigan Math. J., Volume 67 (2018), pp. 253-265
[16] Lack of natural weighted estimates for some singular integral operators, Trans. Amer. Math. Soc., Volume 357 (2005), pp. 385-396
[17] Calderón–Zygmund and Multilinear Operators, Cambridge Studies in Advanced Mathematics, vol. 48, Cambridge University Press, Cambridge, UK, 1997 (translated from the 1990 and 1991 French originals by David Salinger)
[18] Minimal regularity conditions for the end-point estimate of bilinear Calderón–Zygmund operators, Proc. Amer. Math. Soc. Ser. B, Volume 1 (2014), pp. 1-13
Cited by Sources:
☆ The first author was supported by the Simons Foundation (No. 315380). The second author was supported by the NNSF of China (No. 11701583), the Guangdong Natural Science Foundation (No. 2017A030310054) and the Fundamental Research Funds for the Central Universities (No. 17lgpy11).