Weighted estimates for bilinear square functions with non-smooth kernels and commutators

Abstract

Under weaker conditions on the kernel functions, we discuss the boundedness of bilinear square functions associated with non-smooth kernels on the product of weighted Lebesgue spaces. Moreover, we investigate the weighted boundedness of the commutators of bilinear square functions (with symbols which are BMO functions and their weighted version, respectively) on the product of Lebesgue spaces. As an application, we deduce the corresponding boundedness of bilinear Marcinkiewicz integrals and bilinear Littlewood-Paley g-functions.

Keywords

Bilinear square function / non-smooth kernel / weight / commutator / BMO function

References

[1] Bu R, Chen J C. Compactness for the commutator of multilinear singular integral operators with non-smooth kernels. Appl Math J Chinese Univ Ser B, 2019, 34: 55–75

[2] Chuong N M, Hong N T, Hung H D. Bounds of weighted multilinear Hardy-Cesàro operators in p-adic functional spaces. Front Math China, 2018, 13: 1–24

[3] Coifman R R, Meyer Y. On commutators of singular integrals and bilinear singular integrals. Trans Amer Math Soc, 1975, 212: 315–331

[4] Coifman R R, Meyer Y. Au delà des opérateurs pseudo-différentiels. Astérisque, No 57. Paris: Soc Math France, 1978

[5] Dong B H, Fu Z W, Xu J S. Riesz-Kolmogorov theorem in variable exponent Lebesgue spaces and its applications to Riemann-Liouville fractional differential equations. Sci China Math, 2018, 61: 1807–1824

[6] Duong X T, Gong R M, Grafakos L, Li J, Yan L X. Maximal operator for multilinear singular integrals with non-smooth kernels. Indiana Univ Math J, 2009, 58: 2517–2542

[7] Fabes E B, Jerison D, Kenig C. Multilinear square functions and partial differential equations. Amer J Math, 1985, 107: 1325–1368

[8] Garcia-Cuerva J. Weighted H^p Spaces. Dissertationes Math (Rozprawy Mat), 162. Warsaw: Polish Acad Sci Inst Math, 1979

[9] Grafakos L, Torres R H. Multilinear Calderón-Zygmund theory. Adv Math, 2002, 165: 124–164

[10] Grafakos L, Torres R H. Maximal operator and weighted norm inequalities for multi- linear singular integrals. Indiana Univ Math J, 2002, 51: 1261–1276

[11] Hormozi M, Si Z Y, Xue Q Y. On general multilinear square function with non-smooth kernels. Bull Math Sci, 2018, 149: 1–22

[12] Hou X M, Wu H X. Limiting weak-type behaviors for Riesz transforms and maximal operators in Bessel setting. Front Math China, 2019, 14: 535–550

[13] Hu G E. Weighted compact commutator of bilinear Fourier multiplier operator. Chin Ann Math Ser B, 2017, 38: 795–814

[14] Hu G E, Zhu Y. Weighted norm inequality with general weights for the commutator of Calderón. Acta Math Sin (Engl Ser), 2013, 29: 505–514

[15] Lerner A K. Weighted norm inequalities for the local sharp maximal function. J Fourier Anal Appl, 2004, 10: 645–674

[16] Lerner A K, Ombrosi S, Pérez C, Torres R H, Trujillo-González R. New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory. Adv Math, 2009, 220: 1222–1264

[17] Liu F, Xue Q Y. Characterizations of the multiple Littlewood-Paley operators on product domains. Publ Math Debrecen, 2018, 92: 419–439

[18] Mo H X, Wang X J, Ma R Q. Commutator of Riesz potential in p-adic generalized Morrey spaces. Front Math China, 2018, 13: 633–645

[19] Pérez C, Pradolini G, Torres R H, Trujillo-González R. End-points estimates for iterated commutators of multilinear singular integrals. Bull Lond Math Soc, 2014, 46: 26–42

[20] Rao M M, Ren Z D. Theory of Orlicz Space. New York: Marcel Dekker, 1991

[21] Sato S, Yabuta K. Multilinearized Littlewood-Paley operators. Sci Math Jpn, 2002, 55: 447–453

[22] Strömberg J O. Bounded mean oscillation with Orlicz norm and duality of Hardy spaces. Indiana Univ Math J, 1979, 28: 511–544

[23] Xue Q Y, Yan J Q. On multilinear square function and its applications to multilinear Littlewood-Paley operators with non-convolution type kernels. J Math Anal Appl, 2015, 422: 1342–1362

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature