Continuity properties for commutators of multilinear type operators on product of certain Hardy spaces

Abstract

Similar to the property of a linear Calderón-Zygmund operator, a linear fractional type operator I_α associated with a BMO function b fails to satisfy the continuity from the Hardy space H^p into L^p for p≤1. Thus, an alternative result was given by Y. Ding, S. Lu and P. Zhang, they proved that [b, I_α] is continuous from an atomic Hardy space H_b^p into L^p,where H_b^p is a subspace of the Hardy space H^p for n/(n+1)<p≤1. In this paper, we study the commutators of multilinear fractional type operators on product of certain Hardy spaces. The endpoint (H_{b\mathrm{1}}^{p\mathrm{1}}\times \cdots \times H_{bm}^{pm}, L^p) boundedness for multilinear fractional type operators is obtained. We also give the boundedness for the commutators of multilinear Calderón-Zygmund operators and multilinear fractional type operators on product of certain Hardy spaces when b\in {(\mathrm{L}\mathrm{i}{\mathrm{p}}_{\beta })}^m({ℝ}^n).

Keywords

Multilinear fractional type operator / multilinar Calderón-Zygmund operator / commutator / Hardy type space

References

[1] Adams D R. A note on Riesz potentials. Duke Math J, 1975, 42(4): 765-778

[2] Alvarez J. Continuity properties for linear commutators of Calderon-Zygmund operators. Collect Math, 1998, 48: 1037-1055

[3] Chen X, Xue Q. Weighted estimates for a class of multilinear fractional type operators. J Math Anal Appl, 2010, 362(2): 355-373

[4] Christ M, Journé J L. Polynomial growth estimates for multilinear singular integral operators. Acta Math, 1987, 159: 51-80

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

[6] Coifman R R, Meyer Y. Commutateurs d’intégrales singulières et opérateurs multilinéaires. Ann Inst Fourier (Grenoble), 1978, 28(3): 177-202

[7] Ding Y, Lu S. Hardy spaces estimate for a class of multilinear homogeneous operators. Sci China Ser A, 1999, 42(12): 1270-1278

[8] Ding Y, Lu S. Hardy spaces estimates for multilinear operators with homogeneous kernels. Nagoya Math J, 2003, 170: 117-133

[9] Ding Y, Lu S, Zhang P. Weak estimates for commutators of fractional integral operators. Sci China Ser A, 2001, 44(7): 877-888

[10] Ding Y, Lu S, Zhang P. Continuity of higher order commutators on certain Hardy spaces. Acta Math Sin (Engl Ser), 2002, 18(2): 391-404

[11] Grafakos L. On multilinear fractional integrals. Studia Math, 1992, 102: 49-56

[12] Grafakos L, Kalton N. Multilinear Calderón-Zygmund operators on Hardy spaces. Collect Math, 2001, 52(2): 169-179

[13] Grafakos L, Torres R H. Multilinear Calderón-Zygmund theory. Adv Math, 2002, 165(1): 124-164

[14] Kenig C E, Stein E M. Multilinear estimates and fractional integration. Math Res Lett, 1999, (6): 1-15

[15] 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(4): 1222-1264

[16] Li W, Xue Q. Continuity property for the commutators of multilinear Calderón-Zygmund operators. Forum Math, 2013, 25: 771-785

[17] Lin Y, Lu S. Boundedness of multilinear singular integral operators on Hardy and Herz-Hardy spaces. Hokkaido Math J, 2007, 36(3): 585-613

[18] Lu S, Wu Q, Yang D. Boundedness of commutators on Hardy type spaces. Sci China Ser A, 2002, 45(8): 984-997

[19] Moen K. Weighted inequalities for multilinear fractional integral operators. Collect Math, 2009, 60: 213-238

[20] Muckenhoupt B, Wheeden R. Weighted norm inequalities for fractional integral. Trans Amer Math Soc, 1974, 192: 261-274

[21] Paluszyński M. Characterization of Lipschitz sSpaces via Commutator of Coifman, Rochberg and Weiss; A Multiplier Theorem for the Semigroup of Contractions. Ph D Thesis. Washington Univ, 1992

[22] Pérez C. Endpoint estimates for commutators of singular integral operators. J Funct Anal, 1995, 128(1): 163-185

[23] Welland G V. Weighted norm inequalities for fractional integral. Proc Amer Math Soc, 1975, 51(1): 143-148

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Higher Education Press and Springer-Verlag Berlin Heidelberg