Abstract
This paper studies the mixed radial-angular integrability of parametric Marcinkiewicz integrals along “polynomial curves”. Under the assumption that the kernels satisfy certain rather weak size conditions on the unit sphere with radial roughness, the authors prove that such operators are bounded on the mixed radial-angular spaces. Meanwhile, corresponding vector-valued versions are also obtained.
Similar content being viewed by others
References
Al-Salman A, Al-Qassem H, Cheng L C, Pan Y. Lp bounds for the function of Marcinkiewicz. Math Res Lett, 2002, 9(5/6): 697–700
Benedek A, Calderón A, Panzone R. Convolution operators on Banach space value functions. Proc Nat Acd Sci USA, 1962, 48: 356–365
Cacciafesta F, D’Ancona P. Endpoint estimates and global existence for the nonlinear Dirac equation with potential. J Differential Equations, 2013, 254(5): 2233–2260
Cacciafesta F, Luca R. Singular integrals with angular regularity. Proc Amer Math Soc, 2016, 144: 3413–3418
Chen J, Fan D, Pan Y. A note on a Marcinkiewicz integral operator. Math Nachr, 2001, 227: 33–42.
Córdoba A. Singular integrals and maximal functions: the disk miltiplier revisited. Adv Math, 2016, 290: 208–235
D’Ancona P, Lucà R. On the regularity set and angular integrability for the Navier-Stokes equation. Arch Rational Mech Anal, 2016, 221: 1255–1284
Ding Y, Fan D, Pan Y. Lp-boundedness of Marcinkiewicz integrals with Hardy space function kernel. Acta Math Sin (Engl Ser), 2000, 16: 593–600
Ding Y, Lu S, Yabuta K. A problem on rough Marcinkiewicz functions. J Austral Math Soc, 2001, 71: 1–9
Duoandikoetxea J, Rubio de Francia J L. Maximal and singular integral operators via Fourier transform estimates. Invent Math, 1986, 84: 541–561
Fan D, Pan Y. Singular integral operators with rough kernels supported by subvarieties. Amer J Math, 1997, 119: 799–839
Fan D, Sato S. A note on the singular integrals associated with a variable surface of revolution. Math Inequal Appl, 2009, 12(2): 441–454
Fang D, Wang C. Weighted Strichartz estimates with angular regularity and their applications. Forum Math, 2011, 23: 181–205
Grafakos L, Stefanov A. Lp bounds for singular integrals and maximal singular integrals with rough kernels. Indiana Univ Math J, 1998, 47(2): 455–469
Hofmann S. Weighted norm inequalities and vector valued inequalities for certain rough operators. Indiana Univ Math J, 1993, 42(1): 1–14
Hou X, Wu H. Limiting weak-type behaviors for certain Littlewood-Paley functions. Acta Math Sci, 2019, 39B(1): 11–25
Liu F. Weighted estimates for Marcinkiewicz integrals with applications to angular integrability. 2019, preprint.
Liu F, Fan D. Weighted estimates for rough singualr integrals with applications to angular integrability. Pacific J Math, 2019, 301(1): 267–295
Liu F, Wu H, Zhang D. Lp bounds for parametric Marcinkiewicz integrals with mixed homogeneity. Math Inequal Appl, 2015, 8(2): 453–469
Liu F, Liu R, Wu H. Weighted estiamtes for rough singular integrals with applications to angular integrablity, II. Math Inequal Appl, 2020, 23(1): 393–418
Liu R, Liu F, Wu H. Mixed radial-angular integrability for rough singular integrals and maximal operators. Proc Amer Math Soc, 2020, 148(9): 3943–3956
Li W, Si Z, Yabuta K. Boundedness of singular integrals associated to surfaces of revolution on Triebel Lizorkin spaces. Forum Math, 2016, 28(1): 57–75
Stein E M. On the function of Littlewood-Paley, Lusin and Marcinkiewicz. Trans Amer Math Soc, 1958, 88: 430–466
Sterbenz J. Angular regularity and Strichartz estimates for the wave equation. Inter Math Res Not, 2005, 4: 187–231
Tao T. Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation. Comm Partial Differential Equations, 2000, 25(7/8): 1471–1485
Walsh T. On the function of Marcinkiewicz. Studia Math, 1972, 44: 203–217
Wu H. On Marcinkiewicz integral operators with rough kernels. Integral Equations Operator Theory, 2005, 52: 285–298
Wu H. Lp bounds for Marcinkiewicz integrals associated to surfaces of revolution. J Math Anal Appl, 2006, 321(2): 811–827
Wu H, Xu J. Rough Marcinkiewicz integrals associated to surfaces of revolution on product domains. Acta Math Sci, 2009, 29B(2): 294–304
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors were partly supported by the NSFC (11771358, 11701333, 11871101).
Rights and permissions
About this article
Cite this article
Liu, R., Liu, F. & Wu, H. On the mixed radial-angular integrability of Marcinkiewicz integrals with rough kernels. Acta Math Sci 41, 241–256 (2021). https://doi.org/10.1007/s10473-021-0114-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10473-021-0114-4