Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-23T11:07:20.759Z Has data issue: false hasContentIssue false
  • English
  • Français

Maximal functions associated with families of homogeneous curves: Lp bounds for P ≤ 2

Part of: Integral transforms, operational calculus Harmonic analysis in several variables

Published online by Cambridge University Press:  03 February 2020

Shaoming Guo ,
Joris Roos ,
Andreas Seeger  and
Po-Lam Yung
Shaoming Guo
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Dr., Madison, WI53706, USA ([email protected]; [email protected]; [email protected])
Joris Roos
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Dr., Madison, WI53706, USA ([email protected]; [email protected]; [email protected])
Andreas Seeger
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Dr., Madison, WI53706, USA ([email protected]; [email protected]; [email protected])
Po-Lam Yung
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Ma Liu Shui, Shatin, Hong Kong ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Let M(u), H(u) be the maximal operator and Hilbert transform along the parabola (t, ut2). For U ⊂ (0, ∞) we consider Lp estimates for the maximal functions sup u∈U|M(u)f| and sup u∈U|H(u)f|, when 1 < p ≤ 2. The parabolas can be replaced by more general non-flat homogeneous curves.

Keywords

maximal functionsHilbert transforms along curvesmaximal and singular Radon transforms

MSC classification

Primary: 42B15: Multipliers
Secondary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25: Maximal functions, Littlewood-Paley theory 44A12: Radon transform
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 2 , May 2020 , pp. 398 - 412
Copyright
Copyright © Edinburgh Mathematical Society 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

Current address: Mathematical Sciences Institute, Australian National University, Canberra, ACT 2600, Australia; [email protected].

References

1.Carbery, A., Differentiation in lacunary directions and an extension of the Marcinkiewicz multiplier theorem, Ann. Inst. Fourier (Grenoble) 38(1) (1988), 157–168.CrossRefGoogle Scholar
2.Chang, S.-Y. A., Wilson, J. M. and Wolff, T. H., Some weighted norm inequalities concerning the Schrödinger operators, Comment. Math. Helv. 60(2) (1985), 217–246.CrossRefGoogle Scholar
3.Demeter, C. and Di Plinio, F., Logarithmic L p bounds for maximal directional singular integrals in the plane, J. Geom. Anal. 24(1) (2014), 375–416.CrossRefGoogle Scholar
4.Di Plinio, F., Guo, S., Thiele, C. and Zorin-Kranich, P., Square functions for bi-Lipschitz maps and directional operators, J. Funct. Anal. 275(8) (2018), 2015–2058.CrossRefGoogle Scholar
5.Guo, S., Hickman, J., Lie, V. and Roos, J., Maximal operators and Hilbert transforms along variable non-flat homogeneous curves, Proc. Lond. Math. Soc. (3) 115(1) (2017), 177–219.CrossRefGoogle Scholar
6.Guo, S., Roos, J., Seeger, A. and Yung, P.-L., A maximal function for families of Hilbert transforms along homogeneous curves, Preprint, arXiv:1902.00096. Published online in Math. Ann. doi.org/10.1007/00208-019-01915-3.Google Scholar
7.Karagulyan, G. A., On unboundedness of maximal operators for directional Hilbert transforms, Proc. Amer. Math. Soc. 135(10) (2007), 3133–3141.CrossRefGoogle Scholar
8.Marletta, G. and Ricci, F., Two-parameter maximal functions associated with homogeneous surfaces in ℝn, Studia Math. 130(1) (1998), 53–65.Google Scholar
9.Nagel, A., Stein, E. M. and Wainger, S., Differentiation in lacunary directions, Proc. Nat. Acad. Sci. U.S.A. 75(3) (1978), 1060–1062.CrossRefGoogle ScholarPubMed
10.Seeger, A., Tao, T. and Wright, J., Endpoint mapping properties of spherical maximal operators, J. Inst. Math. Jussieu 2(1) (2003), 109–144.CrossRefGoogle Scholar
11.Seeger, A., Wainger, S. and Wright, J., Pointwise convergence of spherical means, Math. Proc. Cambridge Philos. Soc. 118(1) (1995), 115–124.CrossRefGoogle Scholar
12.Seeger, A., Wainger, S. and Wright, J., Spherical maximal operators on radial functions, Math. Nachr. 187 (1997), 241–265.CrossRefGoogle Scholar