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

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

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 (shaomingguo@math.wisc.edu; jroos@math.wisc.edu; seeger@math.wisc.edu)
Joris Roos
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Dr., Madison, WI53706, USA (shaomingguo@math.wisc.edu; jroos@math.wisc.edu; seeger@math.wisc.edu)
Andreas Seeger
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Dr., Madison, WI53706, USA (shaomingguo@math.wisc.edu; jroos@math.wisc.edu; seeger@math.wisc.edu)
Po-Lam Yung
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Ma Liu Shui, Shatin, Hong Kong (plyung@math.cuhk.edu.hk)
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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

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Footnotes

*

Current address: Mathematical Sciences Institute, Australian National University, Canberra, ACT 2600, Australia; polam.yung@anu.edu.au.

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