Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-27T20:19:06.851Z Has data issue: false hasContentIssue false
  • English
  • Français

MAXIMAL OPERATORS AND HILBERT TRANSFORMS ALONG FLAT CURVES NEAR L1

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  15 December 2009

NEAL BEZ
NEAL BEZ*
Affiliation:
School of Mathematics, The Watson Building, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For a class of convex curves in ℝd we prove that the corresponding maximal operator and Hilbert transform are of weak type Llog L. The point of interest here is that this class admits curves which are infinitely flat at the origin. We also prove an analogous weak type result for a class of nonconvex hypersurfaces.

Keywords

maximal operatorHilbert transformweak type estimate

MSC classification

Secondary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 87 , Issue 3 , December 2009 , pp. 311 - 323
Copyright
Copyright © Australian Mathematical Publishing Association, Inc. 2009

References

[1]Bez, N., ‘L p-boundedness for the Hilbert transform and maximal operator along a class of nonconvex curves’, Proc. Amer. Math. Soc. 135 (2007), 151–161.CrossRefGoogle Scholar
[2]Bez, N., Nonisotropic Operators Arising in the Method of Rotations, PhD Thesis, University of Edinburgh (2007).Google Scholar
[3]Bez, N., ‘Maximal operators along piecewise linear curves near L 1’, Indiana Univ. Math. J. 58 (2009), 1639–1658.CrossRefGoogle Scholar
[4]Carbery, A., Christ, M., Vance, J., Wainger, S. and Watson, D., ‘Operators associated to flat plane curves: L p estimates via dilation methods’, Duke Math. J. 59 (1989), 677–700.CrossRefGoogle Scholar
[5]Carbery, A., Ricci, F. and Wright, J., ‘Maximal functions and Hilbert transforms associated to polynomials’, Rev. Mat. Ibero. 14 (1998), 117–144.CrossRefGoogle Scholar
[6]Carbery, A., Vance, J., Wainger, S. and Watson, D., ‘The Hilbert transform and maximal function along flat curves, dilations, and differential equations’, Amer. J. Math. 116 (1994), 1203–1239.CrossRefGoogle Scholar
[7]Christ, M., ‘Weak type (1,1) bounds for rough operators’, Ann. of Math. (2) 128 (1988), 19–42.CrossRefGoogle Scholar
[8]Christ, M., ‘Examples of singular maximal functions unbounded on L p’, Publ. Mat. 35 (1991), 269–279.CrossRefGoogle Scholar
[9]Christ, M. and Stein, E. M., ‘A remark on singular Calderón–Zygmund theory’, Proc. Amer. Math. Soc. 99 (1987), 71–75.Google Scholar
[10]Kim, W., Wainger, S., Wright, J. and Ziesler, S., ‘Singular integrals and maximal functions associated to surfaces of revolution’, Bull. London Math. Soc. 28 (1996), 291–296.CrossRefGoogle Scholar
[11]Nagel, A., Vance, J., Wainger, S. and Weinberg, D., ‘The Hilbert transform for convex curves in ℝn’, Amer. J. Math. 108 (1986), 485–504.CrossRefGoogle Scholar
[12]Seeger, A. and Tao, T., ‘Sharp Lorentz space estimates for rough operators’, Math. Ann. 320 (2001), 381–415.CrossRefGoogle Scholar
[13]Seeger, A., Tao, T. and Wright, J., ‘Singular maximal functions and Radon transforms near L 1’, Amer. J. Math. 126 (2004), 607–647.CrossRefGoogle Scholar