Abstract
We shall give simple sufficient conditions for the Orlicz type bounds for the averaging operators and restriction operators associated with infinitely flat curves in the plane. Our results, obtained by scaling, can be used to recover, up to the endpoints, the results previously obtained in [4], [1], and [2]. We also prove some three dimensional analogs of those results.
research partially supported by NSF grant number DMS97-06825
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J.-G. Bak, Restrictions of Fourier tranforms to flat curves in ℝ2, Illinois J. Math. 38 (1994).
J.-G. Bak, Sharp convolution estimates for measures on flat surfaces, (preprint 1994).
J.-G. Bak, Averages over surfaces with infinitely flat points, J. Func. Anal. 129 (1995).
J.-G. Bak, D. McMichael, and D. Oberlin, Convolution estimates for some measures on flat curves, J. Func. Anal. 101 (1991).
A. Carbery, M. Christ, J. Vance, S. Wainger, and D. Watson, Operators associated to flat plane curves: L P estimates via dilation methods, Duke Math. J. 59 (1989).
L. Caxleson and P. Sjolin, Oscillatory integrals and the multiplier problem for the disc, Studia Math. 44 (1972).
W. Littman, (Lp, Lq ) estimates for singular integral operators, Proc. Symp. Pure Math. 23 (1973).
F. Ricci and E. Stein, Harmonic analysis on nilpotent groups and singular integrals II, J. Func. Anal. 78 (1988).
E. Stein,Harmonic Analysis, Princeton Univ. Press (1993).
R. Strichartz, Convolution with kernels having singularities on a sphere, Trans. Amer. Math. Soc. 148 (1970).
P. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975).
A. Torchinsky, Interpolation of operators and Orlicz classes, Studia Math. 59 (1976)
A. Torchinsky, Real Variable Methods in Harmonic Analysis Academic Press, Orlando (1986).
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Birkhäuser Boston
About this chapter
Cite this chapter
Iosevich, A. (1999). Scaling properties of infinitely flat curves and surfaces. In: Bray, W.O., Stanojević, Č.V. (eds) Analysis of Divergence. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2236-1_14
Download citation
DOI: https://doi.org/10.1007/978-1-4612-2236-1_14
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-7467-4
Online ISBN: 978-1-4612-2236-1
eBook Packages: Springer Book Archive