Scaling properties of infinitely flat curves and surfaces
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 , , and . We also prove some three dimensional analogs of those results.
KeywordsRestriction Operator Orlicz Space Dilation Operator Young Function Dimensional Analog
Unable to display preview. Download preview PDF.
- J.-G. Bak, Restrictions of Fourier tranforms to flat curves in ℝ2, Illinois J. Math. 38 (1994).Google Scholar
- J.-G. Bak, Sharp convolution estimates for measures on flat surfaces, (preprint 1994).Google Scholar
- J.-G. Bak, Averages over surfaces with infinitely flat points, J. Func. Anal. 129 (1995).Google Scholar
- J.-G. Bak, D. McMichael, and D. Oberlin, Convolution estimates for some measures on flat curves, J. Func. Anal. 101 (1991).Google Scholar
- 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).Google Scholar
- L. Caxleson and P. Sjolin, Oscillatory integrals and the multiplier problem for the disc, Studia Math. 44 (1972).Google Scholar
- W. Littman, (Lp, Lq ) estimates for singular integral operators, Proc. Symp. Pure Math. 23 (1973).Google Scholar
- F. Ricci and E. Stein, Harmonic analysis on nilpotent groups and singular integrals II, J. Func. Anal. 78 (1988).Google Scholar
- R. Strichartz, Convolution with kernels having singularities on a sphere, Trans. Amer. Math. Soc. 148 (1970).Google Scholar
- P. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975).Google Scholar
- A. Torchinsky, Interpolation of operators and Orlicz classes, Studia Math. 59 (1976)Google Scholar