Oscillatory hyper Hilbert transforms along general curves

Abstract

We consider the oscillatory hyper Hilbert transform {{\displaystyle H}}_{\gamma ,\alpha ,\beta }f(x)={\displaystyle {\int }_{\mathrm{0}}^{\infty }f(x-\Gamma (t))}{e}^{it-\beta }{t}^{-(\mathrm{1}+\alpha )}\mathrm{d}t, where Γ(t) = (t, γ(t)) in {ℝ}^{\mathrm{2}} is a general curve. When γ is convex, we give a simple condition on γ such that H_γ,_α,_β is bounded on L² when \beta \ge \mathrm{3}\alpha ,\beta >\mathrm{0}. As a corollary, under this condition, we obtain the L^p-boundedness of H_γ,_α,_β when . When Γ is a general nonconvex curve, we give some more complicated conditions on γ such that H_γ,_α,_β is bounded on L². As an application, we construct a class of strictly convex curves along which H_γ,_α,_β is bounded on L² only if β>2α>0.

Key words： Hilbert transform; oscillatory integral; oscillatory hyper Hilbert transform

References

1 Bez N. L^p-boundedness for the Hilbert transform and maximal operator along a class of nonconvex curves. Proc Amer Math Soc, 2007, 135(1): 151–161

2 Chandarana S. L^p-bounds for hypersingular integral operators along curves. Pacific J Math, 1996, 175(2): 389–416

3 Chandarana S. Hypersigular integral operators along space curves. Preprint

4 Chen J, Fan D, Wang M, Zhu X. L^p bounds for oscillatory hyper-Hilbert transform along curves. Proc Amer Math Soc, 2008, 136(9): 3145–3153

5 Chen J, Fan D, Zhu X. Sharp L2 boundedness of the oscillatory hyper-Hilbert transform along curves. Acta Math Sin (Engl Ser), 2010, 26(4): 653–658

6 Fabes E B, Riviére N M. Singular intervals with mixed homogeneity. Studia Math, 1966, 27: 19–38

7 Fefferman C. Inequalities for strongly singular convolution operators. Acta Math, 1970, 124: 9–36

8 Fefferman C, Stein E M. H^p Spaces of several variables. Acta Math, 1972, 129: 137–193

9 Hirschman I I. On multiplier transformations. Duke Math J, 1959, 26: 221–242

10 Laghi A N, Lyall N. Strongly singular integrals along curves. Pacific J Math, 2007, 233(2): 403–415

11 Le H V. Hypersingular integral operators along surfaces. Integral Equations Operator Theory, 2002, 44(4): 451–465

12 Nagel A, Riviére N M, Wainger S. On Hilbert transform along curves. Bull Amer Math Soc, 1974, 80(1): 106–108

13 Nagel A, Riviére N M, Wainger S. On Hilbert transform along curves II. Amer J Math, 1976, 98(2): 395–403

14 Nagel A, Vance J, Wainger S, Weinberg D. Hilbert transforms for convex curves. Duke Math J, 1983, 50(3): 735–744

15 Nagel A, Wainger S. Hilbert transforms associated with plane curves. Trans Amer Math Soc, 1976, 223: 235–252

16 Stein E M.Singular integrals, harmonic functions and differentiability properties of functions of several variables. Proc Sympos Pure Math, 1976, 10: 316–335

17 Stein E M. Harmonic Analysis Real-Variable Methods, Orthogonality and Oscillatory Integrals.Princeton: Princeton Univ Press, 1993

18 Stein E M, Wainger S. Problems in harmonic analysis related to curvature. Bull Amer Math Soc, 1978, 84(6): 1239–1295

19 Vance J, Wainger S, Wright J. The Hilbert transform and maximal function along nonconvex curves in the plane. Rev Mat Iberoam, 1994, 10(1): 93–121

20 Wainger S. Special trigonometric series in k-dimensions. Mem Amer Math Soc, 1965, 59(3): 735–744

21 Wainger S. On certain aspects of differentiation theory. In: Topics in Modern Harmonic Analysis: Proceedings of a Seminar Held in Torino and Milano, May-June 1982, Vol II. Rome: Istituto Nazionale di Alta Matematica Francesco Severi, 1983, 677–706

22 Wainger S. Averages and singular integrals over lower dimensional sets. In: Beijing Lectures on Harmonic Analysis. Ann of Math Stud, Vol 112. Princeton: Princeton Univ Press, 1986, 357–421

23 Wainger S. Dilations associated to flat curves. Publ Mat, 1991, 35: 251–257

24 Zielinski M. Highly Oscillatory Singular Integrals along Curves. Ph D Dissertation. Madison: University of Wisconsin-Madison, 1985