Oscillatory hyper Hilbert transforms along variable curves

Abstract

For n = 2 or 3 and x\in {ℝ}^n, we study the oscillatory hyper Hilbert transform

{{\displaystyle T}}_{\alpha ,\beta }f(x)={{\displaystyle \int }}_{ℝ}f(x-\Gamma (t,x)){\mathrm{e}}^{{-i\left|t\right|}^{-\beta }}{\left|t\right|}^{-\mathrm{1}-\alpha }\mathrm{d}t

along an appropriate variable curve \mathrm{\Gamma }(t,x) in {ℝ}^n (namely, \mathrm{\Gamma }(t,x) is a curve in {ℝ}^n for each fixed x), where \alpha >\beta >\mathrm{0}. We obtain some L^p boundedness theorems of {{\displaystyle T}}_{\alpha ,\beta }, under some suitable conditions on \alphaand \beta. These results are extensions of some earlier theorems. However, {{\displaystyle T}}_{\alpha ,\beta }f(x) is not a convolution in general. Thus, we only can partially employ the Plancherel theorem, and we mainly use the orthogonality principle to prove our main theorems.

Key words： Hyper Hilbert transform; variable curve

References

1 Bennett J M. Hilbert transforms and maximal functions along variable flat plane curves. Trans Amer Math Soc, 2002, 354: 4871–4892

2 Carbery A. A version of Cotlar's lemma for L^p spaces and some applications. In: Marcantognini S A M, Mendoza G A, Moran M D, Octavio A, Urbina W O, eds. Harmonic Analysis and Operator Theory. Contemp Math, Vol 189. Providence: Amer Math Soc, 1995, 117–134

3 Carbery A, Christ M, Vance J, Wainger S, Watson D K. Operators associated to flat plan curves: L^p estimates via dilation methods. Duke Math J, 1989, 59: 675–700

4 Carbery A, Pérez S. Maximal functions and Hilbert transforms along variable flat curves. Math Res Lett, 1999, 6: 237–249

5 Carbery A, Seeger A, Wainger S, Wright J. Classes of singular operators along variable flat curves. J Geom Anal, 1999, 9: 583–605

6 Carbery A, Wainger S, Wright J. Hilbert transforms and maximal functions along variable flat plane curves. J Fourier Anal Appl, 1995, Special Issue: 119–139

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

8 Chen J, Damtew B M, Zhu X. Oscillatory hyper Hilbert transforms along general curves. Front Math China, 2017, 12: 281–299

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

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

11 Li J, Yu H. The oscillatory hyper-Hilbert transform associated with plane curves. Canad Math Bull, 2018, 61: 802–811

12 Li J, Yu H. L^p boundedness of Hilbert transforms and maximal functions along variable curves. arXiv: 1808.01447v1

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

14 Nagel A, Stein E M, Wainger S. Hilbert transforms and maximal functions related to variable curves. In: Weiss G, Wainger S, eds. Harmonic analysis in Euclidean spaces. Proc Sympos Pure Math, Vol 35, Part 1. Providence: Amer Math Soc, 1979, 95–98

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

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

17 Phone D H, Stein E M. Hilbert integrals, singular integrals and Randon transforms I. Acta Math, 1986, 157: 99–157

18 Seeger A. L²-estimates for a class of singular oscillatory integrals. Math Res Lett, 1994, 1: 65–73

19 Seeger A, Wainger S. Singular Radon transforms and maximal functions under convexity assumptions. Rev Mat Iberoam, 2003, 19: 1019–1044

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

21 Wainger S. Averages and singular integrals over lower dimensional sets. In: Stein E M, ed. Beijing Lectures in Harmonic Analysis. Ann of Math Stud, Vol 112. Princeton: Princeton Univ Press, 1986, 357–421

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

23 Zielinski M. Highly Oscillatory Singular Integralsalong Curves. Ph D Dissertation, University of Wisconsin-Madison, Madison, 1985