$L^p$ bounds for oscillatory hyper-Hilbert transform along curves
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- by Jiecheng Chen, Dashan Fan, Meng Wang and Xiangrong Zhu PDF
- Proc. Amer. Math. Soc. 136 (2008), 3145-3153 Request permission
Abstract:
We study the oscillatory hyper-Hilbert transform \begin{equation} H_{n,\alpha ,\beta }f(x)=\int ^1_0 f(x-\Gamma (t))e^{it^{-\beta }}t^{-1-\alpha }dt \end{equation} along the curve $\Gamma (t)=(t^{p_1},t^{p_2},\cdots ,t^{p_n})$, where $p_1,p_2,\cdots ,p_n,\alpha ,\beta$ are some real positive numbers. We prove that if $\beta >(n+1)\alpha$, then $H_{n,\alpha ,\beta }$ is bounded on $L^p$ whenever $p \in (\frac {2\beta }{2\beta -(n+1)\alpha },\frac {2\beta }{(n+1)\alpha })$. Furthermore, we also prove that $H_{n,\alpha ,\beta }$ is bounded on $L^2$ when $\beta =(n+1)\alpha$. Our work improves and extends some known results by Chandarana in 1996 and in a preprint. As an application, we obtain an $L^p$ boundedness result for some strongly parabolic singular integrals with rough kernels.References
- Sharad Chandarana, $L^p$-bounds for hypersingular integral operators along curves, Pacific J. Math. 175 (1996), no. 2, 389–416. MR 1432837, DOI 10.2140/pjm.1996.175.389
- S. Chandarana, Hypersingular integral operators along space curves. Preprint.
- E. B. Fabes and N. M. Rivière, Singular integrals with mixed homogeneity, Studia Math. 27 (1966), 19–38. MR 209787, DOI 10.4064/sm-27-1-19-38
- Charles Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9–36. MR 257819, DOI 10.1007/BF02394567
- C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
- I. I. Hirschman Jr., On multiplier transformations, Duke Math. J. 26 (1959), 221–242. MR 104973
- Alexander Nagel, Néstor Rivière, and Stephen Wainger, On Hilbert transforms along curves, Bull. Amer. Math. Soc. 80 (1974), 106–108. MR 450899, DOI 10.1090/S0002-9904-1974-13374-4
- Alexander Nagel, Néstor Rivière, and Stephen Wainger, On Hilbert transforms along curves, Bull. Amer. Math. Soc. 80 (1974), 106–108. MR 450899, DOI 10.1090/S0002-9904-1974-13374-4
- Alexander Nagel, James Vance, Stephen Wainger, and David Weinberg, Hilbert transforms for convex curves, Duke Math. J. 50 (1983), no. 3, 735–744. MR 714828, DOI 10.1215/S0012-7094-83-05036-6
- Alexander Nagel and Stephen Wainger, Hilbert transforms associated with plane curves, Trans. Amer. Math. Soc. 223 (1976), 235–252. MR 423010, DOI 10.1090/S0002-9947-1976-0423010-8
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- E. M. Stein, Singular integrals, harmonic functions, and differentiability properties of functions of several variables, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) Amer. Math. Soc., Providence, R.I., 1967, pp. 316–335. MR 0482394
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- Elias M. Stein and Stephen Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1239–1295. MR 508453, DOI 10.1090/S0002-9904-1978-14554-6
- Stephen Wainger, Special trigonometric series in $k$-dimensions, Mem. Amer. Math. Soc. 59 (1965), 102. MR 182838
- Stephen Wainger, Dilations associated to flat curves, Publ. Mat. 35 (1991), no. 1, 251–257. Conference on Mathematical Analysis (El Escorial, 1989). MR 1103618, DOI 10.5565/PUBLMAT_{3}5191_{1}1
- Stephen Wainger, Averages and singular integrals over lower-dimensional sets, Beijing lectures in harmonic analysis (Beijing, 1984) Ann. of Math. Stud., vol. 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 357–421. MR 864376
- Stephen Wainger, On certain aspects of differentiation theory, Topics in modern harmonic analysis, Vol. I, II (Turin/Milan, 1982) Ist. Naz. Alta Mat. Francesco Severi, Rome, 1983, pp. 667–706. MR 748880
- M. Zielinski, Highly oscillatory singular integrals along curves. Ph.D. Dissertation, University of Wisconsin-Madison, Madison, WI, 1985.
Additional Information
- Jiecheng Chen
- Affiliation: Department of Mathematics, Zhejiang, University, Hangzhou, Zhejiang, People’s Republic of China
- Email: jcchen@mail.hz.zj.cn
- Dashan Fan
- Affiliation: Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201
- Email: harmonic_analysis@yahoo.com
- Meng Wang
- Affiliation: Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, People’s Republic of China
- Email: mathdreamcn@zju.edu.cn
- Xiangrong Zhu
- Affiliation: Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, People’s Republic of China
- Email: zxr@zju.edu.cn
- Received by editor(s): July 8, 2005
- Received by editor(s) in revised form: May 23, 2007
- Published electronically: May 2, 2008
- Additional Notes: This work was supported by NSFC (10371043, 10571156, 10601046) and PDSFC (20060400336)
- Communicated by: Michael T. Lacey
- © Copyright 2008 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 136 (2008), 3145-3153
- MSC (2000): Primary 42B25
- DOI: https://doi.org/10.1090/S0002-9939-08-09325-8
- MathSciNet review: 2407077