On Certain Estimates Of Rough Oscillatory Singular Integrals

Authors: H. M. Al-Qassem

Abstract:

We obtain appropriate sharp estimates for rough oscillatory integrals. Our results represent significant improvements as well as natural extensions of what was known previously.

Keywords: Oscillatory singular integral, Rough kernel, Singular integral, L^{p} boundedness.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1078945

References:

[1] A. Al-Salman and Y. Pan, Singular integrals with rough kernels, Canad. Math. Bull. 47 (2004), no. 1, 3-11.
[2] H. M. Al-Qassem, L. Cheng, and Y. Pan, On the boundedness of rough oscillatory singular integrals on Trieble-Lizorkin spaces. To appear in Acta Mathematica Sinica.
[3] H. M. Al-Qassem, L. Cheng, A. Fukui and Y. Pan, Bounds for Oscilla¬tory Singular Integrals on Rn. To appear in Math. Nachrichten J.
[4] G. I. Arkhipov, A. A. Karacuba, and V. N. Cubarikov, Trigonometric integrals, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 5, 971-1003, 1197.
[5] A. Carbery and J. Wright, Distributional and Lq norm inequalities for polynomials over convex bodies in Rn, Math. Res. Lett. 8 (2001), no. 3, 233-248.
[6] R. Coifman and G. Weiss, Extension of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645.
[7] L. Grafakos and A. Stefanov, LP bounds for singular integrals and maximal singular integrals with rough kernel, Indiana Univ. Math. J, whi 47 (1998), 455-469.
[8] I. R. Parissis, Oscillatory Integrals with Polynomial Phase, Ph. D Thesis, University of Crete, 2007.
[9] M. Padimitrakis and I. R. Parissis, Singular oscillatory integrals on Rn, Math. Z., to appear.
[10] E. M. Stein, Oscillatory integrals in Fourier analysis, Beijing lectures in harmonic analysis (Beijing, 1984), Ann. of Math. Stud., vol. 112, Princeton Univ. Press, Princeton, NJ, 1986, p. 307-355.
[11] Elias M. Stein and Stephen Wainger, The estimation of an integral arising in multiplier transformations, Studia Math. 35 (1970), 101-104.